Yalla PhD Thesis

LIQUID DAMPERS FOR MITIGATION OF

STRUCTURAL RESPONSE: THEORETICAL DEVELOPMENT AND

EXPERIMENTAL VALIDATION

A Dissertation

Submitted to the Graduate School

of the University of Notre Dame

in Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

by

Swaroop Krishna Yalla, B.Tech, M.S.

________________________________

Ahsan Kareem, Director

Department of Civil Engineering and Geological Sciences

Notre Dame, Indiana

July 2001

LIQUID DAMPERS FOR MITIGATION OF

STRUCTURAL RESPONSE: THEORETICAL DEVELOPMENT AND

EXPERIMENTAL VALIDATION

Abstract

by

Swaroop Krishna Yalla

The current trend toward structures of increasing heights and the use of light-

weight, high strength materials and advanced construction techniques has led to more flex-

ible and lightly damped structures. Understandably, these structures are very sensitive to

environmental excitations such as wind, ocean waves and earthquakes, leading to vibra-

tions inducing possible structural failure, occupant discomfort, and malfunction of eleva-

tors and equipment. Hence, it has made it critical to search for practical and effective

devices to suppress these vibrations.

The most commonly used passive device is the Tuned Mass Damper (TMD),

which is based on the inertial secondary system principle. A Tuned Liquid Damper (TLD)

is a special class of TMD where the mass is replaced by liquid (usually water). Tuned liq-

uid column dampers (TLCDs) are a special type of TLDs that rely on the motion of a liq-

uid column in a U-tube-like container to counteract the forces acting on the structure, with

damping introduced through a valve/orifice in the liquid passage.

The thrust of this dissertation is to study and develop the next generation of liquid

dampers for mitigation of structural response. New modeling insights into the sloshing

Swaroop Krishna Yalla

phenomenon, which incorporate the effect of the liquid slamming/impact on the container

walls, are presented through experimental and analytical studies. The mechanical model-

ing of TLDs is developed using a Sloshing-Slamming (S2) analogy and the use of impact

characteristics functions which can describe with high fidelity the phenomenological

behavior of the damper. A major focus of this study is the design and development of

semi-active control systems which maintain the optimal damping level under different

loading conditions. Experimental validation of such a system was performed in the labora-

tory using a prototype TLCD equipped with a valve controlled by an electro-pneumatic

actuator and positioning system. Finally, the design, implementation, cost and risk-based

decision analysis for the implementation of liquid dampers in structural vibration control

is presented.

DEDICATION

This work is dedicated to my parents who instilled in me the value of learning.

It cannot be stolen by thieves, Nor can it be taken away by kings.

It cannot be divided among brothers and..

It does not cause a load on your shoulders.

If spent.. It indeed always keeps growing.

The wealth of knowledge..

Is the most superior wealth of all!

TABLE OF CONTENTS

LIST OF TABLES...............................................................................................vi

LIST OF FIGURES...........................................................................................viii

ACKNOWLEDGEMENTS................................................................................xiv

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

1.1 Introduction..............................................................................................................1

1.2 Literature Review.....................................................................................................4

1.3 Applications .............................................................................................................7

1.3.1 Ship/Offshore applications...........................................................................7

1.3.2 Structural Applications ..............................................................................11

1.4 Motivation of Present Work ...................................................................................16

1.5 Organization of Dissertation ..................................................................................18

CHAPTER 2: MODELING OF SLOSHING......................................................20

2.1 Introduction............................................................................................................20

2.1.1 Numerical Modeling of TLDs ...................................................................21

2.1.2 Mechanical Modeling of TLDs..................................................................22

2.2 Sloshing-Slamming (S2) Damper Analogy ...........................................................24

2.2.1 Liquid Sloshing..........................................................................................24

2.2.2 Liquid Slamming .......................................................................................25

2.2.3 Proposed Sloshing-Slamming (S2) Analogy .............................................26

2.2.4 Numerical Study ........................................................................................31

2.2.5 Base Shear Force........................................................................................33

2.3 Impact Characteristics model.................................................................................34

2.4 Equivalent Linear Models ......................................................................................37

2.4.1 Harmonic Linearization .............................................................................37

2.4.2 Statistical Linearization .............................................................................38

ii

2.5 Concluding Remarks..............................................................................................40

CHAPTER 3: TUNED LIQUID COLUMN DAMPERS....................................41

3.1 Introduction............................................................................................................41

3.2 Modeling of Tuned Liquid Column Dampers........................................................43

3.2.1 Equivalent Linearization: ...........................................................................44

3.2.2 Accuracy of Equivalent linearization.........................................................45

3.3 Optimum Absorber Parameters..............................................................................47

3.3.1 White Noise excitation...............................................................................50

3.3.2 First Order Filter (FOF) .............................................................................53

3.3.3 Second Order Filter (SOF).........................................................................55

3.3.4 Example.....................................................................................................56

3.4 Multiple Tuned Liquid Column Dampers (MTLCDs) ..........................................57

3.4.1 Effect of Number of dampers....................................................................59

3.4.2 Effect of damping ratio of dampers ..........................................................59

3.4.3 Effect of Frequency range.........................................................................60


CHAPTER 4: BEAT PHENOMENON...............................................................65

4.1 Introduction............................................................................................................65

4.2 Behavior of SDOF system with TLCD..................................................................68

4.2.1 Case 1: Undamped Combined System.......................................................68

4.2.2 Case 2: Linearly Damped Structure with Undamped Secondary System..71

4.2.3 Case 3: Damped Primary and Secondary System.....................................74

4.3 Experimental Verification ......................................................................................79


CHAPTER 5: SEMI-ACTIVE SYSTEMS AND APPLICATIONS...................81

5.1 Introduction............................................................................................................81

5.2 Gain-scheduled Control .........................................................................................82

5.2.1 Determination of Optimum Headloss Coefficient .....................................83

5.3 Applications ...........................................................................................................86

5.3.1 Example 1: SDOF-TLCD system under random white noise ...................86

5.3.2 Example 2: Application to Offshore Structure ..........................................88

5.4 Clipped-Optimal System........................................................................................92

5.4.1 Control Strategies.......................................................................................95

iii

5.4.2 Example 3: MDOF system under random wind loading ...........................99

5.4.3 Example 4: MDOF system under harmonic loading ...............................102

5.5 Concluding Remarks............................................................................................106

CHAPTER 6: TLD EXPERIMENTS...............................................................108

6.1 Introduction..........................................................................................................108

6.2 Experimental Studies ...........................................................................................110

6.3 System Identification ...........................................................................................112

6.3.1 Nonlinear System Identification ..............................................................113

6.3.2 Combined Structure-damper analysis ......................................................116

6.4 Impact Pressure Studies .......................................................................................118

6.4.1 Single-point pressure measurement .........................................................119

6.4.2 Multiple-point pressure measurements ....................................................122

6.4.3 Shallow water versus deep water sloshing...............................................125

6.4.4 Pressure variation along the tank height ..................................................126

6.5 Hardware-in-the-loop Simulation ........................................................................127

6.5.1 Experimental study ..................................................................................129


CHAPTER 7: TLCD EXPERIMENTS.............................................................132

7.1 Introduction..........................................................................................................132

7.2 Experimental Studies ...........................................................................................134

7.2.1 Effect of tuning ratio ................................................................................136

7.2.2 Effect of damping ....................................................................................137

7.2.3 Effect of amplitude of excitation .............................................................138

7.2.4 Equivalent damping .................................................................................140

7.3 Experimental Validation.......................................................................................143


CHAPTER 8: DESIGN, IMPLEMENTATION AND RELIABILITYISSUES..................................................................................................148

8.1 Introduction..........................................................................................................148

8.2 Comparison of various DVAs ..............................................................................150

8.2.1 Implementation comparisons ...................................................................150

8.2.2 Cost comparison.......................................................................................155

8.3 Risk-based Decision Analysis..............................................................................157

iv

8.3.1 Decision analysis framework ...................................................................159

8.3.2 Reliability Analysis..................................................................................162

8.3.3 Cost and Utility Analysis .........................................................................165

8.3.4 Risk-based Decision Analysis..................................................................166

8.4 Design of Dampers ..............................................................................................167

8.4.1 Design Guidelines....................................................................................167

8.4.2 Control Strategy .......................................................................................169

8.4.3 Design Procedure .....................................................................................170

8.4.4 Technology...............................................................................................174


CHAPTER 9: CONCLUSIONS .............................................................................. 177

APPENDIX................................................................................................................... 181

REFERENCES.............................................................................................................184

v

LIST OF TABLES

TABLE 2.1 Parameters of the model.............................................................................32

TABLE 3.1 Example forcing functions.........................................................................49

TABLE 3.2 Comparison of optimal parameters for TMD and TLCD ..........................52

TABLE 3.3 Optimum parameters for white noise excitation for different mass ratios.53

TABLE 3.4 Optimum absorber parameters for FOF for different parameter ν1...........54

TABLE 3.5 Optimum absorber parameters for FOF for various mass ratios................54

TABLE 3.6 Optimum absorber parameters for SOF for different values of b1 ............57

TABLE 3.7 Optimum absorber parameters for SOF for various mass ratios................57

TABLE 3.8 Optimum absorber parameters...................................................................58

TABLE 3.9 Optimum parameters for MTLCD configurations .....................................62

TABLE 5.1 Comparison of different control strategies: Example 1 .............................88

TABLE 5.2 Numerical parameters used: Example 2 ....................................................89

TABLE 5.3 Comparison of various control strategies: Example 3 .............................101

TABLE 5.4 Comparison of various control strategies: Example 4 .............................106

TABLE 6.1 Time lag and impact influence factor for different sensor locations........122

TABLE 7.1 Performance of semi-active system as compared to uncontrolled and

passive system..........................................................................................146

vi

TABLE 8.1 Component comparison of different DVAs..............................................156

TABLE 8.2 Comparison of different systems for varying wind conditions................159

TABLE 8.3 Random Variables used in Reliability analysis........................................164

TABLE 8.4 Probability of Failure under different wind speeds..................................164

TABLE 8.5 Costs and Normalized Utility Analysis....................................................165

TABLE 8.6 Utility analysis based on the decision analysis ........................................166

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LIST OF FIGURES

Figure 1.1 (a) Frahm anti-rolling tank (b) nutation dampers in satellite applications...5

Figure 1.2 (a) Bi-directional TLCD (b) V-shaped TLCD .............................................7

Figure 1.3 Types of passive/ controllable-passive tanks for ships.................................8

Figure 1.4 (a) Free surface damping tanks (b) Semi-active control for structure with

open bottom tanks......................................................................................10

Figure 1.5 Aqua dampers (Courtesy: MCC Aqua damper literature).........................11

Figure 1.6 (a) Schematic of TLDs installed in SYPH (b) Actual installation in the

building (taken from Tamura et al. 1995)..................................................12

Figure 1.7 (a) Liquid damper with pressure adjustment concept (b) photograph of

Hotel Cosima, Tokyo.................................................................................13

Figure 1.8 Millennium tower: passive and active TLCD concept...............................14

Figure 1.9 (a) Shanghai Financial Trade Center (b) 7 South Dearborn Project ..........15

Figure 1.10 TLDs installed in chimneys .......................................................................16

Figure 2.1 (a) Equivalent mechanical model of sloshing liquid in a tank (b) Impact

damper model.............................................................................................26

Figure 2.2 Variation of (a) jump frequency and (b) damping ratio of the TLD with the

base amplitude (taken from Yu et. al 1999)...............................................27

Figure 2.3 Frames from the sloshing experiments video at high amplitudes: a part of

water moves as a lumped mass and impacts the container wall. (VideoCourtesy: Dr. D.A. Reed)...........................................................................28

Figure 2.4 Schematic diagram of the proposed sloshing-slamming (S2) analogy.......29

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Figure 2.5 Comparison of experimental results with S2 simulation results: (a), (b):

experimental results; (c), (d): simulation results for = 1.0 and 0.9.......32

Figure 2.6 (a) schematic of the jump phenomenon (b)Variation of the non-

dimensionalized base shear force with the frequency ratio. (experimentalresults taken from Fujino et al. 1992)........................................................33

Figure 2.7 Non dimensional interaction force curves for different η..........................36

Figure 3.1 Schematic of the Structure-TLCD system .................................................43

Figure 3.2 Exact (Non-linear) and Equivalent Linearization results...........................46

Figure 3.3 Time histories for ξ = 75............................................................................46

Figure 3.4 Variation of dynamic magnification factor with the head-loss coefficient

and frequency ratio for a TLCD.................................................................47

Figure 3.5 Comparison of optimum absorber parameters for a TLCD with varying αand a TMD.................................................................................................51

Figure 3.6 Transfer function of the filters and the primary system: (a) first order filters

(b) second order filters...............................................................................55

Figure 3.7 MTLCD configuration ...............................................................................58

Figure 3.8 Effect of number of dampers on the frequency response of SDOF-MTLCD

system........................................................................................................61

Figure 3.9 Effect of damping ratio of the dampers on the frequency response of

SDOF-MTLCD system..............................................................................61

Figure 3.10 Effect of frequency range on the frequency response of SDOF-MTLCD

system........................................................................................................62

Figure 4.1 Different coupled system (a) Vibration absorber (b) Coupled penduli

system (c) Electrical system (d) Fluid coupling within two cylinders.......66

Figure 4.2 Uncontrolled and Controlled response of a structure combined with (a)

TLD (b) TLCD...........................................................................................67

Figure 4.3 Different combined systems ......................................................................68

Ω

ix

Figure 4.4 Phase plane portraits of the undamped coupled system.............................69

Figure 4.5 Time histories of primary system displacement for α=0 and α=0.6 .........70

Figure 4.6 Variation of ωΑ and ωΒ and as a function of α .........................................72

Figure 4.7 Time histories of response for ζ1=0.005 and ζ1=0.05 ...............................73

Figure 4.8 Anatomy of the damped response signature ..............................................74

Figure 4.9 Time histories of response for ξ= 0.2, 2 and 50.........................................75

Figure 4.10 Modal frequencies and modal damping ratios of combined system as a

function of the damping ratio of the TLCD...............................................76

Figure 4.11 Phase-plane 3D plots (a) uncoupled system (b) case 1: undamped system

(c) case 2: system with damping in primary system only (d) case 3: system

with damping in both primary and secondary systems..............................77

Figure 4.12 Experimental setup for combined structure-TLCD system on a shaking

table............................................................................................................79

Figure 4.13 Experimental free vibration response with different orifice openings (θ = 0

fully open)..................................................................................................80

Figure 5.1 Gain scheduling concept ............................................................................83

Figure 5.2 Flowchart of the two algorithms (a) iterative method (b) direct method...84

Figure 5.3 Iterative method (a) convergence of response quantities (b) optimum

headloss coefficient....................................................................................85

Figure 5.4 Variation of optimum headloss coefficient with loading intensity: white

noise excitation..........................................................................................86

Figure 5.5 Example 1: SDOF system under random excitation..................................87

Figure 5.6 (a) Single degree of freedom idealization of the offshore structure (b)

Concept of Liquid Dampers in TLPs.........................................................89

Figure 5.7 Optimal Absorber parameters as a function of loading conditions............91

x

Figure 5.8 (a) Variation of Optimal headloss coefficient with loading conditions for

different wave spectra (b) Spectra of structural acceleration at U10=20 m/s

for different ξ.............................................................................................92

Figure 5.9 Semi-active TLCD-Structure combined system ........................................93

Figure 5.10 Schematic of the control system ................................................................98

Figure 5.11 Schematic of 5DOF building with semi-active TLCD on top story.........100

Figure 5.12 Wind loads acting on each lumped mass .................................................101

Figure 5.13 Displacements and Acceleration of Top Level under various control

strategies..................................................................................................102

Figure 5.14 Variation of performance indices with maximum headloss coefficient... 104

Figure 5.15 Displacement of Top Floor under various control strategies ...................104

Figure 5.16 Variation of headloss coefficient with time..............................................105

Figure 5.17 Variation of RMS displacements, RMS accelerations, maximum story

shear and maximum inter-story displacements........................................105

Figure 6.1 (a) Schematic of the experimental setup (b) pressure sensor locations... 110

Figure 6.2 Sample time-histories of the shear force at Ae = 0.3 cm and 2.0 cm....... 113

Figure 6.3 Nonlinear Optimization Scheme..............................................................114

Figure 6.4 Curvefitting the parameters of the impact characteristics model.............115

Figure 6.5 (a) Experimental plots of non-dimensional sloshing force as a function of

excitation frequency for different amplitudes (b) Simulated curves after

optimization.............................................................................................116

Figure 6.6 Response of the structure for different amplitudes ..................................117

Figure 6.7 Pressure time histories for various frequency ratios (Ae = 1.0 cm). ........119

Figure 6.8 Probability distribution function of the peak impact pressures ..............120

xi

Figure 6.9 (a) Anatomy of a single pressure pulse (b) wavelet scalogram of the

pressure signal..........................................................................................121

Figure 6.10 (a) Pressure pulses at different locations on the wall (b) Wavelet

coscalograms with sensor 2 as reference.................................................124

Figure 6.11 Typical sloshing wave with pressure pulse and wave mechanism schematic

for (a) shallow water (h/a =0.12) and (b) deep water (h/a = 0.25) case..125

Figure 6.12 Variation of the peak pressure coefficient with height of the tank wall...126

Figure 6.13 Hardware-in-the-loop concept for structure-liquid damper systems .......128

Figure 6.14 Schematic of the experimental setup for the HIL simulation ..................129

Figure 6.15 Hardware-in-the-loop simulation for random loading case .....................130

Figure 7.1 (a) Photograph of the Electro-pneumatic actuator (b) Schematic diagram

of the experimental set-up........................................................................134

Figure 7.2 (a) Transfer functions for different tuning ratios (b) Variation of H2 norm

with tuning ratio.......................................................................................137

Figure 7.3 Transfer functions for different valve angle openings .............................138

Figure 7.4 Variation of transfer functions for different amplitudes of excitation..... 139

Figure 7.5 (a) Optimization of H2 norm (b) look-up table for semi-active control...140

Figure 7.6 (a) Comparison of transfer functions: (a) θ =40 deg, ζf = 9 % (optimal

damping) (b) θ = 60 deg, ζf = 30% (non-optimal damping)....................141

Figure 7.7 3-D plot of transfer function as a function of effective damping and

frequency (a) experimental results (b) simulation results........................142

Figure 7.8 Excitation time histories, valve angle variations and the resulting

accelerations for uncontrolled, passive and semi-active systems for time-

history 1...................................................................................................144

Figure 7.9 Excitation time histories, valve angle variations and the resulting

accelerations for uncontrolled, passive and semi-active systems for time-

history 2...................................................................................................145

xii

Figure 8.1 Implementation ideas for tuned liquid dampers (a) bridge towers (b) tall

buildings...................................................................................................149

Figure 8.2 TMD system installed in the Citicorp Building, New York City (takenfrom Wiesner, 1979).................................................................................151

Figure 8.3 (a) Single-stage (b) multi-stage Pendulum-type TMDs (c) TMDs with

laminated rubber bearings (taken from Yamazaki et al. 1992)................152

Figure 8.4 Equipment schematic for a building-mounted TLCD .............................155

Figure 8.5 Variation of RMS accelerations of the top floor with increasing wind

velocity.....................................................................................................159

Figure 8.6 Elements of Decision analysis .................................................................160

Figure 8.7 Decision Tree for Building Serviceability ...............................................166

Figure 8.8 Semi-active control strategy in tall buildings..........................................170

Figure 8.9 (a) Equivalent white noise concept (b) Variation of equivalent white noise

with wind velocity....................................................................................172

Figure 8.10 Electro-pneumatic valve (courtesy Hayward Controls)...........................174

Figure A.1 (a) Variation of Valve Conductance (b) Variation of headloss coefficient

with the angle of valve opening...............................................................183

xiii

ACKNOWLEDGEMENTS

I would like to first thank my advisor and guru, Prof. Ahsan Kareem, who pro-

vided encouragement, support and friendship throughout the length of my stay at Notre

Dame. The confidence he placed in me has been instrumental in my professional develop-

ment. I would also like to thank my committee members, particularly Prof. Bill Spencer

and Prof. Jeff Kantor, who guided me through many concepts in dynamics and control. I

would also like to thank Prof. Yahya Kurama and Prof. Steven Skaar for their valuable

guidance and constructive comments. I would also like to thank the staff in the Depart-

ment of Civil Engineering and Geological Sciences, particularly Tammy, Molly and Chris.

Our laboratory technician, Brent Bach, helped me in most stages of the experiments.

Next, I would like to thank my family, both in India and the U.S., who have con-

stantly supported me during my years in graduate school. Thank you Amma, Daddy,

Kumar, Chinni and others. I don’t know what I would have done without my friends: Cass,

Vicky, Adrish and all the other long lasting friendships I made at Notre Dame. Finally,

many thanks to the wonderful campus of the University of Notre Dame whose lakes,

Grotto and Fischer graduate apartments provided a home away from home and a wonder-

ful place to grow and learn.

xiv

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xvi

CHAPTER 1

INTRODUCTIONIf they give you ruled paper, write the other way

- Juan Ramon Jimenez

________________________________________________________________________

This chapter begins with a brief literature review in the area of liquid dampers.

Relevant literature is also referenced at appropriate places in later chapters of the disserta-

tion. Some of the applications of these dampers, especially in civil engineering structures

and offshore structures, are described. The motivation of the present research is presented

in the next section. Finally, the organization of the dissertation is laid out in detail.

1.1 Introduction

The current trend toward buildings of ever increasing heights and the use of light-

weight, high strength materials, and advanced construction techniques have led to increas-

ingly flexible and lightly damped structures. Understandably, these structures are very

sensitive to environmental excitations such as wind, ocean waves and earthquakes. This

causes unwanted vibrations inducing possible structural failure, occupant discomfort, and

malfunction of equipment. Hence it has become important to search for practical and

effective devices for suppresion of these vibrations. This has opened up a new area of

research in the last decade, aptly titled structural control (Yao, 1972).

1

The devices used for mitigating structural vibrations are divided into separate cate-

gories based on their system requirements (Housner et al. 1997). Passive control devices

are systems which do not require an external power source. These devices impart forces

that are developed in response to the motion of the structure, for e.g., base isolation, vis-

coelastic dampers, tuned mass dampers, etc. More details of such systems can be found in

Soong and Dargush (1997). Active control systems are driven by an externally applied

force which tends to oppose the unwanted vibrations. The control force is generated

depending on the feedback of the structural response. Examples of such systems include

active mass dampers (AMDs), active tendon systems, etc (Soong, 1990). Owing to the

uncertainty of the power supply during extreme conditions and the large power source

needed to introduce control force, passive systems are generally favored over active ones.

Semi-active systems are viewed as controllable devices, with energy requirements orders

of magnitude less than typical active control systems. These systems do not impart energy

into the system and thus maintain stability at all times, for e.g., variable orifice dampers,

electro-rheological dampers, etc. A recent paper by Symans and Constantinou (1999) pro-

vides a state-of-the-art review on semi-active devices for seismic protection of structures.

Another paper by Kareem et al. (1999) describes the control systems for mitigation of

motion of buildings under wind loading. Alternative systems are being proposed which

derive the useful characteristics of both systems. One of them is hybrid control which

implies the combined use of active and passive systems or passive and semi-active sys-

tems.

The most commonly used passive device is the Tuned Mass Damper (TMD),

which is based on the inertial secondary system principle, and consists of a mass attached

2

to the building through a spring and a dashpot. In order to be effective, its parameters need

to be optimally tuned to the building dynamic characteristics, thus imparting indirect

damping through modification of the combined structural system. Such systems have been

implemented, for example, in the John Hancock tower in Boston and the Citicorp Building

in New York City (McNamara, 1977).

A Tuned liquid damper (TLD)/tuned sloshing damper (TSD) (used interchange-

ably throughout this thesis) consists of a tank partially filled with liquid. Like a TMD, it

imparts indirect damping to the structure, thereby reducing response. The energy dissipa-

tion occurs through various mechanisms: viscous action of the fluid, wave breaking, con-

tamination of the free surface with beads, and container geometry and roughness. Unlike a

TMD, however, a TSD has an amplitude dependent transfer function which is complicated

by nonlinear liquid sloshing and wave breaking.

The TLDs can be broadly classified into two categories: shallow-water and deep-

water dampers. This classification is based on the ratio of the water depth to the length of

the tank in the direction of the motion. A ratio of less than 0.15 is representative of the

shallow water case. In the shallow water case, the TLD damping originates primarily from

energy dissipation through the action of the internal fluid’s viscous forces and from wave

breaking. For the deep-water damper, baffles or screens are needed to enhance damping.

The damping mechanism is therefore dependent on the amplitude of the fluid motion,

wave breaking patterns, and screen configuration. The deep-water damper has one draw-

back in the fact that a large portion of water does not participate in sloshing and adds to

the dead weight. At an intermediate level of fill depth, the container can be utilized for

building water supply. If the existing water tanks are not utilized, the large space occupied

3

by water containers may, in some cases, require a part of the building roof. However, most

practical installations of TLDs use many smaller tanks so as to maximize the effective

mass of liquid engaged in sloshing.

Tuned liquid column dampers (TLCDs) are a special type of TLDs relying on the

motion of the column of liquid in a U-tube-like container to counteract the forces acting

on the structure, with damping introduced through an valve/orifice in the liquid passage

(Sakai et al. 1989). The damping is amplitude dependent since the valve/orifice constricts

the dynamics of the liquid in a non-linear way.

1.2 Literature Review

TLDs were proposed in the late 1800s where the frequency of motion in two

interconnected tanks tuned to the fundamental rolling frequency of a ship was successfully

utilized to reduce this component of motion, as shown in Fig. 1.1 (Den Hartog, 1956). Ini-

tial applications of TLDs for structural applications were proposed by Kareem and Sun

(1987); Modi et al. (1987) and Fujino et al. (1988). In the area of satellite applications,

these dampers were referred to as nutation dampers (Fig. 1.1(b)).

Sakai et al. (1991) proposed a new type of liquid damper which was termed as a

tuned liquid column damper (TLCD) and described an application for cable-stayed bridge

towers. TLCDs were studied for wind excited structures by Honda et al. (1991); Xu et al.

(1992) and Balendra et al. (1995). Studies were also made for determining certain optimal

characteristics of these passive devices by Gao et al. (1997); Chang and Hsu (1999); and

Gao et al. (1999). The performance of TLCDs for seismic applications has been studied

by Won et al. (1996) and Sadek et al. (1998).

4

Figure 1.1 (a) Frahm Anti-rolling tanks (b) Nutation dampers in satelliteapplications

Most of the earlier studies concerned passive versions of TLCDs. This means that

the design involves no control of the damping characteristics. The damper was designed to

be optimal at design amplitudes of excitation but was non-optimal at other amplitudes of

excitation. In order to solve this difficulty, semi-active and active systems were proposed

by Kareem (1994); Haroun et al. (1994); and Abe et al. (1996). A similar active system

was proposed for TLDs by Lou et al. (1994), in which a baffle was placed inside the liquid

damper. The orientation of the baffle changed the effective length of the damper thereby

making it useful as a variable-stiffness damper.

Most structures under the influence of environmental loads experience both lateral

and torsional motions; therefore, one option is to have separate TLCDs each oriented in

particular directions, or to simply have a bi-directional U-tube (Fig. 1.2(a)). This new con-

figuration consists of a box container with vertical tubes like a candelabrum concept, or a

partitioned container, consisting of stacked U-tube sets ranging in both directions with a

common liquid base. The design eliminates the increased weight incurred by stacking two

(a) (b)

5

independent orthogonal U-tubes. One can also have orifices between the partitions

(Kareem, 1993).

Multiple Mass Dampers (MMDs) with natural frequencies distributed around the

natural frequency of the primary system requiring control have been studied extensively

by Yamaguchi and Harnpornchai (1993); Kareem and Kline, (1994); and Yalla and

Kareem (2000). Such systems lead to smaller sizes of TLCDs which would improve their

construction, installation and maintenance, and also offer a range of possible spatial distri-

butions in the structure. The tuned multiple spatially distributed dampers, offer a signifi-

cant advantage over a single damper since multiple dampers, when strategically located,

are more effective in mitigating the motions of buildings and other structures undergoing

complex motions (Bergman et al. 1990).

Shimizu and Teramura (1994) have proposed and reported implementation in

buildings, a new bi-directional tuned liquid damper with period adjustment equipment.

Other adjustments in shape have been proposed by researchers. To help the damper liquid

maintain its column shape, a V-shaped TLCD can be adopted as shown in Fig. 1.2(b) (Gao

et al. 1997). Another variation of TLCD is proposed, which is termed as LCVA, which

allows the column cross-section to be non-uniform. The performance of LCVA is com-

pared to that of TLCD and is found to be as or even more effective. Other advantages

include versatility and architectural adaptability, since its natural frequency is determined

not only by the length of the liquid column but also the area ratio of the horizontal and ver-

tical portion of the tube (Hitchcock et al. 1997; Chang and Hsu, 1998).

6

Figure 1.2 (a) Bi-directional TLCD (b) V-shaped TLCD

1.3 Applications

1.3.1 Ship/Offshore applications

The operation of a ship is affected by the motions and forces induced by rolling,

which can cause cargo damage, discomfort to passengers and reduce crew efficiency. The

use of devices for stabilizing motion in ships dates back to 1862 when W. Froude intro-

duced them followed by a practical application by P. Watts in 1880. In 1911, H. Frahm

proposed the use of a U-shaped tank as a roll stabilizer. Since early installations of such

passive anti roll tanks in the 1950s, this concept has been applied widely on commercial

vessels. The latest ship stabilizers are capable of both heel and roll control using water

tanks. The stabilizer is equipped with a roll indicator which is a microprocessor-based

computer that constantly calculates the root mean square roll, the heel and the average

apparent roll period (Honkanen, 1990) There are three basic types of passive/ controlled

passive tanks, which are used for roll stabilization in ships, as shown in Fig. 1.3, namely:

β

(b)

7

free surface, U-tube tanks and free flooding tanks. Free surface tanks are open tanks and

can have baffles/nozzle plates to provide internal damping. Different rolling frequencies

can be matched by changing the liquid level in the tank. U-tube tanks consist of two tanks

partially filled with liquid, with the air spaces connected by a duct and a crossover duct at

the tank bottom. Damping is provided by restricting the flow of air between the tanks. Free

flooding tanks are not as popular as other tank systems. It is similar to a U-tube tank except

that the tanks are not connected to one another; however, there is an airduct connecting the

top of the tanks. The tank natural period is set by the size of the inlet ducts relative to the

tank’s internal free surface. It is to be noted that all these stabilizers affect only the roll

amplitude and not the roll period (Sellars and Martin, 1992).

Figure 1.3 Types of passive/ controllable-passive tanks for ships

(a) Free-Surface Tank

(b) U-Tube Tank

(c) External Tanks

8

The excitations acting on most offshore structures are mostly due to wind, waves

and ocean currents. The sloshing motion of the liquids in storage tanks on fixed offshore

structures affects its dynamic response. By prudent selection of the tank geometry, plat-

form response may be reduced by using the tanks as dynamic vibration absorbers. There-

fore, no new equipment is required, but only optimum configuration of tankage that is

already required for storage of water, fuel, mud or crude oil (Vandiver and Mitome, 1978).

Passive, active and semi-active motion reduction systems such as fin and tank stabilizers,

variable mooring systems, controlled and uncontrolled air cushions, perforated pontoons

and columns with gas-spring-like tide tanks have been researched and applied to floating

platforms and other offshore structures like semi-submersibles (Ehlers, 1987). For floating

offshore structures like TLPs (tension leg platforms), the system with controllable moor-

ing tension and variable attaching position are considered. The horizontal low frequency

motions of TLPs can be reduced by active control using dynamic positioning system

thrusters. Other mechanisms include active pulse generators, open bottom tanks and pres-

surized passive air cushions. Control of offshore platforms using active mass dampers,

active tendons and thrusters can be found in Suhardjo and Kareem (1997).

Patel et al. (1985) considered a passive open bottom tank system in TLPs relying

upon the oscillations of the water columns in the tanks. A platform which lies on 4-6 col-

umns containing gas-spring-like tank systems is another consideration, (Delrieu, 1994).

Huse (1987) has studied free surface damping tanks to reduce resonant heave, roll and

pitch motions of semi-submersibles and other offshore structures. The damping tanks will

be situated at the water line and will be open to the sea through suitable restrictions

(Fig.1.4(a)). As shown in the figure, the tank is open to the sea and the atmosphere through

9

two openings. As the structure undergoes vertical motion, the sea water will flow in and

out of the tanks. By choosing a suitable opening size relative to the free surface area of the

tank, the water level in the tank will fluctuate with a certain phase lag relative to the verti-

cal motion of the structure. This will produce a damping force which would reduce the

resonant heaving motion of the structure. Ehlers (1987) considers a semi-active control

method for a structure equipped with open bottom tanks, but the valves in the upper part

can be opened or closed (Fig.1.4 (b)). The relative vertical motion between the water col-

umns in the tanks and the structure is influenced by the position of the valves because of

the air which is trapped in the tank when the valve is closed. These systems however, can

be used only for reduction of vertical motions and not horizontal motions. For some appli-

cations, this is very important since damping in the vertical mode is extremely small.

Figure 1.4 (a) Free surface damping tanks (b) Semi-active control for structurewith open bottom tanks

Damping tanks

Elevation

Plan

Detail

Valve

(a) (b)

10

1.3.2 Structural Applications

There have been several applications of TLDs in Japan, an example of which is the

MCC Aqua DamperTM which was installed in the Gold Tower in Chiba, Japan (Fig. 1.5).

The Aqua Damper is a cubic tank filled with water in which steel wire nets are installed

across the water movement. The TLD frequency is adjusted by changing the length of the

tank and the depth of water. The damping, on the other hand, is adjusted by manipulations

of the damping nets. The top floor of the 158 m tall Gold Tower was installed with 16 units

of the Aqua Damper totalling 10 tons of water (approximately 1% of the tower's weight)

and has witnessed a improved response of 50-60% of the original structural response prior

to the installation of the Aqua Damper (MCC Aqua Damper Pamphlet).

Figure 1.5 Aqua dampers (Courtesy: MCC Aqua damper literature)

A battery of TLDs were installed in the Shin Yokohama Prince Hotel (SYPH) in

Yokohama, Japan (Fig. 1.6). The TLD system prescribed was a multi-layer stack of 9 cir-

cular containers each 2 m in diameter and 22 cm high, yielding a total height of 2 m.

11

Details of the system can be found in Tamura et al. (1995). Before and after the installa-

tion of the TLD in March of 1992, full-scale measurements were taken to document the

performance of the auxiliary damping system. It was found that the RMS accelerations in

each direction were reduced 50% to 70% by the TLD at wind speeds over 20 m/s, with the

decrease in response becoming even greater at higher wind speeds. The RMS acceleration

without the TLD for the building was over 0.01 m/s2, which was reduced to less than

0.006 m/s2, defined by the ISO as the minimum perception level at 0.31 Hz. Similar instal-

lations are reported for Nagasaki airport tower, Tokyo international airport tower and

Yokohama marine tower (Tamura et al. 1995).

Figure 1.6 (a) Schematic of TLDs installed in SYPH (b) Actual installation in thebuilding (taken from Tamura et al. 1995)

(a) (b)

12

A TLCD has also been installed in the Hotel Cosima in Tokyo (Fig. 1.7). The hotel

is a 26 story steel building with a height of 106.2 meters. This building has a large height

to width ratio and is therefore wind sensitive. The foundation of the building is firmly con-

nected to the ground using high strength steel pretensioned grout anchors. In addition, a

super structure is adopted as the frame of the building in order to resist earthquakes and

wind loads. The 58 ton TLCD with pressure adjustment, called MOVICS, was installed in

the top floor and has been observed to reduce the maximum acceleration by 50-70% and

the RMS acceleration by 50% (Shimizu and Teramura, 1994). Other MOVICS systems

have been installed in the Hyatt Hotel in Osaka and the Ichida Building in Osaka.

Figure 1.7 (a) Liquid damper with pressure adjustment concept (b) Installed inHotel Cosima, Tokyo

(a) (b)

13

Recently, Liquid Dampers have been planned for the proposed Millennium Tower,

Tokyo Bay, Japan. Due to this supertall building’s exposure to typhoons, external damping

sources are needed to control the wind induced vibrations. In addition to massive steel

blocks at the top, there are water tanks with ducts between them. The water would provide

passive resistance under normal conditions, but under high winds, the sensors trigger a

pumping mechanism, changing the control mode from passive to active (Sudjic, 1993).

Figure 1.8 shows the schematic of the circular TLD concept in this tower.

Figure 1.8 Millennium tower: passive and active TLCD concept

14

A TLD is also planned to limit the wind induced motion of the proposed Shangai

Financial Trade Center in China. This building will have a square shaft with a diagonal

face that is shaved back (Fig. 1.9(a)). An aperture is cut out of the top to relieve aerody-

namic pressure (Engineering News Record, May 1996). Both the TLD and the aerody-

namic aperture will ensure to keep building motion within acceptable limits. TLDs are

also being considered for the newly proposed 2000 ft building in Chicago, namely, the 7

South Dearborn project.

Figure 1.9 (a) Shanghai Financial Trade Center (b) 7 South Dearborn Project

Liquid tanks are being used to reduce the aerodynamic forces, in particular the

torque components, which cause instability during construction of long-span bridges.

(Brancaleoni 1992; Ueda et al. 1992). Liquid vibration absorbers are also used in tall

(a) (b)

15

chimneys. These have been proven to be economical, can be easily adjusted to the physi-

cal and architectural requirements, and are extremely fail-safe. They are usually designed

as a part of the circular gangway or as a coupling body for the connecting forces of a

group of chimneys (Fig. 1.10).

Figure 1.10 TLDs installed in chimneys

1.4 Motivation of Present Work

A recent paper by Hitchcock et al. (1999) describes the full scale installation of a

bi-directional passive liquid column vibration absorber (LCVA) on a 67m steel frame

communications tower. The LCVA is a passive system with no orifice to control the damp-

ing. The authors observed that “At wind speeds less than approximately 10 m/s, the stan-

dard deviation of the tower acceleration before and after SLCVA system installation are

essentially the same due to the motion of the SLCVA liquid being insufficient to dissipate

significant vibrational energy. At wind speeds of approximately 20 m/s, the response of the

tower is reduced by almost 50% after installation of the SLCVA system.” This shows the

inadequacy of the passive systems to perform optimally at all levels of excitation. For e.g.,

16

at low amplitudes, the liquid velocity is insufficient to generate an optimal value of damp-

ing to reduce the motion substantially. On the other hand, at high amplitudes of excitation,

the damping introduced at the orifice may be more than the optimal and again the effi-

ciency of the TLCD decreases. Similar observations were made in both experimental and

full-scale studies of Tuned Sloshing Dampers (TLDs) which rely on the sloshing of the

liquid in a rectangular/cylindrical container to control the vibration of the primary struc-

ture.

In the proposed research, new models for TLDs and TLCDs are developed. It has

been acknowledged by researchers that the sloshing of liquid at high amplitudes is a non-

linear phenomenon. This work presents a new model using sloshing-slamming analogy of

TLDs based on impact characteristics. The main thrust of this research is to develop the

next generation of liquid dampers. Control concepts are introduced in order to correct

some of the problems inherent in the existing dampers, mainly the potential of liquid

dampers not being fully realized due to their damping being dependent on motion ampli-

tudes or the level of excitation. TLCDs are particularly attractive, in this regard, due to the

following reasons:

1. A mathematical model is available for the TLCD, due to which the tuning of the

damper is precise, and makes it amenable for semi-active and active control.

2. The amount of damping needed to suppress a particular vibration can be easily ascer-

tained and controlled through the orifice. The orifice opening ratio affects the headloss

coefficient which in turn affects the effective damping of the liquid damper. Propor-

tional valves can be actuated by a small voltage signal to obtain the required damping.

17

3. Arbitrariness of shape, giving it versatility and adaptability for housing in available

space, and flexibility in architectural and aesthetic appearance.

4. The TLCD can be tuned by changing its frequency of the TLCD by way of adjusting

the liquid column in the tube. This is an attractive feature should the tuning become

desirable in case of a change in the primary system frequency.

The advantages of liquid damper systems include low cost and maintenance

because no activation mechanism is required. The liquid damper systems are easily mobi-

lized at all levels of structural motion, whereas the mechanism activating a TMD must be

set to a certain threshold of excitation. The most important advantage, however is that such

containers can be utilized for building water supply, unlike a TMD where the dead weight

of the mass has no other functional use. A more elaborate cost analysis of the two systems

is presented in Chapter 8.

1.5 Organization of Dissertation

The next chapter discusses new modeling efforts for TLDs. A new sloshing-slam-

ming (S2) damper analogy has been developed for the sloshing dampers. This is based on

two approaches: firstly, numerical simulation of the differential equations involving

impact phenomenon; and secondly, explicitly including the impact characteristics in the

equations of motion. The equivalent linearization techinique is utilized to derive linear

models from the nonlinear ones.

In chapter 3, mathematical model of TLCD is examined in light of the equivalent

linearization technique. The optimum absorber parameters for TLCDs are determined for

various loading cases. The absorber parameters for multiple-TLCDs are also determined.

18

Chapter 4 presents a common phenomenon which occurs in coupled system,

namely, the beat phenomenon. The focus of this chapter is to mathematically understand

the beat phenomenon followed by experimental validation.

Chapter 5 discusses the development of semi-active strategies for TLCDs. The effi-

ciency of the semi-active algorithms is illustrated through the use of appropriate examples.

Chapter 6 discusses some of the experimental studies on TLDs. Impact character-

istics are derived based on experimental studies. A new type of testing method, namely the

hardware-in-the-loop methodology is presented as an new method for testing dampers..

Chapter 7 describes the experiments with TLCDs. Optimum absorber parameters

derived in chapter 3 are compared with experimental results. Experiments conducted to

show the validity of the semi-active scheme are also discussed.

Chapter 8 deals with cost and reliability analysis for a tall building serviceability

under wind loading. Design guidelines and practical considerations are also delineated.

Chapter 9 discusses some of the important conclusions drawn from the present research

and future work to be done in this area.

19

CHAPTER 2

MODELING OF SLOSHING

remember, when discoursing about water,to induce first experience, then reason.

- Leanardo Da Vinci

In this chapter, modeling of liquid sloshing in TLDs is presented. The first

approach is aimed at understanding the underlying physics of the problem based on a

“Sloshing-Slamming (S2)” analogy which describes the behavior of the TLD as a linear

sloshing model augmented with an impact subsystem. The second model utilizes certain

nonlinear functions known as impact characteristic functions, which clearly describe the

nonlinear behavior of TLDs in the form of a mechanical model. The models are supported

by numerical simulations which highlight the nonlinear characteristics of TLDs.

2.1 Introduction

The motion of liquids in rigid containers has been the subject of many studies in

the past few decades because of its frequent application in several engineering disciplines.

The need for accurate evaluation of the sloshing loads is required for aerospace vehicles

where violent motions of the liquid fuel in the tanks can affect the structure adversely

(Graham and Rodriguez, 1952; Abramson, 1966). Liquid sloshing in tanks has also

received considerable attention in transportation engineering (Bauer, 1972). This is impor-

tant for problems relating to safety, including tank trucks on highways and liquid tank cars

on railroads. In maritime applications, the effect of sloshing of liquids present on board,

20

e.g., liquid cargo or liquid fuel, can cause loss of stability of the ship as well as structural

damage (Bass et al. 1980). In structural applications, the effects of earthquake induced

loads on storage tanks need to be evaluated for design (Ibrahim et al. 1988). Recently

however, the popularity of TLDs as viable devices for structural control has prompted

study of sloshing for structural applications (Modi and Welt 1987; Kareem and Sun 1987;

Fujino et al. 1988).

2.1.1 Numerical Modeling of TLDs

The first approach in the modeling of sloshing liquids involves using numerical

schemes based on linear and/or non-linear potential flow theory. These type of models rep-

resent extensions of the classical theories by Airy and Boussinesq for shallow water tanks.

Faltinson (1978) introduced a fictitious term to artificially include the effect of viscous dis-

sipation. For large motion amplitudes, additional studies have been conducted by Lepelle-

tier and Raichlen (1988); Okamoto and Kawahara (1990); Chen et al. (1996) among

others. Numerical simulation of sloshing waves in a 3-D tank has been conducted by Wu

et al. (1998).

The model presented by Lepelletier and Raichlen (1988) recognized the fact that a

rational approximation of viscous liquid damping has to be introduced in order to model

sloshing at higher amplitudes. Following this approach, a semi-analytical model was pre-

sented by Sun and Fujino (1994) to account for wave breaking in which the linear model

was modified to account for breaking waves. Two experimentally derived empirical con-

stants were included to account for the increase in liquid damping due to breaking waves

and the changes in sloshing frequency, respectively. The attenuation of the waves in the

mathematical model due to the presence of dissipation devices is also possible through a

21

combination of experimentally derived drag coefficients of screens to be used in a numeri-

cal model (Hsieh et al. 1988). Additional models of liquid sloshing in the presence of flow

dampening devices are reported, e.g., Warnitchai and Pinkaew (1998). The main disadvan-

tage of such numerical models is the intensive computational time needed to solve the sys-

tem of finite difference equations.

Numerical techniques for modeling sloshing fail to capture the nonlinear behavior

of TLDs. This is due to the inability of theoretical models to achieve long time simulations

due to numerical loss of fluid mass (Faltinsen and Rognebakke, 1999). Moreover, it is very

difficult to incorporate slamming impact in a direct numerical method. Accurate predic-

tions of impact pressures over the walls of the tanks requires the introduction of local

physical compressibility in the governing equations. The rapid change in time and space

require special treatment which is currently unavailable in existing literature. However,

recent work in numerical simulation of violent sloshing flows in deep water tanks are

encouraging and represent the state-of-the-art in this area, e.g, Kim (2001). However, until

the numerical schemes are more developed, one has to resort to mechanical models for

predicting the sloshing behavior. The chief advantages of a mechanical model are savings

in computational time and a good basis for design of TLDs.

2.1.2 Mechanical Modeling of TLDs

For convenient implementation in design practice, a better model for liquid slosh-

ing would be to represent it using a mechanical model. This is helpful in combining a TLD

system with a given structural system and analyzing the overall system dynamics. Some of

the earliest works in this regard are presented in Abramson (1966). Most of these are lin-

ear models based on the potential formulation of the velocity field. For shallow water

22

TLDs, various mechanisms associated with the free liquid surface come into play to cause

energy dissipation. These include hydraulic jumps, bores, breaking waves, turbulence and

impact on the walls (Lou et al. 1980). The linear models fail to address the effects of such

phenomena on the behavior of the TLD.

Sun et al. (1995) presented a tuned mass damper analogy for non-linear sloshing

TLDs. The interface force between the damper and the structure was represented as a

force induced by a virtual mass and dashpot. The analytical values for the equivalent mass,

frequency and damping were derived from a series of experiments. The data was curve-fit-

ted and the resulting quality of the fit was mixed due to the effects of higher harmonics.

Other non-linear models have been formulated as an equivalent mass damper system with

non-linear stiffness and damping (e.g., Yu et al. 1999). These models can compensate for

the increase in sloshing frequency with the increase in amplitude of excitation. This hard-

ening effect is derived from experimental data in terms of a stiffness hardening ratio. How-

ever, none of these models explain the physics behind the sloshing phenomenon at high

amplitudes.

In contrast with the preceding models, Yalla and Kareem (1999) presented an

analogy which attempts to explain the metamorphosis of linear sloshing to a nonlinear

hardening sloshing system and the observed increase in the damping currently not fully

accounted for by the empirical correction for wave breaking. At high amplitudes, the

sloshing phenomenon resembles a rolling convective liquid mass slamming/impacting on

the container walls periodically. This is similar to the impact of breaking waves on bulk-

heads observed in ocean engineering. None of the existing numerical and mechanical

23

models for TLDs account for this impact effect on the walls of the container. The sloshing-

slamming (S2) is described in detail in the following section.

2.2 Sloshing-Slamming (S2) Damper Analogy

The sloshing-slamming (S2) analogy is a combination of two types of models: the linear

sloshing model and the impact damper model.

2.2.1 Liquid Sloshing

A simplified model of sloshing in rectangular tanks is based on an equivalent mechanical

analogy using lumped masses, springs and dashpots to describe liquid sloshing. The

lumped parameters are determined from the linear wave theory (Abramson, 1966). The

equivalent mechanical model is shown schematically in Fig. 2.1(a). The two key parame-

ters are given by:

; n=1, 2........ (2.1)

; n=1, 2...... (2.2)

where n is the sloshing mode; mn is the mass of liquid acting in that mode; ωn is the fre-

quency of sloshing; r = h/a where h is the height of water in the tank; a is the length of the

tank in the direction of excitation; Ml is the total mass of the water in the tank; and mo is

the inactive mass which does not participate in sloshing, given by .

Usually, only the fundamental mode of liquid sloshing (i.e., n = 1) is used for anal-

ysis. This model works well for small amplitude excitations, where the wave breaking and

mn M l8 2n 1–( )πr tanh

π3r 2n 1–( )3

----------------------------------------------- =

ωn2 g 2n 1–( )π 2n 1–( )πr tanh

a------------------------------------------------------------------------=

m0 M l mnn 1=

∞

∑–=

24

the influence of non-linearities do not influence the overall system response significantly.

This model can also be used for initial design calculations of TLDs (Tokarcyzk, 1997).

2.2.2 Liquid Slamming

An analogy between the slamming of liquid on the container walls and an impact

damper is proposed. An impact damper is characterized by the motion of a small rigid

mass placed in a container firmly attached to the primary system, as shown in Fig. 2.1(b)

(e.g., Masri and Caughey, 1966; Semercigil et al. 1992; Babitsky, 1998). A gap between

the container and the impact damper, denoted by d, is kept by design so that collisions take

place intermittently as soon as the displacement of the primary system exceeds this clear-

ance. The collision produces energy dissipation and an exchange of momentum. The pri-

mary source of attenuation of motion in the primary system is due to this exchange of

momentum. This momentum exchange reverses the direction of motion of the impacting

mass. The equations of motion between successive impacts are given by

(2.3)

(2.4)

The velocity of the primary system after collision is given as (Masri and Caughey, 1966)

(2.5)

where e is the coefficient of restitution of the materials involved in the collision, µ=m/Μ is

the mass ratio, x and z represent the displacement of the primary and secondary system,

and the subscripts ac and bc refer to the after-collision and before-collision state of the

M x C x Kx+ + Fe t( )=

mz 0=

xac1 µe–( )1 µ+( )

-------------------- xbcµ 1 e+( )

1 µ+( )-------------------- zbc+=

25

variables. The velocity of the impact mass is reversed after each collision. The numerical

simulation of this model is discussed in the next section.

Figure 2.1 (a) Equivalent mechanical model of sloshing liquid in a tank (b) Impactdamper model

2.2.3 Proposed Sloshing-Slamming (S2) Analogy

The experimental work on the sloshing characteristics of TLDs has been reported

by Fujino et al. (1992); Reed et al. (1998); Yu et al. (1999), etc. The key experimental

results are summarized in Figs. 2.2 (a) and (b), where the jump frequency and the damping

ratio are shown to increase with the amplitude of excitation. The jump phenomenon is typ-

ical of nonlinear systems in which the system response drops sharply beyond a certain fre-

quency known as the jump frequency. These results have been taken from Yu et al. (1999)

where the increase in damping and the change in frequency have been plotted as a function

of non-dimensional amplitude given as , where is the amplitude of excitation

and is the length of the tank in the direction of excitation.

(a) (b)

mo

m1

m2

mn

k1

k2

kn

c1

c2

cn

X1

X2

Xn

X(t)=Aexp(iωt)

m

M

d/2

z

x

Fe(t)

K

C

Ae a⁄ Ae

a

26

Figure 2.2 (a) shows that there is an increase in the jump frequency (κ ) at higher

amplitudes of excitation for the frequency ratios ( = ωe/ωf) greater than 1 suggesting a

hardening effect, where ωe is the frequency of excitation and ωf is the linear sloshing fre-

quency of the damper. It has been noted that as the amplitude of excitation increases, the

energy dissipation occurs over a broader range of frequencies. This feature points at the

robustness of TLDs. The coupled TLD-structure system exhibits certain nonlinear charac-

teristics as the amplitude of excitation increases. Experimental studies suggest that the fre-

quency response of a TLD, unlike a TMD, is excitation amplitude dependent. The

increased damping (introduced by wave breaking and slamming) causes the frequency

response function to change from a double-peak to a single-peak function. This has been

observed experimentally by researchers, e.g., Sun and Fujino, 1994.

Figure 2.2 Variation of (a) jump frequency and (b) damping ratio of the TLD withthe base amplitude (Yu et al. 1999).

γ f

(a) (b)

0.02 0.04 0.06 0.08 0.15

10

15

20

25

Dam

ping

ratio

(%)

Non−dimensional Amplitude Ae/a

0.02 0.04 0.06 0.08 0.10.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

Non−dimensional Amplitude Ae/a

Jump

fre

quen

cy r

atio

, κ

27

Figure 2.3 Frames from the sloshing experiments video at high amplitudes: a partof water moves as a lumped mass and impacts the container wall. (Video Courtesy:

Dr. D.A. Reed)

28

Figure 2.4 Schematic diagram of the proposed sloshing-slamming (S2) analogy

As will be shown herein, the experimental observations that at higher amplitudes,

the liquid motion is characterized by slamming/impacting of water mass (Fig. 2.3). This

includes wave breaking and the periodic impact of convecting lumped mass on container

walls. Some of the energy is also dissipated in upward deflection of liquid along the con-

tainer walls. The S2 damper analogy is illustrated schematically in Fig. 2.4. Central to this

analogy is the exchange of mass between the sloshing and convective mass that impacts.

This means that at higher amplitudes, some portion of the mass m1 (the linear sloshing liq-

uid), is exchanged to mass m2 (the impact mass), which results in a combined sloshing-

slamming action.

The level of mass exchange is related to the change in the jump frequency as

shown in Fig. 2.2(a). A mass exchange parameter is introduced, which is an indicator

M

K

CF(t)

X

Primary system(structure)

Secondary system(linear sloshingmode)

Secondary system(slamming mode)

m 2

z

m o

m 1k1c1

x1

mass exchange betweenthe two sub-systems

SLOSHING-SLAMMING DAMPER ANALOGY

Fe(t)

Ω

29

of the portion of linear mass m1 acting in the linear mode. Since the total mass is con-

served, this implies that the rest of the mass is acting in the impact mode. For example,

=1.0 means that all of the mass m1 is acting in the linear sloshing mode. After the mass

exchange has taken place, the new masses and in the linear sloshing mode and the

impact mode, respectively, are given by

(2.6)

(2.7)

At low amplitudes, there is almost no mass exchange, therefore, the linear theory

holds. However, as the amplitude increases, γ decreases and the slamming mass increases

concomitantly. Moreover, since m1 is decreasing, the sloshing frequency increases, which

explains the hardening effect. The mass exchange parameter can be related to the jump

frequency ratio. Since , therefore using Eq. 2.7, one can obtain

. The empirical relations as shown in Fig. 2.2(a) for relating the mass

exchange parameter to the amplitude of excitation can be introduced to the proposed

scheme. This scheme can be further refined should it become possible to quantify more

accurately the mass exchange between the sloshing and slamming modes from theoretical

considerations. The equations of motion for the system shown in Fig. 2.4 can be written as

(2.8)

Ω

m˜

1m˜

2

m˜

2m2 1 Ω–( )m1+=

m˜

1Ωm1=

ω˜

1

2 k1

m˜

1

------ω1

2m1

m˜

1

-------------= =

κ 1 Ω⁄=

M X C c1+( ) X K k1+( )X c1 x1– k1x1–+ + Fo ωet( )sin=

m1 x1 c1 x1 k1x1 c1 X– k1 X–+ + 0=

m2 z 0=

30

where . After each impact, the velocity of the convecting liquid is changed

in accordance with Eq. 2.5. An impact is numerically simulated at the time when the rela-

tive displacement between m1 and m2 is within a prescribed error tolerance of d/2, i.e.,

. In this study the error tolerance has been assumed as .

Since the relative displacements have to be checked at each time step, a time domain inte-

gration scheme is employed to solve the system of equations. In order to construct the fre-

quency response curves, the maximum steady-state response was observed at each

excitation frequency and the entire procedure was repeated for the complete range of exci-

tation frequencies.

2.2.4 Numerical Study

A numerical study was conducted using the parameters employed in the experi-

mental study (Fujino et al. 1992). These parameters are listed in Table 2.1. It should be

noted that the initial mass ratio, prior to the mass exchange, has been assumed to take on a

very small value, i.e., = 0.01, which is essential to realize the system in Fig. 2.4

described by Eq. 2.8. This assumption is not unjustified since experimental results show

the presence of nonlinearity in the transfer function, albeit small, even at low amplitudes

of excitation (e.g., at Ae = 0.1 cm, κ = 1.02). Figure 2.5 shows the changes that take place

in the frequency response functions as the mass exchange parameter is varied. This can

also be viewed as the amplitude dependent variation in the frequency response function. It

should be noted that the frequency response function undergoes a change from a double-

peak to a single-peak function at higher amplitudes of excitation. This model gives similar

results as Fujino et al. 1992, however, one has to note that this is a mechanical model as

Fo M Aeωe2

=

x1 z– ε± d 2⁄= ε d⁄ 106–

=

m2 m1⁄

31

opposed to a numerical model described in Fujino et al. 1992. These results demonstrate

that the frequency response function of the combined system derived from the sloshing-

slamming model is in good agreement with the experimental data both at low and high

amplitudes of excitation. Note that uncontrolled and controlled cases in Fig. 2.5 refer to

structure without and with TLD.

Figure 2.5 Comparison of experimental results with S2 simulation results: (a), (b):experimental results (Fujino et al. 1992); (c), (d): simulation results for = 1.0

and = 0.9

0.85 0.9 0.95 1 1.05 1.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalised response

Frequency Ratio

Uncontrolled

Controlled

Experimental Resultsfor low amplitudes of excitation

0.85 0.9 0.95 1 1.05 1.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalised response

Frequency Ratio

Uncontrolled

Controlled

Experimental Resultsfor high amplitudes of excitation

(a) (b)

0.85 0.9 0.95 1 1.05 1.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalised response

Frequency Ratio

Uncontrolled

Controlled

Numerical Simulationfor high amplitudes of excitation

Ω = 0.9

(c) (d)

0.85 0.9 0.95 1 1.05 1.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalised response

Uncontrolled

Controlled

Numerical Simulationfor low amplitudes of excitation

Ω = 1.00

Frequency ratio

ΩΩ

32

TABLE 2.1 Parameters of the model

2.2.5 Base Shear Force

It has been said before that the sloshing exhibits the presence of the jump phenom-

enon as the amplitude of excitation increases. This jump phenomenon is typical of most

nonlinear systems, for e.g., duffing, vanderpol oscillators, etc. A typical transfer function

of a nonlinear system is shown in Fig. 2.6(a). The non-dimensionalized experimental base

shear of TLD is plotted for various amplitudes of excitation in Fig 2.6(b) (Fujino et al.

1992). The presence of jump and hardening phenomenon can be clearly observed. Fur-

thermore, the range of frequencies over which the TLD is effective increases as the base

amplitude increases.

The S2 damper analogy cannot be directly applied to the liquid damper alone due

to the way it is formulated since to determine the post-impact velocity, one requires the

knowledge of the dynamics of the primary system. Therefore, in order to formulate a sin-

gle model which explains the experimental results for both damper characteristic and the

coupled structure-damper system, one can take advantage of certain impact characteristics

which describe the effects of nonlinearities imposed by the slamming mass. When repeti-

tive impacts occur as part of the vibratory motion of a linear system, the problem becomes

nonlinear. Having recognized this, one can search for such impact-characteristic functions

Parameter value Parameter value

Main mass M 168 Kg breadth of tank, b 32 cm

Main mass damping 0.32 % height of water, h 2.1 cm

Natural freq. of main mass 5.636 rad/s Coefficient of restitution, e 0.4

Length of tank, a 25 cm Impact Clearance d/(Fo/k) 20

Mass ratio m1/M 0.01 Initial mass ratio m2/m1 0.01

33

which would produce the same nonlinearities in the linear system. This is studied in the

next section.

Figure 2.6 (a) Jump phenomenon in nonlinear systems (b) Variation of the non-dimensionalized base shear force with the frequency ratio (experimental results

taken from Fujino et al. 1992).

2.3 Impact Characteristics model

In earlier section on sloshing-slamming damper analogy, the impact of the liquid

on the container walls was simulated using the solution of differential equations, also

known as the point-wise mapping method. The impact was modeled as a collision between

the slamming (impact) mass and the tank wall as a discontinuous function. However, from

the extensive work done in the area of vibro-impact systems, it is known that the dynamic

model studied is a limiting case of a hardening type of nonlinear system not only in terms

of structure but also function. It is well known in vibro-impact literature that one can

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20

5

10

15

20

25

Non-dimensional Sloshing Force

Ae=0.1cmAe=0.25cmAe=0.5cmAe=1.0cm

ωe/ωf

jumpfrequency

(a) (b)

ω

34

model the impact behavior by considering impact characteristics instead of simulating

impacts by numerical integration schemes (Pilipchuk and Ibrahim, 1997; Babitsky, 1998).

Hence, the basic character of the nonlinear behavior for vibro-impact systems obtained

using “exact” methods are similar to typical nonlinear hardening systems. In fact, a very

simple model can phenomenologically describe the interaction between the liquid mass

and the tank wall with a nonlinear function. Having recognized this, one can search for

such impact characteristic functions which would produce the same effect as the solution

of differential equations. This equivalence was demonstrated for harmonic as well as ran-

dom excitations (Masri and Caughey, 1965). It is to be noted that in this case, we will not

distinguish the liquid mass into impact mass and sloshing mass as done in the previous

section. The nonlinear model is developed for the entire liquid mass. Consider a oscillator

model given as:

(2.9)

where are the impact characteristics of the system, x is the displacement of the

lumped mass; is the velocity of the lumped mass; m, c and k are the mass, damping and

stiffness terms of the oscillator; Fo is the excitation amplitude = . One can assume

the impact characteristics as a combination of different nonlinear functions of the dis-

placement and velocity. In particular, Hunt and Crossley (1975) presented nonlinear

impact characteristics whereby one can interpret the coefficient of restitution as damping

in vibro-impact. They suggest the following form of the impact system:

(2.10)

mx cx kx mΦ x x,( )+ + + Fo ωet( )sin=

Φ x x,( )

x

mωe2Ae

Φ x x,( ) b1xp1 x b2x

p2+=

35

where b1, b2, p1 and p2 are parameters of the model. However, for the sake of keeping the

model simple, we assume the impact characteristics to be dependent on the displacement,

i.e., , while maintaining the damping to be a nonlinear function of the

amplitude of excitation. Accordingly Eq. 2.9 can be expressed in the following non-

dimensional form as:

(2.11)

where is the linear sloshing frequency and is the nonlinear damping of the

TLD. In this study, we will focus exclusively on shallow water TLDs, i.e. h/a < 0.15,

where h = depth of water and a = length of the tank in the direction of the excitation.

Various functions were considered for modeling the impact characteristics, e.g.,

hyperbolic sine function, power law function, and bi-linear hardening type function. Fig-

ure 2.7 shows the power law function used for modeling the impact characteristics. The

power law curve is used in this study since it allows for a finite value of the impact charac-

teristic function at the boundaries of the wall, i.e., . Note that the ordinate is the

non-dimensionalized displacement of the liquid sloshing mass.

The interaction force is written as a function of displacement of the sloshing mass:

(2.12)

(2.13)

where and η are the parameters of the impact characteristic function .

Φ x x,( ) Φ x( )=

x 2ω f ς Ae( ) x Φ x( )+ + ωe2Ae ωet( )sin=

ω f ς Ae( )

x a 2⁄±=

Feff x( ) Flin Fnon lin–+=

Feff x( ) keff x( )x mω f2

1 ϕ Ae( )x+2 η 1–( )[ ] x= =

ϕ Ae( ) Φ x( )

36

Figure 2.7 Non dimensional interaction force curves for different η

2.4 Equivalent Linear Models

Equivalent linear models are useful for initial approximation of the periodic solu-

tion of nonlinear systems. Moreover, one can represent these systems in transfer function

or state-space form to simplify the analysis by utilizing the linear systems theory. In the

next sub-sections we will briefly look at equivalent linear models when the external excita-

tion is harmonic and random.

2.4.1 Harmonic Linearization

The nonlinear impact characteristics can be linearized as,

(2.14)

The basic idea is to first define an error function and minimize it in the mean square sense

over an infinite time interval. One can write the error function as,

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Displacement of sloshing mass, x/(a/2)

Non

−dimensional Force, F

eff(x )

n=1 n=2 n=10

-a/2 a/2

keff(x)

a

x

Φ x x,( ) λ υ x ψ x+ +=

37

(2.15)

One can assume the solution of the form:

and (2.16)

Utilizing the fact that ; and

and recognizing the following properties of the solution:

; and (2.17)

one can arrive at the following equations

(2.18)

(2.19)

(2.20)

where and for harmonic motion.

2.4.2 Statistical Linearization

In this case also, one can define a error functional similar to Eq. 2.15 as:

(2.21)

Θ λ υ ψ, ,( ) 1

T--- Φ x x,( ) λ– υx– ψ x– 2

0

T

∫ dtT ∞→lim=

x t( ) ax ωt( )cos= x t( ) axω ωt( )cos=

λ∂∂ Θ λ υ ψ, ,( ) 0=

υ∂∂ Θ λ υ ψ, ,( ) 0=

ψ∂∂ Θ λ υ ψ, ,( ) 0=

1

T--- x t( )

0

T

∫ dtT ∞→lim 0=

1

T--- x t( )

0

T


1

T--- x t( ) x t( )

0

T


λ 1

T--- Φ x x,( )

0

T

∫ dtT ∞→lim=

υ 1

σx2

------1

T--- Φ x x,( )x t( )

0

T

∫ dtT ∞→lim=

ψ 1

σx2

------1

T--- Φ x x,( ) x t( )

0

T

∫ dtT ∞→lim=

σx ax 2⁄= σ x axω( ) 2⁄=

Θ λ υ ψ, ,( ) E Φ x x,( ) λ– υx– ψ x– 2( )≡

38

where represents the expected value of the random variable function .

Using similar procedure as before and recognizing that ; and

, one can obtain the following expressions:

(2.22)

(2.23)

(2.24)

where it is assumed that and are independent Gaussian processes with probability dis-

tribution function defined by,

(2.25)

and the nonlinear function can be represented in a separable form, i.e.,

(2.26)

In the case of a power law nonlinearity given by , using Eqs. 2.18-2.20,

one can obtain the coefficients of equivalent linearization (for harmonic excitation) as,

; and (2.27)

E g x x,( )( ) g x x,( )

E xx( ) 0= E x2( ) σx

2=

E x2( ) σx

2=

λ Φ u( )w u( ) ud

∞–

∞

∫=

υ 1

σx2

------ uΦ u( )w u( ) ud

∞–

∞

∫=

ψ 1

σ x2

------ uΦ u( )w u( ) ud

∞–

∞

∫=

x x

w u( ) 1

σu 2π-----------------

u2

–

2σu2

---------

exp=

Φ x x,( ) Φ x( ) Φ x( )+=

Φ x( ) x2η 1–

=

λ η2--- 1–

ax2η 1–

= υ η2---

ax2η 2–

= ψ 0=

39

and for random excitation, using Eqs. 2.22-2.24,

; and (2.28)

The range of validity of this equivalent linearizations is discussed in the next chap-

ter in the context of TLCDs.

2.5 Concluding Remarks

In this chapter, a sloshing-slamming (S2) damper analogy of TLD is presented.

This analogy presents insights into the underlying physics of the problem and reproduces

the dynamic features of TLDs at both low and high amplitudes of excitation. At low

amplitudes, the S2 damper model serves as a conventional linear sloshing damper. At

higher amplitudes, the model accounts for the convection of periodically slamming

lumped mass on the container wall, thus characterizing both the hardening feature and the

observed increase in damping.

Next, based on the understanding of the sloshing and impact of the liquid, explicit

impact characteristics are introduced into the equations of motion in order to derive a sim-

pler mechanical model. These impact characteristics introduce the necessary nonlineari-

ties into the system. Such mechanical models will be useful for design and analysis of

TLD systems. Finally, equivalent linearization technique is used to derive linear models

based on the nonlinear TLD models.

λ 0= υ σx2η

2η 2k 1–( )–( )k 1=

η

∏= ψ 0=

40

CHAPTER 3

TUNED LIQUID COLUM DAMPERS

There is nothing more practical than a good theory- T. Von Karman

In this chapter, tuned liquid column dampers (TLCDs) are discussed. First, the

mathematical model of the TLCD is presented and the equivalent linearized model is com-

pared with the nonlinear model. Next, numerical optimization studies are conducted to

determine the important parameters for optimum TLCD performance, namely, the tuning

ratio and the damping ratio. In a later section, similar values of optimal parameters have

been determined for multiple tuned liquid column dampers (MTLCDs).

3.1 Introduction

In the classical work on the Dynamic Vibration Absorber (also known as TMD),

Den Hartog (1956) derived expressions for the optimum damping ratio and tuning ratio

(i.e., ratio of the absorber frequency to the natural frequency of the primary system) for a

coupled SDOF-TMD system subjected to harmonic excitation. The optimum absorber

parameters which minimize the displacement response of the primary system were found

to be simple functions of the mass ratio (ratio of mass of structure and damper).

McNamara (1977) reported design of TMDs for buildings with attention to experimental

studies and design considerations. Ioi and Ikeda (1978) developed empirical expressions

to determine correction factors for optimum parameters in the case of lightly damped

structures. Randall et al. (1981) and Warburton and Ayorinde (1980) further tabulated and

41

developed design charts for the optimum parameters for specified mass ratios and different

primary system damping.

Previous work has been done with the aim of deriving optimum parameters for

TLCDs. Abe et al. (1996) derived optimum parameters using perturbation techniques.

Gao et al. (1997) studied numerically the optimization of TLCDs for sinusoidal excita-

tions. Chang and Hsu (1998) have also discussed optimal absorber parameters for TLCDs

for undamped structure attached to a TLCD. These dampers were found to be effective for

wind loading (Xu et al. 1992; Balendra et al. 1995) and earthquake loading (Won et al.

1996; Sadek et al. 1998).

In this chapter, similar expressions have been developed and parameters have been

tabulated for undamped and damped primary systems equipped with TLCDs. Usually, in

the design of TMDs for wind and earthquake excitations, the optimum parameters are cho-

sen to be those obtained by assuming a white noise random excitation. In this study, in

addition to the white noise excitation, a set of filtered white noise (FWN) excitation has

been considered for evaluating the optimal absorber parameters.

Optimum parameter analysis of MTLCDs is similar to MMDs (multiple mass

dampers), where the important design parameters are the frequency range of the dampers

and the damping ratio of the dampers (Yamaguchi and Harnpornchai, 1993; Kareem and

Kline, 1995). MTLCDs are useful because the efficiency is higher as compared to a single

TLCD and moreover, the sensitivity to the tuning ratio is diminished. Multiple liquid

dampers have also been studied by Fujino and Sun (1993); Sadek et al. (1998) and Gao et

al. (1999).

42

3.2 Modeling of Tuned Liquid Column Dampers

Figure 3.1 shows the schematic of the TLCD mounted on a structure represented

as a SDOF system.

Figure 3.1 Schematic of the Structure-TLCD system

The equation describing the motion of the fluid in the tube is given as (Sakai et al. 1989),

(3.1)

where the natural frequency of oscillations in the tube are given by . The equa-

tion of motion for the primary system (structure) is given as,

(3.2)

where = response of the primary system (structure); = response of the liquid damper

(TLCD); Ms = mass of the primary system; Ks = stiffness of the primary system; Cs=

damping in the primary system = ; = damping ratio of the primary system;

= natural frequency of the primary system; ρ= liquid density; A = cross sectional area

Ms

ξ

Xs

Xf

F(t)

Ks

Cs

b

lheadlosscoefficient

xf

Fe(t)

Xs

ρAl x f t( ) 1

2---ρAξ x f t( ) x f t( ) 2ρAgx f t( )+ + ρAb X s t( )–=

ω f2gl

------=

M s ρAl+( ) X s t( ) ρAbx f t( ) Cs X s t( ) Ks X s t( )+ + + Fe t( )=

X s x f

2M Sζ sωs ζ s

ωs

43

of the tube; l = total length of the liquid column; b = horizontal length of the column; g =

gravitational constant; =coefficient of headloss of the orifice. The two equations can be

combined into the following matrix equation:

, , (3.3)

where α = length ratio = b/l; mf = mass of fluid in the tube = ρAl; cf = equivalent damp-

ing of the liquid damper = ; ζf = damping ratio of TLCD; = natural fre-

quency of the liquid damper; kf is the stiffness of the liquid column = 2ρAg, and is

the external excitation. The constraint on Eq. 3.3 is placed so as to ensure that the liquid in

the tube maintains the U-shape and the water does not spill out of the tube, thereby

decreasing the dampers effectiveness.

3.2.1 Equivalent Linearization

Using the expressions derived in section 2.4, one can obtain equivalent linear

damping for the nonlinear TLCD damping (cf). In particular, using Eq. 2.20 one can

obtain:

(3.4)

where the excitation force is harmonic, , while for random

excitation, using Eq. 2.24:

(3.5)

ξ

M s m f+ αm f

αm f m f

X s

x f

Cs 0

0 c f

X s

x f

Ks 0

0 k f

X s

x f

+ +Fe t( )

0

= x fl b–( )

2---------------≤

2m f ω f ζ f ω f

Fe t( )

c f

4ρAξ Aeωe

3π---------------------------=

Fe t( ) m f Aeωe2 ωet( )sin=

c f2

π---ρAξσ x f

=

44

where is the standard deviation of the liquid velocity. This analytical model will be

used in the rest of the study.

3.2.2 Accuracy of Equivalent linearization

Since the equivalent damping will be used in later studies on TLCDs, it is useful to

study the accuracy of the equivalent linearization method. The two equations, written in

non-dimensional form, are as follows,

Nonlinear System:

(3.6)

Equivalent Linear System:

(3.7)

where µ is the mass ratio = . The nonlinear equations were simulated using the

nonlinear differential equation solver in MATLABTM, while for the linear equation, an

iterative method was used to solve the equivalent linearized equations. In the second case,

one first assumes a value for , simulates the linear system, recalculates the value of

and iterates till the response quantity converges to an acceptable value. In this study,

the main focus is to examine the error between the exact nonlinear and linearized equation

for variations in the parameter ξ. The excitation used is a band-limited Gaussian white

noise with a pulse width of 0.002 seconds and a spectral intensity of 0.01 m2 /sec3/Hz.

σ x f

1 µ+ αµα 1

X s

x f

2ωsζ s 0

0ξ x f

2l------------

X s

x f

ωs2

0

0 ω f2

X s

x f

+ +

Fe t( )M s

-------------

0

=

1 µ+ αµα 1

X s

x f

2ωsζ s 0

0 2ω f ζ f

X s

x f

ωs2

0

0 ω f2

X s

x f

+ +

Fe t( )M s

-------------

0

=

m f M s⁄

σx f

σ x f

45

Figure 3.2 shows the comparison of the response of the structure and damper for various

headloss coefficients. The maximum error between the nonlinear and the equivalent linear

system is about 2%. Figure 3.3 shows the time histories of the various response quantities

for ξ = 75.

Figure 3.2 Exact (Non-linear) and Equivalent Linearization results

Figure 3.3 Time histories for ξ = 75

0 20 40 60 80 1003.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4

Coeff. of headloss0 20 40 60 80 100

0

1

2

3

4

5

6

7

8

9

Coeff. of headloss

Exact (Nonlinear)Equivalent Linear

(ξ) (ξ)

σX s

σ x f

0 20 40 60 80 100−8

−6

−4

−2

0

2

4

6

8

10

time (sec)

X s

0 20 40 60 80 100−3

−2

−1

0

1

2

3

time (sec)

X f

Exact (Nonlinear)Equivalent Linear

Xs xf

46

3.3 Optimum Absorber Parameters

It has been observed from numerical studies that the headloss coefficient affects

the structure’s frequency response curve. As the head-loss coefficient (ξ) increases, the

response curve changes from a double hump curve to a single hump curve (Fig. 3.4).

Numerical studies conducted by the author indicate that an optimal damping level exists

for the TLCD which depends on the excitation level and the head loss coefficient. The first

task, however, is to obtain the optimum damping ratio and tuning ratio of the absorber.

Figure 3.4 Variation of dynamic magnification factor with the head-loss coefficientand frequency ratio for a TLCD

The analytical model was discussed in section 3.2. One can define transfer functions in the

Laplace domain, namely and , where the following

expressions are obtained :

0.80.9

11.1

1.21.3

0

20

40

60

80

3

4

5

6

7

8

9

10

DynamicMagnificationRatio

Coefficient of Head LossFrequency ratio

H X sFs( )

X s s( )Fe s( )--------------= H x f F s( )

x f s( )Fe s( )-------------=

47

and

where for base excitation in which case is the relative displacement, and

for primary system excitation where corresponds to the absolute displacement.

One can compute the response quantities of interest using random vibration analysis. In

particular, we are interested in the variance of the primary system displacement and the

variance of the liquid velocity in the TLCD. The response quantities are obtained as,

(3.8)

(3.9)

where is the power spectral density of the forcing function. Equation 3.9 is useful

in evaluating the equivalent damping of the TLCD from Eq. 3.5. A simplified solution to

the integral for random vibration analysis has been used to evaluate Eqs. 3.8 and 3.9 (see

Appendix A.1 for details). Three representative forcing functions have been studied here,

as listed in Table 3.1. The optimal absorber parameters are derived for each individual case

of white noise and FWN excitations. It will be shown in subsequent sections that typical

wind and earthquake excitations can be approximated through the use of such filters.

H X sFω( )

∆µαω2 ω–2

2ζ f ω f iω( ) ω f2

+ +

ω–2

1 µ+( ) 2ζ sωs iω( ) ωs2

+ +[ ] ω–2


+ +[ ] ω4α2µ+

-----------------------------------------------------------------------------------------------------------------------------------------------------------------------=

H x f F ω( ) αω2 ∆+

ω–2

1 µ+( ) 2ζ sωs iω( ) ωs2

+ +[ ] ω–2


+ +[ ] ω4α2µ+

-----------------------------------------------------------------------------------------------------------------------------------------------------------------------=

∆ 1= X s

∆ 0= X s

σX s

2H X sF

ω( ) 2SFF ω( ) ωd

∞–

∞

∫=

σ x f

2 ω2H x f F ω( )

2SFF ω( ) ωd

∞–

∞

∫=

SFF ω( )

48

TABLE 3.1 Example forcing functions

Based on these three excitation models, optimal parameters have been obtained for

TLCD attached to damped and undamped primary systems. It has been seen that one can

derive an explicit expression for the case of undamped structure-TLCD system subjected

to white noise. However, for damped systems and/or other excitations, the development of

closed-form solutions is challenging. This is because some characteristics of the classical

damper system, like invariance points, do not exist when damping is introduced in the pri-

mary system (Den Hartog, 1956). Therefore, the optimal absorber parameters (i.e., and

) are obtained numerically for these cases. The optimal conditions are

obtained by setting:

; (3.10)

One can obtain and by solving the two conditions given by Eq. 3.10

In the case of tuned mass dampers, a detailed analysis was carried out by Warbur-

ton (1982) to determine optimum damper parameters for the case of random excitations

(represented by white noise), with excitation applied to the structure (as in the case of

Type of Excitation Spectrum Type of excitation

White Noise Excitation primary system excitation

First Order Filter (FOF) primary system excitation

Second Order Filter

(SOF)

primary system excitation

and/or

base excitation

SFF ω( )

S0

S0

ν1

2 ω2+

---------------------

S0 c1

2ω2d1

2+

b1

2 ω2–[ ]

2a1

2ω2+

-------------------------------------------------------

ζ f

γ ωf ωs⁄=

σxs

2∂ζ f∂

------------ 0=σxs

2∂γ∂

------------ 0=

ζopt γopt

49

wind) or as a base acceleration (as in the case of ground motion). The design of TMDs for

wind and earthquake applications, therefore, uses these design expressions for the optimal

parameters. In the next sub-sections, the theory to determine the optimal parameters is

presented for the example forcing functions listed in Table 3.1.

3.3.1 White Noise excitation

The response integral in Eqs. 3.8 and 3.9 can be cast in the following form:

(3.11)

Details of the integration scheme can be found in Appendix A.1.

Undamped Primary System

Solving the two optimization conditions in Eq. 3.10 and setting yields:

; (3.12)

In case, one can assume the tuning ratio to be equal to one, one can obtain a sim-

pler expression for the optimal damping given by,

(3.13)

This is justifiable because for the low mass ratios of the order 1-2% practical for tall build-

ings, the tuning ratio is close to one, and in this case the optimal damping coefficient given

by Eq. 3.13 approximates Eq. 3.12 quite well. Similar expressions exist for an optimal

damping coefficient and tuning ratio of a TMD given by Warburton and Ayorinde (1980),

σxs

2S0

Ξn ω( ) ωd

Λn iω–( )Λn iω( )---------------------------------------

∞–

∞

∫=

ζ s 0=

ζoptα2---

µ 1 µ α2–

µ4---+

1 µ+( ) 1 µ α2µ2

----------–+

----------------------------------------------------= γopt

1 µ 1α2

2------–

+

1 µ+--------------------------------------=

ζopt1

2---

µ µ α2+( )

1 µ+( )------------------------=

50

; (3.14)

Note that in all cases considered, the optimum damping coefficient is independent

of the value of S0, the intensity of white noise excitation. It is noteworthy that Eq. 3.14

reduces to Eq. 3.12 as approaches 1. Comparison of optimal parameters under different

optimization criteria are summarized in Table 3.2 for TMDs and TLCDs. Figure 3.5 shows

the variation of optimum parameters as a function of the mass ratio. As the length ratio

increases, the damping ratio increases because there is more mass in the horizontal portion

of the TLCD. This contributes to indirect damping, which implies that it is better to keep

the length ratio as high as possible without violating the constraints of the TLCD or the

limitations of structural/architectural considerations.

Figure 3.5 Comparison of optimum absorber parameters for a TLCD withvarying α and a TMD.

ζopt1

2---

µ 13µ4

------+

1 µ+( ) 1µ2---+

------------------------------------= γopt

1µ2---+

1 µ+-----------------=

α

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

α =1.0 and TMD curve

α = 0.1

mass ratio, µ

Optimum damping ratio of the absorber

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.93

0.94

0.95

0.96

0.97

0.98

0.99

1

α = 0.1

α =1.0 and TMDcurve

mass ratio, µ

Optimum tuning ratio of the absorber

51

Damped Primary System

As discussed earlier, it is not convenient to obtain a closed-form solution for opti-

mum damper parameters for a damped primary system; therefore, it must be estimated

numerically (Warburton, 1982). These computations have been conducted for = 1, 2

and 5% and µ= 0.5, 1, 1.5, 2 and 5% and optimum absorber parameters are presented in

Table 3.3.

Table 3.3 shows that as the mass ratio increases, also increases. Equation 3.12

verifies this for undamped case, since it is approximately proportional to the square root of

the mass ratio. The tuning ratio also decreases as the mass ratio and the damping in the

primary system increase, which is consistent with the results obtained for tuned mass

TABLE 3.2 Comparison of optimal parameters for TMD and TLCD

Case number andparameter optimized

TMD TLCD

1 Random

Force act-

ing on

Structure

2 Random

accelera-

tion at the

base

3 Random

Force act-

ing on

Structure

same as

case 2

same as case 2 same as case 2 same as case 2

4 Random

accelera-

tion at the

base

same as

case 1

same as case 1 same as case 1 same as case 1

γopt ζopt γopt ζopt

X s2⟨ ⟩ 1

µ2---+

1 µ+----------------- 1

2---

µ 13µ4

------+

1 µ+( ) 1µ2---+

------------------------------------

1 µ 1α2

2------–

+

1 µ+--------------------------------------

α2---

µ 1 µ α2–

µ4---+

1 µ+( ) 1 µ α2µ2

----------–+

----------------------------------------------------

X s2⟨ ⟩ 1

µ2---–

1 µ+---------------- 1

2---

µ 1µ4---–

1 µ+( ) 1µ2---–

------------------------------------

1 µ 13α2

2---------–

+

1 µ+-----------------------------------------

α2---

µ 1 µ– 3α2µ4---+

1 µ+( ) 1 µ 3α2µ2

-------------–+

--------------------------------------------------------

X s2

⟨ ⟩

X s2

⟨ ⟩

ζ s

ζopt

52

dampers. It is observed that for small values of , is not affected; therefore for a

lightly damped system, the optimum absorber parameters derived for an undamped pri-

mary system are valid. For higher levels of damping in the primary system, one can derive

empirical expressions for the optimum damping ratio as a function of the primary system

damping ratio.

TABLE 3.3 Optimum parameters for white noise excitation for different massratios.

3.3.2 First order filter (FOF)

The forcing function for a FOF has a spectrum given by,

(3.15)

This type of function can be used to approximate wind-induced positive pressures for the

alongwind loading. Figure 3.6 (a) shows the transfer functions of the first order filter with

different values of the parameter . Also shown for reference is the transfer function of

the primary system. Table 3.4 gives the optimum absorber parameters for these first order

filters. Note that when =10, the optimum parameters are the same as those obtained for

white noise, since the filter is fairly uniform like white noise excitation around the natural

Undampedprimary system 1% Damping 2% Damping 5% Damping

γopt γopt γopt γopt

µ=0.5% 0.9965 0.0317 0.9962 0.0317 0.9958 0.0317 0.995 0.0317

µ=1% 0.993 0.0448 0.9925 0.0448 0.9921 0.0448 0.9908 0.0448

µ=1.5% 0.9896 0.0547 0.989 0.0547 0.9885 0.0547 0.9869 0.0547

µ=2% 0.986 0.0631 0.9855 0.0631 0.985 0.0631 0.983 0.0631

µ=5% 0.966 0.0986 0.965 0.0986 0.964 0.0986 0.962 0.0986

ζ s ζopt

ζopt ζopt ζopt ζopt

SFF ω( )S0

ν1

2 ω2+

---------------------=

ν1

ν1

53

frequency of the primary system. However, for other cases (e.g., = 0.1 and 1), the opti-

mum parameters are slightly different. The effect is more pronounced in the case of the

tuning ratio and increases as the damping in the primary system increases. Optimum

parameters have been computed for ν1 = 1 and tabulated in Table 3.5. Though the optimal

parameters can be obtained through the simultaneous solution of the two non-linear equa-

tions resulting from Eq. 3.10, the task becomes computationally intensive for the first and

second order filters. In this numerical study, optimal parameters were obtained by utilizing

the MATLAB optimization toolbox (Grace, 1992).

(These values are computed for undamped primary system with µ =1%)

TABLE 3.5 Optimum absorber parameters for FOF for various mass ratios.

TABLE 3.4 Optimum absorber parameters for FOF for different parameter ν1

parameter offirst order filter γopt

ν1 = 0.1 0.991 0.04477

ν1 = 1 0.992 0.04476

ν1 = 5 0.9925 0.04483

ν1 = 10 0.993 0.04482

v1 = 1



µ=0.5% 0.993 0.03197 0.992 0.03190 0.991 0.03185 0.988 0.0317

µ=1% 0.992 0.04476 0.991 0.04474 0.990 0.04470 0.987 0.04456

µ=1.5% 0.986 0.05484 0.985 0.05476 0.984 0.05468 0.979 0.0545

µ=2% 0.984 0.0630 0.983 0.0629 0.9815 0.06287 0.978 0.0626

µ=5% 0.962 0.0980 0.960 0.09795 0.958 0.0978 0.953 0.09727

ν1

ζopt


54

Figure 3.6 Transfer function of the filters and the primary system: (a) first orderfilters (b) second order filters

3.3.3 Second order filter (SOF)

A general second order filter studied here has the following spectral description,

(3.16)

where a1, b1, c1 and d1 are the parameters of the filter. Second order filters can be used to

represent earthquake and wind excitations. For earthquake representation, the excitation

acts at the base of the structure, while for wind representation, the excitation acts on the

structure. The expression in Eq. 3.16 also describes the well known Kanai-Tajimi spec-

trum (Kanai, 1961; Tajimi, 1960):

(3.17)

where is the dominant ground frequency and is the ground damping factor.

10−2

10−1

100

101

10−2

10−1

100

101

102

103

frequency rad/sec

Magnitude of transfer function

Transfer function of the primary system

Filter parameter

b1 =6b1 =10b1 =15b1 =20

10−1

100

101

10−2

10−1

100

101

102

Transfer Function of primary system

Filter parameter

frequency rad/sec

Magnitude of transfer function

v=0.1v=1v=5v=10

Filter Parameter_____ ν1 = 0.1-------- ν1 = 1.......... ν1 = 5_._._. ν1= 10

SFF ω( )S0 c1

2ω2d1

2+

b1

2 ω2–[ ]

2a1

2ω2+

-------------------------------------------------------=

SFF ω( )S0 1 4ζg

2 ωωg------

2

+

1ωωg------

2

–2

4ζg2 ω

ωg------

2

+

---------------------------------------------------------------------=

ωg ζg

55

Similarly, the across-wind excitation can be modeled as a FWN using a second

order filter. Kareem (1984) has proposed the following empirical expression for the spec-

tral density of the across-wind force for square buildings:

for

= for (3.18)

where ; ;

is the shedding frequency = ; B is the breadth of the building; is the mean

speed at height z; S is the Strouhal number; is the mean square value of the fluctuat-

ing across-wind force; is the exponent term in the power law of the wind velocity pro-

file; H is the height of the building; is the band width coefficient = , where I(z) is

the turbulence intensity at height z; and δ = 0.9. Details of this model can be found in

Kareem (1984). This across-wind loading model can also be represented by Eq. 3.16.

The magnitude of the transfer function of the filter given by Eq. 3.16 is shown in

Fig. 3.6 (b) for parameters a1 = 0.01, c1 =1, d1 =10 and varying b1 = 6, 10, 15 and 20.

Table 3.6 shows how the optimal parameters are influenced as the filter parameter b1

changes. As b1 increases, the assumption of purely white noise becomes valid and the

solution approaches that for the white noise case. The other parameters have been kept the

same and optimal parameters have been computed for damped and undamped cases (Table

3.7).

nSFF z n,( )

σ f2

-------------------------- αoβonns-----

δ= n ns≤

αoβonns-----

3.0

n ns≥

αob

1nns-----

2

–2

2bnns-----

2

+

-------------------------------------------------------------= βo 1.321

3α-------

0.5

0.154 1zH-----–

3.5

+= ns

SU z( )B

---------------- U z( )

σFF2

α

b 2I z( )

56

TABLE 3.6 Optimum absorber parameters for SOF for different values of b1

(All the other parameters are kept constant a1 = 0.01, c1 =1, d1 =10, =0.02 and =0.05)

TABLE 3.7 Optimum absorber parameters for SOF for various mass ratios.

As in previous cases, decreases as the damping in the primary system

increases and increases as the mass ratio increases; however, the damping in the primary

system affects more in this case than in the case of white noise. In addition, the tun-

ing ratio slightly departs from γ =1.00 as the damping in the primary system increases.

3.3.4 EXAMPLE

The optimum parameters for a TLCD placed on an eight story structure subjected

to an earthquake excitation are determined in this example using the theory presented in

the previous section. The parameters of the building stories considered are: floor mass =

345.6 tons, elastic stiffness = 34040 kN/m and internal damping coefficient = 2937 tons/

sec, which corresponds to a 2% damping for each vibrational mode of the structure. The

parameter ofSOF γopt

b1 = 6 1.05 0.1111

b1 = 10 1.01 0.0702

b1 = 15 1.00 0.0572

b1 = 20 0.995 0.0524

a1 = 0.01b1 = 36c1=1d1=10



µ=0.5% 1.04 0.1510 1.04 0.1401 1.045 0.1299 1.05 0.0956

µ=1% 1.04 0.1559 1.04 0.1450 1.045 0.1350 1.05 0.1008

µ=1.5% 1.04 0.1606 1.04 0.1498 1.045 0.1399 1.05 0.106

µ=2% 1.04 0.1654 1.04 0.1546 1.045 0.1448 1.05 0.1111

µ=5% 1.04 0.1927 1.04 0.1821 1.045 0.173 1.05 0.1406

ζopt

µ ζ s


ζopt

ζopt

57

computed natural frequencies are 5.79, 17.18, 27.98, 37.82, 46.38, 53.36, 58.53 and 61.69

rad/sec. The base excitation is modeled by the Kanai-Tajimi spectrum given in Eq. 3.17

with the parameters = 10.5 rad/sec and = 0.317. The parameters of the general sec-

ond order filter can be related to these as follows: and . The

mass of the damper has been taken as 2% of the first generalized mass of the structure. In

Table 3.8, the optimum design damper parameters for the TMD have been compared with

TLCD parameters, both under the white noise and the SOF excitations. It is noted that

there are significant differences in the optimum absorber parameters, justifying the inclu-

sion of the anticipated loading in the optimization process for the damper design.

3.4 Multiple tuned liquid column dampers (MTLCDs)

Multiple units of TLCDs can be incorporated in a structural system at one location

or distributed spatially. In this system, the natural frequencies of the TLCDs are distrib-

uted over a range of frequencies. The advantages of a distributed system is that it is more

robust and effective for excitation frequencies distributed over a wide frequency band. In

the following study, MTLCD configuration design parameters are evaluated.

The primary system is represented as a single degree of freedom (SDOF) system

and the secondary system, in this case, is the system of MTLCDs. The equations of

motion of the SDOF-MTLCD system (Fig. 3.7) can be written in a matrix notation as:

TABLE 3.8 Optimum absorber parameters

TMD TLCD ( white noise) TLCD ( SOF)

γopt0.98 0.985 1.027

7.3 % 6.31 % 6.51 %

ωg ζg

a c 2ζg= = b d ωg= =

ζopt

58

(3.19)

where ; ;

; and are (n, n) diagonal matrices similar to .

The transfer function of the primary system is obtained by non-dimensionalising Eq. 3.19,

and the transfer function for each TLCD is given by,

; n=1..N

Figure 3.7 MTLCD configuration

The analysis of MTLCDs is similar to MMDs (multiple mass dampers), where the

important design parameters are the frequency range and damping ratio of the dampers

(Kareem and Kline, 1995). The frequency range is defined as the total frequency span of

ms m fT

m f m

X s

x fn

Cs 0

0 ceqn

X s

x fn

Ks 0

0 keqn

X s

x fn

+ +Fe t( )

0=

ms M s m fnn 1=

N

∑+= m fT α m f 1 m f 2 … m fN=

m

m f 1 0 0 0

0 m f 2 0 0

0 0 … …0 0 … m fN

= ceqn keqn m

H X S F ω( ) 1

ω–2

1 µ fnn 1=

N

∑+

2ζ sωs iω( ) ωs2

+ + α2ω4 µ fn

ω–2

2ζ fnω fn iω( ) ω fn2

+ +[ ]--------------------------------------------------------------------

n 1=

N

∑+

------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------=

H x fnF ω( )α iω2

H X S F ω( )

ω–2

2ζ fnω fn iω( ) ω fn2

+ +[ ]--------------------------------------------------------------------=

Ms

Xs

Fe(t)

Ks

Cs

. . . .

range of MTLCDs

. . . . . .

(∆ω)

ωf1ωfi

ωfN

59

the MTLCDs given as . The central damper (n = (N+1)/2) is tuned

exactly to the natural frequency of the primary system. It is assumed that N is an odd num-

ber in this analysis. The frequency of each damper can be written as,

;

;

;

A numerical study has been conducted to examine the effects of the number of dampers,

frequency range and damping ratio of the dampers. Optimum values of these parameters

have been obtained by minimization of the RMS displacement.

3.4.1 Effect of number of dampers (N)

From Fig. 3.8, one can observe the flattening action of MTLCDs as compared to

the double peaked response due to an STLCD. The effect of increasing dampers is similar

to that of adding damping: i.e., flattening of the frequency response function. However, it

is also noted that the frequency response due to 5, 11 and 21 TLCD groups, for the partic-

ular frequency range of 0.2, are very similar. This suggests that a large number of TLCDs

do not necessarily mean better performance, limiting the advantage of utilizing large num-

ber of MTLCDs for a particular frequency range.

3.4.2 Effect of damping ratio of dampers (ζfn)

The damping ratio of MTLCDs is studied for a group of eleven dampers with a

fixed frequency range of 0.2 (Fig. 3.9). It is noted that at low damping ratios, the ampli-

tude of the response function is spiked. As the damping ratio is increased, the response

ω∆ ω fN ω f 1–=

ω fn ωsω∆

N-------n–= 1 n

N 1+

2--------------<≤

ωs= nN 1+

2--------------=

ωsω∆

N-------n+= N n

N 1+

2-------------->≥

60

function slowly becomes smoother and the amplitude decreases. After an optimal damp-

ing ratio for the dampers is reached, any further increase in the damping ratio results in an

increase in the amplitude. This suggests that there exists an optimum damping ratio for a

particular set of MTLCD configurations.

3.4.3 Effect of frequency range ( )

Figure 3.10 shows the effect of changing the frequency range on the frequency

response function. It is can be seen from the plots that there is an optimum range where

the curve flattens out over a range of frequencies. The frequency response functions of an

STLCD and a MTLCD with a low frequency range (0.02 and smaller) are similar. If the

range is smaller than the optimum, the frequency response of the MTLCD resembles that

of an STLCD, and so in a way, the MTLCD loses its effectiveness. This is intuitive

because there is a practical limit to which one can distribute the MTLCDs over a given fre-

quency range. As this range becomes very small, MTLCDs act almost like an STLCD.

Two types of configurations can be considered for multiple TLCDs: SDOF-

MTLCD configuration (to control single mode of the structure) and MDOF-MTLCD con-

figuration (to control multiple modes). The time frequency analysis of several earthquake

ground motion records utilizing wavelets has revealed the presence of higher frequency

components in the initial stages of the event, e.g., El-Centro (Gurley and Kareem, 1994).

In such cases, the presence of a TLCD or MTLCD tuned to the higher modes will be

essential in controlling motion induced by higher frequency components.

Table 3.9 tabulates the optimum parameters of the different MTLCD system. One

can note that the optimum damping ratio decreases drastically for MTLCD groups as com-

pared to an STLCD.

∆ω

61

Figure 3.8 Effect of number of dampers on the frequency response of SDOF-MTLCD system

Figure 3.9 Effect of damping ratio of the dampers on the frequency response ofSDOF-MTLCD system

0.1 0.15 0.2 0.25−10

−5

0

5

10

15

20

25

30

35

40

Frequency Hz

Magnitude of Transfer function (dB)

No damper

N=1 N=5 N=11N=21

0.1 0.15 0.2 0.25−10

−5

0

5

10

15

20

25

30

35

40

Frequency Hz


, ,

No damper

zs=0.0005zs=0.005 zs=0.05 zs=0.5

damping ratio ζfn....... 0.0005−−−− 0.005−.−.− 0.05_____ 0.5

62

Figure 3.10 Effect of frequency range on the frequency response of SDOF-MTLCD system

TABLE 3.9 Optimum parameters for MTLCD configurations

(These values have been computed for white noise excitation, So=1, ωs =1 rad/s, ζs=1%, µ = 1%)


A method to determine the optimum absorber parameters in the case of TLCDs,

using a simplified solution to the integral occurring in the estimation of the mean square

response, has been presented. SDOF systems subjected to the white noise and filtered

white noise excitations utilizing first and second order filters have been analyzed, and the

optimum absorber parameters for TLCDs have been determined numerically based on the

CasesOptimum dampingratio of each damper

Optimum frequencyrange RMS displacement

No damper - - 12.533

N=1, STLCD 4.5% - 7.226

N=5 1.4% 0.12 6.927

N=11 0.8% 0.145 6.878

N=21 0.6% 0.155 6.864

0.1 0.15 0.2 0.25−10

−5

0

5

10

15

20

25

30

35

40

Frequency Hz


, ,

No damper

range=0.02range=0.05range=0.1 range=0.2

frequency range (∆ω) ...... 0.02 -.-.- 0.05 ---- 0.1 ___ 0.2

63

minimization of the RMS displacement of the primary system. This work can be extended

to MDOF systems for which a state space approach can be used and the response covari-

ance matrix in the case of white noise can be obtained by solving the Lyapunov equation.

In the case of FWN excitations, the procedure remains the same except that the primary

system equations are augmented with the FWN equations.

Explicit expressions for optimal parameters are only feasible for a simple

undamped primary system subjected to white noise. As the systems and forcing functions

become more complex, numerical solutions are needed to evaluate the optimal parameters.

It has been seen that for lightly damped systems, the optimal damping coefficient

of the absorber does not depend on the damping coefficient of the primary system when

the excitation is purely white noise. However, for the first and second order FWN cases, it

is affected by the primary system damping. This suggests that the damping in the primary

system plays a role in determining the optimum damping coefficient of the TLCD.

Although the undamped case may yield an approximate value of the optimal parameters,

the primary system damping and knowledge of the excitation must be included for accu-

rate estimates.

Optimal absorber parameters have been determined in the case of multiple TLCDs.

These parameters include the number of TLCDs, the frequency range and the damping

ratio of each damper. It is seen that there is an upper limit on the number of TLCDs,

beyond which additional TLCDs in the MTLCD configuration do not enhance the perfor-

mance. MTLCDs are more robust as compared to an STLCD and the smaller value of the

optimal damping makes them more attractive for liquid dampers which have a limited

range of damping. The small size of individual TLCDs in a MTLCD configuration offers

convenient portability and ease of installation at different locations.

64

CHAPTER 4

BEAT PHENOMENON

It is far easier to write differential equations than to perceive the nature of their solutions -- if the latter exist at all.

- Anonymous

This chapter examines a phenomenon which occurs very commonly in combined

structure-liquid damper systems. Transfer of energy takes place in the coupled system

which could induce vibrations in the primary structure instead of suppressing them. This

chapter focusses on understanding the phenomenon from a mathematical point of view.

Numerical and experimental results are presented in this chapter to elucidate the beat phe-

nomenon in combined structure-liquid damper systems.

4.1 Introduction

The beat phenomenon has been discussed in many classical texts on vibration

(e.g., Den Hartog, 1956). Figure 4.1 shows coupling present in different mechanical and

electrical systems. It is well known that beats occur when two frequencies are close

together. This usually occurs when the coupling is very soft in comparison to the main

“springs”. In an electrical analogue, this means larger capacitance of the coupling than the

main capacitances. Transfer of energy takes place in the coupled system which could

induce vibration in the primary system instead of suppressing them.

Experimental studies involving a TLCD combined with a simple structure have

provided insightful understanding into the behavior of liquid damper systems. The motiva-

tion of this paper is portrayed in Figs. 4.2 (a) and (b), which show the free vibration decay

65

of a combined structure-TLD and -TLCD in the laboratory. The controlled response

exhibits the classical beat phenomenon characterized by a modulated instead of an expo-

nential decay in the signature.

Figure 4.1 Different coupled system (a) Vibration absorber (b) Coupled pendulisystem (c) Electrical system (d) Fluid coupling within two cylinders

However, beyond a certain level of damping in the TLCD, this beat phenomenon

ceases and the structural response resembles a SDOF decay. Of course, as a limiting case

one might expect this to happen because when the damping is very high in the secondary

system, the combined system essentially behaves as a SDOF system. However, the critical

damping at which this disappearance of beat phenomenon is initiated is not understood.

(a) (b)

(d)(c)

66

This chapter delves into better understanding the beat phenomenon for the combined

structure-TLCD system.

Figure 4.2 Uncontrolled and Controlled response of a structure combined with (a)TLD (b) TLCD

0 1 2 3 4 5 6 7 8 9 10−1.5

−1

−0.5

0

0.5

1

1.5

Time (sec)

Response of Structure

Uncontrolled

controlled with TLCD

(b)

0 1 2 3 4 5 6 7 8 9 10−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time (sec)

Response of Structure

UncontrolledControlled with TSD

(a)

TLD

67

4.2 Behavior of SDOF system with TLCD

In this section, three different cases are considered as shown in Fig. 4.3. These are

undamped combined system; damped primary system with undamped secondary system;

and damped primary and secondary system. We will look at each case in detail. In order to

keep the discussion general, the subscripts 1 and 2 are introduced instead of s for structure

and f for the damper, as in Chapter 3.

Figure 4.3 Different combined systems

4.2.1 Case 1: Undamped Combined System

The coupled equations of motion without damping in the primary and secondary system

(Fig 4.3 (a)) can be obtained from Eq. 3.6 by setting damping in each system equal to zero,

(4.1)

The modal frequencies of this system are given by:

(4.2)

k1

m 1

m 2 k2

x2

x1

k1m 2 m 2k2 k2

m 1 m 1

x1 x1

x2 x2

c2

c1 c1

(a) (b) (c)

k1

1 µ+ αµα 1

x1

x2

ω1

20

0 ω2

2

x1

x2

+0

0=

ϖ1 2,ω1

2 ω2

21 µ+( ) Π±+

2 1 µ α2µ–+( )------------------------------------------------=

68

where

It is obvious from Eq. 4.2 that, for an uncoupled system (i.e., for α=0), the eigenvalues

reduce to:

; (4.3)

The coupling parameter α in the mass matrix is responsible for the beat phenomenon.

Figure 4.4 shows the phase plane portraits for the primary system for different values of α.

Unless mentioned otherwise, all units of displacements, frequencies and velocities are m,

rad/sec and m/sec, respectively. The first portrait shows that with no coupling there is only

one frequency at which the structure responds, and as the coupling parameter increases

there is interference between the two states of the primary system, namely, and .

Figure 4.4 Phase plane portraits of the undamped coupled system

Π2 ω1

2 ω2

21 µ+( )–( )

24ω1

2ω2

2α2µ+=

ϖ1

ω1

1 µ+-----------------= ϖ2 ω2=

x1 x1

−0.01 −0.005 0 0.005 0.01−0.1

−0.05

0

0.05

0.1

x1

dx 1/dt

α=0

−0.01 −0.005 0 0.005 0.01−0.1

−0.05

0

0.05

0.1

x1

dx 1/dt

α=0.1

−0.01 −0.005 0 0.005 0.01−0.1

−0.05

0

0.05

0.1

x1

dx 1/dt

α=0.6

−0.01 −0.005 0 0.005 0.01−0.1

−0.05

0

0.05

0.1

x1

dx 1/dt

α=0.9

69

For all simulations in this chapter, the following parameters have been kept constant, ω1=1

Hz, µ=0.01 and ω2=0.99Hz. Figure 4.5 shows the time histories of the displacement of

the undamped primary system for α=0 and α=0.6. When coupling is present between the

two systems, the displacement signature is amplitude modulated.

Figure 4.5 Time histories of primary system displacement for α=0 and α=0.6

To understand this phenomenon better, one can consider the solution of the system of

equations given in Eq. 4.1. After some mathematical manipulation the displacement of the

primary system for the initial conditions, ; ; and

, is given by:

(4.4)

0 2 4 6 8 10 12 14 16 18 20−0.015

−0.01

−0.005

0

0.005

0.01

0.015

t

x 1

α=0

0 2 4 6 8 10 12 14 16 18 20−0.015

−0.01

−0.005

0

0.005

0.01

0.015

t

x 1

α=0.6

Time (sec)

x1 0( ) x0= x2 0( ) 0= x1 0( ) 0=

x2 0( ) 0=

x1 t( ) x0

ωBt

2---------

ωAt

2---------

coscos=

70

where and , which means that the resulting function is an

amplitude-modulated harmonic function with a frequency equal to and the amplitude

varying with a frequency of . This undamped combined system case has been exam-

ined in texts on vibration (e.g., Den Hartog, 1956).

4.2.2 Case 2: Linearly Damped Structure with Undamped Secondary System

In this section, a linearly damped primary system with undamped secondary system as

shown in Fig. 4.3(b) is considered. Accordingly, the equations of motion are given by:

(4.5)

This system has two complex conjugate pairs of eigenvalues,

and ,

where are the modal frequencies and are the modal damping ratios. The aver-

age frequency and the beat frequency are plotted in Fig. 4.6 for different damping ratios of

the primary system. At α = 0, the beat frequency (i.e. the difference in modal frequencies)

tends to be zero. As the coupling is increased, there is an increase in the beat frequency

which causes the beat phenomenon. From this analysis, one can conclude that there is no

beat phenomenon when the difference in the modal frequencies approaches zero. Figure

4.6 also shows the effect of introducing damping in the primary system. At high levels of

damping ratio, there is a wider range of coupling term α which results in the beat fre-

quency being equal to zero. This means that, over this range of the coupling term, there is

ωA ϖ1 ϖ2+= ωB ϖ2 ϖ1–=

ωB

ωA

1 µ+ αµα 1

x1

x2

2ω1ζ1 0

0 0

x1

x2

ω1

20

0 ω2

2

x1

x2

+ +0

0=

λ1 2, ϖ– 1ζ1 i± ϖ1 1 ζ1˜ 2

–= λ3 4, ϖ– 2ζ2 i± ϖ2 1 ζ2˜ 2

–=

ϖ1 2, ζ1 2,

71

hardly any beat phenomenon. For α = 0.3, beat phenomenon is present when the damping

ratio in the primary system is 0.005, but it disappears when the damping ratio is 0.05. Fig-

ure 4.7 shows the effect of damping in the primary system on the response of the primary

system. As the damping ratio increases, the response dies out in an exponential decay.

However, the beat phenomenon still exists. This poses difficulty in the estimation of sys-

tem damping from free vibration response time histories.

Figure 4.6 Variation of and as a function of α

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Coupling parameter α

Beat Frequency ,

ωB

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

12.5

12.52

12.54

12.56

Coupling parameter α

Average Frequency ,

ωA

ζ1 = 0

ζ1 = 0.005

ζ1 = 0.05

ωA ωB

72

Figure 4.7 Time histories of response for ζ1=0.005 and ζ1=0.05

At this stage, the effect of a decrease in beat frequency on the response signal can

be further examined. Figure 4.8 shows that as ωB approaches zero, TB (the time period of

the beat frequency) becomes very large. The parameter influencing the decay function is

(for a SDOF system, ). As a result, due to the damping in the primary sys-

tem, the response dies out before the next peak of the beat cycle arises. Therefore, the

response resembles that of a damped SDOF system.

0 5 10 15 20 25 30 35 40−0.01

−0.005

0

0.005

0.01

t

x 1

0 5 10 15 20 25 30 35 40−0.01

−0.005

0

0.005

0.01

Time(sec)

x 1

ζ1 = 0.05

= 0.0051ζ

Ψ Ψ ζ1ω1=

73

Figure 4.8 Anatomy of the damped response signature

4.2.3 Case 3: Damped Primary and Secondary System

In this section, the system represented by Fig 4.3 (c) is considered, where now an orifice in

the middle of the U-tube imparts damping to the system. In this case, the following equa-

tions of motion apply:

(4.6)

where ξ is the headloss coefficient and . Equation 4.6 is numerically integrated

at different levels of the headloss coefficient and setting = 0.001 and α=0.3 (Fig 4.9).

The figure shows an interesting behavior of the liquid damper system. In the previous sec-

X

X

=

exp(- Ψt)

cos(ω t/2)A

cos(ω t/2)B

x1= cos(ω t/2)cos(ω t/2)exp(- Ψt)BA

Time (sec)

1 µ+ αµα 1

x1

x2

2ω1ζ1 0

0 ω2

2ξ x2 4g⁄

x1

x2

ω1

20

0 ω2

2

x1

x2

+ +0

0=

c2

1

2---ρAξ=

ζ1

74

tion, the damping simply caused an exponential decay of the beat response. However, in

this case, the beat phenomenon disappears after a certain level of the headloss coefficient.

Since an analytical solution is not convenient for this equation due to the quadratic nonlin-

earity in the damping associated with the secondary system, a linearized version (see sec-

tion 3.2.1) of this system is generally considered. Therefore, Eq. 4.6 is recast as:

(4.7)

Figure 4.9 Time histories of response for ξ= 0.2, 2 and 50

1 µ+ αµα 1

x1

x2

2ω1ζ1 0

0 2ω2ζ2

x1

x2

ω1

20

0 ω2

2

x1

x2

+ +0

0=

0 5 10 15 20 25 30 35 40−0.01

−0.005

0

0.005

0.01

t

x1

0 5 10 15 20 25 30 35 40−0.01

−0.005

0

0.005

0.01

t

x1

0 5 10 15 20 25 30 35 40−0.01

−0.005

0

0.005

0.01

t

x1

Time (sec)

ξ = 0.2

ξ = 2

ξ = 50

75

The modal frequencies and damping ratios of the system defined in Eq. 4.7 are plotted in

Fig. 4.10 as a function of equivalent damping ratio, . Figure 4.10 explains the disap-

pearance of the beat phenomenon due to coalescing of the modal frequencies after a cer-

tain value of the equivalent damping ratio. As seen in the previous chapter, this change in

equivalent damping ratio is realized through changing of the headloss coefficient. The

resulting beat frequency approaches zero and hence beat phenomenon ceases to exist. This

is similar to a previous case where there was no beat phenomenon for coupling term α = 0,

in which case the beat frequency was zero.

Figure 4.10 Modal frequencies and modal damping ratios of combined system as afunction of the damping ratio of the TLCD

ζ2

0 0.01 0.02 0.03 0.04 0.05 0.060.97

0.98

0.99

1

1.01

1.02

Equivalent damping ratio, ζ2

Modal frequencies

0 0.01 0.02 0.03 0.04 0.05 0.060

0.01

0.02

0.03

0.04

0.05

0.06

Equivalent damping ratio, ζ2

Modal damping ratio

ϖ1,2

ζ~

1,2

ζ1 = 0.001α = 0.3µ = 0.01

76

Figure 4.11 shows the three dimensional plots of state space portraits as a function of time.

Figure 4.11(a) shows the evolution for an uncoupled system in which the amplitude of

response is constant. Figures 4.11(b) and (c) show the cases discussed in sections 4.2.1

and 4.2.2. The final plot, Fig. 4.11(d), shows case 3 in which no beat phenomenon occurs

in the coupled system.

Figure 4.11 Phase-plane 3D plots (a) uncoupled system (b) case 1: undampedsystem (c) case 2: system with damping in primary system only (d) case 3: system

with damping in both primary and secondary systems

−0.01

0

0.01

−0.1

0

0.10

10

20

x1

dx1/dt

time (sec)

−0.01

0

0.01

−0.1

0

0.10

10

20

x1

dx1/dt

time (sec)

−0.01

0

0.01

−0.1

0

0.10

10

20

x1

dx1/dt

time (sec)

−0.01

0

0.01

−0.1

0

0.10

10

20

x1

dx1/dt

time (sec)

No coupling CASE 1

CASE 2 CASE 3

(a) (b)

(c) (d)

77

4.3 Experimental Verification

In order to further validate the observations made in section 4.2, a simple experi-

ment was conducted using the experimental setup shown in Fig. 4.12. A TLCD is mounted

on a SDOF structure. The TLCD was designed with a variable orifice, to effectively

change the headloss coefficient. At θ = 0 degrees, the valve is fully opened and the head-

loss is increased with an increase in the angle of rotation, θ. In Fig. 4.13, one can note the

presence of a beat pattern for low headloss coefficients. However, as the headloss coeffi-

cient is increased, the beat phenomenon disappears and an exponentially decaying signa-

ture is obtained. A similar observation was made in Fig. 4.9 for simulated time histories.

Figure 4.12 Experimental setup for combined structure-TLCD system on ashaking table

78

Figure 4.13 Experimental free vibration response with different orifice openings(θ = 0 fully open)


Similar to coupled mechanical systems, the combined structure-liquid damper system

exhibits the beat phenomenon due to the coupling term that appears in the mass matrix of

the combined system. The free vibration structural response resembles an amplitude mod-

ulated signal. The beat frequency of the modulated signature is given by the difference in

the modal frequencies of the coupled system. However, beyond a certain level of damping

in the secondary system (liquid damper), the beat phenomenon ceases to exist. This is

θ is the angle of valve rotation

0 5 10 15

−0.5

0

0.5

Time

θ =0 degrees

0 5 10 15

−0.5

0

0.5

Time

θ =15 degrees

0 5 10 15

−0.5

0

0.5

Time (sec)

θ =60 degrees

x1

x1

x1

79

attributed to the coalescing of the modal frequencies of the combined system to a common

frequency beyond a certain level of damping in the secondary system.

80

CHAPTER 5

SEMI-ACTIVE SYSTEMS AND APPLICATIONS

If you wish to control the future, study the past...- Confucius

This chapter describes different semi-active strategies developed for optimal func-

tioning of TLCDs. These strategies include gain-scheduling and clipped optimal schemes

with continuously-varying and on-off control. It is shown that such systems provide a sig-

nificant improvement over the performance of a passive system. Numerous examples and

applications are provided to elucidate the theory.

5.1 Introduction

Semi-active control systems were first reported in civil engineering structures by

Hrovat et al. (1983). In other fields such as automotive vibration control, considerable

research has been done on semi-active systems (Ivers and Miller, 1991; Karnopp, 1990). A

number of devices are currently being studied in the area of structural control, namely the

variable stiffness devices, controllable fluid dampers, friction control devices, fluid vis-

cous devices, etc. Recent papers in this area provide a state-of-the-art review of semi-

active control devices for vibration control of structures (Spencer and Sain, 1997; Symans

and Constatinou, 1999; Kareem et al. 1999).

Optimization studies discussed in chapter 3 show that there exist optimal damping

and tuning ratio, which lead to high performance of TLCDs. One of the main features of

these dampers is that the damping is nonlinearly dependent on the amplitude of excitation.

81

This chapter proposes two strategies which can improve over the performance of passive

systems. One of them involves gain-scheduling of the damping based on the feedforward

information of the disturbance. The other is a clipped optimal system with continuously-

varying and on-off control, which involves a continuos changing of the damping based on

feedback of the structural response.

5.2 Gain-scheduled Control

This section discusses a semi-active system which is useful for disturbances which

are of long duration and slowly varying (e.g., wind excitations) and where steady-state

response is the controlling objective. The optimal head loss coefficient as a function of the

loading intensity is described as a look-up table. As the loading intensity changes, the

headloss coefficient of the TLCD is changed in real-time in accordance with this look-up

table by changing the valve/orifice opening.

Gain-scheduling is defined as a special type of non-linear feedback, with a non-lin-

ear regulator whose parameters are changed as a function of the operating conditions in a

pre-programmed manner. As shown in Fig. 5.1, the regulator is tuned for each operating

condition. Though gain-scheduling, an open-loop compensation technique, may be time

consuming to design, its regulator parameters can be changed very quickly in response to

system changes. This kind of control is more commonly used in aerospace and process

control applications (Astrom and Wittenmark, 1989).

82

Figure 5.1 Gain scheduling concept

5.2.1 Determination of Optimum Headloss Coefficient

The procedure for estimating the optimum damping coefficient, , for TLCDs

under a host of loading conditions was outlined in chapter 3. In this section, methods to

determine the optimal headloss coefficient (ξopt) is presented. This is the parameter

responsible for introducing damping in the liquid column of the TLCD. The statistical lin-

earization method gives the following expression for the equivalent damping (assuming

the liquid velocity to be Gaussian) as discussed in section 3.2.1:

(5.1)

Equation 5.1 suggests that since increases as the loading increases, therefore, in order

to maintain the optimal damping, must decrease. Hence, there exists an optimal head-

loss coefficient at each loading intensity. These variations define the damping characteris-

tics of the orifice needed at different excitation levels. An iterative method has been used

in previous studies, (Balendra et al. 1995) since the damping term depends on which

Regulator

GainSchedule

condition

Command

Process controlsignal

operating

output

regulatorparameters

ζopt

c f2

π---ρAξσ x f

=

σ x f

ξ

σ x f

83

is not known a priori. An alternative, which is a direct method is developed in this study.

This involves evaluation of ζopt following the procedure outlined in the previous sections.

This value is then substituted into Eq. 3.8 to obtain . One can then determine ξopt using

Eq. 5.1. Figure 5.2 provides a step by step flowchart for the two methods. Figure 5.3 (a)

shows a typical iterative method for an SDOF-TLCD system subjected to white noise

excitation, where and are calculated by Eqs. 3.8 and 3.9. This is repeated for a

range of ξ, and ξopt is determined where the is minimum.

Figure 5.2 Flowchart of the two algorithms (a) iterative method (b) direct method

Explicit expressions to obtain for an undamped SDOF system subjected to

white noise excitation with tuning ratio close to unity, can be obtained. The optimum value

of the damping coefficient for this case reduces to the expression given in Eq. 3.12. After

some manipulation, Eq. 3.12 and 5.1 provide,

σx f

σX sσ x f

σX s

1: Vary ξ over a range of values

2: Assume ζf

4: Recalculate ζf using Eq. 5.1

5: ξopt is the one which gives

iterate until convergence

(a) Iterative Method (b) Direct Method

using Eq. 3.8 and 3.9

min(σXs)

1. Express σxs as a function of

ζ and set

∂(σXs)/∂ζf = 0; ∂(σXs)/∂γ and obtain ζopt

2. Calculate

3. Calculate ξopt using Eq. 5.1

σ x f

3. Calculate σ x f

using Eq. 3.9

σXs and

ξopt

84

100

37.5tain e

(5.2)

Figure 5.3 Iterative method (a) convergence of response quantities (b) optimumheadloss coefficient

For tuning ratios not equal to unity, one can obtain similar expressions. However,

they are cumbersome and can be obtained numerically. It is noteworthy from Eq. 5.2 that

the optimum headloss coefficient is indirectly proportional to the square root of the inten-

sity of white noise. Using some representative values, it can be shown that the direct (Eq.

5.2) and the iterative methods yield the same values (Fig. 5.3 (b)). However, the direct

method is computationally superior, since it does not require iterations, making it more

attractive for on-line semi-active control of the orifice.

Figure 5.4 shows the variation in the optimum headloss coefficient for various

mass ratios of an SDOF-TLCD system under white noise excitation case. It is noted from

these curves that at high loading intensities, very low headloss coefficients are needed. For

ξopt µ 1 µ α2µ–+( )S0

---------------------------------µ α2

+

1 µ+----------------

3 2⁄

glωd µ=

0 2 4 6 8 10 12 14 16 18 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

iterations

Re

sp

on

se

qu

an

titie

s

--- RMS structure’s displacement-.-. RMS liquid velocity__ ζf

0 10 20 30 40 50 60 70 80 900.008

0.0085

0.009

0.0095

0.01

0.0105

0.011

0.0115

0.012

Coefficient of headloss

Rm

s d

isp

lace

me

nt o

f m

ain

ma

ss

Optimum value =Same value is obDirect method.

Parametersµ= 1%So = 1e-06 α = 0.9ωs=1 rad/sl =19.6 m

85

typical orifice characteristics, this corresponds to a hundred percent orifice opening ratio,

i.e., the orifice should be fully open. At high amplitudes of excitation, it is, therefore, bet-

ter to keep the orifice fully open and let the damping be provided by the liquid velocity.

For low amplitudes of excitation, the liquid velocity is inadequate, therefore, the orifice

opening should be decreased (thereby increasing ξ). The relationship between the orifice

opening ratio and the headloss coefficient for standard orifices can be found in the litera-

ture (Blevins, 1984).

Figure 5.4 Variation of optimum headloss coefficient with loading intensity: whitenoise excitation

5.3 Applications

Two examples of semi-active system using gain-scheduling are presented in this

section. The first example is for an SDOF-TLCD under random white noise excitation.

The second example discusses the application of these dampers to an offshore structure.

5.3.1 Example 1: SDOF-TLCD system under random white noise

The efficiency of the gain-scheduled control can be seen from Fig. 5.5. The look-

up table defined in Fig. 5.4 is used to introduce the semi-active control law. The parame-

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10−3

0

10

20

30

40

50

60

70

80

90

Spectral loading intensity So

Opt

imum

Coe

ffici

ent o

f Hea

dlos

s

Mass ratio 5%Mass ratio 2%Mass ratio 1%

Parameters :

ωs=1 rad/secα=0.9ζs=1 %l =19.6 m

86

ters of this system are as shown in Fig. 5.4. The efficiency of the passive TLCD is

improved as the intensity of the white noise excitation changes from So = 10-6 m2/sec3/Hz

to So =10-4 m2/sec3/Hz (Table 5.1). Note that in the first segment of the loading, the per-

formance of the semi-active and the passive system coincide with each other.

Figure 5.5 Example 1: SDOF system under random excitation.

(Numbers in brackets indicate improvement of each control strategy over uncontrolled case)

TABLE 5.1 Comparison of different control strategies: Example 1

Control Case

RMS Displacement of Primarysystem under random excitation

So = 10-6 m2/sec3/Hz(m)

RMS Displacement of PrimarySystem under random excitation

So = 10-4 m2/sec3/Hz(m)

Uncontrolled

System3.2 X 10-3 2.77 X 10-2

Passive

System2.1 X 10-3 (34.4%) 2.7 X 10-2 (2.5%)

Semi-active

System2.1 X 10-3 (34.4%) 2.09 X 10-2 (24.5%)

0 20 40 60 80 100 120 140 160 180 200−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

time

displacement (m)

UncontrolledPassive ControlSemi−Active Control

So=1e-06 So=1e-04

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10−3

0

10

20

30

40

50

60

70

80

90

Spectral loading intensity So

Opti

mum

Coef

fici

ent

of H

eadl

oss

Mass ratio 5%Mass ratio 2%Mass ratio 1%

Parameters :

ωs=1 rad/secα=0.9ζs=1 %l =19.6 m

ξ1

ξ2

Look-up Table for Semi-Active Control

87

5.3.2 Example 2: Application to Offshore Structure

The forces acting on most offshore structures are due to wind, waves and ocean

currents. The motion of offshore structures is highly undesirable as it causes fatigue and

shutdown of operations. In this section, a TLCD is proposed for control of offshore struc-

tures. The offshore structure has been idealized as a SDOF system as shown in Fig. 5.6(a).

It is noteworthy that unlike land-based structures, platforms experiencing motions in

ocean waves acquire additional mass and damping referred to as added mass and hydrody-

namic damping. The mass, stiffness and damping can be written as (Brebbia et al. 1975):

(5.3)

(5.4)

(5.5)

(5.6)

(5.7)

where , lc is the length of the column, , g is the acceleration due to grav-

ity, ω is the frequency, is the assumed shape of the column, EI is the equivalent

stiffness of the column, Ac is the equivalent area of the column, ρ is the density of water,

Mc is the mass of the platform, CD, CM and CA are the drag and inertia coefficients, and

M lcρc Ac f z( )[ ] 2z CM lc f z( )[ ] 2

z M c+d

0

1

∫+d

0

1

∫=

KEI

lc3

------

z2

2

∂∂

f z( ) 2

zd0

1

∫=

ωsKM-----=

C Cs CD8

π--- σ

Vf z( )[ ] 2

zd0

1

∫+=

σV2

SV V

ω( ) ωd0

∞∫ ω2 kzcosh

kDsinh------------------

2

S η η ω( ) ωd0

∞∫= =

zzlc----= k ω2

g⁄=

f z( )

88

is the spectra of wave elevation. The forcing function under the action of linear

waves can be expressed as:

(5.8)

The shape of the deflected platform is approximated as and hence the mass of

the system is calculated using Eq. 5.3 as M = 7.72 X 106 Kg and stiffness, K = 9 X 106 N/

m using Eq. 5.4. This results in a natural frequency of the structure, ωs = 1.07 rad/s. The

total damping ratio of the structure is evaluated using Eq. 5.6 which is equal to 6%. The

drag and inertia coefficients for the equivalent column are: = 5000 Kg/

m2; = 78000 Kg/m and = 78000 Kg/m (with = =1).

Figure 5.6 (a) Single degree of freedom idealization of an offshore structure (b)Concept of Liquid Dampers in TLPs

S η η ω( )

F ω t,( )η CM C A+( )ω2

kD( )sinh-------------------------------------- kz( ) f z( )cosh z

ηCD8

π---

ωkD( )sinh

----------------------- kz( )σV

f z( )cosh zd0

D

∫+

d0

D

∫=

f z( ) z2

=

CD cdρD 2⁄=

CM cmρV W= C A ρV W= cm cd

z

η

l

D

Mc

TLCD

(a) (b)

89

TABLE 5.2 Numerical parameters used: Example 2

The wave spectrum used in this study is the Pierson and Moskowitz (P-M) spectrum,

(5.9)

where U is the wind speed at 10 meters above the sea surface and α1 , β1 are dimension-

less parameters which determine the shape of the spectrum. For the North Sea, the value of

α1 = 0.0081 and β1 = 0.74. In the frequency domain, the expression for the forcing func-

tion can be derived from Eq. 5.8, which can be written as,

(5.10)

Figure 5.6 (b) shows a schematic of the possible design of liquid dampers func-

tioning as pontoon water tanks of the Tension Leg platform (TLP). The wave forcing func-

tion on such platforms may not be ideally described by Eq. 5.8. This is because the size of

the platform in comparison with the wave length of approaching waves is large, which

results in diffraction of waves. Therefore, in this case the first component of the forcing

function is obtained from diffraction analysis (Kareem and Li, 1988).

ParameterNumerical

Value ParameterNumerical

Value

Depth of water, D 75 m EI value 2250 X109 Nm2

Mass of Platform, Mc 2 X106 Kg Density of water, ρ 1000 Kg/m3

Length of Structure, lc 100 m length of liquid damper, l 18 m

Cross sectional Area, Ac 28 m2 Area of damper (with µ=2%), A 8.8 m2

Total Volume of water displaced

per unit length, VW

78 m3/m Density of Concrete, ρc 2500 Kg/m3

S η η ω( )α1g

2

ω5------------ β1

gωU---------

4

– exp=

SFF ω( ) S η η ω( )CM C A+( )2ω4

kD( )sinh2

------------------------------------ kz( ) f z( )cosh zd0

D

∫ 2

8CD2 ω2

π kD( )sinh2

----------------------------- kz( )σV

f z( )cosh zd0

D

∫ 2

+

=

90

1.1

s

s

s

s

Optimal parameters are obtained using numerical optimization, as done previously

in chapter 3, with the objective of minimizing the accelerations (absorber efficiency =

ratio of RMS structural accelerations with and without the damper). As shown from Fig.

5.7, there exists optimum damper parameters, which are found to be independent of the

loading conditions (i.e., different U10). Therefore, under all loading conditions, these

parameters must be maintained at their optimal values, otherwise the performance of the

damper may deteriorate.

Figure 5.7 Optimal Absorber parameters as a function of loading conditions

Next, one can easily apply the gain-scheduled law described in the previous sec-

tions for semi-active control. The look-up table can be generated as shown in Fig. 5.8 (a)

for different loading conditions. Figure 5.8(b) shows the spectra of structural acceleration

as the headloss coefficient is changed. The mass ratio of the damper mass to the main mass

is 2%. The space is very limited on a typical offshore structure and therefore, the pontoon

tanks filled up with water can also be utilized as water supply for occupants. However, this

0 0.02 0.04 0.06 0.08 0.1 0.12

0.8

0.85

0.9

0.95

1

ζf

Absorber Efficiency

U10=50 m/s

U10=40 m/s

U10=30 m/s

U10=20 m/s

Optimum Dampingratio

0.8 0.85 0.9 0.95 1 1.05

0.8

0.85

0.9

0.95

1

ω f /ω s

Absorber Efficiency

U10=20 m/

U10=30 m/

U10=40 m/

U10=50 m/

Optimal Tuningratio

91

may not be always possible as water is used to ballast a platform and is restricted from

sloshing to eliminate unnecessary sloshing forces on the hull.

Figure 5.8 (a) Variation of Optimal Headloss Coefficient with loading conditionsfor different wave spectra (b) Spectra of structural acceleration at U10=20 m/s for

different ξ.

5.4 Clipped-Optimal System

The semi-active system described in this section requires a controllable orifice

with negligible valve dynamics whose coefficient of headloss can be changed rapidly by

applying a command voltage (Fig.5.9). This type of semi-active control is more suitable

for excitations which are transient in nature, for e.g., sudden wind gusts or earthquakes.

Equation 3.3 can be posed in an active control framework as follows:

(5.11)

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.005

0.01

0.015

0.02

0.025

0.03

Frequnecy (rad/s)

Spectra of Acceleration of Structure

ξ = 1

ξ = 50

ξ = 15 (optimal)

Uncontrolled Response

15 20 25 30 35 40 45 5010

20

30

40

50

60

70

80

90

10 (m/s)

Optimum Coefficient of Headloss

β1=10.0β1= 8.0 β1= 3.0

U

(a) (b)

Ms m f+ αm f

αm f m f

Xs

x f

Cs 0

0 0

Xs

x f

Ks 0

0 k f

Xs

x f

+ +Fe t( )

0

0

1u t( )+=

92

where the bold face denotes matrix notation and u(t) is the control force given by:

(5.12)

Figure 5.9 Semi-active TLCD-Structure combined system

The coefficient of headloss is an important parameter which is controlled by vary-

ing the orifice area of the valve. In the case of a passive system, this headloss coefficient is

unchanged. The headloss through a valve/orifice is defined as:

(5.13)

where V is the velocity of the liquid in the tube. The coefficients of headloss for different

valve openings are well documented for different types of valves (Lyons, 1982). The rela-

tionship between the headloss coefficient (ξ) and the valve conductance (CV) is derived in

Appendix A.3.

u t( )ρ– Aξ t( ) x f

2------------------------------ x f=

K s

Cs

M s

F(t)

Semi-activeTLCD

Primary Mass

Controllable Valve

Fe(t)

hlξV

2

2g----------=

93

The damping force of a semi-active TLCD can be written as:

(5.14)

where is the headloss coefficient, which is a function of the applied voltage ,

needed to control valve opening, at a given time t. Equation 5.14 can be re-written as,

(5.15)

where and . In this format, this damper system can be

compared to typical variable damping fluid dampers. Semi-active fluid viscous dampers

have been studied among others by Symans et al. 1997 and Patten et al. (1998). The

damping force in such a system can be written as:

(5.16)

where is the damping coefficient which is a function of the command voltage

and is the velocity of the piston head. The damping coefficient is bounded by a maxi-

mum and a minimum value and may take any value between these bounds.

Comparing Eqs. 5.15 and 5.16, one can some similarity in the fundamental work-

ing of these dampers. However, there are basic differences in the two physical systems. In

variable orifice dampers, the fluid is viscous, usually some silicone-based material, which

is orificed by a piston. In the TLCD case, the liquid is usually water and is under atmo-

spheric pressure. Moreover, the damping introduced by an orifice in a TLCD system is

quadratic in nature, whereas the damping imparted by a fluid damper is linear (Kareem

and Gurley, 1996).

Fd t( ) ρAξ Λ t,( )2

------------------------ x f t( ) x f t( )=

ξ Λ t,( ) Λ

Fd t( ) C Λ t,( ) V V=

C Λ t,( ) ρAξ Λ t,( )2

------------------------= V x f t( )=

Fd t( ) C Λ t,( )V˜

=

C Λ( ) Λ

V˜

94

5.4.1 Control Strategies

Most semi-active strategies are inherently non-linear due to the non-linearities

introduced by the device in use. Therefore, a great deal of research is based on developing

innovative algorithms for implementing semi-active strategies. Some of the common

examples are sliding mode control and nonlinear strategies (Yoshida et al. 1998).

Some innovative algorithms involving shaping of the force-deformation loop in a variable

damper system are reported in Kurino and Kobori, 1998. Other researchers have used

fuzzy control theories to effectively implement semi-active control (e.g., Sun and Goto,

1994; Symans and Kelly, 1999).

The strategy considered in this study is based on the linear optimal control theory.

The negative sign in Eq. 5.12 ensures that the control force is always acting in a direction

opposite to the liquid velocity. In case, the liquid velocity and the desired control force are

of the same sign, then Eq. 5.12, implies that is negative. Since it is not practical to have

a negative coefficient of headloss, the control strategy sets it to a minimum for ξ, i.e.,

. The control force is regulated by varying the coefficient of headloss in accordance

with the semi-active control strategy given as follows:

if

if (5.17)

In a fully active control system, one needs an actuator to supply the desired control

force. In such a case, the control force is not constrained to be in a direction opposite to the

damper velocity. Therefore, the linear control theory is readily applicable to active control

systems. In case of semi-active systems, however, the proposed control law is a clipped

H∞

ξ

ξmin

ξ t( ) 2– u t( ) ρA x f x f( )⁄ ξmax≤= u t( ) x f t( )( ) 0<

ξ t( ) ξmin= u t( ) x f t( )( ) 0≥

95

optimal control law since it emulates a fully active system only when the desired control

force is dissipative (Karnopp et al. 1974; Dyke et al. 1996). Moreover, the actual control

force that can be introduced is dependent on the physical limitations of the valve used and

the maximum coefficient of the headloss it can supply, which implies bounds on the con-

trol force introduced. This bound is given by,

(5.18)

A slight variation of the preceding continuously-varying orifice control is the com-

monly used on-off control. Most valve manufactures supply valves which operate in a bi-

state: fully open or fully closed. These valves require a two-stage solenoid valve. On the

other hand, the continuously-varying control requires a variable damper which utilizes a

servovalve. This servovalve is driven by a high response motor and contains a spool posi-

tion feedback system, and therefore is more expensive and difficult to control than a sole-

noid valve. The on-off control is simply stated as:

if

if (5.19)

ξmin can be taken as zero because this corresponds to the fully opened valve. It can be

expected that a small value of ξmax will result in a lower level of response reduction.

In order to formulate the system in a state space format, Eq. 5.11 is recast as,

(5.20)

which is then expressed in the state-space form,

(5.21)

ρ– Aξmin x f

2------------------------------ x f

u≤ t( )ρ– Aξmax x f

2------------------------------- x f

≤

ξ t( ) ξmax= u t( ) x f t( )( ) 0<

ξ t( ) ξmin= u t( ) x f t( )( ) 0≥

Mx t( ) Cx t( ) Kx t( )+ + E1W t( ) B1u t( )+=

X AX Bu EW+ +=

96

where ; ; ; and and

and are the control effect and loading effect matrices, respectively. The states of

the system are the displacements and velocities of each lumped mass of the structure and

the displacement and velocity of the liquid in the TLCD. Measurements of the structural

response can be expressed as:

(5.22)

where ; ; and in the case of full state feedback. The desired

optimal control force is generated by solving the standard Linear Quadratic Regulator

(LQR) problem. The main idea in LQR problem is to formulate a feedback control law

which would minimize the cost function given as ,

where Q and R are the control matrices for the LQR strategy. The control force is obtained

by,

(5.23)

where is the control gain vector and is given as:

(5.24)

and P is the Riccati matrix obtained by solving the matrix Riccati equation:

(5.25)

A schematic diagram of the control system is depicted in Fig. 5.10.

Xx

x= A

0 I

M 1– K– M 1– C–

= B0

M 1– B1

= E0

M 1– E1

=

E1 B1

Y CX Du FW+ +=

C I[ ]= D 0[ ]= F 0[ ]=

J E ZT

QZ uT

Ru+( ) td

0

T

∫

T ∞→lim=

u KgX–=

K g

Kg R 1– BT P=

PA PB R 1– BT P( )– AT P+Q=0+

97

The control performance of each strategy is evaluated based on a prescribed criterion. For

this purpose appropriate performance indices, regarding the RMS displacements ,

accelerations of the structure , and the effective control force are defined below:

; ; (5.26)

where subscripts unco and co are used to distinguish between uncontrolled and controlled

cases.

Figure 5.10 Schematic of the control system

In actual practice, it is more realistic to consider a few noisy measurements which are then

used to estimate the system states. In this situation, the standard stochastic Linear Qua-

dratic Gaussian (LQG) framework is used for estimation (Maciejowski, 1989). In a sto-

chastic framework, the measurements are given as,

X s⟨ ⟩

X s⟨ ⟩ u⟨ ⟩

J 1

X s⟨ ⟩ unco X s⟨ ⟩ co–( )X s⟨ ⟩ unco

--------------------------------------------------= J 3

X s⟨ ⟩ unco X s⟨ ⟩ co–( )X s⟨ ⟩ unco

--------------------------------------------------= J u u⟨ ⟩=

Plant

W

u

Observer

-KgX

Semi-Active

Strategy

Z

Y

Feedforward

Feedback

Y CX Du FW+ +=

X AX Bu EW+ +=

u=-KgX

98

(5.27)

where is the measurement (sensor) noise which is invariably present in all measure-

ments. The LQG problem is solved using the seperation principle which states that first an

optimal estimate of the states (optimal in the sense that is

minimized) is obtained, and then this is used as if it were an exact measurement to solve

the determinstic LQR problem discussed earlier. From the measurements, the states of the

system can be estimated using a Luenberger observer:

(5.28)

where L is determined using standard Kalman-Bucy filter estimator techniques. The opti-

mal control is then written as:

(5.29)

where Kg is the optimal control gain matrix obtained by solving the standard LQR prob-

lem as discussed previously.

5.4.2 Example 3: MDOF system under random wind loading

The first example is an MDOF-TLCD system, as shown in Fig. 5.11, which is a

high rise building subjected to alongwind aerodynamic loading. The building dimensions

are 31 m X 31 m in plan and 183 m tall. The structural system is lumped at five levels and

natural frequencies of this building are: 0.2, 0.583, 0.921, 1.182, and 1.348 Hz. The corre-

sponding modal damping ratios are 1%, 1.57%, 2.14%, 2.52% and 2.9%. The description

of the wind loading and the structural system matrices for mass, stiffness and damping are

given in Li and Kareem (1990).

Y CX Du FW+ν+ +=

ν

X X E X X–( )T

X X–( )

X

X˙

AX Bu L Y CX– Du–( )+ +=

u KgX–=

99

Figure 5.11 Schematic of 5DOF building with semi-active TLCD on top story

The TLCD is designed such that the ratio of the mass of liquid in TLCD to the first

generalized mass of the building was 1%, the length ratio, α = 0.9 and =15. Using a

multi-variate simulation approach (Li and Kareem, 1993), wind loads were simulated at

the five levels, as shown in Fig. 5.12. Two types of semi-active strategies, namely the con-

tinuously-varying and the on-off type were examined. The LQR method, as described in

the earlier section, was used to determine the control gains. It was assumed that all states

were available to provide the feedback.

The results are summarized in Fig. 5.13 and Table 5.3. As seen from Table 5.3, the

semi-active strategies provide an additional 10-15% reduction over passive systems. Table

5.3 also shows how the two semi-active strategies deviate from the optimal control force.

W 1

W 2

W 3

W 4

W 5

ξmax

100

One can observe the sub-optimal performance of these schemes, which leads to a lower

response reduction than the active case. In a semi-active system, the applied control force

is generated using a controllable valve which can be operated using a small energy source

such as a battery.

Figure 5.12 Wind loads acting on each lumped mass

TABLE 5.3 Comparison of various control strategies: Example 3

Control CaseRMS Disp. (cm)

and (J1) (%)RMS accel. (cm/s2)

and (J3) (%)RMS control force

(kN) Ju

Uncontrolled 7.05 10.61 -

Passive TLCD 5.24 (25.6%) 7.63 (28.0%) -

Continuously varying 4.84 (31.2%) 6.84(35.3%) 79.8 (Eq. 5.12, 5.17)

On-Off control 4.83 (31.2%) 6.84 (35.3%) 79.9 (Eq. 5.12, 5.19)

Active control 2.51 (64.4%) 4.87 (55.0%) 133.8 (Eq. 5.23)

20 40 60 80 100 120 140 160 180 200−60

−40

−20

0

20

40

60

80

Time (sec)

Wind Load (kN)

1st Floor

2nd Floor

3rd Floor

4th Floor

5th Floor

101

200

200

Figure 5.13 Displacements and Acceleration of Top Level using various controlstrategies

5.4.3 Example 4: MDOF system under harmonic loading

In the next example, a multi degree of freedom (5DOF) system is considered

again, but under harmonic loading. This example is taken from Soong (1991). The lumped

mass on each floor is 131338.6 tons and the damping ratio is assumed to be 3% in each

mode. The natural frequencies are computed to be 0.23, 0.35, 0.42, 0.49 and 0.56 Hz. A

vector of harmonic excitation is defined:

(5.30)

0 20 40 60 80 100 120 140 160 180−20

−10

0

10

20

Time (sec)

Displacement of Structure

0 20 40 60 80 100 120 140 160 180−30

−20

−10

0

10

20

30

Time (sec)

Acceleration of Structure (cm/s

2)

Uncontrolled Passive Continuously−varyingOn−off Active

)

Dis

pla

cem

ent

(cm

)

Accele

rati

on (

cm

/s2)

W t( ) a ωt( )cos b 2ωt( )cos c 3ωt( )cos d 4ωt( )sin+ + +=

102

where ω = 1.47 rad/s (= first natural frequency of the structure), and the values of a, b, c

and d and the stiffness matrix of the structure are given in Appendix A.2. The excitation

acts at a frequency equal to the first natural frequency of the structure. The semi-active

TLCD is placed on the top floor of the building with similar parameters as in Example 3.

Two cases of control strategies are considered: (a) full state feedback, and (b) acceleration

feedback using observer based controllers.

Full state-feedback LQR strategy

The first strategy assumes that all states are available for feed-back (total of 12

measurements). The control gains are calculated using Eq. 5.24. Figure 5.14 shows the

parametric variation of J1, J2 and Ju as a function of ξmax. There are small reductions in

the response after a certain value of ξmax is reached. This can be explained by Eq. 5.18 in

which it is implied that the applied control force is constrained by ξmax. This means that

satisfactory control results can be achieved by choosing a valve which may have a limited

range of headloss coefficients.

Figure 5.15 shows the response of the top floor of the structure using various con-

trol strategies. It is noteworthy that the continuously varying and on-off strategies give

similar reduction in response. This can be explained by the results in Fig. 5.16. The pro-

files of variation in the headloss coefficient as a function of time are similar for the two

strategies. The continuously varying control gives flexibility in the headloss coefficient.

However, the saturation bound introduces a clipping effect similar to on-off control and

therefore in this case, the advantage of continuously-varying control strategy is lost. Fig-

ure 5.17 shows the RMS displacement of the floor displacements and accelerations, maxi-

mum story shear and maximum inter-story displacements using various control strategies.

103

Figure 5.14 Variation of performance indices with maximum headloss coefficient

Figure 5.15 Displacement of Top Floor under various control strategies

0 10 20 30 40 50 60 70 80 90 10070

75

80

85

J 1(%)

0 10 20 30 40 50 60 70 80 90 10065

70

75

80

85

J 3(%)

0 10 20 30 40 50 60 70 80 90 1000

5

10x 10

5

ξm ax

J u(%)

Continuously varyingOn−off

0 20 40 60 80 100 120−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

time (sec)

displacement (m)

UncontrolledPassivecontinuously variablOn−OffActive

uncontrolled

passive

104

Figure 5.16 Variation of headloss coefficient with time

Figure 5.17 Variation of RMS displacements, RMS accelerations, maximum storyshear and maximum inter-story displacements

0 20 40 60 80 100 120−5

0

5

10

15

20

Time (sec)

ξ(t) continuously

−varyin

0 20 40 60 80 100 120−5

0

5

10

15

20

Time (sec)

ξ(t) On

−off

0 0.05 0.1 0.150

1

2

3

4

5

RMS displacements (m)

Story Number

0 0.1 0.2 0.30

1

2

3

4

5

RMS accelerations (m/s2)

Story Number

0 2 4 6

x 107

1

2

3

4

5

Maximum Story Shear (N)

Story Number

0 0.05 0.1 0.151

2

3

4

5

Maximum story displacements (m)

Story Number

Uncontrolled Active Continuously variableOn−off Passive

105

Observer-based LQG strategy

In the previous case, it was conveniently assumed that all the states were available

for feedback. However, in practice only a limited number of measurements are feasible. In

this case, we assumed that the floor accelerations and the liquid level (displacement of the

liquid) were measured. This implied that there were a total of six measurements (five

accelerations and one liquid displacement). The measurement noise was modeled as Gaus-

sian rectangular pulse processes with a pulse width of 0.002 seconds and a spectral inten-

sity of 10-9 m2 /sec3/Hz. A comparison of the various strategies using observer-based

LQG control is presented in Table 5.4. The response reduction is similar to the results

obtained using LQR control.

TABLE 5.4 Comparison of various control strategies: Example 4


Two types of semi-active systems were presented in this chapter. The first was

based on a gain-scheduled feedforward type of control which utilized a look-up table for

control action. The second was a clipped-optimal feedback control system with continu-

ously-varying and on-off type of control.

Control Case

RMSDisplacement(cm)/ (J1 %)

RMSacceleration

(cm/s2)/ (J3 %)RMS control force

(kN) Ju

No. ofmeasure-

ments

Uncontrolled 14.21 30.78 - -

Passive TLCD 4.82 (66.08) 10.72 (65.17) - -

Active case 2.92 (79.45 ) 6.67 (78.33 ) 188 (Eq. 5.23) 12

Continuously varying 3.03 (78.68) 6.81 (77.88) 171.6 (Eq. 5.12, 5.17) 12

On-Off control 3.35 (76.43) 7.43 (75.86) 203.1(Eq. 5.12, 5.19) 12

Continuously Varying

OBSERVER BASED

3.21 (77.41) 7.58 (75.37) 70.4 (Eq. 5.12, 5.17) 6

On-Off control

OBSERVER BASED

3.13 (77.97) 8.43 (72.61) 170.7 (Eq. 5.12, 5.19) 6

106

Numerical examples and applications were presented for the gain-scheduled con-

trol. This type of semi-active system leads to 15-25% improvement over a passive system.

An application of these systems for offshore structures was also presented.

Next, the clipped-optimal control was discussed. The efficiency of the state-feed-

back and observer-based control strategy was compared. Numerical examples showed that

semi-active strategies provide better response reduction than the passive system for both

random and harmonic excitations. In the case of harmonic loading, the improvement was

about 25-30% while for the random excitation, the improvement was about 10-15%. It

was also noted that continuously-varying semi-active control algorithm did not provide a

substantial improvement in response reduction over the relatively simple on-off control

algorithm.

107

CHAPTER 6

TLD EXPERIMENTS

It is a capital mistake to theorize before you have all the evidence... It biases the judgment

-Sherlock Holmes (Sir Arthur Conan Doyle)

The sloshing-slamming analogy and impact characteristics for modeling Tuned

Liquid Dampers (TLDs) were introduced in chapter 2. This chapter focusses on experi-

mental studies conducted on TLDs. Shaking table experiments are conducted to obtain the

parameters needed to model the impact characteristics introduced in chapter 2. Impact

pressures due to sloshing are also measured along the height of the container wall. This

helps to glean better understanding regarding the nature of sloshing-slamming noted at

large amplitudes of excitation. Finally, an innovative technique known as Hardware-in-

the-loop is utilized to conduct structure-damper interaction experiments.

6.1 Introduction

Sloshing of liquids has prompted numerous experimental studies in various disci-

plines due to the complexity of the problem and the difficulty in developing an analytical

model. Some of the relevant work done in the area of liquid dampers is briefly reported

here. The earliest experimental studies on TLDs are reported by Modi and Welt, 1987 and

Fujino et al. 1988. A series of experimental studies, summarized in Modi et al. 1995, were

conducted using nutation dampers. These dampers covered different geometries like a tor-

oidal ring, rectangular or circular cross-section cylinders, and in some situations may

include baffles, screens, particle suspensions to manage liquid sloshing. Damper charac-

108

teristics were determined by varying the amplitude and frequency of excitation. Fujino et

al. 1988 carried out parametric studies of cylindrical containers by free-oscillation experi-

ments. Effects of liquid viscosity, roughness of container bottom, air gap between the liq-

uid and tank roof, and container size on the overall TLD damping were studied.

Experimental studies have been carried out for rectangular TLDs in the region of

relatively small to medium vibration amplitudes, where breaking of a wave does not occur,

and the results have been found to be in good agreement with analytical results obtained

by the shallow water theory (Fujino et al. 1992; Sun and Fujino, 1994; Sun et al. 1995).

Similar experiments were done by Koh et al. (1994) who considered earthquake type exci-

tations as opposed to sinusoidal excitations utilized in previous studies. Large amplitude

excitations, which are more representative of earthquakes, were also investigated through

similar shaking table tests and numerical modeling by Reed et al. (1998). Experimental

investigations of TLDs with submerged nets and other flow dampening devices were stud-

ied by Fediw et al. 1993 and Warnitchai and Pinkaew (1998). Chung and Gu, 1999 carried

out experimental verification of the performance of TLDs in suppressing vortex-excited

vibration on a small-scale structural model in a wind tunnel. Experimental verification of

active TLD systems have been conducted by Chang et al. 1997 and Natani (1998). A com-

prehensive review of various analytical and experimental studies for sloshing dynamics is

documented in Ibrahim et al. 2001.

As mentioned earlier, theoretical analyses are not able to predict sloshing pres-

sures and forces in the neighborhood of resonance for large amplitude excitations. In

chapter 2, it was shown that the impact component is an important component of the over-

all sloshing force. Therefore, experimental studies are conducted to better understand the

nature of the liquid impact on the container walls. Previous experimental studies have

109

been conducted, most notably in ship engineering (Bass et al. 1980) and marine engineer-

ing applications (Schmidt et al. 1992; Hattori et al. 1994). However, specific studies of

impact pressures and their relation to the TLD performance have not been studied previ-

ously. The present chapter presents experimental studies conducted on shallow water

TLDs, which shed more light into the nature of sloshing-slamming caused at large ampli-

tude excitations.

6.2 Experimental Studies

In order to derive the impact characteristics of TLDs as discussed in Chapter 2,

experiments were conducted on a rectangular TLD, shown in Fig. 6.1(a). The tank had the

following dimensions: length a = 25.4 cm, width w = 10.64 cm and a liquid height h = 3

cm.

Figure 6.1 (a) Schematic of the experimental setup (b) pressure sensor locations

From the linear wave theory, one can compute the natural frequency of the first sloshing

mode as,

Shaking Table

Baffles

6 DOFLoad Cell

PVC Tank

ADCBoard

SignalConditioner

DACBoard

command signalto shaking table

Pressuresensors

32

456

1

7

(a)(b)

110

(6.1)

Using Eq. 6.1, the first sloshing mode frequency is computed to be 1.05 Hz. The total mass

of water is =0.8 kg. The linear damping is calculated from an expression

given in Abramson, 1966:

(6.2)

where νf is the kinematic viscosity of water, a is the length of the tank in the direction of

the excitation, and g is the gravitational constant. Based on representative values for the

parameters in Eq. 6.2, ζf was found to be equal to 0.004 (0.4%). The water depth ratio is

0.12 which satisfies the shallow water assumption (h/a < 0.15). The excitation amplitudes

considered in this study range from 0.1 to 2 cm, which correspond to Ae/a ratio of 0.004 to

0.08. The excitation frequency ratio ( ) in the sine-sweep tests was in the

range 0.85-1.3.

A six degree of freedom load cell was utilized to measure the base shear due to liq-

uid sloshing. A calibration matrix was used to determine the net shear force in the x-direc-

tion. An accelerometer with a gain equal to the mass of the empty tank estimates the

contribution of the inertial component of the shear force due to the empty tank. This was

verified in the laboratory by testing the tank without water and comparing the value of the

base shear force and the accelerometer reading. The net sloshing force, Fb(t), due to the

liquid sloshing alone is obtained by subtracting the inertial contribution of the empty tank

from the total shear force. Finally, the shear force was expressed in a non dimensional

form as,

ω f1

2π------

gπa

------ πha---

tanh=

m ρawh=

ζ f

ν f

a3 2⁄

g------------------=

γ f ωe ω f⁄=

111

(6.3)

Pressure sensors were also mounted along the wall of the TLD to monitor the

impact pressures generated due to the liquid sloshing. The experimental setup is shown in

Fig. 6.1(b), wherein seven holes at 1.5 cm intervals are made on the side of the tank wall.

The pressures sensors used in this study were piezoelectric transducers with a range of 2

psi and a frequency response of 10,000 Hz. The sensitivity of these sensors is of the order

of 0.15 mV/psi. The sensors were specially fabricated with a silicon gel coating in order to

remove the possibility of any zero-shift problems associated with the change of media the

sensor comes in contact with. In the absence of such a layer, periodic artificial spikes due

to the unbalancing of the bridge resistance are observed which contaminate measure-

ments. The sensor performs this way due to the response of the bridge elements to the

cooling effect of water. Although water is at room temperature, it cools the diaphragm due

to its higher thermal conductivity (Souter and Krachman). Alternating exposure to air and

water during sloshing causes this difficulty, which if not ameliorated can affect measure-

ments significantly.

6.3 System Identification

Time-histories of the non-dimensional base shear force are plotted in Fig. 6.2 for

Ae = 0.3 cm and 2.0 cm. As noted from the figure, the resonant condition occurs at differ-

ent frequency ratios for the two cases, e.g., at Ae =0.3 cm and at Ae

=2.0 cm. Sine-sweep studies were conducted in order to construct the frequency response

curves.

Fb'Fb

mωe2Ae

-----------------=

γ f 1.10= γ f 1.20=

112

Figure 6.2 Sample time-histories of the shear force at Ae = 0.3 cm and 2.0 cm

6.3.1 Nonlinear System Identification

A nonlinear identification scheme was utilized to determine the parameters for the

nonlinear impact characteristics of the TLD. The algorithm used was a nonlinear least

squares constrained optimization algorithm in the MATLAB optimization toolbox (Grace

1992). The objective function evaluates the square of the error between the experimental

0 2 4 6 8 10−0.04

−0.02

0

0.02

0.04

time (sec)

Fb/(mw e2Ae)

γf = 1.10, A

e=0.3 cm

0 2 4 6 8 10−0.04

−0.02

0

0.02

0.04

time (sec)

γf = 1.10, A

e=2.0 cm

0 2 4 6 8 10−0.02

−0.01

0

0.01

0.02

time (sec)

Fb/(mw e2Ae)

γf = 1.15,A

e=0.3cm

0 2 4 6 8 10−0.02

−0.01

0

0.01

0.02

time (sec)

γf = 1.15,A

e=2.0cm

0 2 4 6 8 10−0.02

−0.01

0

0.01

0.02

time (sec)

Fb/(mw e2Ae)

γf = 1.20, A

e=0.3cm

0 2 4 6 8 10−0.02

−0.01

0

0.01

0.02

time (sec)

γf = 1.20, A

e=2.0cm

113

data and the simulated data using the assumed values of the unknown parameters. The

flowchart of the optimization scheme is shown in Fig. 6.3. Figure 6.4 shows the variation

in the impact characteristic function parameters, i.e., ϕ and , introduced in chapter 2, as a

function of the non dimensional amplitude of excitation. After optimization, the following

expressions were obtained:

; ; (6.4)

Figure 6.3 Nonlinear Optimization Scheme

ς

η 2≈ ϕ Ae( ) 2.3

Ae a⁄( )0.78--------------------------≈ ς Ae( ) 1.78 Ae a⁄( )0.68≈

assume initialguess

experimentaldata : F’b, expat amplitude Ae

Define objectivefunctionΣ(F’b(Xo)-F’b, exp)2

and constraintsYu > Y > Yl

Run NonlinearOptimization scheme(lsqnonlin.m)

outputoptimizedvalue ofYo,opt

Yo

114

Figure 6.4 Curvefitting the parameters of the impact characteristics model

Equation 6.4 implies that the damping due to inherent liquid (ζf = 0.4% calculated

using Eq. 6.2) is negligible compared to the total damping, , induced in the TLD

due to sloshing at higher amplitudes. The results of the identification can be seen in Fig.

6.5 where the experimental non dimensional shear force and the analytical shear force

plotted as a function of the frequency ratio are compared. The analytical model success-

fully captures the jump phenomenon and the widening of the frequency band very well.

However, it was noticed that there is a presence of a sub-harmonic resonance at a fre-

quency ratio of 0.96 which is not reflected by the nonlinear model. However, this reso-

nance though present at low amplitudes is more pronounced at some medium amplitudes

and is suppressed at high amplitudes. The current analytical model does not contain these

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

50

100

150

200

non−dim. amplitude Ae/a

Parameter

φ(A e)

Data Non−linear Fit

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

0.05

0.1

0.15

0.2

0.25

0.3

0.35

non−dim. amplitude Ae/a

effective damping

ς(A e)

Data Non−linear Fit

ς Ae( )( )

115

second-order effects. More complex models which include higher order nonlinearities can

model this effect. However, this is not pursued in this study. Figure 6.5 suggests that even

at low amplitudes (0.1 cm), the nonlinear jump phenomenon is present.

Figure 6.5 (a) Experimental plots of non-dimensional sloshing force as a functionof excitation frequency for different amplitudes (b) Simulated curves after

optimization

6.3.2 Combined Structure-damper analysis

Next, combined TLD-structure system is studied. The equations of motion of a structure

represented as a SDOF system and TLD are given by,

(6.5)

(6.6)

where the subscripts s and f refer to the structure and damper respectively, and the rest of

the symbols have been defined earlier. The mass ratio, is equal to 0.01 and

0.9 1 1.1 1.2 1.30

5

10

15

Frequency ratio ωe/ωf

Non

−dimensional Sloshing Force

Ae=0.1 cm Ae=0.25 cmAe=0.5 cm Ae=1.0 cm Ae=2.0 cm

0.9 1 1.1 1.2 1.30

5

10

15

Frequency ratio ωe/ωf

Experimental Analytical

M s X s˙ Cs X s

˙ Ks X s c f X s˙ x f

˙–( ) keff X s( )X s keff x f( )x f–+ + + + Fe t( )=

m f x f c f X s x f–( ) keff X s( )X s keff x f( )x f–+=

µ m f M s⁄=

116

the tuning ratio is equal to 0.99. Solving the equations of motion given in

Eqs. 6.5 and 6.6 numerically and plotting the non-dimensional displacement of the struc-

ture (Xs/Ae) as a function of the frequency, the transfer functions as shown in Fig. 6.6 are

obtained.

Figure 6.6 Response of the structure for different amplitudes

The combined TLD-structure system exhibits certain change in transfer function

characteristics as the amplitude of excitation increases. The frequency response of a TLD,

unlike a tuned mass damper, is excitation amplitude dependent. The increased damping

(introduced by wave breaking and slamming) causes the frequency response function to

change from a double-peak to a single-peak function like an over-damped TMD. This

change in frequency response has also been observed experimentally, e.g., Sun and Fujino,

1994.

γ ωf ωs⁄=

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.250

5

10

15

20

25

30

35

40

ωe/ωs

Non−dimensional displacement of Structure

Ae=0.01 cmAe=1 cm

117

6.4 Impact Pressure Studies

The shallow water theory leads to a hydrostatic pressure description for loads on

the sloshing container walls. This is appropriate when standing waves or small travelling

waves are excited. However, as soon as impacts are recorded at the walls, the pressure dis-

tribution appears very different due to the presence of the impulsive peaks. At this time,

the pressure distribution at the vertical walls is far from hydrostatic. In this section, the

local pressures on the walls of the TLDs arising due to the sloshing impacts of the liquid

are studied in detail.

Seven measurement taps were drilled in the side of the tank for pressure sensors at

intervals of 1.5 cm (Fig. 6.1(b)). Sensor 1 is at 1.5 cm from the bottom of the tank, sensor

2 is at 3.0 cm (static liquid level) and so on. The sampling frequency of the data acquisi-

tion system was maintained at 1000 Hz. This was found to be adequate since the duration

of the peak impact in the resonant pressure trace was found to be of the order of 15-20

milli-seconds. Data acquisition for each case was carried out for about 30 sec which corre-

sponded roughly to 30 cycles of data. The average value of the peak pressure over N cycles

is calculated as follows:

(6.7)

The pressure peak coefficient at a certain height z on the vertical wall is defined as:

(6.8)

P peak[ ]

Pi peak,i 1=

N

∑N

---------------------------=

K Pz

P peak[ ]ρga

------------------=

118

6.4.1 Single-point pressure measurement

Figure 6.7 shows typical pressure traces at different frequency ratios including res-

onant and non-resonant cases, i.e., γf = 0.7, 1.1 and 2.0. As seen from the plots, the impact

peak pulses are present only at the resonant condition. As we know from base shear

results, this resonant condition does not occur at γf = 1.0, but at 1.1 due to the hardening

nature of the sloshing phenomenon.

Figure 6.7 Pressure time histories for various frequency ratios (Ae = 1.0 cm).

It has been observed that these typical pressure time histories are neither harmonic

nor periodic since the magnitude and duration of the peaks vary from cycle to cycle. This

is true even though the excitation experienced by the tank is harmonic. Figure 6.8 shows

the histogram of peak impact pressure for 100 cycles of pressure pulses for sensor at loca-

tion 2. A statistical analysis of the pressure time records was conducted and the data was

fitted with a Lognormal distribution (Fig. 6.8).

0

0.005

0.01

0.015

0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0

0.005

0.01

0.015

0.02

γ = 0.7

γ = 1.4

γ = 1.1

Non-resonant Sloshing Resonant Sloshing

Pressure (psi)

Pressure (psi)

Pressure (psi)

f

f

f

119

Figure 6.8 Probability distribution function of the peak impact pressures

Figure 6.9(a) shows the anatomy of a single pressure profile as it evolves over time

along with corresponding visual photographs of wave sloshing. It is noteworthy that the

impulsive peak is observed at 15 msec which suggests the high frequency slamming

nature of the pressure pulse. After the initial impact caused by the wave, the full sloshing

action of the wave is developed, which can be seen as a second peak of lower magnitude

and longer duration. A wavelet scalogram (using Morlet wavelet) was utilized to study the

time-frequency fluctuations of the pressure time-history. For more details on this tech-

nique, one can refer to Gurley and Kareem (1999). A scalogram is a plot wherein the

square of the coefficients obtained by continuos wavelet transform (CWT) are plotted as a

measure of the signal energy in the time-frequency domain. The scalogram of the pressure

signal reveals the presence of high frequency components at the time of occurrence of the

impulsive peak (Fig. 6.9b). The energy in regular sloshing is concentrated at lower fre-

quencies which occurs after a certain time-lag following the initial impact.

0 0.2 0.4 0.6 0.80

5

10

15

20

25

30

35

pressure (psi)

Probability distribution

lognormal distribution

−2.5 −2 −1.5 −1

0.003

0.01

0.02

0.05

0.10

0.25

0.50

0.75

0.90

0.95

0.98

0.99

0.997

Log of Data

Probability

Normal Probability Plot

120

Figure 6.9 (a) Anatomy of a single pressure pulse (b) wavelet scalogram of thepressure signal

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Time (mSec)

Pressure (psi)

Point A= 15 mSec

Point B= 37 mSec

Point C= 200 mSec

Point D= 285 mSec

(a)

(b)

121

6.4.2 Multiple-point pressure measurements

Next, four sensors were recorded simultaneously to observe the time-lag as the

pulse travels along the tank height and the spatial distribution of the impulsive peak to the

overall slosh pulse. Figure 6.10 shows the simultaneous pressure pulse traces for a single

cycle. The time-lag is measured with respect to sensor 2 (which is at the mean water

level). The impact influence factor (IIF) is defined as:

(6.9)

where Ai is the area under the impulsive peak in the pressure time-history and At is the

total area under the sloshing/slamming trace (including the impulse component). It is

observed that at levels above the water level, the contribution is entirely due to impulsive

slamming. On the other hand, the contribution of slamming at sensor 1, which is below the

water level is only about 10%. This corroborates with topology of wave slamming because

the slamming action is more prevalent in the region above the mean water level. The roll-

ing convective mass of water, which is responsible for the slamming action, is primarily

effective at these locations. The time-lag and IIF for the four locations are documented in

Table 6.1.

Figure 6.10(b) shows the coscalograms of the different sensor measurements. A

coscalogram in wavelet analysis is analogous to the cospectrum in the spectral analysis.

Like the scalogram, it is useful in revealing time varying pockets of high and low correla-

tion in different frequency bands. It is obtained by plotting the product of the wavelet coef-

ficients of two signals as a function of time and frequency. The coscalograms in Fig.

6.10(b) are plotted with reference to sensor 2. The light patches in the coscalograms help

identify areas of correlation. It is noted that the maximum correlation between each sen-

IIFAi

At-----=

122

sors is near the low frequency sloshing component of the pressure signals. The correla-

tions in the high frequency slamming portion is maximum in the sensor 2-1 coscalogram

and drops off progressively in the 2-3 and 2-4 coscalograms due to the time lag of peaks

which was discussed earlier.

A pressure-time integration of the pulses recorded at sensors 2 and 3, averaged

over a number of measurements, yields that the contribution of the impulsive peak is

around 20-30% of the total contribution of the pulse. This is a substantial contribution

which is neglected by most numerical simulations. Moreover, the peak pressures obtained

due to slamming are 5-10 times higher than those obtained from regular sloshing as

observed earlier. The sloshing-slamming damper analogy, described in Chapter 2, also

emphasizes the importance of estimating the effect of the liquid slamming on the overall

system response. Similar concerns have been expressed in the study of impact loading of

vertical structures in the offshore community, where the impact pressures were assumed to

be not important and hence were not considered in the design. However, Schmidt et al.

1992 have demonstrated that this is an inadequate approach to design.

TABLE 6.1 Time lag and impact influence factor for different sensor locations

Time lag of peaks withrespect to sensor 2 (msec)

Impact influence factorIIF (%)

1st sensor -2 10

2nd sensor 0 21

3rd sensor 14 30

4th sensor 42 85

123

Figure 6.10 (a) Pressure pulses at different locations on the wall (b) Waveletcoscalograms with sensor 2 as reference

2 2.05 2.1 2.15 2.2 2.25 2.3 2.35 2.4 2.45 2.50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

time (sec)

Pressure (psi) 1

2 34

Time lag of peaks with respect to 1st sensor:2nd sensor: 2 msec3rd sensor: 14 msec4th sensor: 42 msec

sensor 2-2 sensor 2-1

sensor 2-3 sensor 2-4

(a)

(b)

124

6.4.3 Shallow water versus deep water sloshing

Until now, the results presented were for the shallow water case (h/a < 0.15).

In this case, sloshing at high amplitudes is characterized by travelling waves and hydraulic

jumps (Fig. 6.11a). For deep water cases, i.e, h/a > 0.15, large standing waves are usually

formed at resonance. Figure 6.11(b) shows the difference between the shallow water (h/a

=0.12) slosh pressure trace and deep water (h/a = 0.25) pressure traces for the pressure tap

locations at the mean water level. In the case of shallow water TLD, the pressure is maxi-

mum at the mean liquid level, while for the case of deep water TLD, impact pressures are

also observed in a large part of the ceiling. The impulsive peak is more pronounced in the

shallow water case and reaches peak value at 15 msec as opposed to the deep water case

where the peak value is reached at 50 msec.

Figure 6.11 Typical sloshing wave with pressure pulse and wave mechanismschematic for (a) shallow water (h/a =0.12) and (b) deep water (h/a = 0.25) case

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Time (mSec)

Pres

sure

(ps

i)

Point A= 15 mSec

Point B= 37 mSec

Point C= 200 mSec

Point D= 285 mSec

(a) (b)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Time (mSec)

pres

sure

(ps

i)

Point A= 50 mSec

Point B= 90 mSec

Point C= 220 mSec

Point D= 300 mSec

125

6.4.4 Pressure variation along the tank height

The pressure distribution over the tank walls is important for establishing integral

load effects due to slamming and design considerations of walls under sloshing/slamming-

induced loads. Bass et al. 1980 have provided an idealized distribution for vertical tank

walls based on their experiments in terms of a pressure coefficient which was described by

the following cosine function:

; (6.10)

where KPz is the peak pressure coefficient, KPmax is the maximum pressure coefficient

(which occurs at the mean water level for the shallow water case), z = distance from tank

bottom, h = liquid filling height, H = tank height. As seen from Fig. 6.12, where the maxi-

mum pressure coefficients at Ae = 2.0 cm are plotted along the height of the wall. One can

note that the curve described by Eq. 6.10 envelopes the maximum pressure peak coeffi-

cients obtained from the present studies.

Figure 6.12 Variation of the peak pressure coefficient with height of the tank wall

K Pz1

2---K Pmax 1

5π z h–( )H

----------------------- cos+= h

H5----- z h

H5-----+≤ ≤–

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.5

0

0.5

1

1.5

2

z/h

KPz = [K

Pm ax(1+cos(5π(z−h)/H)]/2

Mean Peak Pressures (Mean+1Std.) Peak Pressures (Mean−1Std.) Peak Pressures Curve described in Bass et al. (1980)

Mean Liquid Level , h = 3 cm

KPz/KPmax

126

6.5 Hardware-in-the-loop Simulation

One of the main areas of investigation in the design of TLDs is the actual perfor-

mance when installed in a structure. Hardware-in-the-loop (HIL) refers to a simulation

technique in which some of the system components are numerically simulated while oth-

ers are physically modeled with appropriate interface conditions. Usually, there are real

hardware characteristics that are unknown or too complex to model in pure simulations. In

these situations, HIL is an extremely useful simulation technique. Hardware-in-the-loop

developed out of a hybrid between control prototyping and software-in-the-loop simula-

tions (Isermann, 1999). HIL is routinely used in aerospace, automotive control and

embedded systems engineering as an inexpensive and reliable rapid-prototyping technique

for product development. Its application in structural testing of damping systems has been

rather limited.

This method is especially applicable to structure-damper experimental studies.

One can specify the external loading and model the structure by appropriate equations,

which are solved in real-time to obtain the displacement response. This displacement is

used to drive the shaking-table on which the damper is mounted. The base-shear force due

to sloshing liquid in the damper is simultaneously measured and feedback into the com-

puter where it is used in the fore mentioned numerical equations. Thus, a real-time

dynamic coupled structure-damper analysis is conducted without the use of an actual

physical structure or heavy actuators to actuate the structure.

Some of the advantages of HIL simulation over conventional testing methods are

the cost and time savings in repeated simulations. Figure 6.13 shows the difference in

scale and the associated costs one can achieve with HIL testing for combined structure-

damper experiments. The dynamic testing of structural systems with nonlinear append-

127

ages require considerable infrastructure involving structural system model, actuators,

reaction wall system, and instrumentation. Often the actuators are limited in their dynamic

capability which restricts these tests to a pseudo-dynamic level. While, in HIL simulation,

one can build a virtual structure in a computer model and the non-linear structural ele-

ments, such as dampers, hysteretic elements and base-isolators, can be included in the

physical structural model. Moreover, one needs a smaller shaking table for component

testing. One of the most useful aspects of the HIL testing is that the user can perform on-

the-fly tuning of important structural and excitation parameters. This can help in identify-

ing important parametric relations between the two systems. A computer controlled sys-

tem, which is standard in most dynamic testing laboratories and an essential component

for implementing controllers for the shaking table, is needed to conduct the test in real-

time. Some of the main issues for the success of this test is the speed of the computer con-

trol system. The disadvantage of the test is that a good system model is needed for the

structure which is not available in all cases.

Figure 6.13 Hardware-in-the-loop concept for structure-liquid damper systems

sensors

TLD

Structure

On the fly tuningof parameters for thevirtual structure

M s

ζsωs

Fe(t)

Fe(t)

xs(t)

128

6.5.1 Experimental study

Figure 6.14 shows a schematic of the experimental setup for verification of the

hardware-in-the-loop concept. It is similar to the experimental setup shown in Fig. 6.1. As

discussed earlier, the net sloshing force, Fb(t), due to the liquid sloshing alone was

obtained by subtracting the inertial contribution of the empty tank from the total shear

force. For the combined structure-damper system, the equation of motion of the structure

can be written as,

(6.11)

The displacement of the structure was calculated using the finite-difference version of Eq.

6.11 and the displacement signal was sent back as a voltage to the shaking table. In this

way a real-time experiment of the combined dynamics of the structure and the damper was

conducted.

Figure 6.14 Schematic of the experimental setup for the HIL simulation

M s X s Cs X s Ks X s+ + Fe t( ) Fb t( )+=

Shaking Table

Baffles

6 DOFLoad Cell

PVC Tank

ADCBoard

DAC

Board


+

-

Accelerometer

Signalconditioner

Signalconditioner

m tank

Ftotal

Fs

Shaking table motor and encoder

Combined equations

of motion solved

Sloshing Force Fb

xs

in the computer

129

An important aspect of the HIL simulation is the real-time integration algorithm.

For real-time simulation one should use fixed-time steps and should require inputs for

derivative calculations that occur at the current time step and earlier. This means that

fourth-order Runge-Kutta method is not applicable in such circumstances. Euler’s first

order algorithm has poor characteristics. The Adams-Basforth second order algorithm

seems to provide much better accuracy yet it is suitable for real-time use. The displace-

ment of the structure for the next time step tj+1 is calculated from displacements and

velocities at current and earlier time steps tj and tj-1 as,

(6.12)

Figure 6.15 Hardware-in-the-loop simulation for random loading case

X s j, 1+ X s j, ∆t3

2--- X s j,

1

2--- X s j, 1––

+=

0 10 20 30 40 50 60−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time (sec)

Disp

lace

ment

of

Stru

ctur

e, x

s (c

m)

Uncontrolled Controlled with TLD

0 10 20 30 40 50 60−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time (sec)

External Excitation, F

e(t) (cm)

130

In the current experiment, a fixed time step of 0.005 (sampling frequency of 200

Hz) was utilized. This was suitable for our application as the frequency range of interest

was less than 5 Hz. The parameters used in the simulation are =1.1 Hz, = 3% and

= 10%. Figure 6.15 shows the excitation time history used which is a random white noise

signal. The figure also shows a comparison of the uncontrolled response and the controlled

response by including the sloshing force due to the TLD. The total reduction in RMS

response with and without the damper is 75%.


A new sloshing model incorporating impact characteristics has been presented.

The model parameters can be obtained from experimental data obtained by an instru-

mented sloshing tank placed on the shaking table. Impact pressure distributions were also

measured along the height of the container. It was noted that the slamming action is

present in shallow water TLDs and has a significant contribution to the overall sloshing

force. These impact pressure studies also indicate the nature of sloshing-slamming along

the height of the container, for e.g., at levels below the static liquid level, the pressure is

dominated by the sloshing component while at levels above the static liquid level, it is

governed by the slamming action. Finally, a new technique, namely the hardware-in-the-

loop testing technique was presented for testing structure-liquid damper systems. This

method promises to be a cheaper alternative to dynamic testing without the use of an

actual structure, its scale model or large high-speed dynamic actuators to induce dynamic

loading.

ωs ζ s µ

131

CHAPTER 7

TLCD EXPERIMENTS

In theory, there is no difference between theory and practice.But, in practice, there is.

- Jan L.A. Van de Snepscheut

In this chapter, different experiments conducted using scale models of structures

along with a prototype semi-active TLCD are presented. First, the dynamic characteristics

of the combined structure-damper system were compared with previously obtained analyt-

ical results reported in Chapter 3. Next, a gain-scheduled control law based on a pre-

scribed look-up table was experimentally verified for achieving the optimum damping

based on a prescribed look-up table.

7.1 Introduction

Experimental studies using tuned liquid column dampers (TLCDs) for evaluating

their control performance have been limited to passive systems. Sakai et al. (1991) verified

the performance of a TLCD installed on a scaled-down model of an actual cable stayed

bridge tower. Balendra et al. (1995) conducted shaking table tests using TLCDs and stud-

ied the effect of different orifice opening ratios on the liquid motion. Experimental studies

have also been reported by Hitchcock et al. (1997) using passive TLCDs with no orifice,

termed as liquid column vibration absorbers (LCVAs). Recently Xue et al. (2000) pre-

sented experimental studies on the application of a passive TLCD in suppressing the pitch-

ing motion of structures and conducted experiments to delineate the influence of different

damper parameters on the TLCD performance.

132

A full scale installation of a bi-directional passive liquid column vibration absorber

(LCVA) on a 67m steel frame communications tower has been reported by Hitchcock et

al. (1999). This device does not have an orifice/valve in the U-tube and hence, it is not

possible to control the damping in the LCVA. The authors also acknowledge that due to

the lack of orifice, the damping ratio of the LCVA was not expected to be optimum. The

authors observed that the LCVA did not perform optimally at all wind speeds. Response

reduction of almost 50% was noted, however, non-optimal performance of the damper was

noted above and below the design wind speed. This observation re-affirms the fact that

passive liquid damper systems are inadequate in performing optimally at all levels of exci-

tation (Kareem, 1994).

This chapter discusses experimental verification of a semi-active system which

may be utilized to overcome the aforementioned shortcomings of a passive TLCD system.

Although researchers have studied the semi-active version of TLCD theoretically (Haroun

et al. 1994; Kareem, 1994; Abe et al. 1996; Yalla et al. 1998), there has been no reported

experimental verification of such a system. In this chapter, different experiments were

conducted using scale models of structures along with a prototype semi-active TLCD. The

dynamic characteristics of the coupled structure-damper system were compared to previ-

ously obtained analytical results presented in Chapter 3. Next, a gain-scheduled control

law for achieving the optimum damping based on a prescribed look-up table was verified

experimentally.

7.2 Experimental Studies

The experimental set-up is shown in Fig. 7.1(a)-(b). It consists of a single story

structure model attached to a TLCD. The TLCD consists of a U-shaped tube made of PVC

133

material with an electro-pneumatic actuator driving a ball valve attached at the center of

the tube.

Figure 7.1 (a) Photograph of the Electro-pneumatic actuator (b) Schematicdiagram of the experimental set-up

The U-tube has a circular cross-section with an inner diameter of 3.8 cm and a

horizontal length of 35.5 cm and a total length of 81 cm. The valve used in this study is a

DACchannels

ADCchannels

signalconditioner

signalconditioner

position transmitter

4-20 mAto positioner

Shaking Table


SigLabSpectrum analyzer

Accelerometer encoder ouput

4-20 mAsignal

BuildingModel

shownin detail

80 psi Pneumatic Air-line

Computer

(a)

(b)

134

ball valve of 3.8 cm (1.5 inches) diameter. A command voltage changes the valve opening

angle (θ), which effectively changes the orifice area of the valve. The details of the valve

characteristics are presented in Appendix A.3, where the valve opening angle is related to

the headloss coefficient( ).

Transfer function measurements were obtained by exciting the shaking table with a

band-limited random white noise (cutoff frequency fc = 2 Hz), at different levels of excita-

tion amplitudes and the acceleration was measured at the top of the structure. The excita-

tion amplitude in these experimental studies is referred to as S0 and it represents an RMS

value of excitation (in volts). The range of feasible RMS excitation displacement ampli-

tudes of the shaking table without spilling water out of the U-tube was varied between

0.05-0.3 volts.

The model structure without the damper is a linear system, which was confirmed

through identification of the transfer function at different amplitudes of excitation. The

effect of the pneumatic actuator used to drive the valve in the TLCD on the dynamics of

the structure was found to be negligible. This was done by comparing the transfer func-

tions with and without the air-supply to the pneumatic actuator. All transfer function mea-

surements were obtained using SigLabTM spectrum analyzer using the average of 15

measurements. From the transfer function and free vibration decay curves, the natural fre-

quency and damping ratio of the uncontrolled building was determined to be 0.92 Hz and

0.6%, respectively. The mass ratio (ratio of the liquid mass in the damper to the first modal

mass of the structure) is kept approximately 10% of the total mass of the structure.

ξ

135

7.2.1 Effect of tuning ratio

The tuning ratio (γ ) is defined as the ratio of the natural frequency of the damper

(= ) to the natural frequency of the structure. In order to determine the optimum

tuning ratio, liquid columns of different lengths were considered. Figure 7.2 (a) shows the

transfer function with different tuning ratios. The norm was used as a measure of

evaluating the performance at each tuning ratio, which is defined as:

(7.1)

where is the acceleration of the structure, is the ground acceleration of the shaking

table, =0.5 Hz and =1.5 Hz. The range of frequencies were limited to 0.5-1.5 Hz

because below 0.5 Hz there was a lot of noise in the system and above 1.5 Hz, there is neg-

ligible change in each transfer function.

Figure 7.2 (b) shows the variation of the H2 norm as a function of the tuning ratio.

A fourth order polynomial fit was used to determine the optimum tuning ratio as equal to

0.953. This corresponds to a liquid length of 25 inches (63.5 cm). One can observe that the

two peaks in the transfer function are almost equal in height at the optimum tuning. This is

consistent with the analytical formulations regarding optimal tuning of two-degree-of-

freedom systems (Den Hartog, 1940).

2g l⁄

H 2

H 2 HX s xg

ω( ) 2 ωd

ωa

ωb

∫≈

X s xg

ωa ωb

136

Figure 7.2 (a) Transfer functions for different tuning ratios (b) Variation of H2norm with tuning ratio

7.2.2 Effect of damping

The effective damping in the TLCD is obtained through changing the orifice open-

ing of the valve. As noted in previous chapters, the effective damping of the TLCD is an

important parameter for optimum absorber performance. The damping is varied by chang-

ing the valve angle, where θ = 0 corresponds to fully-open valve and θ = 90 degrees corre-

sponds to fully-closed valve. In the fully-closed position, no liquid oscillations take place

and the system becomes a SDOF system. An upper limit of θ = 60 degrees is used in this

study. At this position, the valve is almost fully closed. Figure 7.3 shows how the transfer

function changes as the valve opening is changed.

0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

frequency (Hz)

Tran

sfer

func

tion

(Mag

)

γ=1.02γ=1.00γ=0.98γ=0.96γ=0.94

0.92 0.94 0.96 0.98 1 1.02 1.040.18

0.185

0.19

0.195

0.2

0.205

0.21

0.215

0.22

Tuning ratio, γ

H2 n

orm

curve−fittedusing 4th orderpolynomial

(a) (b)

137

Figure 7.3 Transfer functions for different valve angle openings

7.2.3 Effect of amplitude of excitation

It is well known that the damping introduced by valves and orifices is quadratic in

nature. This has been studied experimentally for passive TLCDs (Sakai and Takaeda,

1989; Balendra et al. 1995). The damping force is dependent on the liquid velocity,

(7.2)

This implies that the damping introduced by the valve is non-linear and changes as a func-

tion of the amplitude of excitation. Figure 7.4 shows the transfer functions of the com-

bined system at two different excitation levels, i.e., S0 = 0.1 and 0.3 V with different valve

opening angles. The transfer functions at θ = 0 degrees (fully-open) are virtually identical

0.5 1 1.50

0.5

1

1.5

2

2.5

frequency (Hz)

Transfer function (Magnitude)

0 deg 10 deg20 deg35 deg40 deg45 deg50 deg60 deg

Fd c x f x f=

138

as no nonlinearity is introduced due to the valve. At other valve opening, however, the

non-linearity introduced by the valve can be clearly noted.

Figure 7.4 Variation of transfer functions for different amplitudes of excitation

From Fig. 7.4, one can note the change in effective damping as the excitation

amplitude is varied. Therefore, for the damper to perform optimally at all levels, one needs

to determine the optimum damping required at each amplitude of excitation and organized

in the form of a look-up table. The main idea of a look-up table is to determine the angle of

opening which minimizes the norm of the structural response. This corresponds to

the optimal valve opening for a particular amplitude of excitation, as shown in Fig. 7.5(a)

for S0 = 0.1 V and S0 = 0.3 V. This procedure is repeated for a wide range of amplitudes of

excitation. Using these optimal values, one can construct a nonlinear look-up table as

shown in Fig. 7.5(b).

0.5 1 1.50

0.5

1

1.5

2

2.5

frequency (Hz)

Transfer function (Mag dB)

θ = 0 deg

0.5 1 1.50

0.5

1

1.5

2

2.5

frequency (Hz)


θ = 35 deg

0.5 1 1.50

0.5

1

1.5

2

2.5

frequency (Hz)


θ = 40 deg

S0=0.1 V

S0=0.3 V

H 2

139

0.5

ntr

lts)

Figure 7.5 (a) Optimization of H2 norm (b) Look-up table for semi-active control

7.2.4 Equivalent damping

The equivalent damping of the TLCD is a function of the excitation amplitude and

the valve opening. An MATLABTM program was used to curve-fit the experimental trans-

fer function by minimizing the norm of the error function. The equivalent damping was

found to range from 2% (for fully open, θ = 0 deg) to 30% (for almost closed, θ = 60 deg).

The optimal damping ratio is obtained as 9% (θ = 40 deg at S0 = 0.1 V) as seen in Fig.

7.6(a)). Figure 7.6 (b) shows the transfer function with non-optimal damping (about 30%)

which is realized at θ = 60 deg.

Closed-form equations for the case of white noise excitation applied to the primary

system were presented in Chapter 3. However, as reported in Warburton (1982), it is

known that the optimum absorber parameters that minimize the RMS accelerations of the

primary system for a white noise base excitation are the same as those that minimize the

0 10 20 30 40 50 600.34

0.36

0.38

0.4

0 10 20 30 40 50 600.35

0.36

0.37

0.38

Φ

0.3 V0.1 V

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

10

20

30

40

50

60

70

0

Optimum Valve Opening,

θ

Look-up Tablefor Semi-active co

θ 1 = 30 degrees at So=0.3 Vθ 2 = 40 degrees at So=0.1 V

Valve opening, θ (degrees) Excitation Amplitude, So (Vo

H

2 norm

(a) (b

140

RMS displacements for a white noise excitation applied to the primary system. Therefore,

in this study the equations derived in Chapter 3 are used. In the case of an undamped pri-

mary system, one can write the expressions for optimal damping and tuning ratio as,

; (7.3)

In the case of µ = 0.1 and α = 0.56 and , optimum values of the absorber

parameters obtained from Eq. 7.3 are: = 8.9% and = 0.95, which are close to the

experimental values ( = 9.0% and = 0.953).

Figure 7.6 (a) Comparison of transfer functions: (a) θ =40 deg, = 9 % (optimal

damping) (b) θ = 60 deg, = 30% (non-optimal damping)

ζoptα2---

µ 1 µ α2–

µ4---+

1 µ+( ) 1 µ α2µ2

----------–+

----------------------------------------------------= γopt

1 µ 1α2

2------–

+

1 µ+--------------------------------------=

ζ s 0≈

ζopt γopt

ζopt γopt

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.5

1

1.5

2

2.5

3

frequency (Hz)

Transfer Function (Mag)

Experimental DataSimulated

controlled

uncontrolled

(b)

So = 0.1 V

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.5

1

1.5

2

2.5

3

frequency (Hz)

Transfer Function (Mag)

Experimental DataSimulated

controlled

uncontrolled

So = 0.1 V

(a)

ζ f

ζ f

141

1.5

Figure 7.7 (a) shows the 3-D plot of the magnitude of the experimental transfer

function as a function of the valve opening angle (and effective damping) and the fre-

quency at S0 = 0.1 Volts. One can observe that the double peaked transfer function changes

to a single peak curve as the valve opening angle is increased. Figure 7.7 (b) shows the

simulated 3-D transfer function as a function of frequency and equivalent damping ratio.

A similar curve is obtained by solving the actual non-linear equations of the TLCD and

plotting the dynamic magnification ratio as a function of frequency and the headloss coef-

ficient (for e.g., see Haroun and Pires, 1994). The effect of coalescing of the modal fre-

quencies, from a double peaked curve to a single peaked curve, was also described in

chapter 4 while examining the beat phenomenon of the combined structure-TLCD system.

Figure 7.7 3-D plot of transfer function as a function of effective damping andfrequency (a) experimental results (b) simulation results.

0

0.1

0.2

0.30.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

0

0.5

1

1.5

2

2.5

frequency, Hz ζd

Transfer Function (mag)

0

20

40

600.5

11.5

0

0.5

1

1.5

2

frequency, HzΦ

Tran

sfer

Fun

ctio

n (m

ag)

(a) (b)

ζf

frequency, Hz

ζf

0

θ

0.1

0.2

0.3

2.5

142

The experimental results show that the effective damping is a function of the amplitude of

excitation and valve angle opening, i.e.,

(7.4)

In section 3.2.1, the expression for the equivalent damping was obtained as:

(7.5)

From the Appendix A.3, one can note that the headloss coefficient is a function of the

valve opening angle, i.e.,

(7.6)

while the standard deviation of the liquid velocity is related to the amplitude of excitation

by Eq. 3.9,

(7.7)

Therefore, it follows that,

(7.8)

Note that the damping is dependent on which in turn is dependent on implying

that the relationship in Eq. 7.8 is a nonlinear function.

7.3 Experimental Validation

The next step was the experimental validation of the control strategy outlined in

Chapter 5. The main idea was to benchmark the performance of the semi-active system to

a purely passive system. In the case of a passive system, the headloss coefficient was kept

constant. For the semi-active case, the valve opening was changed according to the look-

up table developed in Fig. 7.5 (b).

ζ f f S0 θ,( )=

ζ f

ξσ x f

2 πgl---------------- f σx f

ξ,( )≡=

ξ f θ( )=

σ x ff S0( )=

ζ f f S0 θ,( ) f ξ σ x f,( )≡=

σ x fζ f

143

Two different loading time-histories were chosen. The first time history, referred to

as case 1, comprised of segments of 20 sec each in length of 0.1 and 0.3 V RMS excita-

tions, while the second time history (case 2) comprises of segments 40 sec each in length

of 0.1 and 0.3 V RMS excitations. The underlying objective was to show that the semi-

active TLCD, which changes the headloss coefficient in response to changes in external

excitation, performed much better than a passive TLCD.

Figure 7.8 Excitation time history, valve angle variations and the resultingaccelerations for uncontrolled, passive and semi-active systems for case 1.

0 5 10 15 20 25 30 35 40

−1

−0.5

0

0.5

1

Time (sec)

Acceleration (m/s

2 )

Uncontrolled Passive System Semi−active System

0 5 10 15 20 25 30 35 40−1

−0.5

0

0.5

1

time (sec)

S 0 (Volts)

0 5 10 15 20 25 30 35 400

10

20

30

40

50

time (sec)

Angle of Valve,

θ

144

Figure 7.9 Excitation time history, valve angle variations and the resultingaccelerations for uncontrolled, passive and semi-active systems for case 2.

From Figs. 7.8, 7.9 and Table 7.1, one can note that for 0.3 V, there is hardly any

response reduction for the case 1, while there is a 76% reduction for case 2. This is

because case 2 record is of a longer duration and hence the steady-state of the response is

established. This increases the liquid damper effectiveness as liquid oscillations are fully

developed. One can also see that at higher levels of excitation, the optimum damping is

close to the passive system damping, therefore the improvement of semi-active system is

0 10 20 30 40 50 60 70 80−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time (sec)

Acce

lera

tion

(m/

s2 )

Uncontrolled Passive System Semi−active System

0 10 20 30 40 50 60 70 80−1

−0.5

0

0.5

1

time (sec)

S 0 (Volts)

0 10 20 30 40 50 60 70 800

10

20

30

40

50

time (sec)

Angle of Valve,

θ

145

not substantial (about 13% improvement over passive system). On the other hand, for

lower levels of excitation, the improvement is more drastic (about 27% improvement over

passive system). The overall RMS response reduction of semi-active system over passive

system was 23% for case 1 and 15% for case 2.

It is noteworthy that the response reduction of 76% is high. This is because the

mass ratio of the damper considered in the scaled-down experiment was 10%. This is a

very high mass ratio since in most typical buildings, a mass ratio of approximately 1% can

be accommodated due to the weight and space requirements. However, in this study, a

comparison of the performance of passive and semi-active systems was performed.

Numerical studies indicate, however, that a 1% mass ratio would provide about 45%

reduction in response. A similar analytical study was performed in Chapter 5 (Section

5.3.1), where an improvement of 20% was noted for a semi-active system over a passive

system using a TLCD mass ratio of 1%.

TABLE 7.1 Performance of semi-active system as compared to uncontrolledand passive system

Case 1

RMS

(cm/s2)

Peak

(cm/s2)

RMS

(cm/s2)

Peak

(cm/s2)

RMS

(cm/s2)

Peak

(cm/s2)

segment 1: First 20 sec segment 2: Next 20 sec Total 40 sec

Uncontrolled 20.17 45.08 46.65 125.57 35.94 125.57

Passive 13.69

(32.1%)

32.0

(29%)

45.30

(2.8 %)

105.25

(16.2 %)

33.46

(6.9 %)

105.25

(16.2 %)

Semi-Active 10.09

(50.0 %)

26.34

(41.6 %)

34.95

(25.08 %)

92.76

(26.1 %)

25.73

(28.4 %)

92.76

(26.1 %)

Case 2 segment 1: First 40 sec segment 2: Next 40 sec Total 80 sec

Uncontrolled 27.69 64.49 125.72 262.67 91.03 262.67

Passive 17.04

(38.5 %)

55.34

(14.2 %)

34.73

(72.3 %)

100.12

(61.8 %)

27.35

(69.95 %)

100.12

(61.8 %)

Semi-Active 12.56

(54.6 %)

40.86

(36.64 %)

30.2

(75.97 %)

95.02

(63.8 %)

23.15

(74.56 %)

95.02

(63.8 %)

146


An experimental investigation to determine the optimal absorber parameters of the

combined structure-TLCD system was presented. The experimental results were com-

pared to the previously obtained analytical results. A control strategy based on a gain-

scheduled look-up table was verified experimentally. It was observed that at low ampli-

tudes of excitation, the TLCD damping was enhanced by constricting the orifice and at

higher amplitudes by dilating the orifice to supply the optimal damping. Experimental

studies have shown that the semi-active TLCD can boost the performance of the passive

TLCD by an additional 15-25% and maintains the optimal damping at all levels of excita-

tion. This justifies the additional costs of using sensors and controllable valves in the semi-

active system. A more detailed cost and implementation comparison is discussed in Chap-

ter 8.

147

CHAPTER 8

DESIGN, IMPLEMENTATION AND RELIABILITY ISSUES

I strive for structural simplicity.... the technical man must not be lost in his owntechnology.

- Dr. Fazlur Khan

In this chapter various aspects dealing with design considerations, implementation

details, cost analysis and reliability issues of liquid dampers are discussed. First, compari-

sons are made among different types of dynamic vibration absorbers (DVAs) in terms of

their implementation and cost. Next, a risk-based decision analysis framework is pre-

sented to measure the risk of unserviceability in tall buildings and to provide a basis for

choosing the optimal decision. Finally, some design guidelines for technology transfer are

laid out in accordance with the research conducted and documented in earlier chapters.

8.1 Introduction

In previous chapters, analytical studies on liquid dampers and experimental valida-

tion on scale models have been discussed. However, the actual implementation of these

dampers in full-scale structures needs careful consideration of certain practical design

constraints. Furthermore, various players including the building owners, designers, archi-

tects and engineers need to be cognizant of the risks and related costs involved regarding

various choices available to them for improving the serviceability of structures due to high

winds and other loading conditions. This chapter addresses the design and implementation

issues and also quantitatively justifies the use of the dampers within a risk-based decision

analysis framework.

148

The full-scale implementation of liquid dampers in airport control towers and

chimney masts was discussed in Chapter 1. However, future implementations in skyscrap-

ers, bridge towers and offshore structures would require their integration into the overall

system. Moreover, the adoption of semi-active TLCDs requires additional equipment and

a more sophisticated set-up as compared to a passive system. Figure 8.1 show some of the

implementation concepts in bridge towers and tall skyscrapers.

Figure 8.1 Implementation ideas for tuned liquid dampers (a) bridge towers (b)tall buildings.

8.2 Comparison of various DVAs

There are various factors which influence the selection of a dynamic vibration

absorber (DVA) for structures, namely: efficiency, size and compactness, capital cost,

(a) (b)

149

operating cost, maintenance, safety, and reliability. In this section, a comparison among

three different types of DVAs, namely, the TMD, TLD and TLCD is made.

8.2.1 Implementation comparisons

Tuned Mass Damper (TMD)

The TMD system installed in the Citicorp building is a sophisticated system with a

linear gas spring, pressure balance system, control actuator, power supply and electronic

control system (Weisner, 1979). The different components used in a building-mounted

TMD include in addition to the mass, gravity support system, and the spring system: a

damping/active force generating system with a servo-valve and a hydraulic actuator;

instrumentation including accelerometers, displacement transducers, pressure and temper-

ature sensors; an electronic control system which turns TMD on and off automatically.

Other parts of the TMD include restraint systems for TMDs including anti-yaw torque

box, over-travel snubber system with reaction guides, and directional guides so that the

mass block does not rotate during travel.

A TMD system needs to be designed in the face of several practical restraints. One

of the main disadvantages in the TMD operation is that although it is theoretically a pas-

sive device, it needs electricity to operate. This is a problem since power could be lost dur-

ing a high wind storm, a time when the TMD is expected to be operational (ENR, 1977).

Figure 8.2 shows the actual TMD system installed in the Citicorp building in New York.

150

Figure 8.2 TMD system installed in the Citicorp Building, New York City (takenfrom Wiesner, 1979)

Modern TMDs, however, have been designed to accommodate these restraints.

Pendulum-type TMDs with single and multi-stage suspensions have been devised. These

pendulum-type dampers do not need power to operate. Multi-stage pendulum-type TMDs

are advantageous for buildings with low frequency as the length of suspension can be

quite large for single-stage pendulum-type TMD as shown in Fig. 8.3(a, b) (Yamazaki et

al. 1992). Pendulum-type TMDs are usually augmented with coil springs for fine tuning.

Mechanically guided slide tables, hydrostatic bearings, and laminated rubber bearings are

used to provide low friction platforms. For TMDs with laminated rubber bearings, the

bearings act as horizontal springs which eliminates the need for spring system. This type

of system is shown in Fig. 8.3 (c). Innovative methods for integrating TMDs into existing

buildings have been proposed by researchers. Mita and Feng (1994) proposed a mega-sub

151

control system which utilize sub-structures in a mega-structure configuration to act as

vibration absorbers. Similarly, researchers are considering the concept of a roof isolation

system in which the top floor or roof of a structure act as mass dampers.

Recent notable TMD applications include the skybridge in the Petronas towers,

Kaula Lumpur, Malaysia, where the legs of the bridge were found to be highly sensitive to

vortex-induced excitations (Breukelman et al. 1998). A good overview of various types of

TMD systems for reduction of wind response in structures is provided by Kwok and

Samali (1995) and Kareem et al. (1999).

Figure 8.3 (a) Single-stage (b) multi-stage Pendulum-type TMDs (c) TMDs withlaminated rubber bearings (taken from Yamazaki et al. 1992)

Tuned Liquid Damper (TLD)

Although the mathematical theory involved in accurately describing sloshing is

complicated, TLDs are the most convenient to install and maintain due to the simplicity of

the device. Furthermore, maintenance costs of these dampers are practically non-existent.

(a) (b) (c)

152

Due to their inherent simplicity, TLDs may be added to existing buildings as retro-fit solu-

tions, even for temporary use if desired, e.g., during construction phases of a structure. A

typical TLD may be designed in a variety of configurations ranging from rectangular tanks

to stacks of circular tanks (Tamura, 1995).

The biggest advantage of liquid dampers is apparent in the case of tall buildings.

In most commercial buildings, water supply is needed for day-to-day usage and for sprin-

kler tanks used for fire-fighting purposes. The maintaining of water pressure can be effec-

tively done by placing water reservoir tanks on roof tops, where the water flows into

plumbing with its own gravity. So, instead of maintaining a high water level using special-

ized pumping equipment, a water tank is an ideal cost-effective solution. On the other

hand, in case of a TMD, the concrete/steel mass has no functional use.

Due to the nature of the system, a small error may be expected when measuring the

still water level, which is the parameter that controls the fundamental sloshing frequency.

However, an important advantage that the liquid damper has over a TMD is that for wide

range of amplitudes of oscillation, particularly at higher levels, the system is not very sen-

sitive to the actual frequency ratio between the primary and secondary systems. Another

major advantage of liquid dampers is that no activation mechanism is needed for their

operation. TMDs, for e.g., are designed to be activated at a certain threshold acceleration.

However, no such activation mechanism is needed for liquid dampers.

Note that for small and medium amplitudes of oscillation, proper tuning of the sys-

tem may considerably influence the response. Some installations of TLDs include baffles

and/or metallic balls to dissipate energy. However, the exact amount of damping cannot be

ascertained with these systems. Moreover, nonlinear frequency and damping characteris-

tics inherent to these systems make them unsuitable for functioning as optimal devices.

153

Tuned Liquid Column Damper (TLCDs)

Some of the main advantages of using TLCDs are the following:

1. The damping in the TLCD can be controlled through the orifice. The orifice opening

ratio affects the headloss coefficient which in turn affects the effective damping of the

liquid damper. Proportional valves can be actuated by a voltage signal obtained from a

battery to obtain the required damping without the use of heavy power.

2. The TLCD can be tuned by changing its frequency by way of adjusting the liquid col-

umn in the tube. This is an attractive feature in case re-tuning becomes desirable in case

of changes in the primary system frequency.

3. A mathematical model, which accurately describes the dynamics of the TLCD, can be

formulated. This is an attractive feature for semi-active and active control.

TLCD has the advantage of convenient mathematical formulation, but suffers from

the need for an appropriate tube length to satisfy the required frequency of oscillation.

Therefore, it may be in conflict with the available space allocated to house it. One way of

avoiding this is to introduce multiple TLCDs as discussed in Chapter 3. Figure 8.4 shows

the schematic of an actual TLCD implementation similar to the prototype studied in the

laboratory. Additional details are water level control system which has been introduced for

tuning control. This means that changes in structural frequency can be compensated by

changes in liquid level measured by a capacitance type wave gauge.

154

Figure 8.4 Equipment schematic for a building-mounted TLCD

8.2.2 Cost comparison

Damping devices are an efficient and cost effective means of reducing motion than

traditional approaches of increasing structural mass and stiffness. The Citicorp building’s

TMD cost was about $1.5 million (costs in 1977, in 2001 this is roughly $5.0 million);

however, it saved an overall structural cost of $4.0 million dollars that would have been

spent to add some 28,000 tons of structural steel to add lateral stiffness to the frame and

additional floor concrete to increase the mass of the structure (ENR, 1977). Typically, the

capital cost of a conventional TMD system is in the vicinity of 1% of the total building

cost. Table 8.1 lists some of the different components used in various systems. A prelimi-

nary analysis of the cost of a fully functional TLCD system has been estimated to be

TLCDCONTROL CONSOLE

AIRSUPPLY80 psi

BatteryPower

PneumaticActuated

Control Unit

4-20 mAcontrolsignal

positionersignal

Capacitanceliquid level

Bearing Surface

Tie-downs

Sensor Readings from Structure

Water

Valve

liquid level con tunit

155

roughly 1/10 times the cost of an equivalent TMD system with similar performance in

response reduction.

TABLE 8.1 Component comparison of different DVAs

Different costcomponents TLDs TLCDs TMDs

Design andconsulting fees

Very limited, simple

design

Specialized design and

consulting services

needed

Specialized design and consult-

ing services needed

AdditionalConstruction

None, easy installation

during construction

stages

None, easy installation

during construction

stages

Local strengthening needed to

support large amounts of spring

and actuator forces

Needs an over-travel snubber

system

Mechanicalcomponents

None Manual/actuated Valve

Water level self-tuning

control system

Nitrogen Springs/ laminated rub-

ber bearings/ Hydraulic bearings

Servo-valve hydraulic actuators

Anti-yaw torque box, linear

guideways

Pendulum-type TMDs

Electroniccomponents

None Computer control sys-

tem needed

Computer control system needed

Sensors Liquid level sensor Liquid level sensor

Accelerometers

Anemometer

Accelerometers

Displacement transducers

Pressure and Temperature sen-

sors

Space Take up a lot of valu-

able space, especially at

the top of skyscrapers

which is prime space,

however water has

functional use at the top

of a skyscraper, in a

TLP, etc.

Take up a lot of valu-

able space, especially at

the top of skyscrapers

which is prime space,

however water has

functional use at the top

of a skyscraper, in a

TLP, etc.

Definite savings in space as

compared to the liquid dampers.

However, Pendulum-type TMDs

also require a large space for

high-rise structures. This could

be alleviated using multi-stage

TMDs.

Power require-ments

None None (battery power) Power required for some designs

of TMDs.

Maintenaceand opera-tional costs

Very limited opera-

tional cost

Regular cleaning of

tanks and change of

water (to prevent algae

and fungi) is required

Control system mainte-

nance

Battery power

Constant air supply

needed for pneumatic

actuator

Cleaning of tanks and

water is required

Control system maintenance

Maintenance of mechanical

components: nitrogen springs,

hydraulic oil bearings, etc.

Power supply needed

Oil Supply needed

Cooling water

156

8.3 Risk-based Decision Analysis

Serviceability is an important factor in the design of tall buildings under wind

loading. There are primarily two types of adverse serviceability conditions caused by

strong winds. The first is that excessive wind may cause large deflections in the structure

causing architectural damage to non-structural members, for e.g., panels, cladding, etc.,

and affect elevator operation. The second is that the oscillatory motion may cause occu-

pant discomfort or even panic. It is generally accepted that acceleration and the rate of

change of acceleration (commonly known as jerk) are the main causes of human discom-

fort. Usually, the risk of unserviceability (i.e., excessive deflections or accelerations) is

calculated assuming that failure occurs when the deflection or acceleration exceeds a cer-

tain specified value.

The example considered in this chapter is merely for illustration purposes. How-

ever, the framework presented is quite general and could be applied to any system. The

building considered is a 60 story, 183 m tall building with a square base of 31 X 31 m. The

spectral characteristics of wind loads are defined in Li and Kareem (1990). In this exam-

ple, designers and building owners are considering the option of adding liquid dampers for

increasing the serviceability of this building under winds. Two types of TLCDs are con-

sidered for application in the along-wind direction. The first is a passive system with the

frequency of oscillation of liquid tuned to the first mode frequency of the building while

the damping is optimized for design level wind speed. The second is a semi-active system,

in which an optimal level of damping is maintained at all levels of vibration.

In the case of passive system, the damping is assumed to be arising due to the fric-

tion in the tube. The headloss coefficient in this case is assumed to be equal to 1, which is

typical of such a system. In the case of semi-active system, the optimal damping ratio of

157

4.5% is maintained at all levels of excitation by means of a controllable orifice using a

gain-scheduled law as outlined in Chapters 5 and 7. The mass ratio (µ) is 1% and the tun-

ing ratio (γ) is 0.99, which corresponds to a total mass of 280 tons and liquid column

length of 12 meters. Multiple units of TLCDs of 1 m diameter can be used to accommo-

date the total weight of the damper and these may be distributed on the building roof.

The RMS acceleration response of the uncontrolled and controlled response using

passive and semi-active systems is plotted as a function of the mean wind velocity at 10

meters height, U10 (Fig. 8.5). It can be seen from Table 8.2 that the dampers are effective

in reducing the structural accelerations and displacements. In this analysis, the effect of

bracing the structure is also examined. It has been assumed that the super-structure stiff-

ness can be increased by a particular bracing system by 20%. Table 8.2 shows that the

bracing system is quite effective in reducing displacement but not equally effective in

reducing acceleration. Moreover, the bracing system increases significantly the overall

building cost due to additional steel required for structural bracing.

From Table 8.2, it can be noted that there is an improvement of 10-25% in RMS

acceleration response over the entire range of wind velocities using a semi-active system.

The semi-active system realizes a 45% improvement over the uncontrolled system. This

improvement justifies small additional cost associated with a semi-active system, for e.g.,

sensors, controllable valves, etc. This analysis is based on the assumption that all the sys-

tem parameters are known with certainty. The parametric uncertainty and the resulting

reliability of structural and loading parameters are treated in the following section.

158

Figure 8.5 Variation of RMS accelerations of the top floor with increasing windvelocity

TABLE 8.2 Comparison of different systems for varying wind conditions

8.3.1 Decision analysis framework

The decision making framework, shown in Fig. 8.6, is commonly composed of the follow-

ing components: objectives of decision analysis; decision variables; decision outcomes;

and associated probabilities and consequences. Each element of the analysis framework is

described briefly here.

RMSdisplacementU10 =15m/s

(cm)


(cm)


(cm)

RMSaccelerationU10=15m/s

(cm/sec2)


(cm/sec2)


(cm/sec2)

Uncontrolled 2.37 5.97 12.19 3.79 9.57 19.56

Stiffened

Structure

1.54 (30.4 %) 3.87 (35.1 %) 7.92 (35 %) 2.95 (22.1 %) 7.44 (22.2 %) 15.23 (22.1 %)

Passive sys-

tem

1.73 (23.4 %) 3.93 (34.1 %) 7.17 (41.2 %) 2.69 (29 %) 6.20 (35.2 %) 11.56 (40.9 %)

Semi-Active

System

1.26 (40.6 %) 3.18 (46.7 %) 6.49 (46.7 %) 2.07 (45.4 %) 5.22 (45.4 %) 10.69 (45.3 %)

14 16 18 20 22 24 26

2

4

6

8

10

12

14

16

18

20

Mean wind velocity at 10m height, U10

m/s

RM

S a

cce

lera

tion

s (c

m/s

2)

Uncontrolled Braced Structure Passive Conrol Semi−Active Control

Maximum permissableRMS accelerations

Annoyance Threshold

159

Objectives of Decision analysis: Decision analysis problems require an objective func-

tion(s) to be clearly defined. In our present example, the objective could be minimizing the

total expected utility value.

Figure 8.6 Elements of Decision analysis

Decision variables: These could be the various decision alternatives available to the deci-

sion maker. In our example, these could be the following alternatives available to the

building owners:

1. Do not take any action to improve building serviceability.

2. Invest in traditional bracing/outrigger systems to increase the lateral stiffness. The net

increase in the effective stiffness of the resulting structure due to the addition of bracing

is given by a factor kf defined as the ratio of the stiffness of the structure with added

bracing to the stiffness of the uncontrolled structure.

160

3. Install passive liquid dampers with optimal tuning ratio and optimal damping at design

wind speed. This is a sub-optimal configuration of the TLCD since the damping is pri-

marily due to friction in the tube and a fixed orifice which cannot be controlled.

4. Install semi-active TLCD system which maintains the optimal damping at all levels of

response.

Decision outcomes: The various decision alternatives described above may have the fol-

lowing outcomes:

1. Building serviceability may be compromised severely leading to building shutdown.

An important cost function to be considered is to account for the associated costs of an

unserviceable structure brought about by business shutdown and loss of reputation.

2. Bracing systems and outrigger systems are expensive and are not as effective in reduc-

ing acceleration which is the primary metric used to assess serviceability problems.

3. The passive liquid damper devices are effective in reducing displacement and accelera-

tion responses, however they perform optimally only at the design wind speed.

4. Semi-active system is more effective than the passive system, however, there are addi-

tional costs for controllable valves, computer control system, sensors and maintenance.

Associated Probabilities and Consequences: In the following sub-sections, methods to

estimate the probabilities of failure and the associated costs/utility values of each decision

are examined. Finally these are integrated into a risk-based decision analysis tree. The risk

of an event is defined by the following traditional relationship:

(8.1)Risk pi H Ci,( )U Ci( )i∑=

161

where is the probability of failure, H is the hazard, is the utility func-

tion and Ci are the consequences. The impact of risk can be improved by either reducing

the occurrence probability through system/component changes (which in our case refers to

adding dampers) or by reducing the potential consequences.

8.3.2 Reliability Analysis

The structural reliability analysis is performed using limit states which are mathe-

matical functions of a combination of random variables that describe whether the structure

performs satisfactorily for the specific criteria it has been designed for. The design of

damping systems needs to consider the model and physical uncertainties, for e.g., struc-

tural mass changes, damage to structure, hardening of concrete, loss of stiffness due to

corrosion and fracture, stiffness changes in foundation, etc. Changes could also be inher-

ent in the loading, for e.g., wind climate, change in surface roughness, etc. The damper is

also not free from uncertainties, for e.g., decrease in its performance due to equipment

wear and tear. Therefore, all these variables need to be considered in probabilistic terms

for the reliability analysis.

For ultimate strength limit states, one is concerned about structural load and resis-

tance, while for serviceability, the limit state represents the evaluation of a performance

criteria. For design of very tall and slender structures under winds, it is usually the service-

ability limit state which often governs the design. The limit state function is usually writ-

ten as,

(8.2)

and the probability of failure Pf for the component is defined as,

(8.3)

pi H Ci,( ) U Ci( )

Z g X 1 X 2 … X n, , ,( )=

P f P Z 0<( ) P g X 1 X 2 … X n, , ,( ) 0<[ ]= =

162

(8.4)

where is the joint probability density function of the n-dimensional vector X

which describes the vector of random variables. In this case, the limit state function is a

hyper-surface in the n-dimensional space and separates the fail and safe regions. Usually,

standard reliability techniques, for e.g., First and second-order reliability (FORM and

SORM) methods are used, wherein the limit state is linearized at the design point on the

failure surface (Ditlevsen, 1999). This procedure involves transformation of the variables

in the limit state equation to reduced normal variates which yields a new limit state equa-

tion in the reduced space. The probability of failure is then determined from the reliability

index ( ), which is defined as the shortest distance from the origin to the failure surface

and is given by,

(8.5)

The limit state equation for drift serviceability is commonly written as:

(8.6)

where is the allowable deflection, usually taken as = where is the height

of the building and is the maximum deflection in the structure.

Similarly, for comfort serviceability, the limit state equation is written as,

(8.7)

where is the maximum allowable RMS accelerations, which lies between 5-10 mg in

the perception threshold range and 10-15 mg in the annoyance level range. In this study

the focus is on the comfort considerations. Therefore, different values of = 8, 10 and

12 mg have been considered. Random variables used in the analysis are listed in Table 8.3.

P f f X X( ) Xdg X( ) 0<∫=

f X X( )

β

P f Φ β–( )=

Z ∆all ∆– max=

∆all H b 400⁄ H b

∆max

Z σma σ x–=

σma

σma

163

The distribution of wind velocity for a well behaved wind climate can be adequately mod-

eled by a Type 1 extreme value distribution. The other variables along with their statistical

characteristics, i.e., probability distribution, and mean and coefficient of variation (COV)

can be found in Rojiani (1978) and Kareem (1990). The probability of failure for the dif-

ferent systems under different mean wind velocities and different is tabulated in

Table 8.4.

TABLE 8.4 Probability of Failure under different wind speeds

TABLE 8.3 Random Variables used in Reliability analysis

Type #. Random VariableProbabilityDistribution

Mean COV

StructuralParameters

1 Mass matrix multiplier,

(non-dimensional)

Normal 1.0 0.1

2Stiffness matrix multiplier,

(non-dimensional)

Normal 1.0 0.25

3 1st mode damping, ζs Log Normal 1 % 0.35

Wind LoadParameters

4 Air density, ρa Log Normal 1.25 kg/m3 0.05

5 Drag coefficient, Cd Log Normal 1.2 0.17

6 Power law exponent, Log Normal 0.3 0.1

7 Wind Velocity, U10 Extreme Value

Type 1

18, 20 m/s 0.1

Liquid DamperParameters

8 Tuning ratio, γ Normal 0.9870 0.1

9 Coefficient of Headloss, ξ Normal 1 0.1

10 Optimal Damping, ζf Log Normal 5.5 % 0.05

Probability of Failure(%)

U10 = 18 m/s U10 = 20 m/s

=8 mg =10 mg =10 mg =12 mg

Uncontrolled 39.34 % 14.21 % 44.43 % 29.87 %

Braced System 33.43 % 11.12 % 40.23 % 24.71 %

Passive System 14.86 % 3.66 % 23.17 % 8.79 %

Semi-Active Case 4.69 % 0.71 % 10.28 % 2.69 %

σma

m

k

α

σma σma σma σma

164

8.3.3 Cost and Utility Analysis

A generalized total expected cost function (for a period of T years) can be written as:

(8.8)

where Cs is the initial fixed cost of the structure, Cd is the initial fixed cost of the damper,

Cm is the maintenance cost per unit year and Cf is the repair/business interruption cost per

unit year. The estimation of these cost functions requires a detailed analysis of the system

at hand. In particular, the cost which is hard to quantify is Cf because it is a function of

several factors, e.g., local market value and real estate demand. For a simplified analysis,

this can be written as:

(8.9)

where C(E) is the cost of repair/ business interruption/ decreased employee productivity

when an event E occurs. In this analysis, C(E) has been assumed to be equal to 10. Table

8.5 tabulates some general costs and utilities of a typical tall building. Most of these values

are arrived at in an empirical way, however, the framework for more market value based

cost analysis would remain the same.

TABLE 8.5 Costs and Normalized Utility Analysis

Type of system Fixed Costs (Cost of structure (Cs) same for all

options)

Dollar values (% ofTotal cost ofStructure Cs)

Utility

Bracing Amount of Steel, construction costs, loss of floor

space

2.5% 5

Passive system Cost of liquid tanks, loss of floor space, maintenance 0.5% 1

Semi-active

system

Costs of liquid tanks, controllable valve, design and

consulting fees, computer controlled system, mainte-

nance

1% 2

Ct Cs Cd Cm t( ) t C f t( ) td

0

T

∫+d

0

T

∫+ +=

C f T P f C E( )=

165

8.3.4 Risk-based Decision Analysis

Figure 8.7 shows a typical decision tree used to examine the given problem in a

systematic format. The decision tree includes decision and chance nodes. The decision

nodes are followed by possible actions which the decision maker takes. The chance nodes

are followed by outcomes that are beyond the control of the decision maker. The total

expected utility for each branch is computed and the decision is selected such that the

expected total utility function is minimized. As seen from Table 8.6, when the probabili-

ties of failure are low, choosing semi-active dampers over passive dampers is not cost

effective. However, in critically unserviceable structures, the semi-active scheme delivers

better cost/utility benefits.

Figure 8.7 Decision Tree for Building Serviceability

TABLE 8.6 Utility analysis based on the decision analysis

Total CostCt

U10 = 18 m/s U10 = 20 m/s

=8 mg =10 mg =10 mg =12 mg

Uncontrolled (CA) 7.86 2.84 8.88 5.97

Braced System (CB) 11.68 7.24 13.08 9.94

Passive System (CC) 3.97 1.73 5.63 2.75

Semi-Active Case (CD) 2.93 2.14 4.05 2.53

DecisionNode

C1

C2

C3

C4

FixedCosts

ChanceNodes

fail

Safe

Cf*Pf

Cs*(1-Pf)

CA

C B

CC

CD

σma σma σma σma

166

8.4 Design of Dampers

8.4.1 Design Guidelines

Liquid

Usually water is the preferred liquid used in TLDs and TLCDs. It has been noted by

Fujino et al. 1988 that the use of high viscosity liquids do not offer any advantage. This is

because, for liquid dampers, there is an optimal level of damping that will provide the

desired level of response reduction, therefore, higher liquid viscosity is not always effec-

tive.

Mass ratio (µ)

The mass ratio is dictated by the efficiency (defined as the ratio of response with

control system to response of uncontrolled structure) of the dampers needed. For e.g., if an

efficiency of 50% is required, then at least a mass ratio of 1% is needed. Practically, no

more than 1% mass ratio is possible to be placed on the top of tall buildings. For example,

TMD mass weighing up to 400 tons was installed in Citicorp Building. In case of TLDs

and TLCDs, this implies more space requirement, therefore innovative schemes to inte-

grate these into water storage tanks and fire-sprinkler tanks need to be designed.

Length ratio (α)

The length ratio determines the horizontal to total length of the liquid column. The

length ratio also needs to be determined from an architectural point of view. For increasing

length ratio, the efficiency of the damper increases. However, two things need to be con-

sidered. The vertical length of the tube should be high enough so that water does not spill

out of the tube. Secondly, water should remain in the vertical portion of the U-tube at all

167

times to provide continuity in the water column in the horizontal segment. This can be

ensured by designing l and b such that,

(8.10)

Tuning ratio (γopt)

Typically, auxiliary devices are tuned to the first modal frequency of the structure.

An acceptable design is obtained by ensuring a tuning ratio of almost unity for mass ratio

of 1%. Exact values are provided for a variety of cases in chapter 3. In case the natural fre-

quency of the structure changes by , the length of the water column in the U-tube

needs to be compensated by the following relation,

(8.11)

Damping ratio (ζopt)

This is the damping ratio of the liquid damper. For a regular TMD, this represents

the linear damping ratio. However, for liquid dampers the damping varies nonlinearly with

amplitude. Based on design curves obtained in Chapter 3, a damping ratio of about 4.5%

for mass ratio of 1% is recommended for optimal damping.

Number of Dampers

The number of dampers depends on various factors such as the available space,

shape and sizing of the damper units. In case of multiple dampers, it was shown in Chapter

3 that by increasing the number of dampers does not necessarily improve better perfor-

max x f l b–( )2

---------------≤

ωs∆

l∆ 4– g

γoptωs( )3----------------------- ωs∆=

168

mance concomitantly. A typical number of 5 units is usually adequate. Kareem and Kline

(1995) conducted numerical studies on multiple dampers with non-uniform mass distribu-

tion and non-uniform frequency spacing. They concluded that such systems did not offer

any useful advantage over systems with uniform mass distribution and frequency spacing.

Orientation of the liquid dampers

For structures with different fundamental frequencies in the two major directions,

tuning may be accomplished by using rectangular tanks or TLCDs. With proper design of

the damper dimensions, fundamental frequencies in both directions may be tuned. This is

important since the theory is based on tanks subjected to only a uni-directional excitation.

For structures with the same fundamental frequency in the two principal directions, a cir-

cular tank may be used.

8.4.2 Control Strategy

As discussed in section 5.2, gain-scheduling is an ideal control policy for main-

taining optimal damping in TLCDs. Sensors on the buildings (accelerometers, liquid level

sensor, or anemometer) estimate the excitation level, which is used to adjust the headloss

coefficient based on a pre-computed look-up table.

Comparing Fig. 5.1 and Fig. 8.8, one can draw analogies wherein the look-up table

is the gain-scheduler, the controllable valve of the TLCD is the regulator, and the head loss

coefficient is the parameter being changed. The external environment is the wind loading

acting on the structure and the process represented by the combined structure-TLCD sys-

tem.

169

Figure 8.8 Semi-active control strategy in tall buildings

8.4.3 Design Procedure

Structural Characteristics

The first step in the design of the dampers is to gather adequate knowledge of the

natural modes and damping of the structure being considered for control. The structural

characteristics are determined either at the design stage by analysis or for existing build-

ings by monitoring full-scale data or a combination of both techniques. The first method

involves a FEM analysis of the structural system. The second relies on analyzing full-scale

measurements from instrumented buildings. The response power spectral density provides

an estimate of the natural frequency and damping in the structure. Usually, it is advisable

to conduct full scale testing in order to obtain ambient or forced building response before

installing dampers. This is because FEM models usually not reliable for accurate esti-

mates of frequencies due to difficulties in modeling accurate boundary conditions, e.g.,

soil-structure interactions, and other nonlinear effects.

Estimate Excitation and loading intensity

U10, S0

Look-up Tableξ = f (So)

Accelerometer/Anemometer

change headloss coefficient (ξ)

170

Loading Characteristics

The wind, earthquake or wave loading characteristics have to be determined from

site characteristics and hazard maps. Wind tunnel experiments are also needed for critical

projects to investigate the characteristics of wind force acting on the building and to esti-

mate the structural response. This analysis is done during the design stages of the struc-

ture. In this section, we will discuss alongwind loading only, although the acrosswind and

torsional directions can be handled accordingly if the spectral information is available

(Aerodynamic load database, www.nd.edu/~nathaz/database/index.html). The loading

spectra for alongwind excitation can be defined as

(8.12)

where ; ; ; zo =

surface roughness length; zd = zero plane displacement; U10 = mean wind velocity at 10m

height. The coherence function required for the cross-spectrum is given as

(8.13)

where (x1, z1) and (x2, z2) are the coordinates of the nodes, Cv and Ch are the coherence

decay coefficients in the vertical and horizontal directions. The multiple-point representa-

tion may be simplified for line-like structures, e.g., buildings, towers, in which the spatial

variation of wind fluctuations are only implemented for one spatial dimension. The wind

force at a certain level j is obtained as,

nSvv z n,( )

uo2

------------------------200 f

1 50 f+( )5

3---

---------------------------=

fnz

U z( )------------= U z( )

z 10m> 2.5uo

z zd–

zo------------- ln= uo U 10 2.5

10 zd–

zo----------------- ln

⁄=

cohn– Cv

2z1 z2–( )2 Ch

2x1 x2–( )2+[ ]

1

2---

1

2--- U z1( ) U z2( )+[ ]

----------------------------------------------------------------------------------

exp=

171

(8.14)

where Abj is the tributary area exposed to wind, CDj is the drag coefficient at the jth floor

and is the air density. From Eq. 8.14, one can also obtain the spectra of the loading,

given as: .

In the last section, the gain-scheduled control was derived for different loading

intensities. In order to extend it to wind excited structures, one needs to find relationship

between the wind force spectra, , and an “equivalent” white noise excitation. For

small values of , one can approximate by a equivalent white noise So, which is

the value of at the natural frequency of the structure (Lutes and Sarkani, 1997).

This is shown schematically in Fig. 8.9(a) where using the following relationship:

(8.15)

The equivalent white noise for an example case where = 1 Hz is given in Fig. 8.9 (b).

Figure 8.9 (a) Equivalent white noise concept (b) Variation of equivalent whitenoise with wind velocity.

F j t( ) 0.5ρaAbjCDj U z j( ) v j+( )2=

ρa

SFF ω z,( ) ρaAbjCDjU z( )( )2Svv ω z,( )=

SFF ω( )

ζ s SFF ω( )

SFF ω( )

So U 10( ) SFF ωs( )=

ωs

0.5 1 1.5 2 2.5 3 3.5 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

SFF(ω)

ωs

So(ω)=S

FF(ω

s)

Frequency (Hz)

Magnitude of Transfer Function

|Hx(ω)|2

"Equivalent" White Noise Excitation

55 60 65 70 75 80 85 90 95 1000

200

400

600

800

1000

1200

1400

1600

U10 (ft/s)

Equivalent Loading intensity S

o (lbf

2s)

(a) (b)

172

Damper Sizing

Once the structural and loading characteristics have been determined, the designer

can begin design of the damper. The optimum design parameters are discussed in Chapter

3. All symbols, unless explained here, refer to the earlier notations. The length of the

water column is given by,

(8.16)

where .

The cross sectional area of the damper can be obtained by,

(8.17)

and for a spatially distributed single TLCD,

(8.18)

where N is the number of units and M1 is the generalized first modal mass of the structure.

In case of multiple TLCDs, the length of liquid column and the cross sectional area of

each unit are given by,

(8.19)

(8.20)

Next, from the wind loading excitation information, the headloss coefficient can be deter-

mined as follows,

(8.21)

where is given by:

l 2g ω f2⁄=

ω f γoptωs=

AµM 1

ρl-----------=

Ai

µM 1

Nρl-----------=

li 2g ω fi2⁄=

Ai

µM 1

Nρli-----------=

ξopt

2ζopt glπσx f

--------------------------=

σx f

173

(8.22)

The valve sizing should be selected such that the entire range of desired values of ξ

can be covered. This can be ensured by relating the headloss coefficient to the valve con-

ductance, CV for different angles of valve opening (see Appendix A.3). Typically, for most

applications a headloss coefficient between the range of 1-100 should be adequate.

8.4.4 Technology

Actuated Valves

Actuated valve technology has

improved in the last few years. Electro-pneu-

matic valves are available with an option of a

position transmitter which can be used for con-

trolling the valve. Figure 8.10 shows the actua-

tor commercially available, which is a

pneumatically actuated ball/butterfly valve with

an additional solenoid valve for modulating the

valve opening. The electro-pneumatic posi-

tioner uses a 4-20mA signal to change the valve

position. The positioner modulates the flow of

supply air (at 80 psi) and converts the input sig-

nal to a 3-15 psi air pressure for proportional

modulation of the valve. The headloss characteristics for the valve are described in Appen-

dix A.3.

σx f

2S0 U 10( ) H x f F ω( ) ωd

0

∞

∫=

Figure 8.10 Electro-pneumaticvalve (courtesy Hayward Controls)

174

Tubing Systems

Clear PVC piping systems are the best choice for the TLCD tube construction.

This is because they are rugged and durable, yet allow easy maintenance and visibility of

the liquid.

Sensors

A capacitance type liquid level sensor is needed to determine the liquid level in the

TLCD. This is important for tuning the TLCD to the building frequency. This needs to be

done on a regular basis because changes in structural frequency may take place due to

aging or stiffness degradation of building characteristics which can lead to mis-tuning of

the system. Additionally, accelerometers and anemometers for estimating the loading

characteristics are needed. These are commercially available from a variety of vendors. It

should be noted that accelerometers chosen should have good frequency characteristics in

the low frequency region (< 1 Hz). This is because the response of tall buildings is prima-

rily concentrated in this low frequency band.

Control System Software and Hardware

With advances in control system implementation hardware, a computer controlled

system running on auxiliary power is quite affordable these days. A typical computer run-

ning a data acquisition and control implementation software can be set up very cheaply.

The system can also be configured to include remote control using TCP/IP system which

enables off-site users to monitor the system, which eliminates the need for an on site oper-

ator.

175


This chapter discussed the design consideration and implementation details of liq-

uid dampers. Different dynamic vibration absorbers, namely TMDs, TLDs and TLCDs are

compared in terms of implementation and costs. Next, a probabilistic framework for deci-

sion analysis concerning the serviceability of a building has been presented. Both deter-

ministic and reliability-based analyses confirm the attractiveness of the passive and semi-

active liquid dampers in reducing acceleration response and the associated probabilities of

failure. The decision analysis framework presented here would facilitate building owners/

designers to ensure adequate life-cycle reliability of the building from a serviceability

viewpoint at a minimum cost. Finally, some design guidelines for technology transfer are

laid out based on research work presented in earlier chapters.

176

CHAPTER 9

CONCLUSIONS

What we call the beginning is often the end And to make an end is to make a beginning.

The end is where we start from. - T.S.Elliot

This research focussed on the development of the next generation of liquid damp-

ers for mitigation of structural response. Two type of liquid dampers, namely the sloshing

dampers (TLDs) and the liquid column dampers (TLCDs) were considered. Firstly, a new

sloshing-slamming analogy was presented for sloshing type dampers. It was noted that the

existing models neglect the effect of impact of liquid on the container walls. The first

approach proposed by the authors is to consider a sloshing-slamming analogy of TLD.

This involves modeling the TLD as a linear system augmented with an impact subsystem.

This analogy captures the essence of the underlying physics behind the complexity of the

sloshing phenomenon at higher amplitudes. The second approach uses certain nonlinear

functions, described as impact characteristic functions, which can succinctly describe the

phenomenological behavior of the TLD. The parameters of this model are derived from

experimental studies. Experiments were also conducted to study the local pressures on the

walls of the container and to better understand the nature of the impact process. It was

observed that the peak pressures occur at the static liquid height. The pressure-time inte-

gration shows that the contribution of the impact pulse to the overall sloshing pulse is

177

approximately 20-25%. This feature may play an important role in future modeling stud-

ies on TLDs.

Next, analytical modeling of tuned liquid column dampers (TLCDs) was consid-

ered. Optimum absorber parameters (i.e., tuning ratio and damping ratio) were derived for

a variety of loading cases ranging from white noise excitation to filtered white noise cases.

The theoretically obtained optimum absorber parameters were compared with experimen-

tal results and the match was found to very be good. The optimum absorber parameters

were also determined for the case of multiple TLCDs (MTLCDs). These parameters

include number of TLCDs, the frequency range and the damping ratio of each damper.

MTLCDs are more robust as compared to single TLCDs with respect to changes in the

primary system frequency. Moreover, the smaller size of MTLCDs offers convenient port-

ability and ease of installation at different locations in the structure.

The beat phenomenon is very common in combined systems like structure-damper

systems. This involves transfer of energy from one system to another and in some

instances could be harmful to the structure. It has been observed that beyond a certain

level of damping in the secondary system (i.e., the damper), the beat phenomenon ceases

to exist. A mathematical and experimental study of the beat phenomenon was conducted.

It was noted that the disappearance of the beat phenomenon is attributed to the coalescing

of the modal frequencies of the combined system. Experimental validation of the beat phe-

nomenon in combined structure-TLCD system was shown in the laboratory.

Various semi-active strategies were developed for the optimal functioning of

TLCDs. These include gain-scheduling and clipped optimal system with continuously-

varying and on-off control. Gain-scheduled control is useful for disturbances which are of

long-duration and slowly-varying (e.g., wind excitation) and where the steady-state

178

response is the control objective. The headloss coefficient is changed adaptively in accor-

dance with a look-up table by changing the valve/orifice opening. This type of semi-active

system leads to 15-25% improvement over a passive system. The application of these sys-

tems for offshore structures was also considered. Experimental validation of the gain-

scheduled system was done in the laboratory using a prototype TLCD equipped with a

valve controlled by a electro-pneumatic actuator and positioning system.

A different semi-active algorithm was also examined, which requires a controlla-

ble valve with negligible valve dynamics and whose coefficient of headloss can be

changed by applying a command voltage. This type of control is more suited for excita-

tions which are transient in nature, for e.g., sudden wind gusts or earthquakes. The effi-

ciency of the state-feedback and observer-based control strategy was compared.

Numerical examples showed that semi-active strategies perform better in terms of

response reduction than the passive systems for both random and harmonic excitations. In

the case of harmonic loading, the improvement was about 25-30% while for random exci-

tation, the improvement was about 10-15% over a passive system. It was also noted that

continuously-varying semi-active control algorithm did not provide a substantial improve-

ment in response reduction over the relatively simple on-off control algorithm.

An experimental technique, namely the hardware-in-the-loop technique, was

developed for testing liquid dampers. The main advantages, namely the cost effectiveness

and repeatability of the test, is realized due to the fact that a virtual structure simulated in

the computer interacts in real-time with the damper.

Finally, the design, implementation, cost and risk-based decision analysis for the

use of liquid dampers in structural vibration control was laid out. Comparisons were made

between different dynamic vibration absorbers (DVAs), namely the TMDs, TLDs and

179

TLCDs. It was estimated that the cost of a fully functional TLCD system has been esti-

mated to be 1/10 times the cost of a TMD system with a similar level of performance. The

risk-based decision analysis framework presented would facilitate building owners/

designers to ensure adequate life-cycle reliability of the building from serviceability view-

point at a minimum cost. It was concluded that when the probabilities of failure are low,

choosing semi-active dampers over passive dampers is not cost effective. However, in crit-

ically unserviceable structures, the semi-active scheme delivers better cost/utility benefits.

The following future studies in this area are recommended:

1. In the sloshing-slamming analogy of TLDs, the mass exchange parameter was deter-

mined from empirical relationships obtained through experiments, which relate the

change in the hardening frequency as a function of excitation amplitude. This analogy

could be further refined should it be possible to quantify more accurately the mass

exchange between the sloshing and slamming modes from theoretical considerations.

2. The sloshing pressures and forces obtained during experiments should be compared to

numerical sloshing studies which incorporate the slamming/impact action of the liquid.

3. Hardware-in-the-loop studies can be experimentally verified by conducting a full-scale

test of the structure-damper system and then verifying it using a HIL simulation.

4. Experiments concerning semi-active TLCDs were done on band-limited white noise

type excitations in order to provide proof of concept for the damping schemes. A more

elaborate experiment in the wind-tunnel using a structure attached to a semi-active

TLCD is needed before installing these dampers on actual structural systems. This will

however, pose serious modeling concerns.

180

REFERENCES

Relevant Author Publications

[1] Kareem, A. and Yalla, S.K. (1997), “Liquid Dampers: Recent Developments and

Applications’, submitted to the monograph on Structural Control.

[2] Yalla S.K., Kareem, A. and Kantor, J.C. (1998), “Semi-Active Control Strategies for

Tuned Liquid Column Dampers to Reduce Wind and Seismic Response of Structures,”

2nd World Conference on Structural Control, Kyoto, John Wiley and Sons, 559-568.

[3] Kareem, A., Kabat, S., Haan, F. Jr., Mei, G. and Yalla, S.K. (1998) “Modeling and

Control of Wind Induced Response of a TV Tower,” 2nd World Conference on Struc-

tural Control, Kyoto, John Wiley and Sons, 2421-2430.

[4] Yalla, S.K. and Kareem, A. (1999) “Modeling of TLDs as Sloshing-Slamming Damp-

ers,” Wind Engineering into the 21st century: Proc. 10th Int. Conf. on Wind Engng.,

Copenhagen, Balkema Press, 1569-1575.

[5] Yalla, S.K and Kareem, A. (2000a) “Optimum Absorber Parameters for Tuned Liquid

Column Dampers ,” ASCE Journal of Structural Engineering, 125(8), 906-915.

[6] Yalla, S.K., Kareem, A. and Kantor, J.C. (2000b), “Semi-Active Variable Damping

Tuned Liquid Column Dampers,” Proc. of the 7th SPIE Conf. on Smart Sructures and

Materials, Newport Beach, CA.

[7] Yalla, S.K. and Kareem, A. (2000c), “On the Beat Phenomenon in Coupled Systems,”

Proc. of the 8th ASCE Speciality Conf. on Probabilistic Mechanics and Structural Reli-

ability, University of Notre Dame, CD-ROM.

[8] Yalla, S.K., Kareem A. and Abdelrazaq, A.K. (2000d), “Risk-based Decision Analysis

for the Building Serviceability,” Proc. of the 8th ASCE Speciality Conf. on Probabilis-

tic Mechanics and Structural Reliability, University of Notre Dame, CDROM.

[9] Yalla, S.K. and Kareem, A. (2001a), “Beat Phenomenon in Combined Structure-Liq-

uid Damper Systems,” 23(6), Engineering Structures, 622-630.

[10] Yalla, S.K. and Kareem, A. (2001b), “Hardware-in-the-loop Simulation: A case study

for Liquid Dampers,” Proceedings of the Mechanics and Materials Summer Confer-

ence, San Diego, CA.

[11] Yalla, S.K., Kareem, A. and Kantor, J.C. (2001c), “Semi-Active Tuned Liquid Col-

umn Dampers for vibration control of structures,” Engineering Structures (in press).

184

[12] Yalla, S.K. and Kareem, A. (2001d), “Sloshing-Slamming -S2- Damper Analogy for

Tuned Liquid Dampers,” ASCE Journal of Engineering Mechanics (in press).

[13] Yalla, S.K. and Kareem, A. (2001e), “Semi-active Tuned Liquid Column Dampers

for mitigation of wind induced vibrations: Experiments,” submitted to ASCE Journal of

Structural Engineering, special issue on Semi-active control.

[14] Yalla, S.K. and Kareem, A. (2001f), “Modeling TLDs using Impact Characteristics:

Experiments and System Identification,” to be submitted to Earthquake Engineering

and Structural Dynamics.

Bibliography

[1]Abe, M., Kimura, S. and Fujino, Y. (1996), “Control Laws for Semi-Active Tuned Liq-

uid Column Damper with Variable Orifice Opening”, 2nd Int. Workshop on Structural

Control, Hong Kong.

[2] Abramson, H.N. (ed.) (1966), “The Dynamic Behavior of Liquids in Moving Contain-

ers”, NASA, SP-106.

[3] Astrom, K.J and Wittenmark, B. “Adaptive Control”, Addison Wesley, 1989.

[4] Babitsky, V.I. (1998), Theory of Vibro-impact Systems and Applications, Springer, N.Y.

[5] Balendra, T., Wang, C.M. and Cheong, H.F. (1995) “Effectiveness of Tuned Liquid

Column Dampers for Vibration Control of Towers”, Engineering Structures, 17(9),

668-675.

[6] Bass, R.L., Bowles, E.B. and Cox, P.A. (1980), “Liquid Dynamic Loads in LNG Cargo

Tanks,” SNAME Transactions, 88, 103-126.

[7] Bauer, H.F. (1972), “On the destabilizing effect of liquids in various vehicles- Part 1,”

Vehicle System Dynamics, 1, 227-260.

[8] Bergman, L.A., Mc Farland, D.M., Hall, J.K., Johnson, E.A. and Kareem, A. (1989),

“Optimal Distribution of Tuned Mass Dampers in Wind Sensitive Structures,” Proceed-

ings of 5th ICOSSAR, New York.

[9] Blevins, R.D. (1984), Applied Fluid Dynamics Handbook, Van Nostrand Reinhold.

[10] Brancaleoni, F. (1992), “The Construction phase and its Aerodynamic issues,”Aero-

dynamics of Large Bridges- Proceedings of the First International Symposium on

Aerodynamics of large Bridges, Copenhagen, Denmark.

[11] Brebbia et al. (1975), Vibrations of Engineering Structures, Lecture Notes in Engi-

neering (10), Springer-Verlag.

[12] Breukelman, B., Irwin, P., Gamble, S. and Stone, G. (1998), “The practical applica-

tion of vibration absorbers in controlling wind serviceability and fatigue problems,”

Proceedings of Structural engineers World Congress, San Francisco, July.

185

[13] Caughey, T.K. (1963), “Equivalent Linearization Techniques,” J. Acoust. Soc. Am.,

35, 1706-1711.

[14] Chang, C.C. and Hsu, C.T. (1999), “Control performance of Liquid column vibration

absorber,” Engineering Structures, 20(7), 580-586.

[15] Chang, P.M., Lou, J.Y.K. and Lutes, L.D. (1997), “Model Identification and Control

of a tuned liquid damper,” Engineering Structures.

[16] Chen, W., Haroun, M. A. and Liu, F. (1996), “Large amplitude Liquid Sloshing in

Seismically Excited Tanks,” Earthquake Engineering and Structural Dynamics, 25,

653-669.

[17] Chung, C.C. and Gu, M. (1999), “Suppression of vortex-excited vibration of build-

ings using tuned liquid dampers,” Journal of Wind Engg. and Ind. Aerodynamics, 83,

225-237.

[18] Delrieu, J.L.(1994), “Cost Effective Deepwater Platform Dedicated to West Africa

and Brazil”, Proc. 7th Int. Conf. on the behavior of Offshore Structures, 2, 805-812.

[19] Den Hartog, J.P. (1956), Mechanical Vibrations, 4th Ed, McGraw-Hill.

[20] Ditlevsen, O. (1999), Structural Reliability Methods, John Wiley and Sons.

[21] Dyke, S.J., Spencer, B.F. Jr., Sain, M.K. and Carlson, J.D. (1996) “Modeling and

Control of Magnetorheological Dampers for Seismic Response Reduction,” Smart

Materials and Structures, 5, 565-575.

[22] Ehlers, J. (1987), “Active and Semi-Active Control Methods in Wave-Structure Inter-

action,” (incomplete reference)

[23] ENR magazine (1977), “Tuned mass dampers steady sway of skyscrapers in Wind”.

[24] Faltinson, O. M. (1978), “A Numerical Nonlinear Method of Sloshing in Tanks with

Two-Dimensional Flow,” Journal of Ship Research, 22.

[25] Faltinsen, O.M. and Rognebakke, O.F. (1999), “Sloshing and Slamming in Tanks,”

Hydronav’99-Manouvering’99, Gdansk-Ostrada, Poland.

[26] Fediw, A.A., Isyumov, N. and Vickery, B.J. (1993), “Performance of a one-dimen-

sional Tuned sloshing water damper,” Wind Engineering, London, 247-256.

[27] Fujino, Y., Pacheco, B.M., Chaiseri, P. and Sun, L.M. (1988), “Parametric studies on

TLD using circular containers by free-oscillation experiments,” Struc. Engng and

Earthquake Engng., 5(2), 381-391.

[28] Fujino,Y., Sun, L.M., Paceno, B. and Chaiseri, P. (1992), “Tuned Liquid Damper

(TLD) for Suppressing Horizontal Motion of Structures,” ASCE Journal of Engineer-

ing Mechanics, 118(10), 2017-2030.

[29] Fujino, Y. and Sun, L.M. (1993), “Vibration Control by multiple tuned liquid damp-

ers,” Journal of Structural Engineering, ASCE, 119(12).

[30] Gao, H., Kwok, K.C.S. and Samali, B. (1997),“Optimization of Tuned Liquid Col-

umn Dampers”, Engineering Structures, 19, 476-486.

186

[31] Gao, H., Kwok, K.C.S. and Samali, B. (1999),“Charcteristics of multiple Tuned Liq-

uid Column Dampers in suppressing Structural Vibration,” Engineering Structures, 21,

316-331.

[32] Grace, A. (1992), MATLAB Optimization Toolbox User’s Guide, MathWorks, Inc.

[33] Graham, E.W. and Rodriguez, A.M. (1962), “The Characteristics of Fuel Motion

which affect Airplane Dynamics,” Journal of Applied Mechanics, ASME, 123(9), 381-

388.

[34] Gurley, K. and Kareem, A. (1994), “On the Analysis and Simulation of Random Pro-

cesses Utilizing Higher Order Spectra and Wavelet Transforms,” Proc. of 2nd Int. Conf.

on Computational Stochastic Mechanics, Athens, Greece, Balkema Publishers.

[35] Hattori, M., Atsushi, A. and Yui, T. (1994), “Wave impact pressures on vertical walls

under breaking waves of various types,” Coastal Engineering, 22, 79-114.

[36] Haroun, M.A. and Pires J.A. (1994), “Active orifice control in Hybrid liquid column

dampers,” Proceedings of the First World Conference on Wind Engineering, Vol.I, Los

Angeles.

[37] Hitchcock, P.A., Kwok, K.C.S., Watkins, R.D. and Samali, B. (1997), “Characteris-

tics of Liquid column vibration absorbers (LCVA) -I and II”, Engineering Structures,

19(2).

[38] Hitchcock, P.A., Glanville, M.J., Kwok, K.C.S., Watkins, R.D. and Samali, B. (1999),

“Damping properties and wind-induced response of a steel frame tower fitted with liq-

uid column vibration absorbers,” Journal of Wind Engineering and Industrial Aerody-

namics, (83), 183-196

[39] Housner, G.W., Bergman, L.A., Caughey, T.K., Chassiakos, A.G., Claus, R.O., Masri,

S.F., Skelton, R.E., Soong, T.T., Spencer, B.F. and Yao, J.T.P. (1997), “Structural Con-

trol: Past, Present, and Future,” Journal of Engineering Mechanics, 123(9), 897-971.

[40] Honkanen, M.G. (1990), “Heel and Roll Control by Water Tank”, Naval Architect,

215-216.

[41] Hrovat, D., Barak, P. and Rabins, M. (1983), “Semi-active versus Passive or Active

Tuned Mass Dampers for Structural Control,” Journal of Engineering Mechanics,

ASCE, 109(3), 691-705.

[42] Hsieh, C.C., Kareem. A. and Williams, A.N. (1988), “Wave Phase Effects on

Dynamic Response of Offshore Platforms,” Proc. of the Offshore Mechanics and Arctic

Engineering , ASME, Houston, TX.

[43] Hunt, K.H. and Crossley, F.R.E. (1975), “Coefficient of restitution interpreted as

Damping in Vibro-impact,” Transactions of the ASME, Journal of Applied Mechanics,

440-444.

[44] Huse, E. (1987), “Free Surface Damping Tanks to reduce Motions of Offshore Struc-

tures,” Proc. of 6th Int. Symp. on Offshore Mechanics and Arctic Engineering, 313-

324.

187

[45] Ibrahim, R.I., Gau, J.S. and Soundarajan, A. (1988), “Parametric and auto-parametric

vibrations of an elevated water tower- Part I: Parametric response,” Journal of Sound

and Vibration, 225(5), 857-885.

[46] Ibrahim, R.A., Pilipchuk, V.N. and Ikeda, T. (2001), “Recent Advances in Liquid

Sloshing Dynamics,” Applied Mechanics Reviews, 54(2), 133-199.

[47] Ioi, T. and Ikeda, K. (1978) “On the Dynamic Vibration Damped Absorber of the

Vibration System”, Bull. of Japanese Society of Mech. Engineers, 21 (151), 64-71.

[48] Iserman, R., Schaffnit, J. and Sinsel, S. (1999), “Hardware-in-the-loop simulation for

the design and testing of engine-control systems,” Control Engineering Practice, 7,

643-653.

[49] Ivers, D.E. and Miller L.R. (1991), “Semi-active Suspension Technology: an evolu-

tionary view, Advanced Automotive Technologies, ASME, 327-46.

[50] Kanai, K. (1961), “An Empirical Formula for the Spectrum of Strong Earthquake

Motions,” Bull. Earthquake Research Inst., Univ. of Tokyo, Japan, 39.

[51] Kareem, A. and Sun, W.J. (1987), “Stochastic Response of Structures with Fluid-

Containing Appendages,” Journal of Sound and Vibration, 119(3).

[52] Kareem, A. and Li, Y. (1988), “Stochastic response of a TLP to wind and wave

fields,” Department of Civil Engineering, University of Houston, Tech. report

UHCE88-18.

[53] Kareem, A. (1990), “Reliability analysis of Wind-sensitive structures,” Journal of

Wind Engg. and Ind. Aerodynamics, 23, 495-514.

[54] Kareem, A. (1993), “Liquid Tuned Mass Dampers: Past, Present and Future,” Pro-

ceedings of the Seventh U.S. National Conference on Wind Engineering, Vol. I, Los

Angeles.

[55] Kareem, A. (1994) “The next generation of Tuned liquid dampers,” Proceedings of

the First World Conference on Structural Control, Vol.I, Los Angeles.

[56] Kareem, A. and Kline, S. (1995), “Performance of Multiple Mass Dampers under

Random loading,” Journal of Structural Engineering, ASCE, 121(2), 348-361.

[57] Kareem, A. and Gurley, K. (1996), “Damping in Structures: Its evaluation and Treat-

ment of Uncertainity,” Journal of Wind Engineering and Structural Aerodynamics, 59,

131-157.

[58] Kareem, A., Kijewski, T. and Tamura, Y. (1999), “Mitigation of motion of Tall build-

ings with recent applications,” Wind and Structures, 2(3), 201-251.

[59] Karnopp, D., Crosby, M.J. and Harwood, R.A. (1974), “Vibration Control using

Semi-Active Force Generators,” ASME Journal of Engineering for Industry, 96(2),

619-626.

[60] Karnopp, D. (1990), “Design Principles for vibration control systems using Semi-

active dampers,” Journal of Dynamic systems, Measurement and Control, 112, 448-55.

188

[61] Koh, C.G., Mahatma, S. and Wang, C.M. (1994) “Theoretical and experimental stud-

ies on rectangular liquid dampers under arbitrary excitations,” Earthquake Engng. and

Struc. Dynamics. 23, 17-31.

[62] Kim, Y. (2001), “Numerical simulation of Sloshing flows with impact loads,” Applied

Ocean Research, 23, 53-62.

[63] Kurino, H. and Kobori, T. (1998), “Semi-Active Structural response control by opti-

mizing the Force-deformation loop of Variable Damper,” Proceedings of Second World

Conference on Structural Control, Kyoto, John Wiley and Sons, 407-417.

[64] Kwok, K.C.S. and Samali, B. (1995), “Performance of Tuned Mass Dampers under

wind loads,” Engineering Structures, 17(9), 655-667.

[65] Lepelletier, T.G. and Raichlen, F. (1988), “Nonlinear Oscillations in Rectangular

Tanks,” Journal of Engineering Mechanics, ASCE, 114(1).

[66] Li, Y. and Kareem, A. (1990), “Recursive Modeling of Dynamic systems”, Journal of

Engineering Mechanics, ASCE, 116, 660-679.

[67] Li, Y. and Kareem, A. (1993), “Simulation of Multi-Variate Random Processes:

Hybrid DFT and Digital Filtering Approach,” Journal of Engineering Mechanics,

ASCE, 119(5), 1078-1098.

[68] Lou, Y.K., Lutes, L.D. and Li, J.J. (1994), “Active Tuned Liquid Damper for Struc-

tural Control,” Proceedings of the First World Conference on Wind Engineering, Vol.I,

Los Angeles.

[69] Lutes, L.D. and Sarkani, S. (1997), Stochastic Analysis of Structural and MechanicalVibrations, Prentice-Hall.

[70] Lyons, J.L. (1982), Lyon’s Valve Designers Handbook, Van Nostrand Reinhold Co.

[71] Maciejowski, J.M. (1989), Multivariable Feedback Design, Addison-Wesley, Inc.

[72] Masri, S.F. and Caughey, T.K. (1966), “On the Stability of the Impact Damper,” Jour-

nal of Applied Mechanics, ASME, 586-592.

[73] McNamara, R. J. (1977), “Tuned Mass Dampers for Buildings,” Journal of Structural

Division, ASCE, 103, 9,1785-1789.

[74] Mita, A. and Feng, M.Q. (1994), “Response control strategy for tall buildings using

interaction between Mega and Sub-structures,” Proc. Int. workshop on Civil Infrastruc-

ture systems, Taipei, Taiwan, PRC, 329-341.

[75] Modi, V.J. and Welt F. (1987), “Vibration control using nutation dampers,” Proceed-

ings Int. conf. on Flow-induced vibrations, England, 369-376.

[76] Modi, V.J., Welt, F. and Seto, M.L. (1995), “Control of Wind-Induced Instabilities

though application of Nutation Dampers: A brief overview,” Engineering Structures,

17(9), 626-638.

[77] Natani, S. (1998), “Active Sloshing Damping System with Inverse Sloshing Genera-

tion,” Proceedings of 2nd World conf. on Struc. Control, Kyoto, John Wiley and Sons.

189

[78] Okomoto, T. and Kawahara, M. (1990), “Two-dimensional Sloshing analysis by

Lagrangian Finite element method,” Int. Journal of Numerical methods in Engng., 453-

477.

[79] Patel, M. H. and Witz, J.A., (1985),“On Improvements to the Design of Tensioned

Buoyant Platforms”, Behavior of Offshore Structures, 563-573.

[80] Pilipchuk, V.N. and Ibrahim, R.A. (1997), “The Dynamics of a nonlinear system sim-

ulating liquid sloshing impact in moving structures,” Journal of Sound and Vibration,

205(5), 593-615.

[81] Rana, R. and Soong, T.T. (1998), “Parametric Study and Simplified Design of Tuned

Mass Dampers,” Engineering Structures, 20(3), 193-204.

[82] Randall, S.E., Halsted, D.M. and Taylor, D.L. (1981), “Optimum Vibration Absorbers

for Linear Damped Systems,” Journal of Mechanical Design, ASME, 103, 908-913.

[83] Reed, D., Yu, J., Yeh, H. and Gardarsson, S. (1998), “An Investigation of Tuned Liq-

uid Dampers under large amplitude excitation,” Journal of Engineering Mechanics,

ASCE, 124, 405-413.

[84] Roberson, R.E. (1952), “Synthesis of a Nonlinear Dynamic Vibration Absorber,”

Journal of Franklin Institute, 205-220.

[85] Roberts, J.B. and Spanos, P.D. (1990), Random Vibration and Statistical Lineariza-

tion, Wiley, New York.

[86] Rojiani, K.B. (1978), “Evaluation of Steel Buildings to Wind Loadings,” Ph.D. The-

sis, University of Illinois, Urbana-Champaign.

[87] Sadek, F., Mohraz, B. and Lew, H.S. (1998) “Single- and Multiple-Tuned Liquid Col-

umn Dampers for Seismic Applications,” Earthquake Engng. and Struc. Dyn., 27, 439-

463.

[88] Sakai, F. et al. (1989), “Tuned Liquid Column Damper - New Type Device for Sup-

pression of Building Vibrations,” Proc. Int. Conf. on High Rise Buildings, Nanjing,

China, March 25-27.

[89] Sakai, F.,and Takaeda. S. (1991), “Tuned Liquid Column Damper (TLCD) for cable

stayed bridges,” Innovation in Cable-stayed Bridges, Fukonova, Japan.

[90] Schmidt, R., Oumeraci, H. and Partenscky, H.W. (1992), “Impact loads induced by

plunging breakwaters on vertical structures,” Coastal Engineering, 1545-1558.

[91] Sellars, F. H., and Martin, P.M. (1992), “Selection and Evaluation of Ship Roll Stabi-

lization Systems,” Marine Technology, 29(2), 84-101.

[92] Semercigil, S.E., Lammers, D. and Ying, Z. (1992), “A new Tuned Vibration

Absorber for Wide-band Excitations,” Journal of Sound and Vibration, 156 (3), 445-

459.

[93] Shimizu, K. and Teramura, A. (1994), “Development of vibration control system

using U-shaped tank”, Proceedings of the Ist International Workshop and Seminar on

Behavior of Steel Structures in Seismic Areas, Timisoara, Romania, 7.25-7.34.

190

[94] Soong, T.T. (1991), Active Structural Control-Theory and Practice, Longman, Lon-

don & Wiley, New York.

[95] Soong, T.T. and Dargush, G.F. (1997), Passive Energy Dissipation Systems in Struc-tural Engineering, Wiley, New York.

[96] Spencer, B.F., Jr. and Sain, M.K. (1997), “Controlling Buildings: A New Frontier in

Feedback,” IEEE Control Systems Magazine: Special Issue on Emerging Technolo-

gies,”17( 6), 19-35.

[97] Sudjic, D. (1993), “Their love keeps lifting us higher,” Telegraph magazine, May, 17-

25.

[98] Suhardjo, J. and Kareem, A. (1997), “Structural Control of Offshore Platforms,” Pro-

ceedings of the 7th International Offshore and Polar Engineering Conference IOSPE-

97, Honolulu.

[99] Suhardjo, J., Spencer, Jr., B.F. and Kareem, A. (1992a), “Active Control of Wind

excited Buildings: A Frequency Domain based Design Approach,” Journal of Wind

Engineering and Industrial Aerodynamics, 41-44.

[100] Suhardjo, J., Spencer, Jr., B.F. and Kareem, A. (1992b), “Frequency Domain Opti-

mal Control of Wind Excited Buildings,” Journal of Engineering Mechanics, ASCE,

118(12).

[101] Suhardjo, J. and Kareem, A. (2001), “Feedback-feedforward control of offshore

platforms under random waves,” Earthquake Engng Struct. Dyn., 30, 213-235.

[102] Sun, L.M., Fujino, Y., Paceno, B. and Chaiseri, P. (1991), “Modeling Tuned Liquid

Damper,” Proc. of the 8th International Conference on Wind Engineering, Elsevier.

[103] Sun, L.M. and Fujino, Y. (1994), “A Semi-analytical model for Tuned Liquid

Damper (TLD) with wave breaking,” Journal of Fluids and Structures, 8, 471-488.

[104] Sun, L. M., Fujino,Y., Paceno, B. and Chaiseri, P. (1995), “The properties of Tuned

Liquid Dampers using a TMD analogy,” Earthquake Engng and Struc. Dynamics, 24,

967-976.

[105] Symans, M.D. and Constantinou, M.C. (1999), “Semi-active control systems for

seismic protection of structures: a state-of-the-art review,” Engineering Structures (21),

469-487.

[106] Symans, M.D. and Constantinou, M.C. (1997), “Experimental testing and Analyti-

cal modeling of Semi-active fluid dampers for seismic protection, “ Journal of Intelli-

gent Material Systems and Structures, 8(8), 644-657.

[107] Symans, M.D. and Kelly, S.W. (1999), “Fuzzy Logic Control of Bridge Structures

using Intelligent Semi-Active Seismic Isolation Systems,” Earthquake Engineering and

Structural Dynamics, 28, 37-60.

[108] Tajimi, H. (1960), “A Statistical Method of Determining the Maximum Response of

a Building Structure During an Earthquake,” Proceedings 2nd World Conference on

Earthquake Engng, Vol. II, Tokyo and Kyoto, Japan, 781-798.

191

[109] Tamura, Y., Fujii, K., Ohtsuki, T., Wakahara, T. and Kohsaka, R. (1995), “Effective-

ness of Tuned Liquid Dampers under Wind Excitation,” Engineering Structures, 17 (9),

609-621.

[110] Tokarcyzk, B.L. (1997), “The Mathematical Modeling of a Tuned Liquid Damper,”

M.S. Thesis, Department of Civil Engineering, Texas A&M University, College Sta-

tion, TX.

[111] Ueda. T., Nakagaki, R. and Koshida, K. (1992) “Suppression of wind-induced vibra-

tion by dynamic dampers in tower-like structures,” J. of Wind Engg. and Ind. Aerody-

namics, 41-44, 1907-1918.

[112] Vandiver, J. K. and Mitome, S. (1978), “Effect of Liquid Storage Tanks on the

Dynamic Response of Offshore Platforms,” Dynamic Analysis of Offshore Structures:

Recent Developments.

[113] Venugopal, M. (1990), “A Toroidal Hydrodynamic Absorber for Damping Low Fre-

quency motions of Fixed and Floating Offshore Platforms,” Marine Technology, 27(1),

42-46.

[114] Warburton, G.B. and Ayorinde, E.O. (1980), “Optimum Absorber Parameters for

Simple Systems”, Earthquake Engineers and Structural Dynamics, 8, 197-217.

[115] Warburton, G.B. (1982), “Optimal Absorber Parameters for Various Combination of

Response and Excitation parameters,” Earthquake Engineering and Structural Dynam-

ics, 10, 381-401.

[116] Warnitchai, P. and Pinkaew, T. (1998), “Modeling of Liquid Sloshing in Rectangular

Tanks with Flow Dampening Devices,” Engineering Structures, 20(7), 593-600.

[117] Weisner, K.B. (1979), “Tuned Mass Dampers to reduce building sway,” presented at

the ASCE Boston Convention, April.

[118] Won, A.Y.J., Pires, J.A. and Haroun, M.A. (1996) “Stochastic seismic performance

evaluation of tuned liquid column dampers”, Earthquake Engng & Structural Dynam-

ics, 25, 1259-1274.

[119] Wu, G.X., Ma, Q.W. and Taylor, E.R. (1998), “Numerical Simulation of Sloshing

Waves in a 3D tank based on a Finite Element Method,” Applied Ocean Research, 20,

337-355.

[120] Xu, Y.L, Samali, B. and Kwok, K.C.S. (1992), “Control of Along-wind Response of

Structures by Mass and Liquid Dampers,” Journal of Engineering Mechanics, 118(1),

20-39.

[121] Xue, S.M., Ko, J.M. and Xu, Y.L. (2000), “Tuned Liquid Column Damper for sup-

pressing pitching motion of structures,” Engineering Structures, 23, 1538-1551.

[122] Yamaguchi, H. and Harnpornchai, N. (1993), “Fundamental Characteristics of Mul-

tiple Tuned Mass Dampers for Suppressing Harmonically Forced Oscillation,” Journal

of Earthquake Engineering and Structural Dynamics, 22.

192

[123] Yamazaki, S., Nagata, N. and Abiru, H. (1992), “Tuned active dampers installed in

the Minato Mirai (MM) 21 Landmark tower in Yokohama,” Journal of Wind Engng and

Industrial Aerodynamics 41-44, 1937-1948.

[124] Yang, J.N., Akbarpour, A. and Ghaemmaghami, P. (1987), “New Control algorithms

for Structural Control,” ASCE Journal of Engineering Mechanics, 113, 1369-86.

[125] Yao, J.T.P. (1972), “Concept of Structural Control,” ASCE, Journal of Structural

Division, 98, 1567-74.

[126] Yoshida, K., Yoshida, S. and Takeda, Y. (1998), “Semi-active control of Base Isola-

tion using Feedforward information of Disturbance,” Proceedings of Second World

Conference on Structural Control, Kyoto, John Wiley and Sons, 377-386.

[127] Yu, J., Wakahara, T. and Reed, D.A. (1999), “A Non-linear Numerical Model of the

Tuned Liquid Damper,” Earthquake Engng and Structural Dynamics, 28, 671-686.

193

APPENDIX

A.1 Evaluation of Response Integral

In order to evaluate the response statistics of systems subject to random excita-

tions with rational power spectra, the integrals are of the following form,

(A. 1)

where

and

This integral can be written in a matrix form as (Roberts and Spanos, 1990),

(A. 2)

where | | denotes determinant of the matrix.

I n

Ξn ω( ) ωd

Λn iω–( )Λn iω( )---------------------------------------

∞–

∞

∫≡

Ξn ω( ) χn 1– ω2n 2– χn 2– ω2n 4– … χ0+ + +=

Λn iω( ) λn iω( )n λn 1– iω( )n 1– … λ0+ + +=

I nπλn-----

χm 1– χm 2– … … … χ0

λm– λm 2– λm 4–– λm 6– … …

0 λm 1–– λm 3– λm 5–– … …

… 0 … … … …0 0 … … λ2– λ0

λm 1– λ– m 3– λm 5– λm 7–– … …

λm– λm 2– λm 4–– λm 6– … …

0 λm 1–– λm 3– λm 5–– … …

… 0 … … … …0 0 … … λ2– λ0

--------------------------------------------------------------------------------------------------=

181

A.2 Building and Excitation Parameters (Example 4 in Chapter 5)

The building stiffness matrix is given by,

kN/m

and the excitation parameters in Eq. 5.30 are given as:

a = kN; b = kN; c = kN; d = kN

A.3 Relation between Cv and

Most valve suppliers provide a different measure of flow characteristic than the

headloss coefficient (ξ) used thoroughout this dissertation. The commonly used measure is

the valve conductance which is defined as the mass flow of liquid through the valve, given

by,

(A. 3)

where Q is the mass flow (Kg/s); CV is the valve conductance (m2); ρ is the specific den-

sity of the liquid (Kg/m3); is the pressure drop across the valve (Pa).The valve conduc-

tance is usually supplied in British rather than S.I. units. The parameter in gall/min/

(psi)1/2 can be related to (in S.I. units) by the conversion factor,

(A. 4)

K=4.5

0.0254----------------

2000 1000– 0 0 0

1000– 4800 1400– 0 0

0 1400– 6000 1600– 0

0 0 1600– 6600 1700–

0 0 0 1700– 7400

4.5

675.45

700.45

615.15

555.25

475.05

4.5

0.3

375

284.5

175.3

15.1

4.5

735.5

655.15

564.45

690.15

18.6

4.5

180.5

35.5

425.0

280.0

650.05

ξ

Q CV ρ ∆p( )=

∆p

CV

CV

CV 2.3837 105–× CV=

182

A 1.5 inch ball valve has been used for the experimental study described in chapter 7. The

valve manufacturer provided the valve conductance values as a function of the valve open-

ing angle (Fig. A.1 (a)). The headloss across a valve/orifice can be written as,

(A. 5)

Equation A.5 can be rewritten as follows:

(A. 6)

The flow through the pipe of diameter D is given by:

(A. 7)

Comparing Eqs. A.3 and A.7, we obtain:

(A. 8)

Equation A.8 has been plotted for the 1.5 inch ball valve as a function of the angle of valve

opening.

Figure A.1 (a) Variation of Valve Conductance (b) Variation of headloss coefficientwith the angle of valve opening

∆pρξV

2

2-------------=

∆pQ

2

ρCV2

-------------=

Q ρAVπρD

2

4--------------V= =

ξ π2D

4

8CV2

-------------=

0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

Angle of valve opening, Φ

C v v

alue

s (g

al/m

in/p

si1/2 )

0 20 40 600

5

10

15

20

25

30

35

40

45

50

Angle of valve opening, Φ

Head

loss

Coe

ffic

ient

ξ= f (θ )

θ = 0deg

θ = 90deg

θ = 25deg

θ θ

183

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