structural control

SEISMIC RESPONSE CONTROL OF BUILDING

USING PASSIVE VISCOELASTIC DAMPER AND

ACTIVE TUNED MASS DAMPER

A DESSERTATION

Submitted in partial fulfillment of the requirements for the award of degree

of

MASTER OF TECHNOLOGY

in

EARTHQUAKE ENGINEERING

(With Specialization in Structural Dynamics)

Submitted By

KAMAL KRISHNA BERA

M.Tech (II Year)

Under the guidance of

Prof. D. K. Paul

DEPARTMENT OF EARTHQUAKE ENGINEERING

INDIAN INSTITUTE OF TECHNOLOGY ROORKEE

ROORKEE 247667, INDIA

MARCH, 2014

i

CANDIDATES DECLARATION

I hereby certify that the work which is being presented in this Dissertation, entitled

Seismic Response Control of Building using Passive Viscoelastic damper and Active

Tuned Mass Damper in partial fulfilment of the requirements for the award of the

degree of the Master of Technology in Earthquake Engineering with specialization

in Structural Dynamics submitted to the Department of Earthquake Engineering,

Indian Institute of Technology Roorkee is the authentic record of my own work

carried out under the supervision of Prof. D. K. Paul, Emeritus Fellow, Department

of Earthquake Engineering, Indian Institute of Technology Roorkee, Roorkee, India.

The matter embodied in this seminar report has not been submitted by me for the

award of any other degree or diploma.

Place: IIT Roorkee

Dated: 8th March, 2014 Kamal Krishna Bera

CERTIFICATE

This is to certify that the above statement made by the candidate is correct to the

best of my knowledge.

Prof. D.K. Paul

Emeritus Fellow

ii

ACKNOWLEDGEMENT

I wish to express my deep sense of gratitude to Prof. D.K. Paul, Emeritus Fellow,

Department of Earthquake Engineering, Indian Institute of Technology, Roorkee, for

his timely suggestions and constant encouragement throughout the course of this

work. His effort in thoroughly reading the manuscript and invaluable suggestions

are greatly acknowledged.

Place: IIT Roorkee

Dated: 8th March, 2014 Kamal Krishna Bera

iii

ABSTRACT

This study focuses mainly on the effectiveness of active control system in seismic response

reduction. Here an Active Tuned Mass Damper (ATMD) is considered as the active control

device and its performance in response reduction is compared with that of a Passive Tuned

Mass damper system for a 20 storey building with and without shear wall. Linear Quadratic

Regulator (LQR) algorithms is used to determine the required control force. It is observed

that the ATMD system is able to reduce the peak response to the desired limit, however

the peak value control force requirements and the maximum displacement of auxiliary

mass are quite large for only ATMD controlled structure. Therefore, a combination of

passive Viscoelastic Damper (VED) and Active Tuned Mass damper is proposed. Maxwell

model of Viscoelastic Damper is considered and the state-space formulation for structure

equipped with both the VEDs and the ATMD is developed. Numerical result shows great

reduction in both the control force requirement and auxiliary mass movement. Finally, it is

concluded that a combination of passive VEDs and an ATMD system may be a practically

feasible option to control seismic response of large-scale buildings subjected to strong

ground motion.

iv

CONTENTS

TITLE PAGE NO

CANDIDATE DECLARATION i

ACKNOWLEDGEMENT ii

ABSTRACT iii

CONTENTS iv

LIST OF FIGURES vi

LIST OF TABLES viii

NOTATIONS ix

1. INTODUCTION

1.1. Active Control System 1

1.2. Configuration of Active Control System 2

1.3. From TMD to ATMD 3

1.4. Structure Equipped With Passive Damper and ATMD 4

1.5. Real Life Structures With Active Control System 5

2. LITERATURE REVIEW 9

3. ANALYTICAL MODELLING OF CONTROLLED STRUCTURE

3.1. Equation of Motion for Structure with ATMD System 12

3.2. State Variable representation of Equation of Motion 13

3.3. Equation of Motion for Structure with ATMD and

Maxwell Viscoelastic Damper system 14

v

4. CONTROL ALGORITHMS

4.1. Classical Linear Optimal Control 17

5. ANALYSIS AND RESULTS

5.1. Random Earthquake Groungd Accelerogram 20

5.2. Some Useful Formulation 21

5.3. Response Control Using Active Tuned Mass Damper 24

5.4. Response Control Using Combination of Passive VEDs

and Active Tuned Mass Damper 37

6. CONCLUSIONS

6.1. Conclusions 50

6.2. Future Scope of work 51

REFERENCES 52

vi

LIST OF FIGURES

Fig.

No. Title

Page

No.

1.1 Schematic diagram of an active control system 2

1.2 Uncontrolled, Passive and Active Control Systems 3

1.3 (a) Typical Viscoelastic Damper Configuration

(b) Structure with both VEDs and ATMD system

5

3.1 Structure with an ATMD system 12

3.2 The Maxwell rheological model 14

5.1 Selected Ground Motions 21

5.2 Plan and elevation of the 20 storey building without shear wall 25

5.3 Plan and elevation of the 20 storey building with shear wall 29

5.4 Comparison of top-storey displacement time history of flexible

building for five different input ground excitation

34

5.5 Comparison of top-storey displacement time history rigid


35

5.6 Typical Layout of VEDs in flexible Building and application as X-

bracing

38

5.7 Comparison of control force requirement between only ATMD

and VEDs with ATMD controlled flexible building for four

different input ground excitation

41

5.8 Comparison of auxiliary mass displacement between only ATMD

and VEDs with ATMD controlled flexible building for four

different input ground excitation

42

5.9 Comparison of control force requirement between only ATMD

and VEDs with ATMD controlled rigid building for four different

input ground excitation

43

vii

Fig.

No.

Title

Page

No.

5.10 Comparison of auxiliary mass displacement between only ATMD

and VEDs with ATMD controlled rigid building for four different

input ground excitation

44

5.11 Comparison of (a) Peak value of Control Force and (b) Maximum

displacement of auxiliary mass for building without shear wall

45

5.12 Comparison of (a) Peak value of Control Force and (b) Maximum

displacement of auxiliary mass for building with shear wall

46

5.13 Comparison of Inter Storey Drifts for building without shear wall 47

5.14 Comparison of Inter Storey Drifts for building with shear wall 48

viii

LIST OF TABLES

Table

No. Title

Page

No.

1.1 Summary of Actively Controlled Buildings or Towers 6

5.1 Comparison study on the Maximum displacement at the top

floor of uncontrolled, TMD (with different mass) and ATMD

controlled flexible structure without shear wall

32


floor of uncontrolled, TMD (with different mass) and ATMD

controlled rigid structure with shear wall

33

5.3 Properties and Number of VEDs for both the building 37


floor of uncontrolled, TMD, ATMD and ATMD with Passive VEDs


39


floor of uncontrolled, TMD, ATMD and ATMD with Passive VEDs


40

ix

NOTATIONS

open-loop system matrix

closed-loop system matrix

coefficient matrix for control force vector

damping matrix

location matrix of control force

force in viscoelastic damper

feedback gain matrix

, gain matrices for displacements and velocities

coefficient vector for earthquake excitation

imaginary number = 1

identity matrix

performance index

stiffness matrix

location coefficient matrix of viscoelastic dampers

mass matrix

controllability matrix

, , lumped mass, damping coefficient and stiffness of i th floor

, , lumped mass, damping coefficient and stiffness of damper mass

number of structure stories

null matrix

() Riccati matrix

! state weighting matrix

" control weighting matrix

# the i th system pole

time variable

$ initial time instant

final time instant

x

% transformation matrix

&() control command vector

'() relative displacement vector

( state vector

) Rayleigh damping coefficient in C = M + K

0 Rayleigh damping coefficient in C = M + K

2() relative displacement of VED

3 relaxation time of VED

4 damping ratio of the i th mode

5 the i th mode natural frequency

6 loss factor

2 time interval

% active tuned mass damper

% tuned mass damper

!" Linear quadratic regulator

78 Viscoelastic damper

# inter-storey drift

1

Chapter-1

INTRODUCTION

1.1 ACTIVE CONTROL SYSTEM

In structural/earthquake engineering, one of the constant challenges is to find new and

better means of designing new structures or strengthening existing one so that they,

together with their occupants and contents, can be better protected from the damaging

effects of destructive environmental forces such as earthquake and wind. As a result new

and innovative concepts of structural protection have been advanced and are at various

stage of development. Structural control system can be broadly divided into three groups

namely Seismic Isolation, Passive Energy Dissipation & Active Control System.

A Seismic Isolation System is typically placed at the base of a structure which by means of

its flexibility and energy absorption capacity, partially absorb and partially reflects some of

the earthquake input energy before it is transmitted to the structure. The net effect is a

reduction of demand on the structural system. The basic role of Passive Energy Dissipation

system is to absorb a portion of the input energy, thereby reducing energy dissipation

demand on primary structural members and minimizing possible structural damage.

On the other hand Active Structural Control has a more recent origin. In Active Structural

Control, the response of structure is controlled or modified by means of the action of

control system through some external energy supply. The control force is determined by

some predefined control algorithms with a measured response of structure and/or

excitation. The control force is applied by actuator. Example of active control system

include Active Tendon system, Active Tuned Mass Damper, Pulse systems etc. As

compared to passive system, an active control system has the following advantages, (1)

control effectiveness is enhanced; (2) it covers a wide frequency range, i.e. all significant

modes of structure. Hence effective for both wind and earthquake excitations; (3) an

active control system can sense the ground motion and then adjust its control efforts.

2

1.2 CONFIGURATION OF ACTIVE CONTROL SYSTEM

An active control system mainly consists of the following three components

(1) Sensors: Sensors are equivalent to the sensing organs of human body. These are used

to measure external excitation and/or system responses such as displacement, velocity,

acceleration.

(2) Controller: It is similar to the human brain. It is an information processor that provides

signal to actuator by a feedback function of sensor measurements.

Fig. 1.1 Schematic diagram of an active control system (Soong, 1990)

(3) Actuators: These are equivalent to hand and feet of human body. Actuator produces

the required control force according to the control signal / control command send by the

controller.

Control forces

Controller

Control signal

Power supply Actuators

Structure

Measurements Measurements

Earthquake excitation

Sensors Sensors

Structural response

3

1.3 TUNED MASS DAMPER TO ACTIVE TUNED MASS DAMPER

In the simplest form, tuned mass damper (TMD) consists of an auxiliary mass-spring-

dashpot system usually attached at the top of the structure. When TMD responds to

structural vibration, part of the vibration energy of structure is transferred to TMD system.

Thus relieving the structure from excessive vibration.TMD frequency is generally tuned to

the fundamental frequency of the structure. Other important parameters are damping

ratio and mass ratio (ratio of mass of TMD to that of main structure). But TMDs

effectiveness is limited as these are suitable in response reduction for a particular mode

(e.g. fundamental mode in case of wind-induced vibration), making them less effective for

seismic response control in which response is governed by several modes. Again TMDs are

very sensitive to mistuning. A solution of this problem is to add active control mechanism

with TMD system- leads to the development of Active TMD (ATMD). A conceptual model

of an ATMD controlled building is shown in Fig. 1.2 along with an uncontrolled structure

and structure with TMD system.

a) Uncontrolled b) Passive TMD c) Active TMD

Fig. 1.2 Uncontrolled, Passive and Active Control Systems

() ()

4

In case of ATMD one actuator is placed between the main structure and the TMD mass to

control the motion of this auxiliary mass.

Although effectiveness of ATMD system is mainly felt at fundamental frequency and

comparatively less at higher frequencies, it is well established through various numerical

and experimental studies (with different control algorithms) that ATMD systems are more

effective in reducing structural seismic response as compared to passive TMD.

1.4 STRUCTURE EQUIPPED WITH PASSIVE DAMPER AND ATMD

The real life structures with active control systems are primarily designed to sustain heavy

wind and moderate earthquake induced vibration. However, active control system can

reduce any types of external excitation to the desired degree if there is no constrained in

backup power supply and or actuator capacity. That is practically not possible. In order to

reduce severe earthquake induced vibration of tall buildings a combination of passive

dampers and active control system may be a practically feasible option. Here, Viscoelastic

damper (VED) is considered as passive energy dissipation device. Typically, copolymer or

glassy substances that dissipates energy when subjected to shear deformation are used as

viscoelastic material in structural application. Figure 1.3(a) shows a typical VED which

consists of viscoelastic layers bonded with steel plates. When structural vibration induces

relative motion between the outer flanges and the centre plate, shear deformation and

hence energy dissipation takes place. In the following sections, theories of ATMD

controlled structure and VEDs with ATMD controlled structures are discussed.

5

ATMD

VED

(a) (b)

Fig. 1.3(a) Typical Viscoelastic Damper Configuration; (b) structure with both VEDs and

ATMD system

1.5 REAL LIFE STRUCTURES EQUIPPED WITH ACTIVE CONTROL SYSTEM

During the last two decades active control systems are implemented in a number of tall

buildings, towers and bridges. Although most of the applications are concentrated in Japan

but slowly it is gaining popularity among other countries like U.S.A., China, Taiwan, Korea

etc. The role of the active system is to reduce the structural vibration under strong wind

and moderate earthquake and consequently to increase the comfort of occupants of the

building. Still there are number of serious challenges remain to be resolved before this

technology can gain general acceptance by the engineering and construction professionals

at large. The following table describes the summary of the actively controlled buildings or

towers. A summary of actively controlled structures are given in Table 1.1.

Steel Plate

VE Material

6

Table 1.1: Summary of Actively Controlled Buildings or Towers. [24, 25]

Structure Name

Location

Year Scale of Building

Control

System

AMD/HMD

No Mass (tons)

Mechanism

of Actuation

Kyobashi Seiwa Building Tokyo, Japan 1989 11 stories, 33m, 400 ton AMD 2 5 Hydraulic

Kajima Research Institute KaTRI

No.21 Building

Tokyo, Japan 1990 3 stories, 33m, 400 ton SVSS

(6 nos.)

Hydraulic

Osaka Resort City Osaka, Japan 1992 50 stories, 200m, 56980 ton HMD 2 200 Servomotor

Kansai Int. Airport Control Tower Osaka, Japan 1992 7 stories, 86m, 2570 ton HMD 2 10 Servomotor

Hankyu Chayamachi Building Osaka, Japan 1992 34 stories, 161m, 13943 ton HMD 1 480 Hydraulic

Sendayaga INTES Tokyo, Japan 1992 11 stories, 58m, 3280 ton AMD 2 72 Hydraulic

ORC 200 Bay Tower Osaka, Japan 1992 50 stories, 200m, 56680 ton HMD 2 230 Servomotor

Ando Nishikicho Tokyo, Japan 1993 14 stories, 54m, 2600 ton HMD

(DUOX)

1 22 Servomotor

Long Term Credit Bank Tokyo, Japan 1993 21 stories, 129m, 40000 ton HMD 1 195 Hydraulic

Hamamatsu ACT Tower Hamamatsu

Shizuoka, Japan

1994 212m, 107500 ton HMD 2 180 Servomotor

MHI Yokohama Building Yokohama,

Kanagawa, Japan

1994 34 stories, 152m, 31800 ton HMD

1 60 Servomotor

7

Hotel Nikko Kanazawa

Kanazawa,

Ishikawa, Japan

1994 29 stories, 131m, 27000 ton HMD 2 100 Hydraulic

RIHGS Royal Hotel Hiroshima, Japan 1994 35 stories, 150m, 83000 ton HMD 1 80 Servomotor

Osaka WTC Building Osaka, Japan 1994 55 stories, 255m, 80000 ton HMD 2 100 Servomotor

Riverside Sumida Tokyo, Japan 1994 33 stories, 134m, 52000 ton AMD 2 30 Servomotor

Hikarigaoka J-City Tokyo, Japan 1994 26 stories, 110m, 29300 ton HMD 2 44 Servomotor

Miyazaki Phoenix Hotel Ocean 45 Miyazaki, Japan 1994 43 stories, 154m, 83650 ton HMD 2 240 Servomotor

Dowa Kasai Phoenix Tower Osaka, Japan 1995 28 stories, 145m, 28000 ton HMD

(DUOX)

2 84 Servomotor

Rinku Gate Tower North Building Osaka, Japan 1995 56 stories, 255m, 65000 ton HMD

2 160 Servomotor

Herbis Osaka Osaka, Japan 1997 40 stories, 190m, 62450 ton HMD 2 320 Hydraulic

Itoyama Tower Tokyo, Japan 1997 18 stories, 89m, 9025 ton HMD 1 48 Servomotor

Nisseki Yokohama Building Yokohama, Japan 1997 30 stories,133m, 53000 ton HMD 2 100 Servomotor

TC Tower Kau-Shon, Taiwan 1997 85 stories,348m,221000 ton HMD 2 100 Servomotor

Yokohama Bay Sheraton Hotel

and Towers

Yokohama, Japan 1998 27 stories,

115m, 33000 ton

HMD 2 122 Servomotor

Bunka Gakuen New Building Tokyo, Japan 1998 20 stories, 93m, 43488 ton HMD 2 48 Servomotor

Kaikayo Messe Dream Tower Yamaguchi, Japan 1998 153m, 5400 ton HMD 1 10 Servomotor

Nanjing Tower Nanjing, China 1999 310 m AMD 1 60 Hydraulic

8

Century Park Tower Tokyo, Japan 1999 54stories,170m, 124540 ton HMD 4 440 Servomotor

Shin-Jei Building Taipei, Taiwan 1999 22 stories, 99 m AMD 3 120 Servomotor

ATC Tower, Incheon Int. Airport Incheon, Korea 2000 100 m HMD 2 12 Servomotor

ATC Tower, Osaka Int. Airport Osaka, Japan 2001 5 stories, 69 m, 3600 ton HMD 2 10 Servomotor

Cerulean Tower Hotel Tokyo, Japan 2001 41 stories,188 m, 65000 ton HMD 2 210 Hydraulic

Hotel Nikko Bayside Osaka Osaka, Japan 2002 33 stories,138 m, 37000 ton HMD 2 124 Servomotor

Dentsu New Headquarter, Office

Building

Tokyo, Japan 2002 48 stories,210 m, 130000 tn HMD 2 440 Servomotor

AMD=Active Mass Damper; HMD=Hybrid Mass Damper; SVSS=Semi-active Variable Stiffness System

9

Chapter-2

LITERATURE REVIEW

Till date numerous work has been done on various structural control systems like passive,

active, semi-active and hybrid control systems. Although passive control of structure using

base isolation, tuned mass damper and additional passive energy dissipation devices are

quite extensively studied as well as implemented in real life structures, the concept of

active structural control is relatively new especially for civil engineering structures. A

systematic study on active control research started, when Yao [1972] presented a control-

theory based concept of structural control. In 1989, Professor Kobori and his associates

launched the active control movement with the installation of active mass driver system in

Kyobashi Seiwa Building in Tokyo, Japan. Since then great studies have been made in

advancing the theory and application of active structural control technology over the last

30 years. Excellent state-of-the-art review on active structural control are available in the

papers of Soong [1988], Spencer Jr.and Sain [1997], Soong and Spencer [2000], Datta

[2003], Spencer Jr. and Nagarajaiah [2003], Fisco and Adeli [2011].

Wang and Amini [1983] proposed a simple way to reduce the multi-degree of freedom

structure to an equivalent single degree of freedom system and pole assignment

technique is adopted for determination of control force. An empirical formula is suggested

to determine the desired pole locations systematically so that the peak response remain

within desirable limit.

An experimental verification on active structural control was demonstrated by Chung et al.

[1988]. In laboratory they created a single degree of freedom model structure which was

controlled using pre-stressing tendons. Optimal closed loop feedback control scheme is

applied to reduce the structural responses under base excitation. A good agreement

between analytical and experimental results are observed, though compared to analytical

results, a larger control force is required but less reduction of responses are obtained

experimentally because of less than 100% efficiency.

10

Yang et al. [1987] proposed a new control algorithms called Instantaneous Optimal

Control in which the time dependent quadratic performance index is minimized at every

instant of time over the entire time interval.

Many researcher has studied detail analysis and effectiveness of active tuned mass under

different seismic excitations over the years. They have used different control algorithms to

check their suitability in the application of civil engineering structures.

Loh and Cao [1995] performed a comparative study on effectiveness of response control

through TMD and Active TMD systems in detail with application on both flexible and rigid

structures. As expected, the performance of the ATMD system is found to be quite better

than the passive one in structural response reduction at all floor levels. In addition, the

control force requirement for stiff structure is found to be greater than that of the soft

structure for almost equal percentage of response reduction. A systematic and reliable

way to calculate the weighting matrix Q in the control algorithms is proposed.

Cao et al. [1998] describe the design of an active mass damper system for wind response

reduction in Nanjing TV tower, China. Several practical limitations are encountered during

design and implementation of the system.

Singh et al. [1997] performed a detail comparative study on the effectiveness of active

tendon system and active tuned mass damper system under four different earthquakes.

Several sets of numerical results are obtained for a 10-storey shear building controlled by

active or semi-active control schemes. In this paper the sliding-mode control approach is

used as the control algorithms. Active control performs very effectively to reduce the

structural responses, but the required control force values can be quite large and thus its

application in large and massive buildings may be impractical.

Rasouli and Yahyai [2001] compared the performance of a 25-storey building with Passive

and Active Tuned Mass Damper under El-Centro and Tabas earthquakes. Advantage of the

ATMD system over the other active control system lies in the fact that it can be operated

in passive mode when moderate reduction in response is encountered and in active mode

when higher reduction of response is desired.

11

There are several literature available on passive control systems. Here a few of those,

which are required in the present study, are described briefly.

Rana and Soong [1998] performed a parametric study to understand some important

characteristics of tuned mass damper. They proposed a simplified method to use Den

Hartogs formulation of optimal damper parameters for multi-degree of freedom

structure. Also investigation on multi-tuned mass dampers (MTMD) are made in

controlling multiple structural modes.

Shukla and Datta [1999] proposed a strategy for optimal placement of viscoelastic damper

to control the seismic response of a 20-storey shear-frame building. Three different

mathematical models (Kelvin model, Linear Hysteretic model and Maxwell model) of

viscoelastic damper are considered. It is shown that the optimal placement of viscoelastic

damper provide more response reduction as compared to the other scheme of placement.

Singh et al. [2003] present an optimal design procedure of viscoelastic dampers,

represented by a Maxwell model. In case of Fluid Orifice Dampers, the Maxwell model can

captures the frequency dependence of the damping and stiffness coefficients. Optimal

damper parameters and distribution are obtained through a Gradient-based optimization

scheme. The effectiveness of supplemental damping is evaluated in terms of the reduction

of the various response quantities.

Lewandowski and Chorazyczewski [2007] discussed about the frequently used models

(e.g. simple Kelvin model, simple Maxwell model, Generalized Maxwell model and

Fractional Maxwell model of dampers) to represent viscoelastic dampers. Dynamic

properties of various rheological models and their ability to reflect the dynamic

characteristics of VE dampers are presented. The constants of the models are determined

by using the method of fitting the respective model with experimental data.

More details on the mechanics and working principles of various passive energy dissipation

devices can be found in excellent treatises by Soong and Dargush [1997] and

Constantinou et al. [1998].

12

Chapter-3

ANALYTICAL MODELLING OF CONTROLLED STRUCTURE

3.1 EQUATION OF MOTION FOR STRUCTURE WITH ATMD SYSTEM

Deriving equation of motion of the ATMD-structure system using theory of structural

dynamics is the first step of system modelling. Figure 2.1 shows a storey building

equipped with an ATMD system at the roof level and subjected to ground acceleration

).(tx g&&

Fig. 3.1 Structure with an ATMD system

Let ;........,,,;..,..........,,;...,..........,,;.,..........,, 21212121 nnnn xxxkkkcccmmm are the

lumped masses, coefficients of damping, storey stiffnesses and relative displacements for

storey number one to n respectively. While dddd xkcm ,,, are the corresponding values

of the auxiliary mass system. The equation of motion in condensed matrix-vector form is

written as follows.

)()()()()( txtuttt g&&&&& MIDxKxCxM =++ (3.1)

() ()

)(txg&&

13

where KM , and C respectively represent the mass, stiffness and damping matrices of the

structure-TMD system; I is the ground motion influence coefficient vector; D is the

location vector for control force given as

;}1,1....,..........,0,0,0{}{T=D and

;)}(),(.......,..........),(),({)}({ 21T

dn txtxtxtxt =x

For linear feedback control )(tu can be expressed as

)()()( tttu xGxG vd &+= (3.2)

where vd GG and are the gain matrices for displacements and velocities respectively.

Substituting (3.2) in (3.1) we obtain

)()()()()()( txttt g&&&&& MIxGDKxGDCxM dv =++ (3.3)

Comparing (3.3) with (3.1) in the absence of control, it is clearly seen that the effect of

control (closed-loop) is to modify the structural parameters so that it can respond more

favourably to the external excitation. The selection of vd GG and depends on the control

algorithms adopted.

3.2 STATE VARIABLE REPRESENTATION OF EQUATION OF MOTION

State variables representation is important in the sense that it is central to the

development of modern control theory. The state or set of system variables provide us

with the status of a particular system at any instant of time. Since we are usually

concerned with how system behaviour changes with time, we find that the most useful

state variables are often the rate of change of variables within the system or combination

of these variables and their derivatives.

So for the second order differential equation of motion, if )(t1z and )(t2z are the two

state variables then )()( tt xz 1 = and )()( tt xz 2 &= , as knowing )(tx and )(tx& are sufficient

to know the states of the system at any time. The state variable representation of

equation of motion (3.1) is given by

)()()()( txtutt g&&& HBzAz ++= (3.4)

14

where

=

(t)x

x(t)z

&)(t

=

CMKM

IOA

11;

=

DM

OB

1;

=

I

OH

The size of state vector )(tz is N2 , where 1+= nN . Oand I denote, respectively the null

and identity matrix of appropriate dimensions. This representation converts N second

order differential equations to N2 first order differential equations.

Now for linear feedback control )()( ttu zG= , substituting this in state equation (3.4) we

get

)()()()( txtt g&&& HzGBAz += (3.5)

Again, it is clear that the effect of closed-loop control is one of structural modification

where the system matrix is changed form A to )( GBA .

3.3 EQUATION OF MOTION FOR STRUCTURE WITH ATMD AND MAXWELL VISCO-ELASTIC

DAMPER (VED) SYSTEM

The Simple Maxwell Model of Viscoelastic Damper

Fig.3.2 The Maxwell rheological model

Although Kelvin model, which consists of a linear spring and a damper, connected in

parallel is the simplest model to represent a viscoelastic damper, here simple Maxwell

model, which can represent the behaviour of VED more accurately, is considered. A

Maxwell model consists of a linear spring with constant k and a linear viscous dashpot

with constant dk in series. d is known as the relaxation time. Figure 3.2 shows a typical

)(tf d

)(td

kdk

)(tf d

15

configuration of Maxwell damper. From the free body diagram, the Maxwell model is

given by the following first order differential equation

)()()( tktftf ddddd =+ && or

)(1

)()( tftktf dd

dd = && (3.6)

where )(tf d is the dampers force, )(td is the damper relative displacement, k

cd = is

the relaxation time. If the damper is harmonically excited, i.e. )exp()( 0 titd = , where

1=i and is the excitation frequency. So for steady state vibration the damper force

can be expressed as )exp()( 0 tiftf dd = . Substituting these two expressions in equation

(3.6) we have

020 )(1

+

+=

d

d

dd

ikf (3.7)

Now defining storage modulus2

2

)(1

)(

d

dkK+

= ; loss modulus 2

)(1

d

dkK+

= and loss

factor

dK

K 1=

= . It is quite clear that both storage and loss modulus are dependent

on excitation frequency, which is not the case of simple Kelvin model of VED [in Kelvin

model loss modulus is linearly dependent on the frequency and storage modulus is

independent of the frequency, which is not an accurate representation for rubber and

polymer like materials]. According to Maxwell model the storage modulus increases with

and approximately equal to k for higher values of d . The loss modulus has an

extremum value at d

e

1= and

2)(

kK e = . The loss factor )( always decreases with

which is consistent with some experimental results.

Equation of Motion and State-Variable Representation

The equation of motion of a multi-storey building equipped with both ATMD and VED

system can be described by the following differential equation:

)()()()()( txtuttt gT

&&&&& MIDfLxKxCxM d =+++ (3.8)

16

where df is the vector of forces in the energy dissipation devices and L is the location

coefficient matrix of size Nn e , en being the number of energy dissipation devices. Now

equation (3.6) for a single damper can be assembled and written in a matrix form as

)()()( tttd

ddf

IxLPf

= && (3.9)

where local deformation of the devices are related to that of the main structure by the

following expression

)()( tt xL d = ; P is the diagonal matrix with diagonal terms as ik .

Now equations (3.8) and (3.9) may be written in state space as:

)()()()( txtutt g&&& HBzAz ++= (3.10)

Where

=

)(

)(

)(

)(

t

t

t

t

df

x

x

z &

=

I

PLO

LMCMKM

OIO

AT111 ;

=

O

DM

O

B1

;

=

O

I

O

H

The size of state vector )(tz is enN +2 . Oand I denote, respectively the null and identity

matrix of appropriate dimensions. Again, in a similar way we can obtain the controlled

state equation for linear state feedback control as

)()()( txtt g&&& HzAz +=

(3.11)

Where )( BGAA =

. Now we can solve equation (3.11) to obtain the structural

responses as well as damper forces for the structure equipped with both the ATMD and

VED systems.

17

Chapter-4

CONTROL ALORITHAMS

CONTROL ALGORITHAMS

There are number of algorithms developed for control force determination. These

algorithms derive control force by minimizing some approximate performance index or

based on some stability criteria or some other different consideration. Optimal control

force obtained as a linear function of the state vector and hence called linear optimal

control algorithms. Here Classical Linear Optimal Control (LQR method) is used to obtain

the optimal control force.

4.1 CLASSICAL LINEAR OPTIMAL CONTROL

In classical linear optimal control, the control force is assumed to be a linear function of

the state vector and is obtained by minimizing a quadratic performance function.

Therefore, it is also known as linear quadratic Regulator method. Usually the form of

performance index chosen for structural control purpose is quadratic in )(tz and )(tu

dttttt

ft

T ])()()()([0

uRuzQzJT+=

(4.1)

The time interval ],0[ ft must be longer than the duration of external excitation. The two

matrices Qand R called as the weighting matrices for system states and control force. For

single ATMD, as shown in Fig.3.1, R reduces to a scalar number since there is only one

control force. So performance index J , represents a weighted balance between structural

response and control energy. When elements of Qare large, system response is reduced

at the expense of increased control force and the opposite is true when elements of R are

large as compared to those ofQ .

18

In this algorithm earthquake excitation is neglected as future earthquakes are not known a

priori. Also it makes the control algorithm much simpler. Thus J in equation (4.1) is

minimized subject to the constraining equation

0zzuBzAz =+= )0(),()()( ttt& (4.2)

Define the Hamiltonian as

})()()(){(})()()()({ tttttttt zuBzAuRuzQz TTT &+++= (4.3)

where )(t is the vector of time dependent Lagrangian multipliers.

The necessary conditions of optimality are

0)}({)}({

=

tdt

d

t zz & (4.4)

0)}({)}({

=

tdt

d

t uu & (4.5)

Substituting equation (4.3) in equation (4.4) and (4.5), we get respectively

zQAT 2=& , 0)( =ft (4.6)

BRu

T1=2

1)(t (4.7)

when the control vector is regulated by the state vector (i.e. for closed loop control), we

have

)()()( ttt zP = (4.8)

Substituting (4.8) in equations (4.7), (4.6) and (4.2) and rearranging we obtain

02)()(2

1)()( =+++ QPAPBRBPAPP TT1 tt(t)tt& , 0)( =ftP (4.9)

In optimal control theory, the above equation is referred to as Matrix Riccati Equation

(MRE) and )(tP is the Riccati matrix. For most structural problem, it is observed that P

19

remains constant over the control interval and dropping to zero rapidly near ft . Hence,

)(tP in most of the cases can be approximated by a constant matrix P and MRE reduces to

Algebraic Riccati Equation (ARE) as follows

02

2

1=++ QPAPBRPBAP TT1 (4.10)

Solution of the above equation gives Riccati matrix P . Then the control force can be given

by equation (4.7) as

)()(2

1)( ttt zGzPBRu T1 == (4.11)

where PBRG T1=2

1is called the control feedback gain matrix. Now substituting

equation (4.11) in state equation (3.4), the controlled response can be determined by

solving the following equation

)()()()( txtt g&&& HzGBAz += (4.12)

20

Chapter-5

ANALYSIS AND RESULTS

5.1 RANDOM EARTHQUAKE GROUND MOTION

Total four numbers of random accelerogram named the N-S component of 1940 EL-Centro

earthquake (PGA=0.318g), the 1994 Northridge earthquake (PGA=0.568g), the 1979

Imperial Valley earthquake (PGA=0.454g) and the compatible time history as per spectra of

IS 1893 (Part-I) :2002 for 5% damping at medium soil are taken into consideration for time

history analysis of the 20 storey building considered. These four time histories are shown

in Fig. 5.1(a), 5.1(b), 5.1(c), and 5.1(d) respectively.

Fig. 5.1 (a) El-Centro Earthquake

Fig. 5.1 (b) Northridge Earthquake

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 5 10 15 20 25 30 35

Acc

ele

rati

on

(g

)

Time (sec)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 5 10 15 20 25 30

Acc

ele

rati

on

(g

)

Time (sec)

21

Fig. 5.1 (c) Imperial Valley Earthquake

Fig. 5.1 (d) IS 1893-2002 spectra compatible time history

Fig. 5.1. Selected Ground Motions

5.2 SOME USEFUL FORMULATION

5.2.1 SOLUTION OF DYNAMIC EQUILIBRIUM EQUATION BY NUMERICAL INTEGRATION

The analytical solution of the dynamic equilibrium equation of the structure is not possible

if the applied force or ground acceleration varies arbitrarily with time or the system is

nonlinear. The most general approach to tackle such problem is the direct numerical

integration of the equation of motion. There are many different numerical techniques

which are fundamentally classified as implicit or explicit method are available. Most

methods use equal time intervals at ...........3,2, tNttt usually only single-step, implicit

and unconditionally stable methods are used for the step-by-step seismic analysis of

practical structures. Here Newmarks Beta method has been used for solution of

differential equation of motions.

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 5 10 15 20 25 30 35 40

Acc

ele

rati

on

(g

)

Time (sec)

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 5 10 15 20 25 30 35 40 45 50

Acc

ele

rati

on

(g

)

Time (sec)

22

NEWMARKS BETA METHOD

In 1959, Newmark developed a family of time stepping methods considering the following

two equations:

11 )(])1[( ++ ++= iiii tt XXXX &&&&&&

1

22

1 ])[(])()5.0[()( ++ +++= iiiii ttt XXXXX &&&&&

Typical selection of and :

Average acceleration method 4

1;

2

1==

Linear acceleration method 6

1;

2

1==

ALGORITHM

1. Initial calculations

a) Specify initial conditions 000 ,, XXX&&&

b) Form mass matrix M , stiffness matrix K and damping matrix C

c) Specify integration parameters and

d) Select t

e) Calculate )( 0001

0 XKXCFMX = &&&

f) Calculate modified stiffness

MCKK2)(

1

tt +

+=

g) Calculate the constants

CMCM )12

(2

1;

1+=+

=

t

t

2. Calculations for each time step

a) iiii XXFF&&& ++=

b) iF)KX

= 1(i

23

c) iiii tt

XXXX &&&& )2

1(

+

=

d) iiii

ttXXXX &&&&&

211

)(

12

=

e) iiiiiiiii XXXXXXXXX&&&&&&&&& +=+=+= +++ 111 ,,

3. Repetition for the next time step

Implement steps 2(a) to 2(e) for the next time step by replacing i by 1+i .

5.2.2 STATIC CONDENSATION METHOD

It is a popular method used to reduce or condensed the large size stiffness matrix by

identifying those degrees of freedom to be condensed as dependent or secondary degrees

of freedom and express them in the term of the primary or independent degrees freedom.

This relationship is obtained by establishing the static relation between secondary and

primary degrees of freedom, hence called as static condensation method. If the secondary

degrees of freedom to be reduced are arranged as the first s coordinate and the remaining

primary degrees of freedom as the last p coordinates, then by using partition matrices the

stiffness equation of the structure may be written as

=

ppp

ss

ppps

spss

FX

X

KK

KK 0 (5.1)

Expanding the above equation we get the following to matrix equations

0=+ ppspssss XKXK (5.2)

pppppssps FXKXK =+ (5.3)

Extracting ssX from equation (4.2) and substituting it in equation (4.3) we get

pppspsspspp FXKKKK = )( 1

i.e. ppp FXK =

(5.4)

where

K is the condensed or reduced stiffness matrix given by

spsspspp KKKKK1

= (5.5)

24

5.3 RESPONSE CONTROL USING ACTIVE TUNED MASS DAMPER

Since the primary objective of this work is to control the response of (usually massive) civil

structures subjected to seismic motions, a 20 storey shear building model subjected to

recorded earthquake induced excitation, is considered as an example problem. Two

separate cases are considered (i) 20 storey building without any shear wall (fundamental

period=3.234 sec.) and (ii) same building having shear wall placed symmetrically at four

corners to make it relatively rigid (fundamental period=2.268 sec.). The floor area of both

the building are nearly 900 m2. Figure 5.2 and 5.3 show the typical plan and elevation of

both the buildings along with first two natural frequencies.

Building details are as follows

1. Grade of Concrete used is M30 and grade of steel used is Fe 415.

2. Floor to floor height is 3.5 m.

3. Slab thickness is 150 mm.

4. External wall thickness is 230 mm and no internal walls are provided.

5. Size of columns are 700 mm X 700 mm and size of beams are 300 mm X 500 mm.

6. Live load on floor is 3 kN/m2 and live load on roof is 1.5 kN/m2.

7. Floor finishes is 1 kN/m2 and roof treatment is 1.5 kN/m2.

8. Building frame is special moment resisting frame.

9. Density of concrete is 25 kN/m2 and density of masonry wall is 20 kN/m2.

For shear wall the thickness of wall is taken as 250 mm and placed symmetrically at four

corners as shown in Fig. 5.3.

25

6 bay @ 5 m

A VIEW- A

PLAN ELEVATION

Mode No. Natural frequency (rad/sec)

First 942.1

Second 979.5

Fig.5.2 Plan and elevation of the 20 storey building without shear wall

6 b

ay @

5 m

20

sto

rey @

3.5

m

26

The mass matrix of the above building is obtained by considering the masses are lumped at

each floor level i.e. lumped mass matrix of dimensions 20 by 20. The building is modelled

in SAP 2000 and 2D analysis (3 degrees of freedom per node) is performed. The stiffness

matrix obtained is of dimensions 1980 X 1980. This stiffness matrix is then condensed for

20 primary translation (in the X-direction) degrees of freedom through Static

Condensation. The C -matrix for the bare frame is obtained by Rayleigh damping of the

structure with the help of first two natural frequencies, i.e. KMC += ; where and

are calculated

as

)(

2&

)(

2

2121

21

+

=+

= . (5.6)

1 and 2 are the first two natural frequencies of the structure. For the first two modes

the modal damping ratio is taken as %5 of the critical.

Optimal Damper Parameters

Den Hartogs formula for optimal damper parameters was based on the SDOF undamped

structure under harmonic load. According to Den Hartog the optimum tuning frequency (

structureTMDopt /= ) can be expressed as

+

=1

1opt

(5.7)

The optimum-damping ratio of the damper dopt is formulated as

)1(8

3

+

=dopt (5.8)

is the mass ratio of damper.

27

To use the formula in MDOF structure, first of all it is to be converted to an equivalent

SDOF structure following the procedure of Soong and Dargush by normalizing the mode

shape at the location of TMD to be 1 unit.

For 20-storey building without shear wall, the resulting first mode is:

T

1 =[ 0.029 0.089 0.159 0.233 0.307 0.380 0.451 0.519 0.585 0.647 0.705 0.759

0.808 0.852 0.891 0.924 0.951 0.973 0.989 1.0 ]

The first modal mass: 7.96461T

11 == MM t

Considering TMD mass as 1% of total mass of building i.e. 206.64 t (case-I), the mass ratio

is:

0214.01

==M

md

So 979.01

1=

+=

opt

From which we can obtain

.sec/902.1.,1 radstroptd ==

N/m. 747338.6682 == ddd mk

0887.0)1(8

3=

+=

dopt

s/m.-N 69697.472 == dddd mc

28

The following two tables represent the optimal damper parameters for two cases.

Case-I: when TMD mass is 1% of the total mass of the building.

Mass(ton)

Stiffness(N/m)

Co-efficient of

damping(N-s/m)

9646701.3 0.0214 0.979 0.088 206.63 747338.66 69697.47

Case-II: when TMD mass is 1.5% of the total mass of the building.

Mass(ton)

Stiffness(N/m)

Co-efficient of

damping(N-s/m)

9646701.3 0.0321 0.968 0.108 309.95 1097863.89 126054.62

,

,

29

A 250 thick shear wall (typ) VIEW- A

PLAN ELEVATION

Mode No. Natural frequency (rad/sec)

First 77.2

Second 32.10

Fig.5.3 Plan and elevation of the 20 storey building with shear wall

6 bay @ 5 m

6 b

ay @

5 m

20

sto

rey @

3.5

m

30

The mass, stiffness and damping matrices of the structure are obtained by following the

similar procedure as that of building without shear wall.

Again the optimal damper parameters are obtained through the similar procedure as

discussed as in the case of 20 storey flexible building without shear wall. The following two

tables represent the optimal damper parameters for two cases.

Case-I: when TMD mass is 1% of the total mass of the building.

Mass(ton)

Stiffness(N/m)

Co-efficient of

damping(N-s/m)

7032243.12 0.02954 0.971 0.1037 207.80

1504797.17

116021.3592

Case-II: when TMD mass is 1.5% of the total mass of the building.

Mass(ton)

Stiffness(N/m)

Co-efficient of

damping(N-s/m)

7032243.12 0.04432 0.957 0.126 311.68

2193782.28

208637.83

El-Centro (1940), Northridge (1994) and Imperial Valley (1979) earthquakes and IS

1893:2002 response spectra (for Zone-IV, medium type soil and damping ratio 0.05)

compatible time history (as shown in Fig. 5.1(a) to 5.1(d)) along with one sinusoidal ground

acceleration having frequency equal to the fundamental frequency of the structure are

considered for time history analysis.

,

,

31

In active control case, the Classical Linear Optimal Control algorithm is used to obtain the

required control force. The weighting matrixQ is considered to be diagonal with non-zero

value assigned to first 20 diagonal terms and the remaining diagonal terms are taken as

zero(here the weightage is given only to the displacement response reduction of structural

degrees of freedom). Q matrix is adjusted suitably to obtain the control force required to

get the desired level of top floor displacement response. R in this case is a scalar and is

assigned a value of 1. Now the gain matrix is obtained with the help of Eqn. (4.11) and

solving Eqn. (4.10) in MATLAB. The response quantities are determined either through

solving equation of motion by Newmarks Beta method or through solving state-space

equation.

In the following two table, the top floor peak displacement responses for uncontrolled,

TMD controlled and ATMD controlled structures are compared for different ground

motion for both the flexible and rigid structures. The limiting value of top floor peak

displacement is set to 120 mm (

32

Table 5.1-Comparison study on the Maximum displacement at the top floor of uncontrolled, TMD (with different mass) and ATMD


Type/Name

of Loading

Top Storey

Uncontrolled

Disp.(m)

TMD Controlled Displacement(m) ATMD Controlled

Displacement(m)

Peak Disp. of

Auxiliary

Mass(m)

Peak Control

Force(kN)

Case-I

With TMD

Mass=206

ton

%

reduction

Case-II

With TMD

Mass=310

ton

%

reduction

Limiting

Disp.(m)

%

reduction Case-I Case-II Case-I Case-II

Sinusoidal 0.397 0.252 36.52 0.227 42.82 0.120 69.77 1.93 1.31 384 379

El-Centro

Earthquake

1940

0.411 0.389 5.35 0.383 6.81 0.120 70.80 6.09 4.11 4338 4394

Northridge

Earthquake

1994

0.267 0.257 3.75 0.253 5.24 0.120 55.06 4.58 3.05 8064 8105

IS

1893:2002

Compatible

Time History

0.343 0.337 1.75 0.328 4.37 0.120 65.01 9.7 6.5 6904 7044

Imperial

Valley

Earthquake

1979

0.415 0.402 3.13 0.397 4.34 0.120 71.08 9.9 6.68 7960 7880

33

Table 5.2-Comparison study on the Maximum displacement at the top floor of uncontrolled, TMD (with different mass) and ATMD


Type/Name

of Loading

Top Storey

Uncontrolled

Disp.(m)

TMD Controlled Displacement(m) ATMD Controlled

Displacement(m)

Peak Disp. of

Auxiliary Mass(m)

Peak Control

Force(kN)

Case-I

With TMD

Mass=206

ton

%

reduction

Case-II

With TMD

Mass=310

ton

%

reduction

Limiting

Disp.(m)

%

reduction Case-I Case-II Case-I Case-II

Sinusoidal 0.238 0.126 47.06 0.113 52.52 - - - - - -

El-Centro

Earthquake

1940

0.326 0.08 5.52 0.306 6.14 0.120 63.19 4.03 2.74 4981 4876

Northridge

Earthquake

1994

0.347 0.332 4.32 0.325 6.36 0.120 65.42 5.32 3.9 8870 8930

IS

1893:2002

Compatible

Time History

0.263 0.244 7.22 0.242 7.98 0.120 54.37 5.00 3.34 6693 6632

Imperial

Valley

Earthquake

1979

0.243 0.233 4.12 0.228 6.17 0.120 50.62 7.15 4.87 9062 8976

34

Fig. 5.4-Plot showing comparison of top-storey displacement time history of flexible


-0.50

-0.30

-0.10

0.10

0.30

0.50

0 2 4 6 8 10 12 14 16 18 20

Dis

p.(

m)

Time (sec)

Sinusoidal

Uncontrolled TMD II ATMD

-0.50

-0.30

-0.10

0.10

0.30

0.50

0 5 10 15 20 25 30

Dis

p.(

m)

Time (sec)

El-centro


-0.50

-0.30

-0.10

0.10

0.30

0.50

0 5 10 15 20 25 30Dis

p.(

m)

Time (sec)

Northridge


-0.50

-0.30

-0.10

0.10

0.30

0.50

0 5 10 15 20 25 30Dis

p.(

m)

Time (sec)

IS 1893 compatible


-0.50

-0.30

-0.10

0.10

0.30

0.50

0 5 10 15 20 25 30

Dis

p.(

m)

Time (sec)

Impareial Valley


35

Fig. 5.5-Plot showing comparison of top-storey displacement time history rigid building for

five different input ground excitation

-0.50

-0.30

-0.10

0.10

0.30

0.50

0 2 4 6 8 10 12 14 16 18 20

Dis

p.(

m)

Time (sec)

Sinusoidal

Uncontrolled TMD II

-0.50

-0.30

-0.10

0.10

0.30

0.50

0 5 10 15 20 25 30

Dis

p.(

m)

Time (sec)

El-centro


-0.50

-0.30

-0.10

0.10

0.30

0.50

0 5 10 15 20 25 30

Dis

p.(

m)

Time (sec)

Northridge


-0.50

-0.30

-0.10

0.10

0.30

0.50

0 5 10 15 20 25 30

Dis

p.(

m)

Time (sec)

IS 1893 compatible


-0.50

0.00

0.50

0 5 10 15 20 25 30

Dis

p.(

m)

Time (sec)

Imperial valley


36

Interpretation of results

From Table 5.1 it can be concluded that the top storey peak displacement is quite

large irrespective of the type of loading. Therefore, some response control strategy

must be adopted.

It is clear that only passive tuned mass damper is not able to control the peak

response to the desirable limit even after increasing its mass. Although for

sinusoidal loading, TMD can reduce the peak response substantially. So in case of

seismic loading passive TMD is not a good option to control the building response

to desirable value.

For both the cases and all types of loading active tuned mass damper can

effectively reduce the peak response. However, the peak value of control force

requirements are relatively large especially for Northridge and Imperial Valley

earthquake. In addition, the maximum movement of the auxiliary mass is quite

large which is a concern of practical difficulties. The movement of the auxiliary

mass can be reduced by increasing its mass.

From Table 5.2 it can be concluded that even after making the building relatively

stiffer by adding a shear wall of 250 mm thick at four corner of the building

symmetrically, the response control performance is not improved. Moreover, in

ATMD cases the control force requirement is increased.

Theoretically, active control system can reduce the response to any desirable limit

provided there is no difficulties or constrained in providing required control force

through the actuator.

37

5.4 RESPONSE CONTROL USING COMBINATION OF PASSIVE VEDs AND ACTIVE TUNED

MASS DAMPER

In order to overcome the above-mentioned difficulties, a combination of passive

viscoelastic dampers and active tuned mass damper is proposed for the same 20 storey

flexible as well as rigid buildings. The Maxwell model of VED and the state-space

formulation of the structure equipped with both the ATMD and VEDs are already

elaborated in Section 2.3. The response of the structure and forces in the dampers are

obtained by solving equation (3.11) in MATLAB. Again, the Classical Linear Optimal Control

algorithm is used and the weighting matrices (Q andR ) are judiciously adjusted to

determine the control force required to achieve the desired degree of response reduction.

The number and properties of the VEDs are given in the following table

Table-5.3: Properties and Number of VEDs for both the building

Properties and Numbers Building Without

Shear Wall

Building With Shear

Wall

k N/m100.18 N/m100.1 8

dc sec/m-N1027 sec/m-N102 7

d (Relaxation Time) sec20.0 sec20.0

Number of Damper per floor 24 16

Total Number of VED 480 320

38

VED (typ.)

(a) VED (b)

(c)

Fig.5.6-Typical Layout of VEDs in (a) flexible building, (b) rigid building and (c) application

as X-bracing

It should be noted that the number of damper considered here are required for only one

orthogonal direction analysis purpose. So consequently, the total number no VEDs in each

floor and hence in the building will be twice that of the value given in Table 5.3 i.e. 640 and

960 numbers of dampers for building with and without shear wall respectively.

For ATMD the auxiliary mass chosen is that of case-II i.e. having weight of 310 tonnes. In

the following table, comparison is made on peak displacement of top floor as well peak

value of control force requirements and maximum movement of auxiliary mass for

different control strategies. Again, the peak value of top floor displacement is limited to

120 mm irrespective of the external excitation.

39

Table 5.4: Comparison study on the Maximum displacement at the top floor of uncontrolled, TMD, ATMD and ATMD with Passive

VEDs controlled flexible structure without shear wall

Type/Name of

Loading

Top Storey

Uncontrolled

Disp.(m)

ATMD CONTROLLED STRUCTURE ATMD and PASSIVE VEDs(480 no) CONTROLLED STRUCTURE

Top Storey

Limiting

Peak disp.

(m)

Peak

Disp. Of

Auxiliary

Mass(m)

Peak

Control

Force(kN)

Top

Storey

Limiting

Peak disp.

(m)

Peak Disp.

of

Auxiliary

Mass(m)

%

reduction

w.r.t. only

ATMD

case

Peak

Control

Force(kN)

%

reduction

w.r.t.

only

ATMD

case

Max.

Force in

VED(kN)

El-Centro

Earthquake

1940

0.411 0.120 4.11 4394 0.120 1.20 70.80 998 77.29 434

Northridge

Earthquake

1994

0.267 0.120 3.05 8105 0.120 1.36 55.41 3279 59.94 790

IS 1893:2002

Compatible

Time History

0.343 0.120 6.50 7044 0.120 2.81 57.74 3600 48.89 430

Imperial Valley

Earthquake

1979

0.415 0.120 6.68 7880 0.120 3.09 53.74 2930 62.82 385

40

Table 5.5-Comparison study on the Maximum displacement at the top floor of uncontrolled, TMD, ATMD and ATMD with Passive

VEDs controlled rigid structure with shear wall

Type/Name of

Loading

Top Storey

Uncontrolled

Disp.(m)

ATMD CONTROLLED STRUCTURE ATMD and PASSIVE VEDs(320 no) CONTROLLED STRUCTURE

Top Storey

Limiting

Peak disp.

(m)

Peak

Disp. Of

Auxiliary

Mass(m)

Peak

Control

Force(kN)

Top

Storey

Limiting

Peak disp.

(m)

Peak Disp.

of

Auxiliary

Mass(m)

%

reduction

w.r.t. only

ATMD

case

Peak

Control

Force(kN)

%

reduction

w.r.t.

only

ATMD

case

Max.

Force in

VED(kN)

El-Centro

Earthquake

1940

0.326 0.120 2.74 4876 0.120 1.40 48.91 2368 51.44 451

Northridge

Earthquake

1994

0.347 0.120 3.90 8930 0.120 2.12 45.64 3925 56.05 629

IS 1893:2002

Compatible

Time History

0.263 0.120 3.34 6632 0.120 1.44 56.89 2453 63.01 392

Imperial Valley

Earthquake

1979

0.243 0.120 4.87 8976 0.120 1.70 65.09 2360 73.71 347

41

-9000

-6000

-3000

0

3000

6000

9000

0 5 10 15 20 25 30

Co

ntr

ol

Fo

rce

(kN

)

Time (sec)

Northridge

ATMD VED & ATMD

Fig. 5.7-Plot showing comparison of control force requirement between only ATMD and VEDs

with ATMD controlled flexible building for four different input ground excitation

-6000

-4000

-2000

0

2000

4000

6000

0 5 10 15 20 25 30

Co

ntr

ol

Fo

rce

(kN

)

Time (sec)

El-centro

ATMD VED & ATMD

-9000

-6000

-3000

0

3000

6000

9000

0 5 10 15 20 25 30

Co

ntr

ol

Fo

rce

(kN

)

Time (sec)

IS 1893 compatible

ATMD VED & ATMD

-9000

-6000

-3000

0

3000

6000

9000

0 5 10 15 20 25 30

Co

ntr

ol

Fo

rce

(kN

)

Time (sec)

Imperial Valley

ATMD VED & ATMD

42

Fig. 5.8-Plot showing comparison of auxiliary mass displacement between only ATMD and

VEDs with ATMD controlled flexible building for four different input ground excitation

-6.00

-4.00

-2.00

0.00

2.00

4.00

6.00

0 5 10 15 20 25 30Dis

p.(

m)

Time (sec)

El-centro

VED & ATMD ATMD

-4.00

-2.00

0.00

2.00

4.00

0 5 10 15 20 25 30

Dis

p.(

m)

Time (sec)

Northridge

VED & ATMD ATMD

-9.00

-6.00

-3.00

0.00

3.00

6.00

9.00

0 5 10 15 20 25 30

Dis

p.(

m)

Time (sec)

IS 1893 compatible

VED & ATMD ATMD

-10.00

-5.00

0.00

5.00

10.00

0 5 10 15 20 25 30Dis

p.(

m)

Time (sec)

Imperial Valley

VED & ATMD ATMD

43

Fig. 5.9-Plot showing comparison of control force requirement between only ATMD and VEDs

with ATMD controlled rigid building for four different input ground excitation

-7500

-2500

2500

7500

0 5 10 15 20 25 30

Co

ntr

ol

Fo

rce

(kN

)

Time (sec)

El-centro

ATMD VED & ATMD

-10000

-5000

0

5000

10000

0 5 10 15 20 25 30

Co

ntr

ol

Fo

rce

(kN

)

Time (sec)

ATMD VED & ATMD

-8000

-4000

0

4000

8000

0 5 10 15 20 25 30

Co

ntr

ol

Fo

rce

(kN

)

Time (sec)

IS 1893 compatible

ATMD VED & ATMD

-9000

-6000

-3000

0

3000

6000

9000

0 5 10 15 20 25 30

Co

ntr

ol

Fo

rce

(kN

)

Time (sec)

Imperial Valley

ATMD VED & ATMD

44

Fig. 5.10-Plot showing comparison of auxiliary mass displacement between only ATMD and

VEDs with ATMD controlled rigid building for four different input ground excitation

-4.00

-2.00

0.00

2.00

4.00

0 5 10 15 20 25 30

Dis

p.(

m)

Time (sec)

El-centro

VED & ATMD ATMD

-6.00

-4.00

-2.00

0.00

2.00

4.00

0 5 10 15 20 25 30

Dis

p.(

m)

Time (sec)

VED & ATMD ATMD

-4.00

-2.00

0.00

2.00

4.00

0 5 10 15 20 25 30Dis

p.(

m)

Time (sec)

IS 1893 compatible

VED & ATMD ATMD

-4.00

-2.00

0.00

2.00

4.00

6.00

0 5 10 15 20 25 30Dis

p.(

m)

Time (sec)

Imperial Valley

VED & ATMD ATMD

45

(a)

(b)

Fig. 5.11-Comparison of (a) Peak value of Control Force and (b) Maximum displacement of

auxiliary mass for building without shear wall

4394

8105

7044

7880

998

32793600

2930

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

El-Centro Northridge IS 1893

Compatible

Impareial Valley

Co

ntr

ol

Fo

rce

(k

N)

Comparison of Peak Value of Control Force

Only ATMD Controlled VEDs and ATMD Controlled

4.11

3.05

6.65 6.68

1.21.36

2.813.09

0

1

2

3

4

5

6

7

8


Compatible

Impareial Valley

Dis

pla

cem

en

t (m

)

Comparison of Max. Displacement of Auxiliary Mass


46

(a)

(b)

Fig. 5.12-Comparison of (a) Peak value of Control Force and (b) Maximum displacement of

auxiliary mass for building with shear wall

4876

8930

6632

8976

2368

3925

2453 2360

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000


Compatible

Impareial Valley

Co

ntr

ol

Fo

rce

(k

N)

Comparison of Peak Value of Control Force


2.74

3.90

3.34

4.87

1.40

2.12

1.441.70

0

1

2

3

4

5

6


Compatible

Impareial Valley

Dis

pla

cem

en

t (m

)

Comparison of Max. Displacement of Auxiliary Mass


47

Fig. 5.13-Comparison of Inter Storey Drifts envelope for building without shear wall

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.01 0.02 0.03 0.04

Sto

rey

No

Drift value(m)

El-Centro

UNCONTROLLED ATMD+VED

Only ATMD

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.005 0.01 0.015 0.02 0.025

Sto

rey

No

Drift value(m)

Northridge


Only ATMD

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.01 0.02 0.03

Sto

rey N

o

Drift value(m)

IS 1893 Compatible TH


Only ATMD

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.01 0.02 0.03

Sto

rey N

o

Drift value(m)

Imperial Valley


Only ATMD

48

Fig. 5.14-Comparison of Inter Storey Drifts envelope for building with shear wall

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.005 0.01 0.015 0.02 0.025

Sto

rey

No

Drift value(m)

El-Centro


Only ATMD

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.01 0.02 0.03

Sto

rey

No

Drift value(m)

Northridge


Only ATMD

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.005 0.01 0.015 0.02

Sto

rey N

o

Drift value(m)

IS 1893 Compatible TH


Only ATMD

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.005 0.01 0.015

Sto

rey N

o

Drift value(m)

Imperial Valley


Only ATMD

49

Interpretation of results

It is quite clear from the above results that the introduction of passive VEDs not

only reduces the control force requirement of actuator but also reduces the

maximum displacement of the auxiliary mass substantially for both the flexible and

rigid building. For all the four types of earthquake loading more than 50% reduction

in maximum control force requirement and displacement of auxiliary mass is

observed. The maximum reduction is observed in case of El-Centro earthquake in

which maximum control force requirement reduced by nearly 78% and auxiliary

mass displacement has been reduced to 1.20 m from 4.11 m (Table 5.4).

Furthermore as shown in Fig.5.13 and 5.14, ATMD and VEDs controlled structure

exhibit better performance in terms of Inter Storey Drifts (ISD) control as compared

to the only ATMD controlled structure especially for flexible building.

Since only passive VEDs can reduce the response significantly, one can argue that

by increasing the number of damper desired level of response reduction may be

achieved. However being a passive system have the limitations of not being able to

adapt to structural changes and to varying uses pattern and loading condition for

which active control strategy is necessary. In addition, application of excessive

numbers of dampers may practically hamper the structure functioning.

Therefore, it can be concluded that a combination of passive VEDs and an ATMD

systems are practically feasible option to control seismic response of large-scale

buildings subjected to strong ground motion.

50

Chapter-6

CONCLUSIONS

5.1 CONCLUSIONS

Active control system has been a popular area of research in recent decades and

significant progress has been made. In this system motion of the structure is controlled or

modified by means of the action of a control system through some external energy supply.

The most commonly used active control device for civil engineering structures is the active

tuned mass damper (ATMD). The ATMD system is a hybrid combination of passive and

active systems. It consists of a tuned mass damper with a control force actuator, which

means it can supply control passively as well as actively. The high efficiency is the major

advantage of ATMD, in which a relatively small mass can be used to reduce structural

response. On the other hand, unlike some other active control devices, ATMD can be

installed in many kinds of structures: buildings, towers and bridges.

An ATMD system effectively reduces the structural response, but the required control

force could be extremely large in the case of massive and large buildings subjected to

severe earthquakes. Moreover, this type of system needs continuous power supply and

digital computer system during earthquake, which may be difficult to provide during

strong earthquakes. As a result of this limitations active control system are not used as

widely in practice as passive one.

Viscoelastic dampers are quite effectively used to reduce structural vibration due to wind

and seismic load. For example, there were 10000 viscoelastic dampers installed in each of

the twin towers of the World Trade Centre in New York. Therefore, a combination of

passive VEDs and ATMD is considered as the control devices. Numerical results shows that

the control force requirement as well as the maximum displacement of auxiliary mass

reduced substantially as compared to that of the only ATMD controlled case. In addition,

ATMD and VEDs controlled structure exhibit better performance in terms of Inter Storey

Drifts (ISD) control. Also during earthquake, if the active system is not functioning due to

51

unavailability of power then the remaining passive viscoelastic dampers can at least

protect the structure from being excessively damaged.

Classical linear feedback control algorithm has been used for structural control problem

over the past three decades. This algorithm is among the most popular feedback control

algorithms mainly due to its simplicity and ease of implementation. However, if suffers a

number of fundamental shortcomings. Thus different new and innovative control

algorithms and optimization techniques (e.g. Neural network, fuzzy logic and genetic

algorithms, sliding mode control, wavelet based approach etc.) are being proposed to

higher control of responses with reasonable control force (Fisco et al. 2011).

Finally the idea of active control itself is not only attractive, but potentially revolutionary,

since it elevates structural concepts from a static and passive level to one of dynamism

and adaptability (Soong, 1990).

5.2 FUTURE SCOPE OF STUDY

1. The passive energy dissipation device considered here is as viscoelastic damper.

There are other devices like friction damper, metallic yield damper, viscous fluid

damper that can also be used along with ATMD for structural response reduction.

2. In current study the base of the building is taken as fixed to the ground. However

the soil-structure interaction effect can also be considered in the analysis of

actively controlled structure.

3. Optimal control force can also be obtained by using different control algorithms

and optimization techniques.

4. A future study can be done on response control with semi-active devices like semi

active hydraulic damper, Electro-rheological damper, Magneto-rheological damper

etc., which requires very less amount of power as compared to the active control

system.

52

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53

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54

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