Theoretical and experimental study of Tuned Liquid Column Dampers

UNIVERSIDAD POLITÉCNICA DE MADRID

ESCUELA TÉCNICA SUPERIOR DE INGENIERÍA

AERONÁUTICA Y DEL ESPACIO

GRADO EN INGENIERÍA AEROESPACIAL

TRABAJO FIN DE GRADO

Theoretical and experimental study of

Tuned Liquid Column Dampers

AUTOR:

Ángel Manuel GARCÍA GARCÍA

ESPECIALIDAD:

Ciencias y Tecnologías Aeroespaciales

TUTOR DEL TRABAJO:

Marcos CHIMENO MANGUÁN

2

Contents

1 Introduction 11

1.1 Vibration attenuation in engineering . . . 11

1.2 Scope of the project . . . 14

2 Formulation 15 2.1 Basic formulation of a MDOF oscillator . . . 15

2.2 Tuned Mass Damper . . . 16

2.3 Structural modeling and dynamic analysis . . . 19

2.4 Formulation of Tuned Liquid Column Dampers . . . 21

2.4.1 Equivalent linear damping coefficient . . . 23

2.5 2-DOF approximation for an U-shaped TLCD . . . 25

2.5.1 Damping modeling of liquid sloshing . . . 25

2.5.2 Analysis of a simplified 2-DOF system . . . 26

2.6 Finite Element approximation for an U-shaped TLCD . . . 28

2.7 Sensitivity to the parameters of the model . . . 30

3 Vibration Tests Analysis 33 3.1 Equipment used . . . 33

3.2 Characterization of the structure . . . 34

3.3 TLCD performance . . . 36

4 Experimental Correlation 41 4.1 Model adjustment . . . 41

4 CONTENTS

5 Concluding remarks 45

5.1 Future work . . . 46

Bibliography 47

Appendix A Discrete modeling of a 1D beam 49

Appendix B Experimental Results 53

B.1 Non-correlated simulations . . . 53 B.2 Optimized η simulations . . . 57

List of Figures

1.1 Taipei 101 exterior (a) and interior images (b). . . 12 1.2 The huge pendulum mounted on the structure of the Taipei 101. 12 1.3 One Rincon Hill South Tower in San Francisco, California, 2008.

The water tank is visible from the left picture’s point of view. . 13

2.1 Schema of a beam with a pendulum attached to its tip. . . 16 2.2 Schema of the structure used in the simulation. . . 19 2.3 The stiffness matrix assembly. Kf represents the stiffness

ma-trix of each floor which is compounded of multiple beam sub-matrices. Mass matrix assembly follows same structure. . . 20 2.4 Schema of a U-shaped tuned liquid columns damper. y

rep-resents the mean displacement of the liquid surface around its equilibrium point whilexand θ are the lineal and rotating mo-tion of the base. . . 21 2.5 Schema of the 2-DOF system. . . 26 2.6 The stiffness matrix assembly with the liquid damper mounted

on the rooftop of the building. Kf loor represents the stiffness

matrix of each floor which is compounded of multiple beam sub-matrices. Mass matrix assembly follows same structure. . . 29 2.7 Normalized transmisibility ˆT against frequency ratio Ω

ωd under

variations ofµ around the reference. . . 30 2.8 Normalized transmisibility ˆT against frequency ratio Ω

ωd under

6 LIST OF FIGURES

2.9 Normalized transmisibility ˆT against frequency ratio Ω

ωd under

variations ofωd around the reference. . . 31

3.1 Test setup. . . 34 3.2 Normalized transmisibility of the structure, ˆT, against the

non-dimensional excitation frequency _ωΩ₁. . . 35 3.4 TLCD 1 with the experimental setup schema. . . 37 3.5 TLCD 1.B: Normalized transmisibility of the structure, ˆT, against

the non-dimensional excitation frequency _ωΩ₁. . . 39 3.6 TLCD 4.B: Normalized transmisibility of the structure, ˆT, against

the non-dimensional excitation frequency _ωΩ₁. Oscillations be-tween 0.4 and 0.6 are due controller’s punctual errors. . . 40 4.1 Comparison between the 2 different sloshing types identified by

visual observation in the tests. . . 42 4.2 TLCD 1: Normalized transmisibility of the structure, ˆT, against

the non-dimensional excitation frequency _ωΩ₁ for the structure test, simulation of equivalent inert mass at rooftop and TLCD tests. . . 43 A.1 Schema of the 4 DoF FEM beam. . . 49 B.1 TLCD 1: Normalized transmisibility of the structure, ˆT, against

the non-dimensional excitation frequency Ω

ω1. . . 53 B.2 TLCD 2: Normalized transmisibility of the structure, ˆT, against

the non-dimensional excitation frequency _ωΩ₁. . . 54 B.3 TLCD 3: Normalized transmisibility of the structure, ˆT, against

the non-dimensional excitation frequency _ωΩ₁. . . 54 B.4 TLCD 4: Normalized transmisibility of the structure, ˆT, against

the non-dimensional excitation frequency Ω

ω1. . . 55 B.5 TLCD 2.B: Normalized transmisibility of the structure, ˆT, against

the non-dimensional excitation frequency Ω

ω1. . . 55 B.6 TLCD 3.B: Normalized transmisibility of the structure, ˆT, against

LIST OF FIGURES 7 B.7 TLCD 4.B: Normalized transmisibility of the structure, ˆT, against

the non-dimensional excitation frequency _ωΩ₁. . . 56 B.8 TLCD 1: Normalized transmisibility of the structure, ˆT, against

the non-dimensional excitation frequency Ω

ω1 for the correlated simulation corresponding to its optimumη. . . 57 B.9 TLCD 2: Normalized transmisibility of the structure, ˆT, against

the non-dimensional excitation frequency _ωΩ₁ for the correlated simulation corresponding to its optimumη. . . 57 B.10 TLCD 3: Normalized transmisibility of the structure, ˆT, against

the non-dimensional excitation frequency _ωΩ₁ for the correlated simulation corresponding to its optimumη. . . 58 B.11 TLCD 4: Normalized transmisibility of the structure, ˆT, against

the non-dimensional excitation frequency _ωΩ₁ for the correlated simulation corresponding to its optimumη. . . 58 B.12 TLCD 1.B: Normalized transmisibility of the structure, ˆT, against

the non-dimensional excitation frequency _ωΩ₁ for the correlated simulation corresponding to its optimumη. . . 59 B.13 TLCD 2.B: Normalized transmisibility of the structure, ˆT, against

the non-dimensional excitation frequency Ω

ω1 for the correlated simulation corresponding to its optimumη. . . 59 B.14 TLCD 3.B: Normalized transmisibility of the structure, ˆT, against

the non-dimensional excitation frequency Ω

ω1 for the correlated simulation corresponding to its optimumη. . . 60 B.15 TLCD 4.B: Normalized transmisibility of the structure, ˆT, against

the non-dimensional excitation frequency Ω

ω1 for the correlated simulation corresponding to its optimumη. . . 60

8 LIST OF FIGURES

C.2 TLCD 2: Normalized transmisibility of the structure, ˆT, against the non-dimensional excitation frequency _ωΩ₁ for the structure test, simulation of equivalent inert mass at rooftop and TLCD tests. . . 62 C.3 TLCD 3: Normalized transmisibility of the structure, ˆT, against

the non-dimensional excitation frequency _ωΩ₁ for the structure test, simulation of equivalent inert mass at rooftop and TLCD tests. . . 62 C.4 TLCD 1.B: Normalized transmisibility of the structure, ˆT, against

the non-dimensional excitation frequency Ω

ω1 for the structure test, simulation of equivalent inert mass at rooftop and TLCD tests. . . 63 C.5 TLCD 2.B: Normalized transmisibility of the structure, ˆT, against

the non-dimensional excitation frequency _ωΩ₁ for the structure test, simulation of equivalent inert mass at rooftop and TLCD tests. . . 63 C.6 TLCD 3.B: Normalized transmisibility of the structure, ˆT, against

the non-dimensional excitation frequency _ωΩ₁ for the structure test, simulation of equivalent inert mass at rooftop and TLCD tests. . . 64 C.7 TLCD 4.B: Normalized transmisibility of the structure, ˆT, against

the non-dimensional excitation frequency Ω

List of Tables

3.1 Parameters that define the mock-up structure used in the tests. h is the assumed damping ratio which mainly corresponds to the first vibration mode. rf is the stiffness reduction coefficient

used. . . 34 3.2 TLCD principal parameters comparison. . . 37 3.3 Principal frequency domain response parameters for all the TLCD

setups. . . 38

Chapter 1

Introduction

1.1

Vibration attenuation in engineering

From spacecrafts to bridges, vibration behavior has to be studied in order to achieve an adequate design of every engineering work. In most situations it could seem nonsense as we tend to think of the structures that we build as completely solid objects, furthermore, design approach based on applying static load distributions reinforces this belief.

However, mechanical vibrations are a matter of importance of almost ev-ery physical system. A vibration damper is an auxiliary system whose main function is to reduce the motion of the structure in which it is integrated. This section is a brief introduction on how to perform that task.

It is well known that the current tendency of higher and lighter build-ings has led to increasingly flexible structures that are sensitive to the outer conditions such as wind and earthquakes. The reason of that is that certain combinations of the inertia, stiffness and damping of the structure makes it oscillate. The vibrations induced by them are totally undesirable and could lead to structural damage and discomfort.

12 CHAPTER 1. INTRODUCTION

Therefore, over the past few years, there has been a substantial increase in the implementation of additional damping devices in a wide variety of struc-tures such as buildings, fuel deposits of aircraft and even spacecraft.

(a) Skyline of Taiwan with the Taipei 101

shining at the left.

(b) Signpost that indicates where the

su-per big wind damsu-per is located.

Figure 1.1: Taipei 101 exterior (a) and interior images (b).

Devices used for mitigating structural vibrations can be divided into two different groups, passive and active dampers. The difference among them is essentially the need of external power source. While active dampers are usually more effective, they use, as we have just stated, a power source. Un-fortunately, during natural disasters power outages are more easily to occur. Passive dampers are a strong asset in this case. They are reliable and have little dependability with maintenance.

An example of those devices is the Tuned Mass Damper (TMD) mounted on the Taipei 101, located in Taiwan, currently the 10th _{tallest building on}

Earth, with a height of 508m and 101 floors above the ground which has a huge pendulum inside that efficiently mitigates vibrations.

1.1. VIBRATION ATTENUATION IN ENGINEERING 13 A Tuned Liquid Damper is a passive device that uses a liquid confined in certain geometries to take advantage of the energy dissipation of liquid sloshing. Their main advantage against TMDs is that they dissipate much more energy. It has been a recent increase in their use in the past years.

A good example is the Tuned Liquid Damper (TLD) that was mounted on One Rincon Hill South Tower, a 213 m tall residential building located in San Francisco, California that has a hidden water tank in his interior’s top floor that contains around 190.000 liters of water. The mentioned building has also steel buckling-restrained braced frames designed to absorb energy through hysteretic behavior. Both systems work together to avoid undesired motion of the tower.

Figure 1.3: One Rincon Hill South Tower in San Francisco, California, 2008. The water tank is visible from the left picture’s point of view.

A Tuned Liquid Columns Damper is a type of TLCD which uses columns of liquid connected at their base.

14 CHAPTER 1. INTRODUCTION

1.2

Scope of the project

The purpose of this project is to formulate a simplified model that repre-sents the equations that govern the liquid sloshing of Tuned Liquid Column Dampers, test a scaled mock-up with various configurations and correlate the results with the predictions of the initial model.

In order to achieve it, Chapter 1 will begin by defining and introducing the Multiple Degree or Freedom Oscillator and apply those concepts to simple cases until reaching a simplified formulation for a TLCD in Chapter 2. Also, we will create a FEM model of the future mock-up that will be used in the tests that will be followed by a sensitivity analysis.

Chapter 2

Formulation

2.1

Basic formulation of a MDOF oscillator

The differential equations that govern the dynamics of a multiple degree of freedom oscillator can be described as follows

[M]¨q+ [F] ˙q+ [K]q=p (2.1.1)

    

q(t= 0) =q0

˙

q(t= 0) = ˙q0

in which q represents the generalized coordinates, p the generalized forces applied to the system, [M] the mass matrix, [F] the damping matrix and [K] the stiffness matrix.

Equation (2.1.1) describes a 2nd_{order system of ordinary differential}

equa-tions which, in the best case, will have constant coefficients, that is, [M],[F] and [K] will be conformed of constant values.

Those equations have been derived by applying Lagrange’s equations d

dt

∂L ∂q˙

−∂L_∂q + ∂D

∂q˙ =Fnc (2.1.2)

16 CHAPTER 2. FORMULATION

potential energy respectively, D is the non-conservative potential and Fnc is the force done by the forces that do not derive from any potential.

As have been proved, in most situations, specially analyzing systems with a moderate-to-high number of degrees of freedom, the Lagrangian approach is much more powerful than the Newtonian one, i.e.,PF=ma.

2.2

Tuned Mass Damper

To illustrate this as well as maintaining simplicity, a clamped-free beam with a pendulum on its tip will be modeled as a 2-Degree of Freedom system. The generalized coordinates will be x, the movement of the tip of the beam; and θ, the angular motion of the pendulum. A known movement z of the base will be considered.

Figure 2.1: Schema of a beam with a pendulum attached to its tip.

The beam subject of analysis will have an equivalent mass1 _{M =} 33

140ρAL and a stiffness 2

K = 3EI

L3 . E represents the Young’s Modulus of the material,I the moment of inertia of the beam’s section,ρthe density of the material,Athe section of the beam and Lthe longitude of the mentioned beam. The pendulum attached to its tip, in turn, will have a punctual massmattached to a massless string of longitude l.

According to the stated on the previous sec-tion, we will begin by deriving its kinetic and po-tential energy around the equilibrium point

T = 1 2Mx˙

2₊1

2m( ˙x+lθ˙)

2 _(2.2.1)

1_{Applying Rayleigh’s method.}

2.2. TUNED MASS DAMPER 17

U = 1 2

3EI L3

(x₋z)2+ 1 2mglθ

2 _(2.2.2)

and, applying Lagrange’s equations leads to

    

(M+m)¨x+mlθ¨+ 3EI L3

(x₋z) = 0 mlx¨+ml2_θ¨_{+mglθ= 0}

(2.2.3)

which rearranging the imposed movement term and using matrix notation, we reach the equation described in (2.1.1) being [F] = 0,

[M] =



M +m ml

ml ml2



 [K] =

 

3EI L3 0

0 mgl   q=    x θ    p=    3EI L3 z

0

  

(2.2.4)

Now, we are going to suppose harmonic movement of the base with the form: z =z0cos(Ωt) =Re{z0eiΩt}. It is understandable to suppose q∝eiΩt.

There-fore

(₋Ω2[M] + [K])

| {z }

[H]−1_(Ω)

q=p (2.2.5)

where H(Ω) is usually called transfer function as it gives the generalized co-ordinates response q of the system given any harmonic entry force. In this case

H(Ω) = 1 ∆(Ω)



ml(g−lΩ

2_{) ml}

ml K ₋(M +m)Ω2



 (2.2.6)

where

∆(Ω) =(H(Ω)=ml[(M+m)lΩ4−[(M+m)g+Kl]Ω2+Kg−ml] (2.2.7)

Finally, horizontalx movement is given by

x= (g−lΩ

2_)Kz

0cos(Ωt)

(M +m)lΩ4_{−[(M +m)g+Kl]Ω}2 _+Kg−ml (2.2.8)

18 CHAPTER 2. FORMULATION

Viscously damped TMD

Now, we are going to add more complexity to the modelby adding a dissipation function. We are going to suppose that the pendulum has a viscous damping F added to the yet existent gravitational force that acts as an elastic spring. That being said, we define

D= 1 2Fθ˙

2 _(2.2.9)

The equations of small movements of the system described are then, by direct application of Lagrange’s equations, in matrix form



M +m ml

ml ml2

     ¨ x ¨ θ    +  0 0

0 F      ˙ x ˙ θ    +   3EI L3 0

0 mgl      x θ    =    Kz 0    (2.2.10) Proceeding exactly as before, we are going to suppose harmonic movement of the base following: z =z0eiΩt. It is also understandable to suppose q ∝eiΩt.

Therefore:

(₋Ω2[M] +iΩ[F] + [K])

| {z }

[H]−1_(iΩ)

q=p (2.2.11)

The transfer function is then:

H(iΩ) = 1 ∆(iΩ)



ml(g−lΩ

2_{) +iΩF ml}

ml K₋(M +m)Ω2



 (2.2.12)

where

∆(iΩ) =H(iΩ)= (2.2.13)

=ml[(M +m)lΩ4_{−[(M +m)g+Kl]Ω}2_{+Kg−ml] +iF[KΩ + (M +m)Ω}3_]

Note the iΩ dependency denoting the complex character of the matrix. Hori-zontal x movement is given by

x=R_e

(

[ml(g₋lΩ2_{) +iΩF]Kz} 0eiΩt

∆(iΩ)

)

(2.2.14)

denoting ∆(iΩ) = α(Ω) +iβ(Ω) =∆(iΩ)eiδ, where δ = −tan−1₍β

α) as well

2.3. STRUCTURAL MODELING AND DYNAMIC ANALYSIS 19

tan−1 ΩF

ml(g−lΩ2)

; the above expression can be rewritten as

x=

p

[ml(g₋lΩ2_)]2_{+ Ω}2_F2_Kz

0eiΩt+ϕ+δ

∆(iΩ) (2.2.15)

It is important to take into account that phase angle δ is negative so the system response occurs after the excitation.

As we can see, it is impossible to totally suppress the vibration of the tip of the beam as we did before, imposing x to be zero. That phenomenon is common to all damped systems where the delay between the actual excitation and the response is clearly apparent.

2.3

Structural modeling and dynamic analysis

The structural behavior has been modeled using Finite Element Method as it represents a powerful tool joining together calculus velocity and precision. In one hand, a Material Resistance model could have been used sacrificing accuracy whereas, in the order hand, a ODE solver would have needed so much more time to run obtaining similar results. FEM has demonstrated to be a great asset in structural engineering.

Figure 2.2: Schema of the structure used in the simulation.

20 CHAPTER 2. FORMULATION

a sub-structure ofN₋k nodes numbered in ascending order in a right-handed spiral. Figure 2.6 will clarify the explanation.

The stiffness of the plate located under and above each floor will be taken as infinitely large compared with the flexural stiffness of the beams so that, to our analysis, the whole plate will represent a single node.The kth and N th nodes will be shared among the four beams on every joint.

Each beam mounted on the sub-structure has been discretized using 7 nodes, which has been proven to represent accurately enough the transversal vibration phenomenon. A more profound discussion can be found on L. Cremer et al. [5].

Figure 2.3: The stiffness matrix assembly. Kf represents the stiffness matrix of

each floor which is compounded of multiple beam sub-matrices. Mass matrix assembly follows same structure.

That being said, the whole structure will consist, after applying clamped-base boundary conditions, on 26 nodes corresponding to each of the sub-structures that simultaneously share 1 of the nodes with the one above. That makes (26₋1)_·3 = 75 nodes.

2.4. FORMULATION OF TUNED LIQUID COLUMN DAMPERS 21 stiffness matrix of theith _{floor is determined by}

Kj,k =Kj,k+Kf loor (2.3.1)

where j and k follow a succession of the form

ji+1 =ki−1 (2.3.2)

ki+1 = 2(r(1 +i)−i) (2.3.3)

where r is the number of nodes of each substructure, j1 = 1 and k1 =r.

The procedure to assemble the mass matrix is identical.

2.4

Formulation of Tuned Liquid Column Dampers

In this section the formulation of an U-shaped circular section Tuned Liquid Column Damper (TLCD) as shown in the Figure 2.4 will be derived under the hypothesis of small oscillations.

Figure 2.4: Schema of a U-shaped tuned liquid columns damper. y represents the mean displacement of the liquid surface around its equilibrium point while x and θ are the lineal and rotating motion of the base.

22 CHAPTER 2. FORMULATION

similar formulation as Jong-Cheng Wu, 2005 [3] letting α=Lh/Ll and calling

Ll= 2Lv+Lh, as the sum of this four contributions

T_ltrans = 1

2ρlAlLl( ˙x

2 _{+ ˙y}2_{+ 2αx˙y˙) (2.4.1)}

Tlrot =

1 2

"

Il,h+ 2

Il,v+

ρlAlLv

4 (L

2

h+L2v)

#

˙ θ2 = 1

2Ilθ˙

2 _(2.4.2)

T_strans = 1 2msx˙

2 _(2.4.3) Trot s = 1 2 "

Is,h+ 2

Is,v+

ms

4 (Lh

2_+L∗

v

2₎

#

˙ θ2 ₌ 1

2Isθ˙

2 _(2.4.4)

where3 _A

l =πRl2, Il,k = ρlA₁₂lLk 3R2l +L2k

and Is,k =

ρsAlL∗kts

6Rl 6R

2

l +L∗k

being ts the thickness of the tube, L∗v = κLv noting that κ will prevent the

liquid spilling so κ > 1, L∗

h = Lh, Rl the internal radius of the tube, ρl and

ρs the densities of the liquid and the material used to build the structure of

the damper respectively, ms the mass of the structure of the damper and k

denoting either the subindex l or v. ∂T

∂x˙ =ρlAlLl( ˙x+αy˙)→ d dt

∂T ∂x˙

=ρlAlLl(¨x+αy¨) +msx¨

∂T

∂y˙ =ρlAlLl( ˙y+αx˙)→ d dt

∂T ∂y˙

=ρlAlLl(¨y+αx¨) (2.4.5)

∂T

∂θ˙ = (Il+Is) ˙θ→ d dt ∂T ∂θ˙

= (Il+Is)¨θ

The potential energy, taking as the zero potential point the equilibrium liquid surface, is

U =ρlAlgy2 (2.4.6)

similarly, the potential term is ∂U

∂y = 2ρlAlgy (2.4.7) The non-conservative force, fd, caused mainly by the pressure-loss in the

el-bows, being η the head-loss coefficient, can be expressed as4

fd=

1

2ρlAlη sgn( ˙y) ˙y

2 _(2.4.8)

3_{The subindex ”l” denotes liquid properties.}

2.4. FORMULATION OF TUNED LIQUID COLUMN DAMPERS 23 The non-linearity of the dissipation force will be managed by defining an equiv-alent force which, essentially, will dissipate the same amount of energy per cycle. We can define an equivalent energy dissipation coefficient Ceq.

2.4.1

Equivalent linear damping coefficient

As our aim is to define a linearized damping coefficient, we will firstly find out how the linear damping coefficient is obtained given a linear force and then apply the same procedure to a quadratic force, which is the case that is being faced, to obtain a linearized coefficient.

Linear damping

Given harmonic motion of a generalized coordinate y attached to a linear damper in which its dissipation force has the form ofFlinear

d =Cy˙, its motion

can be written as

y=y0sin(Ωt) (2.4.9)

˙

y=y0Ωcos(Ωt) (2.4.10)

that way, taking into account that dy_dt = ˙y _→dy = ˙ydt, the energy dissipated per cycle Wd is

Wd=

I

Fddy=

I

Cy˙2dt =Cy2₀Ω2

Z 2π

Ω

0

cos2(Ωt)dt=Cy₀2Ωπ (2.4.11) therefore,

C = Wd y2

0Ωπ

(2.4.12)

Quadratic damping

24 CHAPTER 2. FORMULATION

thereby, a similar procedure can be followed to obtain the energy loss per cycle due to a force with the form of (2.4.8)

Wd=

I

Cquady˙2dy=Cquady30Ω3 4

Z π

2Ω

0

cos3_(Ωt)dt= 8

3Cquady

3

0Ω2 (2.4.13)

Equaling (2.4.13) and (2.4.11) we obtain Ceq =

4

3πρlAlηΩy0 (2.4.14) Therefore, (2.4.8) can be rewritten as

fd=

4

3πρlAlηΩy0y˙ (2.4.15)

Linearized damping coefficient

Arranging (2.4.5), (2.4.7) and (2.4.15) and taking x as an outer displacement imposed to the system, we obtain

ρlAlLly¨+

4

3πρlAlηΩy0y˙+ 2ρlAlg y=−ρlAlLlαx¨ (2.4.16) factoring out ρA, dividing the whole expression byLΩ2 _{and identifying ω}

d=

q

2g

L as the natural frequency of the TLCD

1 Ω2 y¨+

4η 3πΩ

y0 Ll ˙ y+ ωd Ω 2

y=₋α

Ω2x¨ (2.4.17)

Applying the condition of harmonic motion x= ˜xeiΩt

→y˜= ˜yeiΩt

˜

y= _ω αx˜

d

Ω

2

−1 + 4ηy0

3πLi

(2.4.18) equaling their magnitude and noting that_|y˜_|=y0 and|x˜|=x0

y0 =

αx0 r h ωd Ω 2

−1i2+4ηy0

3πLl

2 (2.4.19)

This expression indicates that the amplitude of the motion of the liquid surface is a function of itself. Rearranging the previous expression, we get

y0 =

1₋ ωd Ω 2

3πLl

4√2η

v u u u u u u t v u u u u u t1 +

   

8ηαxˆ0

3πh1₋ ωd

Ω

2i2

   

2

2.5. 2-DOF APPROXIMATION FOR AN U-SHAPED TLCD 25 while, under the hypothesis thatηαxˆ0 and noting that the order of magnitude

of α and η is at most 1, it simplifies as y0 ≃

αxˆ0Ll

1− ωΩd

2

(2.4.21)

So, finally, equation (2.4.15) can be expressed as fd≃

4

3πρlAlLl

Ωηαxˆ0

1− ωΩd

2

˙

y (2.4.22)

which is a reasonably well estimation if the excitation frequency, Ω, is not too close to ωd ass well as the product ηαxˆ0 remains close to unity.

2.5

2-DOF approximation for an U-shaped TLCD

While the remaining of this work will be centered on FEM analysis, this 2-Degree of Freedom model is still interesting as same conclusions can be ob-tained from it. The reader can consult specialized literature that follows this approach such as Swaroop K. Yalla & Ahsan Kareem [2] and Jong-Cheng Wu [3].

2.5.1

Damping modeling of liquid sloshing

We can define a proportional damping matrix [F]dbeing a linear combination

of the mass and stiffness matrices with the form

[F]d=γl[M]d+βl[K]d (2.5.1)

In our case, we will suppose that the damping term will be proportional only to the stiffness term,γl= 0

[F]d=

4

3πρlAlLl

2ρlAlg

Ωηαxˆ0

1− ωΩd

2

[K]d =

4 3π

ηαxˆ0

ωd

Ω

ωd

1− ωΩd

2

26 CHAPTER 2. FORMULATION

This way, using (2.1.1) and (2.5.1), the equations that describe the dynamics of the damper are

[M]q¨+βl[K]q˙ + [K]q=p (2.5.3)

After transforming the previous expression into the frequency domain, it is easy to obtain

(iΩβl + 1) [K] =

    

4ηαxˆ0

3π Ω ωd 2 1− Ω ωd 2 + 1     

[K] (2.5.4)

which will be named as the complex stiffness matrix of the damper.

2.5.2

Analysis of a simplified 2-DOF system

Figure 2.5: Schema of the 2-DOF system.

Given an harmonic excitation of the base of mag-nitudez =z0cos(Ωt), the equations of motion are,



m+ml αml

αml ml

     ¨ x ¨ y    +  k 0

0 kl

     x y    =    kz 0    (2.5.5) where m is the effective mass of the beam yet de-fined,ml=ρlAlLl the mass of the liquid damper,

α is the well known aspect ratio of the horizontal and vertical columns of the liquid, k = 3EI

L3 is the stiffness of the clamped-free beam, kl = 2ρlAlg

is the stiffness of the TLCD due to gravitational force.

The same expression, in non-dimensional form is _

1 +µ αµ

α 1      ¨ x ¨ y    +  ω 2 n 0

0 ω2

d      x y    =    ω2 nz 0    (2.5.6)

where µ = ml

2.5. 2-DOF APPROXIMATION FOR AN U-SHAPED TLCD 27 unity;ωn is the natural frequency of the unaltered beam andωd is the natural

frequency of the Tuned Liquid Column Damper (TLCD).

The generalized eigenvalue problem described by (2.5.6) leads to

ω1,2

ωd

2

=

(1 +µ) +

ωn ωd 2 ± v u u t " 1₋ ωn ωd

2#2

+ 2µ(2α2₋₁₎

ωn

ωd

2

+µ(µ+ 2) 2(1 +µ₋µα2₎

(2.5.7)

Damped 2-DOF model

Now, a proportional damping model will be assumed, as it has been done through this whole text. The damping terms can be separated into the inherent structural damping and the added by the liquid damper itself, that is,

[F] =



βk 0

0 βlkl



 (2.5.8)

where β, is the constant of proportionality of the structural damping and βl,

firstly mentioned in (2.5.1), is its homologue in the liquid damper in spite of the fact that it is not constant, whose expression has been derived.

Therefore, the equations of the system after adding the damping terms, are



1 +µ αµ

α 1      ¨ x ¨ y    +  βω 2 n 0

0 βlωd2

     ˙ x ˙ y    +  ω 2 n 0

0 ω2

d      x y    =    ω2 nz 0    (2.5.9) which, if harmonic motion is supposed

        

−Ω2



1 +µ αµ

α 1

 +iΩ

     βω2 n 0

0 4ηαˆx0

3π Ω ωd 2

1−(ωdΩ) 2      +  ω 2 n 0

0 ω2

d           

| {z }

H−1_(iΩ)

   ˜ x ˜ y    =    ω2

nz˜

0

  

28 CHAPTER 2. FORMULATION

thus, the transfer function of the system will be

H(iΩ) = 1 H−1(iΩ)

     ω2

d(1 +i

4ηαxˆ0

3π Ω ωd 2

1−(ωdΩ) 2

)₋Ω2 _Ω2_αµ

Ω2_{α ω}2

n(1 +iΩβ)−Ω2(1 +µ)

     (2.5.11) finally, the motion of the generalized coordinates x and y will be, being the vector of nodal forces, p= (ω2

nz˜ 0)T

q=    x y   

=H(iΩ)p (2.5.12)

that is

x(t) z0

=R_e

             

ωd2

   1 +i

4ηαxˆ0

3π Ω ωd 2

1− ωΩd

2    −Ω 2      ω2

neiΩt

H(iΩ)          (2.5.13)

y(t) z0

=R_e

(

αΩ2_ω2

neiΩt

H(iΩ) )

(2.5.14)

2.6

Finite Element approximation for an

U-shaped TLCD

In the previous section, an approximated expression for kinetic energy, poten-tial energy and dissipation force were obtained. Those expressions in matrix form are

Td=

1 2

n

˙ x θ˙ y˙

o

ρlAlLl

    

1 + ms

ρlAlLl 0 α

0 Il+Is

ρlAlLl 0

α 0 1

              ˙ x ˙ θ ˙ y          (2.6.1)

Ud =

1 2

n

x θ y

o

2ρlAlg

    

0 0 0 0 0 0 0 0 1

2.6. FINITE ELEMENT APPROXIMATION FOR AN U-SHAPED TLCD29

Dd ≃

1 2

n

˙ x θ˙ y˙

o ₄

3πρlAlLl

Ωηαxˆ0

1− ωΩd

2     

0 0 0 0 0 0 0 0 1

              ˙ x ˙ θ ˙ y          (2.6.3)

This approach will let us assemble the TLCD described with three degrees of freedomx, θ andyinto the structure. This shows that the three parameters that affect the behavior of the U-shaped liquid damper are, having fixed the liquid nature5_{: α, ω}

d and ml =ρlAlLl or any other combination of those.

Figure 2.6: The stiffness matrix assembly with the liquid damper mounted on the rooftop of the building. Kf loor represents the stiffness matrix of each floor

which is compounded of multiple beam sub-matrices. Mass matrix assembly follows same structure.

[M]d=ρlAlLl

    

1 + ms

ρlAlLl 0 α

0 Il+Is

ρlAlLl 0

α 0 1

    

[K]d= 2ρlAlg

    

0 0 0 0 0 0 0 0 1

    

[F]d=

4 3π

ηαxˆ0

ωd

Ω

ωd

1− ωΩd

2

[K]d (2.6.4)

30 CHAPTER 2. FORMULATION

2.7

Sensitivity to the parameters of the model

At this point, the model developed is known to be dependent of mainly 3 different parameters but nothing has been stated about how weak or strong is their dependence. With the intention of clarifying it, the following figures have been made by iterating over µ, α and ωd during the simulations.

A reference TLCD, which is the one that will serve as reference in the vibration tests, has been taken to run the following simulations and study the variation of the response.

The ratio of masses shows the strongest dependency with both the natural frequency and the transmisibility, whereas the natural frequency of the TLCD also affects them but in a slightly lower amount.

The influence of the ratio of lengths of liquid in the tube seems negligible compared to the others.

Figure 2.7: Normalized transmisibility ˆT against frequency ratio _ωΩ

d under

2.7. SENSITIVITY TO THE PARAMETERS OF THE MODEL 31

Figure 2.8: Normalized transmisibility ˆT against frequency ratio _ωΩ

d under

variations ofα around the reference.

Figure 2.9: Normalized transmisibility ˆT against frequency ratio _ωΩ

d under

Chapter 3

Vibration Tests Analysis

3.1

Equipment used

In the tests, the following equipment was used:

• Shaker: LDS 406

• Control system: LDS LaserUSB

• Accelerometer: Br¨uel & Kjaer 4533

The profile of all the tests was a constant acceleration of 0.4g. The range of frequencies analyzed in all the tests was between 5Hz to 500Hz in an ascending frequency rate of 1oct/min. The setup can be seen in the Figure below.

34 CHAPTER 3. VIBRATION TESTS ANALYSIS

Figure 3.1: Test setup.

3.2

Characterization of the structure

E

[GP a]

I

[m4_]

L

[m] h rf

m

[g] 48 3.41_·10−10 _{0.309 3.55·10}−4 _{0.184 304}

Table 3.1: Parameters that define the mock-up structure used in the tests. h is the assumed damping ratio which mainly corresponds to the first vibration mode. rf is the stiffness reduction coefficient used.

The structure used in the tests, as it has already been explained in Section 2.3 is a three-floor building-like mock-up built with aluminum squared section rods. Because of the symmetry of the model, a single leg of the structure is sufficient to show the results.

colloqui-3.2. CHARACTERIZATION OF THE STRUCTURE 35 ally speaking, is an indicator of how well the motion is transmitted. Take into account that, for a majority of applications, a low value of transmisibil-ity is preferred as it isolates the part of interest from the possible source of vibrations.

The transmisibility of the structure, defined as the ratio between the dis-placement applied at the base and the disdis-placement of its rooftop, for both the the measured from the tests and the simulated using the model previously described, is shown in Figure 3.2.

Figure 3.2: Normalized transmisibility of the structure, ˆT, against the non-dimensional excitation frequency _ωΩ₁.

As we can see, the first natural frequency of the structure fits correctly in the model. The magnitudes are shown dimensionless, that it, ˆT = T

TM AX and

Ω

ω1 where ω1 is the fundamental frequency of the mock-up andTM AX = 18.31 is its maximum transmisibility.

36 CHAPTER 3. VIBRATION TESTS ANALYSIS

frequency of the model with the used in the tests (ω1 = 34.61 Hz), a reduction

of the stiffness of the FE’s around the linking nodes has been made as well as an overall reduction. This reduction is mainly because of 2 factors. Firstly, the mock-up structure was already built when the tests were going to take place so exact materials used are unknown and, secondly but in relation with the first reason, the structure was build with screw connected beams whose tightening torque was both unknown and imprecise.

The proportional damping efficient of the structure h, was set to match the peak transmisibility of the structure. Because of this crude simplification, the loss of accuracy in the peak response of successive natural frequencies is inevitable, but, since the main focus of this text is to characterize the funda-mental frequency of the structure, it seems sufficient.

3.3

TLCD performance

The 4 different TLCDs tested are presented below. Two different diameters of tube have been used. The TLCDs were mounted on a expanded polystyrene support and tightly knotted to the rooftop of the structure to ensure that the base of the TLCD moved just as the rooftop of the building.

3.3. TLCD PERFORMANCE 37

Figure 3.4: TLCD 1 with the experimental setup schema.

µ α ωd

[rad/s]

TLCD 1.A 1.95 0.25 8.37

TLCD 2.A 1.26 0.25 8.37

TLCD 3.A 1.96 0.21 8.37

TLCD 4.A 2.05 0.25 6.55

TLCD 1.B 1.83 0.30 9.17

TLCD 2.B 1.05 0.30 9.17

TLCD 3.B 1.40 0.30 9.90

TLCD 4.B 1.71 0.30 7.17

38 CHAPTER 3. VIBRATION TESTS ANALYSIS

f₁expn

[Hz]

fsim

1n

[Hz]

Ef1n

[%] T

exp

M AX TM AXsim

TLCD 1.A 10.22 10.84 6.07 3.41 33.20

TLCD 2.A 11.36 13.08 15.14 3.25 28.17

TLCD 3.A 9.42 10.74 14.01 2.42 33.04

TLCD 4.A 9.20 9.13 0.76 3.64 27.94

TLCD 1.B 10.97 12.24 11.58 3.35 32.07

TLCD 2.B 12.05 14.68 21.83 3.98 27.55

TLCD 3.B 10.97 13.33 21.51 2.67 31.07

TLCD 4.B 10.47 10.70 2.20 4.70 28.39

Table 3.3: Principal frequency domain response parameters for all the TLCD setups.

Recalling expression (2.6.4) and noting the its dependence withη, we can estimate the order of magnitude of the averaged velocity over the frequency domain of interest (which will be between 5 to 50Hz) of the liquid in the tube and, hence, the Reynolds number. The averaged velocity is:

< v >=

R100π

10π v(Ω)dΩ

R100π

10π dΩ

≃0.032m

s (3.3.1)

being v(Ω) = 0._Ω4g. This gives, assuming that the peak transmisibility of the structure + TLCD is order unity, which seems logical as TM AX was 18.31; a

Reynolds number of

Re= < v > Dh

ν ∼

0.032Di

10−6 →O(Re)∼10

3 _(3.3.2)

where D1 = 33.6mm and D2 = 43mm. That value corresponds with a

3.3. TLCD PERFORMANCE 39

Figure 3.5: TLCD 1.B: Normalized transmisibility of the structure, ˆT, against the non-dimensional excitation frequency Ω

ω1.

40 CHAPTER 3. VIBRATION TESTS ANALYSIS

Figure 3.6: TLCD 4.B: Normalized transmisibility of the structure, ˆT, against the non-dimensional excitation frequency Ω

ω1. Oscillations between 0.4 and 0.6 are due controller’s punctual errors.

Chapter 4

Experimental Correlation

4.1

Model adjustment

In order to adjust the model to the experimental results, some simplifications must be undone.

Firstly, we will take equation (2.4.15) in its complete form, that is, taking y0from (2.4.20) and arrange them together. Then, a new proportional damping

coefficient will be formulated following the same steps that in (2.5.2) to obtain1

βlns=

1₋ ωd Ω 2 Ω √

2ω2

d v u u u u u u t v u u u u u t1 +

   

8ηαxˆ0

3πh1₋ ωd

Ω

2i2

   

2

−1 (4.1.1)

which is the full proportional damping linearized coefficient.

The best η approximation possible have been obtained using a Quasi-Newton optimization algorithm applied to the objective functionF(η) =Tˆsim

−Tˆexp

where sim stands for simulation and exp stands for experimental data. The results can be viewed in Appendix B.2. While those results are clearly unsat-isfactory in our aim to adjust the peak transmisibility, they reveal that some

1_{superscriptnsstands forno simplifications.}

42 CHAPTER 4. EXPERIMENTAL CORRELATION

other energy dissipation mechanism have been acting and it has not been taken into account in the formulation. Visual observation reveals that extremely vi-olent water sloshing was occurring during that frequency interval has broken the hypothesis of small oscillations. The expanded polystyrene support could also have contributed to the increasing of damping.

(a) Violent oscillations near the

funda-mental frequency of the structure.

(b) Smooth oscillations far away from the

fundamental frequency of the structure.

Figure 4.1: Comparison between the 2 different sloshing types identified by visual observation in the tests.

4.1. MODEL ADJUSTMENT 43

Figure 4.2: TLCD 1: Normalized transmisibility of the structure, ˆT, against the non-dimensional excitation frequency Ω

Chapter 5

Concluding remarks

Although motion elimination seems the perfect scenario for most situations when a damper is required, it has been proven that it cannot be carried out since all systems, even the considered as conservative, posses some amount of damping, however small it is.

On one hand, a powerful tool as FEM is, has proved to perform brilliantly meeting both speed calculation and precision. Only by using unidirectional elements such as beams a whole mock-up building has been modeled success-fully.

On the other hand, a relatively simple formulation for a complex problem such as liquid sloshing has been used with a moderate degree of success.

We can conclude that, while the initial model represented correctly the fundamental frequency of vibration of the structure, it fails in predicting its behavior near its new first eigenfrequency, where violent oscillations occur over a centered interval of the natural frequency with an amplitude of 30% of it. In spite of the inaccuracy, the model adjusts fairly enough the eigenfrequencies with a mean error of around 11%. Cannot say the same for the transmisibility. The simplifications made to the dissipation function of the water were too

46 CHAPTER 5. CONCLUDING REMARKS

excessive making the model inaccurate when damping got high.

Even when most of the simplifications were undone, the model did not represent the physical scenario. Absurd values of η during the optimization suggest that another energy dissipation mechanism could have been taking place.

Water sloshing has been proved to be an efficient way to absorb energy. That is certainly the main reason why this methods to create supplemental damping of structures have been recently seen an increasing interest in use.

5.1

Future work

Due to the limited period of time and time dedication that this end-of-degree project was designed to, both in the theoretical and experimental part a lot of things have been left behind that could have increased the accuracy as well and utility of this work.

Firstly, since a greater shaker was not available, nonsense values ofµhave been tested while for practical uses, that parameter should be near 1%. It is left for the future to build and test an structure that represents better the reality.

Secondly, a deeper understanding of sloshing, being able to analyze the modal forms of the free surface of the liquid could be a subject of future studies. That knowledge could be applied to enhance the damping model. The use of Computational Fluid Dynamics makes necessary.

Bibliography

[1] Srikant Bhave. Mechanical Vibrations: Theory and Practice. Pearson Ed-ucation, 2010.

[2] Swaroop K. Yalla & Ahsan Kareem. Optimum Absorber parameters for Tuned Liquid Column Dampers. Journal of Structural Engineering, 2000. [3] Jong-Cheng Wu. Experimental Calibration and Head Loss Prediction of

Tuned Liquid Column Damper.Department of Civil Engineering, Tamkang University, Tamsui, Taiwan 251, R.O.C., 2005.

[4] Abramson, Norman, H. The Dynamics of Liquids in Moving Containers.

University of Illinois, 1966.

[5] L. Cremer, M. Heckl, B.A.T. Petersson Structure-Borne Sound. Struc-tural Vibrations and Sound Radiation at Audio Frequencies. Chapter 3.3. Springer, 3rd _{Edition, 2005.}

[6] Wameedh T.M. Al-Tameemi Pressure-loss coefficient of 90o

sharp-angled mitre elbows. Department of Mechanical Engineering, University of Sheffield, Sheffield, UK. 2018.

[7] Thomas J.R. Hughes The Finite Element Method. Linear Static and Dy-namic Finite Element Analysis.Prentice-Hall, 1987.

[8] Daniel S. Stutts Equivalent Viscous Damping. 2009.

Appendix A

Discrete modeling of a 1D beam

In this section we will focus on the derivation of the FEM model used to describe a one directional beam subject to pure flex, which is known as the Euler-Bernoulli Beam.

We will begin by assuming a continuous displacement field w(x, t) along the longitudinal direction of the beam.

Figure A.1: Schema of the 4 DoF FEM beam.

Firstly, we will interpolate that dis-placement field using shape functions. Therefore:

w(x, t) = (w1(x, t) θ1(x, t) w2(x, t) θ2(x, t) )

(A.0.1) that field will be approximated as

w(x, t) = [N]T(x)q(t) =hN1(x) N2(x) N3(x) N4(x)

i               

q1(t)

q2(t)

q3(t)

q4(t)

50 APPENDIX A. DISCRETE MODELING OF A 1D BEAM

Secondly, we will assume a generic displacement field1

    

w(x, t) =a0(t) +a1(t)x+a2(t)x2 +a3(t)x3 =xTa(t)

w,x(x, t) =a1(t) + 2a2(t)x+ 3a3(t)x2 =xT,xa(t)

(A.0.3)

where xT = ([1 x x2 _x3_{) and a = (a}

0 a1 a2 a3)T. Matching (A.0.2)

with (A.0.3) and noting the fact that shape functions on their own nodes are evaluated as 1 and 0 on any other node, we obtain

                  

w(x= 0, t) = q1 =a0

w,x(x= 0, t) = q2 =a1

w(x=L, t) = q3 =a0+a1L+a2L2+a3L3

w,x(x=L, t) = q4 =a1+ 2a2L+ 3a3L2

(A.0.4)

accordingly, the above expressions can be rewritten in a compact way as

q=                q1 q2 q3 q4                =        

1 0 0 0 0 1 0 0 1 L L2 _L3

0 1 2L 3L2

                       a0 a1 a2 a3               

= [C]a(t) _→ a(t) = [C]−1_{q (A.0.5)}

using the previous deductions we can then write equation A.0.3 as

w(x, t) =xTa(t) =

[N]T z }| {

xT[C]−1_q w,x(x, t) =xT,xa(t) = xT,x[C]−1

| {z }

[N]T ,x

q (A.0.6)

This paraphernalia could seem nonsense but, as always, our main target is to obtain the expression of kinetic and potential (strain) energy, in this case, of the described beam.

Thereupon, following the schema on Figure A.1, we can compute its ki-netic energy as

T = 1 2ρA

Z L

0

˙

w2(x, t)dx (A.0.7)

1_{The ”, x” subscript indicates} ∂(·)

51 In order to obtain the strain potential energy, we must remind that, in general

U = 1 2

Z

V

¯¯

σ: ¯¯ε dV = 1 2

Z

V

C_{: ¯¯ε : ¯¯ε dV (A.0.8)}

Where C _{is the fourth order tensor of elastic constants, ¯¯σ and ¯¯ε are}

re-spectively the second order tensors of tensions and strains and the symbol ”:” refers to the double scalar product2 of two tensors. A deeper explanation of those is far beyond the scope of this text.

In particular, for a one directional isotropic beam, the expression simplifies nicely

U = 1 2

Z L

0

EI w2

,xxdV (A.0.9)

Now, we are prepared to rewrite expressions A.0.7 and A.0.8

T = 1 2ρA

Z L

0

˙

qT[N][N]T_q˙dx= 1

2q˙

T

"

ρA

Z L

0

[N][N]T _dx

#

˙

q (A.0.10)

U = 1 2EI

Z L

0

qT[N],xx[N]T,xxqdx=

1 2q T " EI Z L 0

[N],xx[N]T,xxdx

#

q (A.0.11)

It is important to remember that q=q(t) and [N] = [N](x).

At this state it is trivial to obtain the mass and stiffness matrices. We just have to make an observation on equations A.0.10 and A.0.11. Finally, we obtain:

[M] =ρA

Z L

0

[N][N]T dx=ρA

Z L

0

[C]−T_{x x}T_[C]−1_{dx (A.0.12)}

[K] =EI

Z L

0

[N],xx[N]T,xxdx =EI

Z L

0

[C]−T_x

,xxxT,xx[C]−1dx (A.0.13)

Definitely, the mass and stiffness matrices are:

2_{For example: ¯A¯: ¯B¯ = (A}

ijˆeiˆej) : (Bklˆekˆel) =AijBkl(ˆeiˆej) : (ˆekˆel) =AijBklδikδjl =

52 APPENDIX A. DISCRETE MODELING OF A 1D BEAM

[M]beam =

ρAL 420

       

156 22L 54 ₋13L 22L 4L2 _{13L −3L}2

54 13L 156 ₋22L −13L ₋3L2 _{−22L 4L}2

       

(A.0.14)

[K]beam =

EI L3

       

12 6L ₋12 6L 6L 4L2 _{−6L 2L}2

−12 ₋6L 12 ₋6L 6L 2L2 _{−6L 4L}3

       

Appendix B

Experimental Results

B.1

Non-correlated simulations

Figure B.1: TLCD 1: Normalized transmisibility of the structure, ˆT, against the non-dimensional excitation frequency _ωΩ₁.

54 APPENDIX B. EXPERIMENTAL RESULTS

Figure B.2: TLCD 2: Normalized transmisibility of the structure, ˆT, against the non-dimensional excitation frequency _ωΩ₁.

Figure B.3: TLCD 3: Normalized transmisibility of the structure, ˆT, against the non-dimensional excitation frequency Ω

B.1. NON-CORRELATED SIMULATIONS 55

Figure B.4: TLCD 4: Normalized transmisibility of the structure, ˆT, against the non-dimensional excitation frequency _ωΩ₁.

Figure B.5: TLCD 2.B: Normalized transmisibility of the structure, ˆT, against the non-dimensional excitation frequency Ω

56 APPENDIX B. EXPERIMENTAL RESULTS

Figure B.6: TLCD 3.B: Normalized transmisibility of the structure, ˆT, against the non-dimensional excitation frequency _ωΩ₁.

Figure B.7: TLCD 4.B: Normalized transmisibility of the structure, ˆT, against the non-dimensional excitation frequency Ω

B.2. OPTIMIZEDη SIMULATIONS 57

B.2

Optimized

η

simulations

Figure B.8: TLCD 1: Normalized transmisibility of the structure, ˆT, against the non-dimensional excitation frequency Ω

ω1 for the correlated simulation cor-responding to its optimum η.

58 APPENDIX B. EXPERIMENTAL RESULTS

Figure B.10: TLCD 3: Normalized transmisibility of the structure, ˆT, against the non-dimensional excitation frequency _ωΩ₁ for the correlated simulation cor-responding to its optimum η.

B.2. OPTIMIZEDη SIMULATIONS 59

Figure B.12: TLCD 1.B: Normalized transmisibility of the structure, ˆT, against the non-dimensional excitation frequency _ωΩ₁ for the correlated sim-ulation corresponding to its optimum η.

60 APPENDIX B. EXPERIMENTAL RESULTS

Figure B.14: TLCD 3.B: Normalized transmisibility of the structure, ˆT, against the non-dimensional excitation frequency _ωΩ₁ for the correlated sim-ulation corresponding to its optimum η.

Appendix C

TLCD performance comparison

Figure C.1: TLCD 1: Normalized transmisibility of the structure, ˆT, against the non-dimensional excitation frequency _ωΩ₁ for the structure test, simulation of equivalent inert mass at rooftop and TLCD tests.

62 APPENDIX C. TLCD PERFORMANCE COMPARISON

Figure C.2: TLCD 2: Normalized transmisibility of the structure, ˆT, against the non-dimensional excitation frequency _ωΩ₁ for the structure test, simulation of equivalent inert mass at rooftop and TLCD tests.

63

Figure C.4: TLCD 1.B: Normalized transmisibility of the structure, ˆT, against the non-dimensional excitation frequency _ωΩ₁ for the structure test, simulation of equivalent inert mass at rooftop and TLCD tests.

64 APPENDIX C. TLCD PERFORMANCE COMPARISON

Figure C.6: TLCD 3.B: Normalized transmisibility of the structure, ˆT, against the non-dimensional excitation frequency _ωΩ₁ for the structure test, simulation of equivalent inert mass at rooftop and TLCD tests.