Optimization of Tuned Mass Dampers: Using Active and Passive Control (Studies in Systems, Decision and Control, 432) 3030983420, 9783030983420

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Studies in Systems, Decision and Control 432

Gebrail Bekdaş Sinan Melih Nigdeli   Editors

Optimization of Tuned Mass Dampers Using Active and Passive Control

Studies in Systems, Decision and Control Volume 432

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output. Indexed by SCOPUS, DBLP, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science.

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Gebrail Bekda¸s · Sinan Melih Nigdeli Editors

Optimization of Tuned Mass Dampers Using Active and Passive Control

Editors Gebrail Bekda¸s Department of Civil Engineering Istanbul University-Cerrahpa¸sa Avcılar, Istanbul, Turkey

Sinan Melih Nigdeli Department of Civil Engineering Istanbul University-Cerrahpa¸sa Avcılar, Istanbul, Turkey

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-030-98342-0 ISBN 978-3-030-98343-7 (eBook) https://doi.org/10.1007/978-3-030-98343-7 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Humans have designed mechanical systems over time. They encountered some problems in the use of these systems, and they designed the systems they created by examining and analyzing the behavior in order to bring these systems better. Thus, despite the safe use of these systems, some problems continued. Additional systems have been designed to eliminate these problems as well. In addition to these, control systems have been developed to eliminate human need or for situations where human intervention is insufficient. Vibration is an undesirable and problematic situation for mechanical systems. Although these vibrations cause the desired movement of the machine and robot elements to not be achieved, in some cases they cause a movement that is not desired and vibrations occurring in the structures are in this type. These situations necessitated the use of control systems in buildings. Especially wind, earthquake, and traffic effects cause unwanted vibrations. Tuned mass dampers (TMDs) are vibration damping systems that can be designed as both active and passive control systems. These systems are added to various systems including structures, minimizing the effect of the vibration mode in which they are adjusted, and also providing more quick damping of vibration. TMDs were added to various structures for various reasons or were included in their initial design. Optimum design is a complex problem, since many different factors and limitations are effective in the tuning of TMDs, and it is still a active field of study today. In addition, various new types of mass dampers have been proposed as development. This book has been created with the aim of showing this development and introducing control systems. In this title, it is possible to find the new and advance developments in design of mass dampers. After an introduction chapter for TMDs, reviews about passive and active structural control systems are presented. Then, a metaheuristic approach for tuning of TMDs is explained. Then, robustness of different techniques is evaluated. Afterwards, Tuned Mass Damper Inerter, Tuned Liquid Dampers, Tuned

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Liquid Column Dampers, and Active Tuned Mass Dampers are presented. Finally, an artificial neural networks prediction model is given for TMDs. Avcılar, Istanbul, Turkey 2022

Gebrail Bekda¸s [email protected] Sinan Melih Nigdeli [email protected]

Contents

Introduction and Overview: Structural Control and Tuned Mass Dampers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gebrail Bekda¸s, Sinan Melih Nigdeli, and Aylin Ece Kayabekir

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Passive Control via Mass Dampers: A Review of State-Of-The-Art Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ayla Ocak, Sinan Melih Nigdeli, and Gebrail Bekda¸s

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Introduction and Review on Active Structural Control . . . . . . . . . . . . . . . . Serdar Ulusoy, Sinan Melih Nigdeli, and Gebrail Bekda¸s Metaheuristics-Based Optimization of TMD Parameters in Time History Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Melda Yücel, Gebrail Bekda¸s, and Sinan Melih Nigdeli Robust Design of Different Tuned Mass Damper Techniques to Mitigate Wind-Induced Vibrations Under Uncertain Conditions . . . . . Javier Fernando Jiménez-Alonso, Jose Manuel Soria, Iván M. Díaz, and Andrés Sáez Optimal Seismic Response Control of Adjacent Buildings Coupled with a Double Mass Tuned Damper Inerter . . . . . . . . . . . . . . . . . . . . . . . . . . Salah Djerouni, Mahdi Abdeddaim, Said Elias, Dario De Domenico, and Rajesh Rupakhety

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Optimization of Tuned Liquid Dampers for Structures with Metaheuristic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Ayla Ocak, Gebrail Bekda¸s, and Sinan Melih Nigdeli Semi-active Tuned Liquid Column Dampers with Variable Natural Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Behnam Mehrkian and Okyay Altay Optimum Tuning of Active Mass Dampers via Metaheuristics . . . . . . . . . 155 Aylin Ece Kayabekir, Gebrail Bekda¸s, and Sinan Melih Nigdeli vii

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Machine Learning-Based Model for Optimum Design of TMDs by Using Artificial Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Melda Yücel, Sinan Melih Nigdeli, and Gebrail Bekda¸s

Introduction and Overview: Structural Control and Tuned Mass Dampers Gebrail Bekda¸s, Sinan Melih Nigdeli, and Aylin Ece Kayabekir

Abstract In this chapter, an introduction is presented for system dynamics and control. The main aim in the control of structures is to reduce vibrations and the vibration is defined then, the main concern of system dynamics and control such as control, automatic control, system, transfer function and control system types are given. Then, a short brief for structural control types such as active, passive, semiactive and hybrid are defined. Lastly, the content of the book “Optimization of Tuned Mass Dampers-Using Active and Passive Control” is presented. Keywords Structural control · System control · Tuned mass damper · Active control · Vibration

1 Introduction Various control systems have been developed to minimize the dynamic effects caused by some undesirable external effects. With the development of technology and computers, control systems have become easier to implement. Control systems are applied in many mechanical systems, including structures. Today, in big cities, the height of the buildings has increased to meet the needs of the people and long bridges have been built for easy transportation. Various factors, especially earthquakes, necessitated control systems in these structures. In addition, it is not enough for these structures to be safe and reliable alone. Structures should be exposed to less vibration under the influence of earthquakes and strong winds.

G. Bekda¸s (B) · S. M. Nigdeli Department of Civil Engineering, Istanbul University-Cerrahpa¸sa, 34320 Avcılar, Istanbul, Turkey e-mail: [email protected] S. M. Nigdeli e-mail: [email protected] A. E. Kayabekir Department of Civil Engineering, Istanbul Geli¸sim University, 34320 Avcılar, Istanbul, Turkey © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. Bekda¸s and S. M. Nigdeli (eds.), Optimization of Tuned Mass Dampers, Studies in Systems, Decision and Control 432, https://doi.org/10.1007/978-3-030-98343-7_1

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In general, although there are two types of control systems, active and passive control systems, there are also semi-active and hybrid systems. Active control systems are applied to very few buildings, while passive control systems are more widely used. Passive control systems are easy to use and maintain but have limited impact. These systems are not only applied to new buildings, but also added to existing structures.

1.1 Vibrations Motion that changes its direction more than once in a given direction is called vibrational motion. Vibration is called harmonic vibration if the movable system re-passes from a specified position at simultaneous intervals. If it passes through the given position at different time intervals, the vibration is non-harmonic vibration. In a dynamic system, the system is stable if the vibrational movements dampen with time. The system is metastable if the vibratory motion continues unabated, or unstable if it increases over time [1].

2 Systems Dynamics and Control Automatic control systems occur outside of human will, in accordance with the laws of nature. Controls that take place without any human involvement are called natural control. Controls created by human thought are called artificial controls. The natural control system generally concerns non-engineering sciences. Adjusting the blood pressure in the body, controlling the amount of sugar in the blood, keeping the body temperature constant in case of changes in the ambient temperature and reflexes are examples of natural control in the human body. Natural controls are also called physiological control systems and are the subject of physiologists. Artificial control systems developed in engineering applications are divided into two groups. In the first group, the human may have undertaken the system control and command as the control element. This group is called manual control system and it is a primitive system. The other system is automatic control systems that are realized without a human will, thanks to technological elements [2].

2.1 Control Operations made to ensure that the data in a system meet the determined sizes or occur within the predicted changes are called control.

Introduction and Overview: Structural Control …

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2.2 Automatic Control It is called the realization of control operations in a system by a system that controls the quantities that need to be controlled, without human intervention. The purpose of the system is determined and during the realization of this purpose, a new order is added to the system that will control and dampen the deviations that will occur due to the disruptive factors. The purposes and benefits of automatic control systems are explained in the following items [2]. (1) (2)

(3)

People have moved away from repetitive monotonous work and used their time in areas where their skills, knowledge and experience are necessary. In some machines used in industry, and especially in mass production, the necessary controls cannot be realized by human will. Immediate notification, corrections in a very short time and at sensitive levels cannot be done by human beings. In some high-powered controls, manpower is insufficient. Similarly, human power cannot affect structures. Automatically controlled tools save significant time, energy and material while manufacturing with high precision. The widespread use of computers in engineering applications has provided great convenience in automatically controlled systems. The features expected from an automatic control system are given below [2].

(1)

(2)

(3)

Stable operation: When the system moves away from the reference value due to disturbance factors, it can make a stable transition to the steady-state value. If the system output reference quantity is parametric, the system can show stability for each parametric value. Sensitivity: If the difference between the physical quantities obtained at the system output and the reference values is zero, the sensitivity is complete. While the disruptors are time-dependent, the amount of error also depends on time. Therefore, it is aimed to keep the error amount within a certain limit. The narrower the error boundary region, the higher the sensitivity of the system. Quick answer: Since the reference values at the system output cannot be obtained due to disturbance factors, the system should be able to switch to a steady state as soon as possible.

2.3 System A set of interrelated and interactive elements to achieve a specific purpose is called a system. A system forms a whole with its elements within its boundaries. Outside of this limit is called the perimeter of the system. The system receives influence from its environment and affects it. In Fig. 1, a general scheme for a system concept that is controlled via sensing output and modifying input is given.

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Fig. 1 A general scheme for system concept

There are systems related to various fields. These are biological, physiological, socioeconomic and engineering systems. Engineering systems are technological systems that consist of human-made elements. In these systems, thanks to technological control, while the production is carried out economically under the desired conditions, the dangers that may arise during the work are prevented.

2.3.1

Transfer Function

The ratio of the Laplace Transform of the output function to the Laplace Transform of the input function is called the transfer function. It is a unitless function, but twenty times of base ten logarithms of it is called Db.

2.4 Control Systems The system that is established to keep the magnitudes at the output of a system at the predicted values despite the effect of the disturbing variables is called the control system. The control system is added to the main system and forms a whole with it, and an automatic controlled system is formed. During the operation of an installed technological system, the set of elements that remain outside the system, only control the system and make the necessary commands, is called the control mechanism. In this way, the predicted sizes related to a working system are protected against disruptive effects. These physical quantities are called control quantities. Measurement, comparison and control operations are performed in the control system. With the measurement, the system values are taken under surveillance. Measuring instruments are used to determine the physical quantities of the system output. Physical quantities can be measured either directly or indirectly. The measured values obtained depend on the precision of the instruments. The amount of error is determined as a result of comparing the measured quantities with the quantities originally thought to be obtained.

Introduction and Overview: Structural Control …

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In order to correct the determined error, there are elements that affect the system input and provide the change. The elements used in the control circuit must work in harmony in order to transmit the effects they receive to the system input. When necessary, auxiliary elements that increase the effect are also used among these elements. Linear Quadratic Regulator (LQR), Sliding Mode (SM) and Proportional Integral Derivative (PID) type controllers are widely used in mechanisms and buildings.

2.4.1

Open-Loop Control System

In an automatically controlled system, if the physical quantities obtained from the system output do not affect the system input, such systems are called open-loop control systems. Examples of such systems are various refrigerators and pressure booster systems used in high-rise buildings.

2.4.2

Closed-Loop Control System

In the closed-loop control system, the quantities obtained at the output of the system are continuously measured, compared with the reference value and the ± error amount is determined. In order to dampen this difference, the necessary effect is applied to the system input. Thus, in this system, the size of the system output influences the input of the system through the intermediate elements. The block diagram of a closed-loop automatic control system is shown in Fig. 2. Closed-loop automatic—float hydraulic circuit and automatically controlled heating system can be given as examples of a closed loop control system. This type also includes active structural control systems.

Fig. 2 Closed loop control system [2, 3]

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3 Structural Control Systems 3.1 Active Control Systems In active control systems, the activator controlled by the external power source applies force to the building. This force can be used to add or dissipate energy to the building. In active feedback control systems, the function of the response of the system measured by physical sensors is sent to the control actuator as a signal. These systems can control responses from internal and external influence. Safety and comfort are the most important elements in these systems. The most important active control systems are active tuned mass damper (ATMD) and active tendon system. In addition, buildings can be actively controlled from the ground. There are active support systems as an alternative to active tendon systems.

3.1.1

Active Tuned Mass Dampers

Active Tuned Mass Damper (ATMD) consists of an activator mass damper. During the physical modeling, it is analyzed as if it consists of a damper and a spring element as seen in Fig. 3. Although it has excellent vibration damping capacity, it requires a lot of operational power in high-rise buildings. Due to its high cost and expense, it can only be used in important high-rise buildings today. There are also active mass dampers in the form of levers [4]. In Table 1, the practical applications of ATMDs are shown for structures. Fig. 3 Active tuned mass damper on a structure model [3]

Introduction and Overview: Structural Control …

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Table 1 The practical applications of ATMDs [5] Structure

Type

High

# of stories

Place

ORC 2000 Symbol tower

Building

188 m

50

Osaka, Japan 1992

2 ATMD Frequency = 0.21 Hz Mass = 200 ton

Rokko Island Procter and Gamble

Building

117 m

36

Kobe, Japan

ATMD Frequency = 0.33–0.62 Hz Mass = 270 ton

Kansai International airport

Tower

86 m

7

Osaka, Japan 1993

2 ATMD Frequency = 0.8 Hz Mass = 10 ton

C Office Tower

Building

130 m

32

Tokyo, Japan 1993

ATMD Frequency = 0.34 Hz Mass = 200 ton

KS Project

Tower

121 m

–

Kanasawa, Japan

ATMD Mass = 100 ton

Ando Nishikicho

Building

68 m

14

Tokyo, Japan 1993

TMD and ATMD Frequency = 0.68–0.72 Hz

MKD8 Hikarigaoka

Building

100 m

30

Tokyo, Japan 1993

ATMD pendulum Frequency = 0.44 Hz

Shinjuku Park Tower

Building

227 m

33

Tokyo, Japan 1994

3 ATMD Mass = 330 ton

Triton Square office complex

Building

195 m

–

Tokyo, Japan 2001

4 ATMD Mass = 4 × 35 ton

Incheon International Airport Control Tower

Tower

100.4 m

22

Incheon, Kore

2001

2 ATMD and TMD Frequency = 0.71 Hz Mass = 11 and 13 ton Placement = 19th story

Air Traffic Control Tower

Tower

57 m

–

Edinburgh, Scotland

2005

ATMD Frequency = 1.7–2.0 Hz Mass = 14 ton

492 m

101

Shanghai, China

2008

ATMD Placement = 90th story

Shanghai World Building Financial Center

Year

1993

1993

ATMD properties

(continued)

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Table 1 (continued) Structure

Type

Canton Tower Tower (Guangzhou TV Tower)

3.1.2

High

# of stories

Place

Year

ATMD properties

600 m

85

Guangzhou, China

2010

ATMD and TMD Mass = 50 and 600 tons Placement = 85th story

Active Tendons

Active tendon control consists of four prestressed cables, two activators and control elements (Fig. 4). Two of these cables are on one side of the building, connected diagonally to the activator, while the other two are on the other side. With the displacement given by the activator, the tension in the cables changes and the control of the floors is ensured by the effect of these stresses on the floors. If it is desired to control the building in both horizontal directions, eight cables and four activators are needed per floor.

3.2 Passive Control Systems Passive control systems do not require an external power source. These systems take the building under control with mechanical forces. Passive control depends on the design of the building and the viscoelastic materials added to the building. The final need is perfect optimization for efficiency. Bending steel dampers (Fig. 5) absorb energy as a result of inelastic deformation. Worn-out tools after deformation can be replaced later. These elements provide a ductile design for the structure. Base isolation systems are also widely used today. These systems allow the structure to be separated from the foundation and thus the movement of the ground is Fig. 4 Active tendon control on a frame [3]

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Fig. 5 Section of bending steel dampers [3]

allowed. In that case, the critical period of the structure increases, and structures are less affected by earthquakes. In this case, as a result of the dynamic effect, the acceleration coming to the structure decreases. Due to this feature, these systems are helpful to protect sensitive equipment and valuable objects. As passive control systems, mass damper systems such as tuned mass dampers (TMD), tuned liquid damper (TLD), multiple tuned mass dampers (MTMD) and tuned mass-damper-inerter (TMDI) are effective ones in structures in cases of winds, earthquakes and traffic.

3.2.1

Passive Tuned Mass Damper (TMD)

Passive tuned mass dampers generally consist of mass, spring-like stiffness elements and viscous dampers. These can be in a form of a pendulum or series of isolation systems. Viscoelastic dampers act on the principle of converting kinetic energy into heat energy. They aim to dampen the energy provided by the sliding movement of the visco-elastic material used in these devices with the help of steel plates that act like a piston, by converting it to damping energy. It has been applied in multistory buildings since the 1980s, and studies on applications related to seismic loads started in the 1990s and generally showed an effective performance in damping the wind-induced load. TMD has been installed in many buildings since 1971. Some of these are Citicorp Center (New York City), John Hancock Tower (Boston) and Fukuoka Tower (Japan). An example of a pendulum-type mass damper is the mass damper in Taipei 101, the tallest structure in the world. In addition, this damper is the largest and heaviest damper in the world. Weighing 730 tons, this mass damper spans five floors. In Table 2, the practical applications of TMDs are shown for structures. As examples, there are two types of TMD. TMD-Horizontal is used to dampen horizontal vibration movement (wind and earthquake motions) and TMD-Vertical is used to dampen vertical vibration (traffic in bridges). For a classical TMD, the elements are listed below [6].

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Table 2 The practical applications of TMDs [5] Structure

Type

High

# of stories

Place

Year

TMD properties

Citycorp Center

Building

278 m

59

New York, USA

1978

TMD Frequency = 0.16 Hz Mass = 370 ton

Al Khobar

Chimney

120 m

–

Al Khobar, Suudi Arabia

1982

TMD Frequency = 0.44 Hz Mass = 7 ton

Kyobashi Center

Building

33 m

11

Tokyo, Japan

1989

2 TMD Mass = 5 ton

Sendagaya INTES

Building

58 m

11

Tokyo, Japan

1992

2 TMD Mass = 72 ton

Chifley Tower

Building

209 m

53

Sydney, Australia

1993

1 TMD Mass = 400 ton

Al Taweelah

Chimney

70 m

–

Abu Dabi, UAE

1993

1 TMD Frequency = 1.4 Hz Mass = 1.35 ton

Hotel Burj-Al-Arab (7-star)

Building

321 m

60

Dubai, UAE

1997

11 TMDs Frequency = 0.8–2 Hz Mass = 11 × 5.00 ton

Stakis Metropole

Building

60 m

20

London, UK

2000

7 TMD Frequency = 4.4 Hz Mass = 4.5 ton

Regensburg Siemens Building

Building

–

–

Regensburg, Germany

1996

11 TMD Mass = 11 × 0.17 ton

Hangzhou Bay Tower Bridge Tower

130 m

–

Jiaxing, China 2009

1 TMD Frequency = 0.3 Hz

Lanxess, Chemical Plant

Building

–

2

Ontario, Canada

2009

4 TMDs Mass = 4 × 3 ton

Estela de la Luz

Tower

104 m

–

Mexico City, Mexico

2010

8 TMD Frequency = 0.3 Hz Mass = 3 ton (continued)

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Table 2 (continued) Structure

Type

High

# of stories

Place

Year

TMD properties

Tokyo Skytree

Tower

634.0 m

–

Tokyo, Japan

2012

1 TMD Mass = 100 ton

(1) (2) (3) (4) (5)

Adjustable damping element Mass guide system Adjustable steel mass Vertical or horizontal running steel springs Adjustable floor plate

Passive tuned mass dampers should be placed where there is the most vibration in buildings. This place is generally top of the building, where the first mode shape amplitude is the maximum. The efficiency of TMD is determined by three ratios. These ratios are mass (μ), frequency ratio (f) and damping ratio of TMD (ξd ). After the invention of Frahm [7] as a dynamic vibration absorber, the idea of using mass dampers for civil structures was proposed by Den Hartog and Ormondroyd [8] by adding damping to gain efficiency on random vibrations. Afterward, several tuning formulations [9–12] were proposed as seen in Table 3. These are basic expressions that have limited behavior consideration of structures, and more advanced methods have been proposed including machine learning techniques [13]. Table 3 The frequency and damping ratio expressions for TMD Method

f

Den Hartog [9]

1 1+μ √ 1−(μ/2) 1+μ

Warburton [10] Sadek et al. [11]

Leung and Zhang [12]

   μ 1 1+μ 1 − ξ 1+μ √ 1 − (μ/2) 1+μ √ √ +(−4.9453 + 20.2319 μ − 37.9419μ) μξ √ √ + (−4.8287 + 25.0000 μ) μξ 2

ξd  

3μ 8(1+μ) μ(1−μ/4) 4(1+μ)(1−μ/2)

ξ 1+μ



+



μ 1+μ

μ(1 − μ/4) 4(1 + μ)(1 − μ/2)

−5.3024ξ 2 μ

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3.3 Hybrid Control Systems As the name defines, these are systems that have both active and passive damping features. The only difference with active mass dampers is the amount of external energy required. In addition, when the power supply of the building is cut off, the mixed control system continues to function with the help of its mechanical elements. Generally, there are versions of hybrid mass dampers (HMDs) that use an AC servomotor and a linear-induction servomotor. There are two types, A and B in shape and the only difference between A and B is that the auxiliary mass is placed in different directions. Which model to use is determined by the shape of the building [14].

3.4 Semi-active Control Systems Semi-active control systems can be considered as a class of active control systems that require less energy than active control systems. These systems can be operated with batteries and are not affected by power cuts in the building. Magnetorheological dampers are the general type of this control type, it is a controllable device using magnetorheological liquid. By a magnetic field, the damping characteristics are controlled by changing the power of the electromagnet, and the fluid viscosity of the damper increases when electromagnet intensity increases.

4 Content of the Book “Optimization of Tuned Mass Dampers-Using Active and Passive Control After this introduction chapter, two state of art studies are presented as chapters “Passive Control via Mass Dampers: A Review of State-of-the-Art Developments” and “Introduction and Review on Active Structural Control” for passive and active structural control techniques, respectively. Then, a metaheuristic-based optimization methodology is presented for passive TMDs as chapter “Metaheuristics-Based Optimization of TMD Parameters in Time History Domain”. Chapter “Robust Design of Different Tuned Mass Damper Techniques to Mitigate Wind-Induced Vibrations Under Uncertain Conditions” includes a robust design for different TMD techniques used for wind-induced vibrations. Chapter “Optimal Seismic Response Control of Adjacent Buildings Coupled with a Double Mass Tuned Damper Inerter” is related to multiple TDMI. Also, metaheuristic-based optimization of TLDs is presented as chapter “Optimization of Tuned Liquid Dampers for Structures with Metaheuristic Algorithms”. A semi-active application of tuned liquid column dampers is presented in chapter “Semi-active Tuned Liquid Column Dampers with Variable

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Natural Frequency”. In chapter “Optimum Tuning of Active Mass Dampers via Metaheuristics”, ATMDs optimized via metaheuristics are presented. As the last, chapter “Machine Learning-Based Model for Optimum Design of TMDs by Using Artificial Neural Networks” presents an artificial neural network model for the prediction of TMD parameters.

References 1. Yalçın, K.: Mekanik Titre¸simler. Lecture Notes. Istanbul University, Department of Mechanical Enginnering (2002) 2. Yalçın, K.: Sistem Dinami˘gi (Otomatik-Kontrol). Lecture Notes. Istanbul University, Department of Mechanical Enginnering (2002) 3. Nigdeli, S.M.: Yapıların Aktif Tendonlar ˙Ile Kontrolü. MSc Thesis, Istanbul Technical University (2007) 4. Li, C.: Evaluation of the lever-type active tuned mass damper for structures. Struct. Control. Health Monit. 11, 259–271 (2004) 5. Soto, M.G., Adeli, H.: Tuned mass dampers. Arch. Comput. Methods Eng. 20(4), 419–431 (2013) 6. Maurer Tuned Mass Dampers—Technical Information and Products, Maurer Söhne 7. Frahm, H.: Device for Damping Vibration of Bodies. US Patent 989958 (1909) 8. Den Hartog, J.P., Ormondroyd, J.: Theory of the dynamic vibration absorber. ASME J. Appl. Mech. 50(7), 11–22 (1928) 9. Den Hartog, J.P.: Mechanical Vibrations, 3rd edn. McGraw-Hill, New York (1947) 10. Warburton, G.B.: Optimum absorber parameters for various combinations of response and excitation parameters. Earthq. Eng. Struct. D 10, 381–401 (1982) 11. Sadek, F., Mohraz, B., Taylor, A.W., Chung, R.M.: A method of estimating the parameters of tuned mass dampers for seismic applications. Earthq. Eng. Struct. D 26, 617–635 (1997) 12. Leung, A.Y.T., Zhang, H.: Particle swarm optimization of tuned mass dampers. Eng. Struct. 31(3), 715–728 (2009) 13. Yucel, M., Bekda¸s, G., Nigdeli, S.M., Sevgen, S.: Estimation of optimum tuned mass damper parameters via machine learning. J. Build. Eng. 26, 100847 (2019) 14. Saito, T., Shiba, K., Tamura, K.: Vibration control characteristics of a hybrid mass damper system installed in tall buildings. Earthq. Eng. Struct. Dyn. 30(11), 1677–1696 (2001)

Passive Control via Mass Dampers: A Review of State-Of-The-Art Developments Ayla Ocak, Sinan Melih Nigdeli, and Gebrail Bekda¸s

Abstract Passive control systems are practical systems that use the system’s energy to absorb its energy to control the dynamic effects on the structure. In the control of these systems, which store energy with the help of a spring and mass, the effect of mass and spring is great. Based on this logic, different types of passive control systems are derived according to the type of material used. Tuned mass dampers (TMD) and tuned liquid dampers (TLD) in the passive control group are often used to solve various engineering problems. In these two systems, which have the same properties, a solid mass is usually chosen for TMD, while this mass is liquid for TLDs. In this study, passive control systems, which have important effects on building control, are explained in general terms, the historical development of TMDs and TLDs from the early times when the concept of structural control emerged until today, and the studies that have been done are included. Keywords Structural control · Passive control · Tuned mass damper · Tuned liquid damper · Tuned mass-damper-inerter

1 Introduction Structural control is the process of keeping the movement of the building at the desired level by minimizing the effect of any dynamic load that a building will be exposed to by external factors, with the help of elements added to the structure during the construction phase or added later. English geologist and mining engineer John Milne, who served as a foreign consultant to the Japanese empire and invented the first horizontal pendulum seismograph by working on earthquakes that took place here, carried out the first structural control operation using the elements added to the A. Ocak · S. M. Nigdeli (B) · G. Bekda¸s Department of Civil Engineering, Istanbul University - Cerrahpa¸sa, 34320 Avcılar, Istanbul, Turkey e-mail: [email protected] G. Bekda¸s e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. Bekda¸s and S. M. Nigdeli (eds.), Optimization of Tuned Mass Dampers, Studies in Systems, Decision and Control 432, https://doi.org/10.1007/978-3-030-98343-7_2

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structure towards the end of the nineteenth century. Milne used metal balls to dampen the movement of a house made of wood, which would be caused by the earthquakes she frequently encountered in Japan and placed the house on these balls [1, 2]. The first theoretical infrastructure related to this, and similar intelligent control elements was developed in 1972 by J.P. Yao created [2, 3]. In an article he wrote, Yao explained that the theories created for control, which are used in engineering fields, are also applicable for structures, and he included the concept of “Structural Control” in his article. Thus, with the use of the concept of “Structural Control”, the term “intelligent structures”, which has been adopted by everyone, has left itself to structural control [2].

2 Passive Control Systems They are control systems in which the system’s energy is used for damping without the need for any external energy. Passive control systems, based on the law of conservation of energy, collect the energy of external forces acting on the structure and then use it for damping. In damping devices, they accumulate energy with the help of a mass or spring. From this point of view, the fact that they are not dependent on foreign sources shows these systems as self-sufficient systems. By absorbing the initial energies of the forces acting on the structure, they reduce the energy dissipation desire [4]. The mechanical properties determine the damping rate of the structure, and an increase in this rate reduces the vibration amplitude in the structure and minimizes the damage to occur [5]. When we examine Passive Control systems, it is seen that they do not need any energy from external sources, they are more economical than other control systems, and they provide the highest level of isolation to structures under dynamic loads as a result of their design, making them more advantageous than other systems. Passive control systems are also known as passive energy absorber and base (seismic) isolation systems [6]. Tuned mass damper, tuned liquid damper, metallic damper, viscous fluid damper, and viscoelastic damper are examples of passive control devices.

2.1 Metallic Damper Metals are produced from steel and undergo plastic deformation due to their structure. Considering this flow and deformation of the metal, metallic damping systems have been developed. In mechanisms arranged with these dampers, they are commonly produced from triangular, rectangular, or X-shaped steel plates. The main reason why it is produced in the form of X and triangular plates is that a uniform flow is desired. In such mechanisms, attention is paid to the equal distribution of the stress on the

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Fig. 1 X shaped metallic damper placed structure and damper detail

material. Before choosing the metallic damper type, it should be checked whether the damper properties are suitable from a theoretical and experimental point of view and its suitability should be examined. The use of such dampers in steel structures is more convenient than other types of structures [7]. Among the commonly used shapes of metallic dampers are x-shaped plate dampers. Added damping and stiffness (ADAS) metallic damper model can be given as an example of this type of damper. Figure 1 shows the X-shaped ADAS metallic damper and the damper detail placed in the structure. When placing the members, they are positioned to be parallel to each other.

2.2 Friction Damper Friction devices use kinetic energy resulting from dynamic movements and convert it into heat. When dynamic loads come to the structure, the material is stretched in the friction region is dragged due to grip and slip. Meanwhile, another of the crossshaped couplings twists [2]. Figure 2 shows the damper example developed by Pall based on this. They are inserted into the structure similar to X-shaped steel bars, with a damper in the center. Considering the possibility of corrosion of the device, attention should be paid to the material to be used and steel material with low carbon content should be avoided [5]. Otherwise, the intended life of the friction damper for control is shortened.

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Fig. 2 Friction type damper developed by Pall

2.3 Visco-Elastic Damper Visco-elastic dampers act on the principle of converting kinetic energy into heat energy. They aim to dampen the energy provided by the sliding movement of the visco-elastic material used in these devices with the help of steel plates that act like a piston, by converting it to damping energy. It has been applied in multi-story buildings since the 1980s, and studies on applications related to seismic loads started in the 1990s and generally showed an effective performance in damping the wind-induced load [8]. With the implementation of the company named 3 M, it took its place in the list of control devices and aimed to reduce the vibration effect on the wind-sourced structure [9]. Visco-elastic dampers were first used in around 10,000 of the twin towers of the World Trade Center, which were destroyed by the September 11 terrorist attacks [10]. An example of the application is shown in Fig. 3. Apart from this, it is also used in the World Center, The Number Two Union Square Building, and the Columbia Seafirst Building to prevent wind-induced vibration. When visco-elastic dampers were examined, it was seen that they were simpler to use in steel designed structures, and based on this, this type of damper was placed in a steel designed structure in Taipei, Taiwan, and an experiment was conducted for its behavior under a temperature tuned to 30 °C and earthquakes of different values [2, 11]. Considering the results, the application of this type of damper to the structure provided an increase of up to 12% on the damping rate [2, 11].

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Fig. 3 a Visco-elastic damper b Location of visco-elastic dampers

2.4 Viscous Damper Viscous dampers have some similar properties to visco-elastic dampers. One of the most important similarities is that the working principles proceed with the same logic. Here, unlike the visco-elastic damper, the visco-elastic solid material used is replaced by the viscous fluid material, and in the same way, the steel plates are replaced by a piston, a push–pull relationship is created between the viscous material and the piston. Mechanical energy is obtained from this movement. This energy is converted into heat energy and damping of the structure is provided. This type of dampers, like visco-elastic dampers, are speed-dependent as they use the energy arising from the sliding motion. These dampers were first put into practice with the development of Constantinou in 1992 and during the construction phase, a cylindrical part made of a metal material containing silicon oil, a piston made of steel, and an accumulator was used to store the energy [2]. The viscous damper and its section are shown in Fig. 4.

2.5 Tuned Mass Damper (TMD) Tuned mass dampers provide damping of vibrations in the structure by using the mass connected by a spring and damper attached to the structure. The frequency of the damper is tuned by choosing the value closest to the natural frequency of the structure. In such devices, oscillations occur depending on the frequency set between the spring and the mass. The phase difference caused by this oscillation causes

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Fig. 4 Viscous liquid damper section

displacements in opposite directions, and the vibration movement of the structure is reduced with the damping arising from this opposite movement. TMDs belong to the oldest known group of controllers. The term vibration control dates back to the 1909 invention of what Frahm called the ‘dynamic vibration damper’ [12, 13]. Its inclusion in architecture as a design element is based on the article published by researcher Mcnamara in 1977 [14, 15]. Compared to the mass of the building, the structure has only about 1% of its mass [5]. Figure 5 shows a schematic drawing of the working principle of tuned mass dampers. Tuned mass dampers have many applications. Some of these are those; John Hancock Tower in 1975 and City Corp Center in 1978, Canadian National Tower

Fig. 5 Tuned mass damper operating principle

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in 1976 in Canada, Chiba Port Tower in 1988, and Crystal Tower in 1990 in Japan, 2000 in England London Millennium Bridge was built [2].

2.6 Tuned Liquid Damper (TLD) Tuned liquid dampers are devices that aim to absorb the energy generated in the structure by providing the movement of the fluid in a structure under dynamic load, with the adjustments made by taking into account the natural frequency of the structure. In other words, TLD is a system in which containers filled with a certain level of liquid are used to absorb the vibration caused by the dynamic movements that will occur in the structure [16]. The movement of the mass, which is used similarly to the tuned mass damper, provides this damping. In this type of device, the liquid is used as a mass. It also includes a period tuner. The oscillation effect created by dynamic loads such as wind or earthquake coming to the building aims to absorb the energy of the building by circulating the liquid, which replaces the mass, in the opposite direction to the vibration movement, by using the pre-set natural frequency during the oscillation. This liquid mass is stored in cylindrical or rectangular-like tanks. Fluctuations occur with the movement of the liquid in the tank. The nature and size of the fluctuations affect the natural vibration period. The fluctuations that occur in the tank during this movement determine the natural vibration period according to the liquid height, the geometry of the tank, its dimensions, and the gravitational acceleration. Housner developed a mechanical model for agitation in the tank. Figure 6 shows this mechanical model with the spring and the masses attached to it [17]. Tuned liquid damping devices are divided into two groups. One of them is oscillating dampers and the other is tuned liquid column (arm) dampers. Figure 7 shows schematically the oscillating plate and rod damper and the tuned liquid column damper. Fig. 6 Mechanical model for sloshing problem

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Fig. 7 a Oscillating plate and rod damper, b tuned liquid column damper (TLCD)

In oscillating dampers, the vibration period is tuned depending on the dimensions of the container in which the liquid is located, and steel bars or a plate are placed in the liquid in the container to increase the damping capacity [2]. In tuned liquid column dampers (TLCD), on the other hand, air pressure is created by the movement of the liquid in the U-shaped mechanism and this pressure change in the tank allows the period of the structure to be tuned. Looking at the basic principle of these dampers, it is seen that they aim to eliminate the kinetic energy that occurs in the structure, similar to the mass dampers. Bridge, tower, etc. of tuned liquid dampers. data on its use in structures were recorded by Fuji and Noji in 1988, by Yoneda in 1989, by Yeda in 1990, and by Wakahara in 1991 [18, 19]. In 1989, an experimental setup for Tuned column liquid dampers (TLCD), which is a type of tuned liquid damper, was established. and this idea was first brought to the literature by the work of Sakai et al. [7]. TLCD dampers started to be applied in public buildings such as Shin Yokohoma Prince Hotel, which was completed in 1992, and Sakitama Bridge, another different building group, which was opened in 1992 [20]. When looking at TLD control devices in general, it is seen that Japanese engineers frequently use many examples in different structures such as the tower used for control in Narita Airport in Japan [21, 22]. We can list some advantages of liquid dampers as follows; • Since they have a simple setup and are easy to install, they do not require much cost. • Maintenance and repairs are easy. • They can be easily placed not only in new structures but also in old structures. • The frequency of the system can be easily tuned by factors such as fluid movement and pressure change in the system. • Considering the disadvantages; • It can give small errors (like measuring still water level).

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Fig. 8 A schematic example of the inerter device

• It may be insufficient for strong vibrations of larger amplitude (mostly used for wind vibrations). Today, examples can be seen in many buildings. Examples of TLD applications are structures such as Nagasaki Airport Tower in Japan, Tokyo Sofitel Hotel, Yokohoma Marine Tower, Shin Yokohoma Prince Hotel, Sakitama Bridge, Narita Airport Control Tower, and Emley Moor Tower in England. Apart from these, we can show the British Airways i360 structure, the thinnest and tallest tower in the world compared to its diameter, in England, which was opened in 2016, as an example of one of the important structures. Although it is on the coastline and exposed to a lot of wind load, TLDs placed in the structure ensured that these vibrations were kept under control.

2.7 Tuned Mass-Damper-Inerter The tuned mass-damper-inerter device is a type of mass damper derived from conventional TMD. It was developed by adding and modifying inerter to TMDs. The added inertial device aims to outperform TMDs by enabling mass augmentation. Provides damping for higher modes under the influence of an inerter device [23]. A schematic example of the inerter device is shown in Fig. 8.

3 Tuned Mass Damper (TMD) in Civil Engineering In this section, the emergence of the tuned mass dampers as an idea and its historical development, its performance on dynamic loads such as wind and earthquakes in different areas, and their integration into different structures, hybridization, etc.

3.1 Historical Development of TMDs In 1909, Herman Frahm made an invention to prevent vibrations in ship machinery and presented a study on vibration control, which laid the foundations of tuned

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mass dampers [12, 13]. By 1911, he took the patent of this invention and named it a tuned vibration damper. Ormondroyd and Den Hartog introduced the beginning of theoretical studies on modulated mass dampers in 1928 [24, 25]. In his work in 1909, Frahm developed a system in which the mass does not have natural damping. Den Hartog, in his study, determined that tuned mass dampers with damping can be more effective because they are insufficient in terms of energy conversion in an undamped system [26]. In 1952, Bishop and Welbourn also developed a model with mass extinction and researched it [27]. In 1956, Hartog made progress on the most suitable damping parameters for the damper with his book titled “Mechanical Vibrations” and assumed that the TMD has damping and that the main mass is undamped [28]. In the same year, Hartog continued to work on the undamped system to examine the performance differences between the damped and undamped systems, which he had observed before, and obtained the optimum parameters of TMDs under harmonic excitation [29]. In 1967, Falcon et al. aim to optimize the damper by suggesting a method for various structural vibrations by adding a damper with a constant damping ratio value [30]. In 1973, Wirsching and Campbell observed that the mass of the TMD had a remarkable effect on the system response after the first mode by placing the damper on the top floor of two structures with different coefficients [31]. Ioi and Ikeda conducted studies on the existing damping parameters in 1978 and tried to optimize these values by using different functions [32]. In 1980, Warburton and Ayorinde stated that by optimizing the values of TMD parameters such as the main mass of the system, the mass of the damper, and the mass ratio, the dampers connected to different regions in the structure can be provided with a simpler SDOF structure [33]. Kaynia et al. (1981) analyzed the response of SDOF structures with and without TMD in different earthquakes and showed that the duration of the earthquakes was related to the design constants [34]. In 1983, Vickery et al worked with the main mass with 5% damping rate and obtained the most appropriate damping ratio that should be added to bring the system control of TMDs to a sufficient level [35]. Iwanami and Seto (1984) studied single tuned and bi-tuned TMD and observed that bi-tuned TMDs performed better [13, 36]. Villaverde (1985) conducted numerical and experimental studies on the responses of seismic stimuli in various structures with different dimensions and obtained data that is effective in reducing the response of the structure [37]. In 1992 and 1994, Xu and Igusa studied and optimally designed tuned mass dampers (MTMD) and found that they gave better results than a single TMD system [13, 38]. Tsai and Lin (1993) investigated a structure exposed to harmonic excitation and developed experimental equations by obtaining optimum design parameters [39]. In the same year, Villaverde and Koyoama investigated the effect of TMD used in a 10-story building using the Mexico earthquake acceleration data of 1985 [40, 41]. Yamaguchi and Harnpornchai (1993) compared TMD and MTMDs by keeping the mass constant in their study and conducted research on the effect of MTMDs on performance and their general characteristics [42]. Igusa and Xu (1994) studied MTMDs and examined their performance for the frequency value they determined and found that they had a stronger effect compared to TMDs [43]. Abe and Fujino (1994) examined the effectiveness of MTMDs by giving harmonic stimuli and found that it was more efficient when at least one of the MTMDs was

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selected and tuned according to one of the modes of the structure [44]. Kareem and Klie (1995) showed that MTMDs performed better than TMD as a result of the analyzes made in a wide frequency range using TMDs that can be tuned to different frequency values in an SDOF designed system [45]. Jangid (1995), in his study, used MTMDs to minimize the response that will occur in the structure by using ground vibrations and found that MTMDs gave better results than a single TMD equal to the mass of the whole [46]. Joshi and Jangid (1997) developed a method for obtaining optimum design parameters using MTMDs to reduce the response of the structure affected by ground vibrations [47]. Although Rana and Soong (1998) placed TMDs on each floor and created an MTMD system in a system with three degrees of freedom (3DOF) to reduce the response of the building by using harmonic stimuli and earthquake load, it was understood that the presence of MTMD on each floor did not have a strong effect on reducing displacement [48]. Jangid (1999) used MTMD in an undamped structure exposed to harmonic ground vibrations and obtained the most suitable design parameters by numerical methods [49]. Lin et al. (1999) proposed an effective method for solving vibration problems in which some of the undamped modes of the structure are taken by using TMDs for offshore platforms as well as lattice beam systems [50]. Chang (1999) compared control performance by exposing 3 different mass dampers tuned mass damper (TMD), vibration absorber (LCVA), and tuned liquid column dampers (TLCD) to white noise and found that the efficiency levels depend on a value mentioned as the efficiency index [51]. Takewaki (2000) operated TMDs together with a different passive control device, viscous dampers, and observed that they performed well in structure control [52]. Agrawal (2000) added TMD to reduce the response of the system by using ground vibrations defined as white noise and found that the torsion connection of the system greatly affects the response to the stimuli in the system [53]. Lin et al. (2000) used TMDs to reduce the response of long-span bridges to torsion and impact and made recommendations for TMD design for bridge-type structures subject to wind load [54]. Gu et al. (2001) using MTMDs for collision control of the Yangpu Bridge in Hong Kong, where typhoons are frequent, found that its performance in the control showed sensitivity to frequency-related data [55]. Chen and Wu (2001) showed in their study that single TMDs were insufficient in reducing the response of a structure exposed to seismic warnings, and then when the MTMDs they proposed were analyzed numerically, they played an active role in reducing the maximum acceleration [56]. Jensen et al. (2002) rain, wind etc. TMDs have been proposed for the response control of cable bridges exposed to vibrations, and a remarkable reduction in vibration response has been observed with the optimization [57]. Chen and Wu (2003) used MTMDs to reduce the response in a structure of different scales where seismic excitations are given and observed that MTMDs are more effective than single designed TMDs by applying forced vibration as well as free vibration [58]. Gerges and Vickery (2003) applied various wind tests to non-linear TMD to prevent wind oscillations in a thinly designed square-section structure and investigated the performance difference with linear TMD [59]. Pinkaew et al. (2003) placed TMD in a 20-story SDOF structure in a structure exposed to seismic warnings using the 1985 Mexico earthquake data and observed that after the flow, TMD reduced the damage

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to the structure even though it could not reduce the displacement [60]. Kwon and Park (2004) dealt with the problem of bridge flutter and found that the performance is better for MTMD (IMTMD) and single and multiple TMDs (STMD and MTMD) with irregular frequency values, which they propose to increase the durability of TMDs against aerodynamic effects for cable and suspension bridge types demonstrated [61]. Chen and Huang (2004) made a series of recommendations on TMD design by making a structural analysis of the TMD using the Hartog method in a two-degree-of-freedom (2DOF) Timoshenko beam exposed to harmonic vibrations [62]. Zuo and Nayfeh (2004) applied TMD for a multi-degree-of-freedom (MDOF) system as well as for viscous systems and gave numerical examples of the damping performance of TMDs for the frequency range selected upon estimation [63]. Yau and Yang (2004) investigated the dynamic response of the bridge using MTMD tuned to some of the frequencies of the system in continuous truss bridges subjected to moving train load and demonstrated its efficiency in vibration control [64]. Gerges and Vickery (2005) proposed a weighted average method by examining the response of TMD with wire rope springs by exposing the structure to white noise [65]. Wang and Lin (2005) examined the effectiveness of TMD in irregular structures where torsion shows the effect and observed that MTMDs are more efficient than STMDs when the interaction between the two is high, considering the connection between the ground and the structure [66]. Zuo and Nayfeh (2005) investigated the effect of values such as mass, the number of dampers, the mass ratio on damping in MTMDs and suggested an algorithm for the optimization of the stiffness and damping coefficient values of STMDs [67]. Lee et al. (2006) used a numerical method to find the parameters for the optimum design of structures with TMD and determined its accuracy for different numbers of TMDs and degrees of freedom [68]. Li and Zhu (2006) showed that by using a double-tuned mass damper (DTMD), one larger than normal and the other smaller, in the ground acceleration structure, it can be more durable than MTMD in some cases [69]. Li and Qu (2006) added MTMD to the structure to reduce the translational and torsion effects in the structures where ground acceleration is effective, tested the efficiency of the damper with simulations, and investigated the normalized eccentricity ratio (NER) that affects the damping capacity [70]. Du et al. (2007) made a new design proposal by working on infinite-multiple modulating mass dampers (IMTMD) and determined the accuracy by making numerical calculations on more than 20 TMDs used to investigate the obtaining of the most suitable design parameters [71]. Wu and Cai (2007) investigated the effectiveness of the tuned mass damper-magnetorheological (TMD-MR) damper anywhere in a cable exposed to free vibration in tension, and examined its effectiveness for two different damping conditions, and provided the basic infrastructure for the design of TMD-MR dampers [72]. Almazán et al. (2007) investigated the performance of a bidirectional and homogeneous mass damper (BH-TMD) supported by cables by carrying out various tests with a shaking table and suggested that it can be an efficient damping solution in vibration control by investigating its independent parameters [73]. Guo and Chen (2007) investigated the dynamic responses of MTMDs in vibration control in three-dimensional designed structures and observed that they affect the natural frequency of the structure and reduce non-long-term responses [74]. Hoang et al.

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(2008) modeled a system that acts as a much larger TMD than normal dimensions for a long-span truss type bridge in Japan, using an isolation system that works with the floor slabs. demonstrated [75]. Liu et al. (2008) designed a mathematical model to investigate the effects of the interaction between the ground and the structure on oscillations caused by wind load, using TMD in multi-story buildings [76]. Chen and Wu (2008) parametrically investigated the performance of the system by developing an analytical model using TMD for the control of vibrations such as wind, earthquake, and vehicle traffic in bridges with long spans [77]. Marano et al. (2008) proposed design optimization to minimize parametric variations in TMD design [78]. Matta and Stefano (2009) have shown that in vibration control of buildings, the area above the building can be transformed into an environmentally friendly TMD control system by using, for example, a roof garden [79]. Leung and Zhang (2009) used TMD in a viscous damped SDOF system and made use of algorithm problems to minimize displacement to reach optimum values with particle swarm optimization (PSO) [80]. Hijmissen et al. (2009) investigated the effect of bending stiffness and bending stiffness on damping control in vibration control in a tensioned cable exposed to vertical vibrations with TMD placed [81]. Alexander and Schilder (2009) examined the efficiency of the nonlinear tuned mass damper (NTMD) in a cubic designed nonlinear 2DOF system and suggested that the resonance effect is the weakness of the passively designed NTMD, so it would be better to design a semi-active NTMD [82]. Ok, et al. (2009) conducted a study on dual-tuned mass dampers and showed that the durability of dual-tuned mass dampers is quite good thanks to the design formulation they developed [83]. Marano et al. (2010) investigated the optimization of TMDs as dampers in vibration control, proposed a different approach for the predetermined mass value, and worked on the optimization of the mass ratio apart from parameters such as damper stiffness and damping [84]. Brownjohn et al. (2010) used TMD to reduce the effect of wind from the stack for the power plant structure using coal and observed the performance of TMD with the monitoring system [85]. Li et al. (2010) investigated pedestrian bridges and investigated the effect of MTMDs on the prevention of vibration caused by crowded crossings that will occur in this structure [86]. Weber and Feltrin (2010) investigated the long-term efficiency of TMDs with more than 20 years of installation on two different pedestrian bridges [87]. Bekda¸s and Ni˘gdeli (2011) tested TMDs for a model structure with seismic evcitations on time-defined simulation on Matlab with the harmony search algorithm, ensuring that optimal TMD parameters are obtained, and investigated the difference between them with several previously applied methods [88]. Almazán et al. (2012) investigated the response of structures using one or two tuned TMDs under seismic stimuli, observing that they are effective in reducing-edge deformation, and found that it is appropriate to place TMDs in the region where the response of the structure without TMD is higher [89]. Bae et al. (2012) designed an improved concept of TMDs, based on the knowledge that electromagnetic forces would reduce structural vibrations, and used it on cantilever beams, and it was observed that TMDs could significantly increase damping even if the tune of the concept was not done as desired [90]. Chen and Georgakis (2013) used tuned rolling ball dampers made of single or multiple steels as TMD in turbines to provide vibration control in wind turbines and showed that increasing

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the number of balls could not increase the efficiency further after a certain level [91]. Zhang and Balendra (2013) proposed an optimization approach to minimize the response corresponding to the nonlinear seismic frequency band of the TMD in buildings under seismic warnings [92]. Lee et al. (2013) made various field measurements to investigate the cause of vibration movement in the 39-story TechnoMart steel structure in 2011 and determined that the design with TMD in the structure can reduce the vibration of the structure by 25% compared to the size of the TMD [93]. Daniel and Lavan (2014) proposed an optimization method to ensure the optimum design of multi-mode MTMDs against various seismic warnings for use in the control of 3D irregular buildings [94]. Viet and Nghi (2014) investigated the performance of single-mass TMDs with bi-frequency oscillation and translation in controlling structures subjected to horizontal vibration [95]. Longarini and Zucca (2014) investigated the response of a hydraulic plant in Italy by placing a TMD in a chimney built in the 20th century to reduce its response to seismic warnings [96]. Marian and Giaralis (2014) proposed a device called a tuned mass damper-inerter (TMDI) and found that it outperforms conventional TMDs in the excitation of undamped SDOF structures with white noise [97]. Domizio et al. (2015) investigated the effect of TMDs, which are effective against distant fault motion, on near-fault motions [98]. Mrabet et al. (2015) found an approach for optimum TMD parameters, researched failure probabilities, and proposed the continuous optimization nested loop (CONLM) method [99]. Frans and Arfiadi (2015) compared the response of MTMDs under seismic stimulation with other studies using a hybrid coded genetic algorithm (HCGA) for optimization [100]. Lievens et al. (2016) found a design optimization approach that also considers modal parameters for TMDs used to prevent vibration in pedestrian bridges [101]. Miguel et al. (2016) presented a new study on the use of TMDs and MTMDs and the application of design optimization in controlling vehicle-bridge vibration [102]. Elias et al. (2016) used multimodal TMDs to control the ground motion of chimneys and investigated the performance of randomly located TMDs and single TMDs (STMDs) when exposed to seismic stimulations [103]. Mokrani et al. (2017) conducted analyzes on bridge decks, using TMD as the number of critical modes, aiming to minimize bending and twisting [104]. Carmona et al. (2017) used TMDs to prevent ground vibrations and tested the performance of the device by giving free and forced vibration to the platform-shaped system [105]. Lu et al. (2017) conducted various experiments on the device they called a particle modulated mass damper (PTMD) [106]. Giaralis and Petrini (2017) numerically optimized the TMDI device for predetermined mass values, using a tuned mass damper inerter (TMDI) modified by the addition of inerter to the classical TMD, regarding a 74story building, and observed its effect on the lighter design of high-rise structures [107]. Chen and Yang (2018) obtained a hybrid device by using another passive control device, tuned fluid dampers (TLD), together with TMD, and showed that the system has a very good performance by analyzing different mass ratios and different fluid depths to investigate the performance of the hybrid device [15]. Lu et al. (2018) examined the operation of this system by placing eddy current regulated mass dampers (EC-TMD) developed from TMDs on a structure with an MDOF system [108]. Domenico and Ricciardi (2018) investigated the effectiveness of TMDI

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devices by considering nonlinearity, unlike damper studies with linearized behavior assumptions [109]. Wang et al. (2019) By placing the TMDI and tuned inerter damper (TID) device in a high-rise building, the response of the structure as a result of experimental wind tunnel tests was compared for the two devices [110]. Shi et al. (2019) developed the adaptive-passive variable mass tuned TMD (APVM-TMD) and tested this self-tuned system in the control of human-induced vibrations, unlike TMDs that normally cannot re-tune [111]. Mrabet et al. (2019) proposed acoustic optimization for TMDs in providing multimodal vibration control of indoor sound and observed remarkable reductions in resonance modes [112]. Zhang (2020) has derived formulas for optimal tuning by adding TMD on a 2DOF model for the control of edge vibrations on blades in wind turbines [113]. Khodaire (2020) investigated the performance of TMD tapering in super-high buildings in preventing wind vibration [114]. Pietrosanti et al. (2020) tested a TMDI device-attached SDOF structure by exposing it to harmonic excitation, with the help of a shaking table, and pioneered the development of a TMDI tuning approach [115]. Li et al. (2020) proposed a variable-tuned mass damping inerter (VTMDI) model and found that VTMDI and TMDI devices were more effective in vibration reduction than VTMD and TMDs when TMD, VTMD, and TMDI devices were under harmonic excitation for performance research in vibration control [116]. Petrini et al. (2020) proposed a new TMDI formulation to convert some of the wind’s energy into electrical energy in tall buildings susceptible to wind-induced eddy scattering (VS) effects and found that TMDIs provide this energy conversion by adding electromagnetic motors to TMDI devices with more floor-inert dispersion [117]. Sarkar and Fitzgerald (2021) investigated the optimum vibration reduction properties for a TMDI device placed on a tower using the Kane method [118]. De Angelis et al. (2021) investigated the optimal design of a grounded tuned mass damper inerter (TMDI) in the amelioration of human-induced vibration in pedestrian bridges and observed that TMDI devices are an efficient control system for pedestrian bridges [119]. Wang et al. (2021) wind-tested TMDI and multi-tuned mass damper inerter (MTMDI) devices for the control of wind-induced vibrations of super-tall buildings, found to significantly reduce displacement response [120]. Bai et al. (2021) applied the newly developed tuned inertial damper (TID) and tuned viscous mass damper (TVMD) systems in the Euler-Bernoulli beam model for vibration control, comparing these two new systems with the classical TMD and found that they provide more durable control for the specified situations [121]. Jahangiri et al. (2021) aimed to control bidirectional vibration by using a three-dimensional pendulum-tuned mass damper (3d-PPTMD) in wind turbines and confirmed that 3d-PPTMDs outperform TMDs in some conditions [122].

4 Tuned Liquid Damper (TLD) in Civil Engineering In this section, the emergence and historical development of tuned liquid dampers, their performance in different structural groups subjected to dynamic loads, and their application to different structures, hybridization, etc. work will be discussed.

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4.1 Historical Development of TLDs Tuned fluid dampers, like TMDs, were first used to control vibration on ships. [19]. It took an active role in the use of ships in the 1950s [123, 124], was used in satellite control in the 1960s [124, 125], and in the 1980s, Bauer conducted research on the use of these systems to prevent structural vibration [124, 126]. Apart from these, information about some of the studies carried out so far is given below: Modi and Seto (1997) completed their study by using the agitation energy of the liquid to control the vortex-shaped oscillations caused by the wind, with TLDs, which several researchers at that time called agitation damper or tuned liquid damper [127]. Won et al. (1997) investigated the damping performance of tuned liquid column dampers using different earthquake data for the control of SDOF and MDOF structures under randomly selected seismic loads [128]. Reed et al. (1998) examined rectangular tank TLDs with the help of a shaking table by constructing a numerical model for large amplitude vibrations and obtained experimental results [129]. Chang and Hsu (1998) investigated the effectiveness of liquid column vibration absorbers (LCVA), a type developed from tuned liquid column dampers, in vibration control and compared their performance with TLCD and TMDs [130]. Yamamoto and Kawahara (1999) analyzed the damping of oscillations of Yokohoma Marine Tower with TLDs using the ALE method [131]. Chang and Gu (1999) placed a rectangular-based TLD in a tall building subject to eddy excitation and observed that when the frequency is optimally tuned, rectangular-based TLDs are highly effective in controlling eddy vibration [132]. Gao et al. (1999) investigated the performance of multi-tuned liquid column dampers (MTLCD) in vibration control and examined the effect of parametric values such as TLCD number, frequency range, head loss coefficient on the control and determined that MTLCDs could be more efficient than a single TLCD for the situations specified in the study [133]. Balendra et al. (1999) presented a numerical study showing that a second damper gives even better results by applying TLCD to different structures designed such as shear walls and rigid frame systems to examine the effectiveness of TLCDs in the control of wind vibrations in high-rise buildings [134]. Xue et al. (2000) examined the effect of TLDs on damping the rolling motion in bridges with wide spans by giving harmonic stimuli [135]. Xue et al. (2000) used TLCDs to control the pitch motion vibration of structures, gave free and forced vibrations to the system, conducted an experimental study for different TLCD parameters, and showed that TLCD was efficient in controlling the pitch motion [136]. Yalla et al. (2001) investigated the response of a system with MDOF structure under wind stimulation using a semi-activated TLCD device, deriving the so-called optimal control theory [137]. Li et al. (2002) created a model for rectangular tank TLDs with a shallow liquid and suggested solving the wave motion equations that will occur in the containers by using the FEM method [138]. Wu and Hsieh (2002) studied the effect of some important TLD parameters on vibration control by deriving the equations of motion for a U-type TLD mounted plate [139]. Casciati et al. (2003) used conical vessels as an alternative to cylindrical and rectangular liquid tanks and subjected an SDOF model to several different

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excitations for structures with small amplitudes, comparing the damping effect of cylindrical TLDs and conical TLDs, and again using conical TLDs on the shaking table analyzed by adding it to the load cell [140]. Shum and Xu (2004) discussed a wide-span bridge and conducted a study to prevent lateral vibration and torsional vibration by randomly giving random excitations to the bridge with harmonic excitations using MTLCD [141]. Felix et al. (2005) simulated and analyzed the performance of TLCDs for vibration control when a non-ideal stimulus was received [142]. Min et al. (2005) used TLCDs for the control of wind vibration in a 76-story building, investigated the performance difference between the nonlinear and linear system, and evaluated the effect of various parameters of multiple TLCDs on the control and their durability [143]. Taflanidis et al. (2005) investigated the control performances at optimum values in a structure exposed to white noise by using TLCD and liquid column vibration absorbers (LCVA) for the control of vibrations arising from structural rotational motion [144]. Lee et al. (2006) took the use of TLDs to another point and used TLCDs to counteract wave vibrations on a floating, open, unstable sea platform [145]. Taflanidis et al. (2007) examined the damping performance, durability, and reliability of TLCDs and LCVAs, SDOF, and MDOF systems in vibration control resulting from earthquake excitations, in excitations using white noise [146]. Lee et al. (2007) investigated the efficiency of the system with the help of a realtime hybrid shaking table using TLD for vibration control in a building exposed to seismic warnings [147]. Pirner and Urushadze (2007) obtained experimental results by exposing TLDs to various vibration stimuli with a two-degree-of-freedom (2DOF) system to control the horizontal and vertical motion of ships [148]. Wu et al. (2008) examined the control of TLCD devices on wind-induced motion in a bridge deck using forced vibration as well as free vibration [149]. Wu et al. (2008) and Tait (2008) developed a linear mechanical design that considers the energy dissipated by the damping effect in the vibration control of TLDs in high-rise buildings [16]. Samantaray (2009) obtained mathematical models using a preloaded passive fluid damper and examined its effect on damping [150]. Love and Tait (2010) tested their proposed TLD models with an expansion method by giving random excitations with the help of a so-called sinusoidal shaking table and compared their damping performance [151]. Marsh et al. (2010) based on TLD, in general terms, when liquid swash absorbers are partially filled with liquid, various experimental studies were conducted and investigated the performance of swash waves in the damper in structural control [152]. Lee and Min (2011) analyzed the damping capacities of two-way TLCDs with the help of a shaking table in the wind-induced vibration movement that will occur in the SDOF structure [153]. Banerji and Samanta (2011) showed that a hybrid mass liquid damper (HMLD), which consists of a TLD attached to a structure and a rigid mass connected to the structure with the help of a spring, performs better than conventional TLD against broadband seismic warnings [154]. Lee et al. (2011) developed a new TLD derivative tuned liquid column and agitation damper (TLCSD) and investigated the effects of the TLD + TLCD system (TLCSD) on the control of horizontal and vertical structure movement [155]. Crowley and Porter (2012) analyzed the response of TLDs exposed to various stimulations by performing a Multi-screen arrangement for rectangular tank TLDs and compared them with a previous study to

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validate the numerical predictions [22]. Zahrai et al. (2012) examined the effectiveness of a derived TLD model with rotating partitions against seismic and dynamic warnings in the example of a 5-story building, showing that using partitions worked well in preventing misalignment from affecting control [156]. Hamelin et al. (2013) modeled a new type of TLD that varies according to the Kulegan-Carpenter number (KC) and investigated the effect of TLD on damping performance by changing the KC numbers with the help of a shaking table [157]. Sarkar and Gudmestad (2013) produced a pendulum-type liquid column damper (PLCD) derived from TMD and TLD and compared experimental and theoretical results for TMD and TLD [158]. Di Matteo et al. (2014) developed a statistical linearization method (SLT) that proposes the appropriate parameters of TLCD by applying random dynamic loads to a structure using TLCDs, making use of the equations of motion of the SDOF structure [159]. In 2015, Di Matteo et al. developed a motion model for the TLCD device and proposed a formulation, and then subjected this formulation to some experiments in U-type tubes to test its accuracy and worked on the proof of the proposed model [160]. Zhang et al. (2015) investigated the damping effect by running the model in 2 different modes with numerical simulations using TLDs for the control of edge vibrations in wind turbine blades [161]. Sonmez et al. (2016) proposed a new model for sTLCD and developed control algorithms. In this model, to analyze the algorithms proposed with sTLCD, random excitations were given and their performance was examined, and their response to stimulation was investigated by adding sTLCD and applying pTLCD without using TLCD [162]. Zhang et al. (2016) used TLDs for the control of lateral vibrations in the tower in wind turbines and investigated the dynamic effect of TLD by applying the real-time hybrid test (RTHT) [163]. Ong et al. (2017), considering masonry structures that are insufficient in the face of earthquake excitations, proposed a new building material that will work in harmony with TLD and investigated the effectiveness of this combined system by subjecting it to various numerical tests [164]. Zhu et al. (2017) used RTHS, which is called a hybrid simulation, in the control of dynamic motion in MDOF structures, examining the application of TLCDs one by one and their contribution to damping [165]. Chen and Yang (2018) created a hybrid system for structural vibration control by using TMD device and TLD and tested this system with the help of a shaking table by creating 4 different experimental models with an empty tank, TLD only, only TMD and TLD + TMD added. [15]. Fu et al. (2019) tested an SDOF structure under seismic loads on a shaking table using Particle damping systems (PD) and TLDs and compared the damping capacities based on the damper parameters obtained with various algorithms [166]. Pandit and Biswal (2020) moved away from the traditional TLD application, used a new shape for liquid tanks, gave 3 different real earthquake excitations, and examined the damping effect in an MDOF structure. They made changes in TLD parameters with the tuning ratio along with the mass and investigated their effects on preventing vibration [167]. Cavalagli et al. (2021) made comparisons between the curved and round-shaped and rectangular-shaped TSDs of different geometries and the TMD of partially water-filled tuned shock absorbers (TSDs) derived from TLDs [168]. Li et al. (2021) investigated the damping capacities of a 243 m high

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solar tower in various wind tunnel tests using six types of small TLDs of different diameters to control vibrations [169].

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Introduction and Review on Active Structural Control Serdar Ulusoy, Sinan Melih Nigdeli, and Gebrail Bekda¸s

Abstract The number of stories of structures has been increased to meet the needs of people or long-span bridges are built for easier transportation. However, various destructive dynamic loads such as earthquakes and strong winds have negative effects on the structure without control systems. Therefore, control systems are necessary for these constructions. Furthermore, it is not enough for these constructions to be just safe and reliable. Constructions should be less exposed to vibrations under the influence of earthquakes and strong winds. For this reason, an external power source (activator) that can handle the large horizontal loads in these structures and a system that controls this source are needed. This study summarizes different active control systems, control techniques and studies on active structural control. Keywords Active control · Activator · Vibration · Earthquakes · Winds

1 Introduction In recent years, with the developing technology, actively controlled structures have gained importance, which has arisen as a result of the works of civil and control engineering to eliminate damage of structural elements caused by earthquakes or strong winds. This collaborative work aims to increase safety through the control systems in the structure and to implement it automatically without human intervention. In the literature, two types of control systems are mentioned as open loop and closed loop based on the working principle. A control mechanism that only controls S. Ulusoy Department of Civil Engineering, Turkish-German University, Beykoz, 34820 Istanbul, Turkey e-mail: [email protected] S. M. Nigdeli (B) · G. Bekda¸s Department of Civil Engineering, Istanbul University-Cerrahpa¸sa, Avcılar, 34320 Istanbul, Turkey e-mail: [email protected] G. Bekda¸s e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. Bekda¸s and S. M. Nigdeli (eds.), Optimization of Tuned Mass Dampers, Studies in Systems, Decision and Control 432, https://doi.org/10.1007/978-3-030-98343-7_3

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Fig. 1 The working schema of active-controlled structures

the given source variable, in which the output variables do not influence the input, is referred to as open control loop. In the opposite case, that is, if the output of the system directly affects the input, these systems are called closed-loop systems. The closed-loop control system (feedback control system), which has an important place in terms of use in structural engineering, measures the structural reactions (output) with measuring instruments such as accelerometer, compares it with the reference value (0) and then generates a control signal to actively control the structures. Also, the working schema of active-controlled structures is given in Fig. 1. The main reasons for using this control system compared to other systems (base isolation or passive control system) include stabilization of the structure that is exposed to high vibrations, high protection in structures such as hospitals and nuclear power plants with high costs, less space in structures and cost reduction appropriate use of building materials [1].

2 Active Control System Active control systems are placed in the structure to ensure that the structure, which is exposed to vibration, reaches its equilibrium position quickly. In general, two systems have been developed to provide active control. One of these is the active tendon system in Fig. 2a. and the other is the active tuned mass damper (ATMD) as shown in Fig. 2b. While the control forces generated by the activators according to the control signal are provided by adding active mechanisms to passive mass dampers in ATMD, the control force is applied to the structural model via pre-tensioned cables in active tendon systems. The performance of active control systems on structural reactions against earthquakes of very high magnitude has an important place in control engineering. However, in practice, these systems have some disadvantages. Some of these disadvantages include obtaining high energy to be applied to the structure, installation and maintenance costs [2]. In addition, control technique deficiencies, modeling errors, time delay, non-linearity of the structure, uncertainty of the structure parameters, limited sensors and control elements can be shown among the disadvantages of

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Fig. 2 Active control systems. a Active tendon control b Active tuned mass damper

active control systems [1]. In order to avoid similar disadvantages, different control techniques are used for different purposes. There are two different control techniques, linear and intelligent. Proportional–integral–derivative (PID), H∞ , and linear optimum controllers (LQR and LQG) are among the controllers used in linear control of structures, while artificial neural networks, fuzzy logic, sliding mode and genetic algorithm controllers are used in intelligent control of structures. (a)

Proportional–Integral–Derivative (PID) Control

The PID controller, which is widely used in the industry, has improved its performance against strong earthquake excitation in structural active control with many academic studies [3, 4] on single and multi degree of freedom systems. The equation of the control signal of PID is given in Eq. 1. E(t) is the time dependent error signal and is equal to the difference between the structural reaction and the reaction of the structure at equilibrium (reference value = 0). Kp , Ti and Td are the proportional coefficient, integral time coefficient and derivative time coefficient, respectively. U(t) is the time dependent control signal and time dependent control force is produced according to this signal. ⎢ ⎥ ⎢ ⎥ t ⎢ 1 de(t) ⎥ ⎦ u(t) = Kp ⎣e(t) + e(t)dt + Td Ti dt

(1)

0

(b)

Linear Quadratic Optimum (LQR) Control

The linear quadratic optimal controller (LQR) is defined as minimizing the classical performance index J given in Eq. 2 and it is one of the most common control

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algorithms for controlling structures. However it have practical application, the main disadvantage of LQR is the need of all the displacements and velocities for the feedback. tf J=



 zT (t)Qz(t) + uT (t)Ru(t) dt

(2)

0

Q and R represent the weighting matrix, tf is the time interval longer than the external excitation duration and z(t) is a 2n-dimensional state vector given in Eq. 3. The solution of Riccati Matrix for shear building model described in the studies of Yang and Akbarpour [5] lead the control vector u(t) given in Eq. 4. Also, the linear quadratic gaussian (LQG) controller formed by adding the Kalman filter (an observer for the structural response) to LQR is used in some cases. Z(t) =

x(t) v(t)

(3)

1 U(t) = F z(t) = − R −1 B  Pz(t) 2 (c)

(4)

H∞ Control

The H∞ controller is one of the robust control techniques that has a wide range of uses in active vibration control of structures. On the other hand, the application of H∞ controller in structures is impractical because of the requirement of a higherorder system which leads to some problems such as numerical difficulties and high computational cost [6, 7]. This controller aims to minimize the H∞ norm transfer functions which define the relationship between the external disturbances such as earthquake ground motion and the controlled output (structural reactions). The H∞ norm transfer function is accepted as the performance index and calculated for the active-controlled structures from the study of Arfiadi and Hadi [8] as follows: (i) (ii) (iii) (iv) (v) (vi)

Define Hamiltonian Matrix H given in Eq. 5 Select a positive number γ Calculate the Eigenvalue of Hamiltonian Matrix containing the matrices such as Acl , E and Cz mentioned in the article Check the Eigenvalue of Hamilton (if it is imaginary or not) Reduce/increase the number γ Find the final value of number γ which is the H∞ norm transfer function

Acl γ 2 E E T H= −C z C zT −AclT (d)

Sliding Mode Control

(5)

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The sliding mode controller is a robust control technique. This controller performs the vibration control of the structure by bringing the structural reactions (error) on the sliding surface using a switching control law. Once the structural reaction is brought on the sliding surface, the controlled structure becomes independent from structural uncertainties, parameter variation and external excitations. This behavior gives priority to actively control the structures but an unwanted oscillation phenomenon with high frequency and amplitude called ‘chattering’ may occur through the switched control signal and other factors. Chattering is a negative effect that shortens the life of the control system and causes unnecessary energy consumption. The Eqs. (6)–(8) for the control force are taken from the studies of Adhikari and Drazenovre [9] and Yakut and Alli [10]. S is a (1 × 2) dimensional constant matrix, ag (t) is ground motion and δ is earthquake effect vector.   Ueq (z, t) = −(S D1 )−1 S Az(t) + S D2 ag (t) D1 =

(e)

0 M −1 B





, D2 =



 T , B = −1 0 0 . . . n , H = −Mδ M −1 H

0 I A= −M −1 K − M −1 C 0

(6)

(7)

(8)

Fuzzy Logic Control

Fuzzy logic control (FLC) is preferred in structural control implementations due to its simplicity, robustness and satisfactory performance on the structural responses by taking into account both the input/output variables and the heuristic behavior of the system. FLC is also more flexible compared to other classic control techniques. However, FLC does not offer optimal control and its fuzzification, as well as defuzzification, are pretty complex. The rules of structural control with FLC in the studies of Wand and Mendel [11] and Aldawod et al. [12] are as follows: (i) (ii) (iii) (iv) (v)

The input and output spaces of the given data are separated data into fuzzy small, medium and big regions. The fuzzy rules are created from the given data. A degree of each of the created rules is assigned to resolve the conflicts among the created rules. A combined fuzzy rule base is generated based on both the created rules and linguistic rules assigned by human experts. Mapping on the combined fuzzy rule base is determined with the use of a defuzzification process.

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Neural Network Control

The use of neural networks as a controller in many studies [13–16], especially for the control of non-linear structures, has made important progress in recent years. The neutral controller is realized with the training of the emulator neural network which is trained to experience the transfer function between the sensor measurings and the signal of the activator. Thus, an emulator neural network can be able to estimate the future values of sensor measurements. The structure control with neural networks aims to minimize the cost function J given in Eq. 9, where k denotes the sampling number and Nf total number of sampling times. N f −1

J=



T z k+1 Qz k+1

k=0

+

u kT

Ru k

N f −1 1  = Jk 2 k=0

(9)

3 Literature Research The practical applicability of active control systems such as active tendons or active tuned mass dampers was investigated by Yang and Samali (1983) in high-rise buildings subject to strong winds. The acceleration values of both structures were significantly improved and the magnitude of the control force was within the applicable limits [17]. The structural reactions of a tall structure with active tendons and an active tuned mass damper under the influence of wind are compared by Abdel-Rahman and Leipholz (1983). With the active tendon, better results were achieved in the design for displacement and acceleration values, but the required control force was calculated to be greater than with the active mass damper with the same damping [18]. Samali et al. (1985) stated that in the previous studies, when designing active control systems in high-rise buildings, earthquake vibration acting in one direction was taken into account and the torsion caused by earthquake motion was neglected. Therefore, The eight-story irregular structure in plan was investigated under the influence of strong earthquakes with active tendons and active tuned mass dampers to reduce structural reactions depending on the control force [19]. The frame and bridge samples were modeled by Manolis ve Song (1987) first as a passive system with braces and then as an active tendon system to explain the basic theory of active-controlled structure. Based on the results obtained from the active tendon system, It was concluded that optimization is possible in the geometry of the structure and for the use of materials [20]. The structural responses of active tendon-controlled structures obtained with analytical and simulation techniques and a real structure model that takes into account

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the time delay are studied together. Chung et al. (1988) determined that the experimental and analytical results showed similar behavior on the structure and that it would be applicable in real systems [21]. Also, the active tendon-controlled structure was reanalyzed analytically and experimentally by turning the previous single degree of freedom (SDOF) system into multiple degrees of freedom (MDOF). The results of the system (MDOF) controlled by the two different algorithms (linear global optimum feedback control and instantaneous optimum control algorithm) were different. Therefore, modifications were necessary for the algorithms to eliminate the time delay problem in the real system [22]. The structural responses of a 23-storey active tendon-controlled structure which is strengthened with only steel frames and only shear walls are calculated by LópezAlmansa ve Rodellar (1989) under the influence of earthquake and wind loads. The control force required for the active tendon-controlled structure with the shear wall is less and the structure performs better [23]. Four different algorithms (Linear quadratic optimum control, Instantaneous optimal closed-loop control, Instantaneous optimal open-loop control and Instantaneous optimal closed-open-loop control) are used for the active-controlled structure. As a result of the study by Akbarpour et al. (1990), the time delay sensitivity is effective on the performance of active-controlled structure [24]. A full-scale dedicated test structure was designed, simulated and installed to evaluate the performance of an actively controlled structure under different types of vibration[25, 26]. The mathematical models of the active-controlled structures can be insufficient because of the nonlinear behavior. For this reason, a new neural network-based control algorithm is proposed by Ghaboussi and Joghataei (1995) instead of conventional control algorithms. It is concluded that this algorithm is a good candidate for solving nonlinear problems [14]. The support displacements of structures such as bridges, pedestrian overpasses, highways and pipelines that extend horizontally depending on the soil properties may differ under the external excitation. It is emphasized that active tendons play a key role in this type of structural model [27]. Acceleration feedback, which is a more robust and effective feedback strategy, is suggested for full-scale application because the velocity and displacement feedback is difficult to obtain [28]. A control theory for nonlinear structures during seismic excitation is developed by Wong and Hart (1997) so that it can be used in practical applications of active control. Significant reductions in the structural response of the non-linear active-controlled structure are achieved [29]. A modified instantaneous control algorithm with a time delay effect is formulated by Chung et al. (1997) to demonstrate the feasibility and effectiveness of a three degree of freedom structure. The control system is still useful despite the existence of time delays [30]. An evaluation model created with 20 states that represent the real reactions of a three-story structure model described in Chung et al. (1989) is addressed in the

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study of Deoskar et al. (1998). The aim is to achieve a high degree of reality with 10 different control algorithms [31, 32]. A modified predictive control algorithm is developed by Chung (1999) to make the control force convenient in real systems. The proposed control system is dynamically stable based on the results obtained from the analyzes in the eigenvalue, time and frequency domains [33]. The objective function of H∞ , H2 and L1 norms are optimized using a genetic algorithm in three-dimensional active control structures excited by earthquakes. According to the study of Arfiadi ve Hadi (2000), a genetic algorithm (GA) is more suitable to minimize the structural responses of the actively controlled structure model by assuming the floor as a rigid diagram [8]. In the active control of the structures, a modified linear quadratic regulator (MLQR) containing a parameter (α) is employed instead of the second-order regulator (LQR) to maintain stability against unknown earthquake effects. The relative displacement in the structure decreased significantly with the increase of the α value, but there was no significant increase in the control force is observed by Aldemir and Bakioglu (2001) [34]. The model predictive control scheme (MPC) using acceleration feedback is applied by Mei et al. (2002) to experimentally and analytically analyze various structures equipped with active devices. The success of the MPC scheme is verified by the results of numerical and experimental samples to actively control the structures subjected to earthquake excitation [35]. The purpose of the paper of Lu et al. (2002) is to develop a control strategy for an active-controlled benchmark structure subjected to wind incorporating H∞ attenuation constraints into the H2 design methodology. Flexibility in improving structural reactions, especially acceleration values is offered with this study [36]. Various numerical studies on active structural controls are summarized by Datta (2003) to demonstrate the developments and implementation in practice. The following points are emphasized; (a) (b) (c) (d)

the effective performance of active control in the structure may vary according to the control method; time delay causes instability in the structure; active tendons are more effective against wind load than active tuned mass dampers; Soil-Structure Interaction (SSI) should be taken into account in activecontrolled structures [37].

Sliding mode fuzzy control (SMFC) with a disturbance estimation filter is designed by Kim et al. (2004) to estimate the wind load on the structure considered by Young (1998). The structural reaction performance of SMCF is better than the evaluated model with Linear Quadratic Gaussian (LOG) under the limitation of Control force [38]. A fuzzy sliding mode control algorithm (FSMC) is suggested by Alli and Yakut (2004) to eliminate the chattering problem that occurs in the conventional sliding mode control system and causes a negative impact on the earthquake-exited structure.

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The robustness of FSMC against the structural and nonstructural uncertainty is better than the conventional sliding mode control [39]. A new energy-based technique which is the combination of energy balance equation and LQR is presented by Alvina San and Moharrami (2006) to select an appropriate gain matrix and reduce the control force along with the structure reactions. The new method has better outcomes compared with the other LQR studies available in the literature in terms of displacement and control force [40]. The results of the PID controller, which is easy to implement and widely used, and sliding mode control systems with non-chattering are discussed by Guclu (2006). SMC with non-chattering is superior to PID controller in the reduction of the amplitudes in case of earthquake [41]. In another study by Guclu and Yazıcı (2008), fuzzy logic controller (FLC) and PD controller are performed to cope the destructive effect of earthquake. In this case, FLC is illustrated better performance than PD to decrease the structural responses [42]. The dynamic fuzzy wavelet neuromodulator which is responsible for the prediction of the future structural reactions and floating-point genetic algorithm which calculates the optimum control force are combined by Jiang and Adeli (2008) to investigate nonlinear behavior (material or geometrical nonlinearity or both) of three dimensional (3D) structures. The structural reactions of actively controlled structures with plan and elevation irregularities are reduced with this new control methodology under various earthquake loads [43]. A simple active control algorithm is implemented by Aldemir (2010) for the active-controlled structure in order to minimize the performance index or to calculate the optimum control force. A comparison of the earthquake-induced controlled vibrations of structures shows the superiority of this algorithm over the classical linear optimal control [44]. H∞ direct output feedback control for an irregular structure excited by earthquake is utilized considering soil-structure interaction to emphasize the effects of the different types of soil conditions. The structural reactions are decreased, but this decrease is less influenced in the soft soil type model than in the fix-based model [45]. Artificial neural networks, genetic algorithms and sliding-mode control have been integrated by Yakut and Alli (2011) to create a suitable controller for the earthquakeresistant structure with active tendons located on the first floor. The structure has exhibited high performance under earthquake records with different characters [46]. Fuzzy controller (FC), hedge-algebras-based fuzzy controller (HAFC) and optimal hedge-algebras-based fuzzy controller (OHAFC) are used in the structure to suppress the vibration of the structure because of the ground motion such as earthquake via active tendons. Using OHAFC has produced a superior outcome to the other controller concerning the control performance [47]. A case study is made by Nigdeli and Boduroglu (2013) for different active tendon orientations using a PID controller, whose parameters are tuned with a numerical algorithm. The vibration amplitudes of torsionally irregular structures which are controlled by active tendons are mitigated under the impulsive motions known as near-fault ground motions [48].

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Active control of structures according to different feedback strategies such as velocity, displacement and acceleration using PID controller are provided by Nigdeli (2014) to evaluate the structural responses under earthquake excitations. It has emerged that all feedback types are sufficient in terms of robustness due to change of structural features, but acceleration feedback is one step ahead [49]. The block pulse function is employed by Ghaffarzadeh and Younespour (2014) for a shear building model with active tendons in order not to solve Riccati matrix equation as in the LQR algorithm and accordingly to reduce the computational effort of analytical approaches. It has been observed that the proposed method has less computation time and fewer structural responses in comparison with LQR [50]. Studies on structural control with neural controller techniques are expanded by Bigdeli and Kim (2014) examining the nonlinear behavior of three-dimensional structures. The observable structural reactions are obtained in the shorter training time [51]. A wavelet-based pole assignment (WPA) controller which is created from the modification of conventional pole assignment (PA) controller is designed by Amini and Samani (2014) to offer the protection in active-controlled structure against the far and near-fault ground motions. More successful results are obtained from WPA compared to LQR [52]. An adaptive model-space reference-model tracking fuzzy control technique (MRFC) is evolved by Park and Ok (2015) to obtain the target structural responses in case of some activator failures during earthquake. The proposed method is adaptive and robust to realize its mission both in the normal operation and in case of some activator failures [53]. In the paper of Sun et al. (2015), a benchmark model for active-controlled structure using wireless sensor network (WSN) is introduced as an alternative method to alleviate the structural reactions. It stated that the model is a good example for the wireless network-structural system in structural engineering considering the results of the evaluation criteria [54]. A new PDPI + PI type fuzzy logic controller is applied to the structures with ATMD to absorb the vibration against wind or earthquake excitation. From the experimental and simulation results that are compatible with one another, it is concluded that the proposed method is useful to effectively control the structures [55]. The effect of soil type on seismic responses of irregular structures equipped with active tendons is investigated by Nazarimofrad and Zahrai (2016) using LQR controller. The effect of active tendons on structural reactions is low if the structure is located on a soft soil condition [56]. The stability and performance of decentralized LQR based networked switched control technique proposed by Bakule et al. (2016) is checked for active-controlled structures subjected to earthquakes. It is emphasized that this new control method can be appropriate to use in wireless structural controls in structural engineering [57]. A multiobjective genetic algorithm is used in the study of Nazari Mofrad (2018) to optimize the placement and number of active tendons in 2D and 3D active-controlled

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structures considering SSI effect. In these structures with an LQR controller, both the cost of active control devices and structural reactions are reduced [58]. Six different metaheuristic algorithms such as BEE algorithm, firefly, harmony search, BAT algorithm, and imperialist competitive algorithm are used to improve the performance of LQR controller in active-controlled structures. The optimization process and results of the objective function of six different algorithms and their effect on the structural responses are interpreted by Katebi et al. (2020) under several historical earthquakes [59]. Metaheuristic-tuned PID controller is proposed by Ulusoy et al. (2020) for the active tendon control of structures considering SSI under 56 near-fault ground motions with and without a pulse. The metaheuristic tuned PID controller reduces the structural reactions of all earthquake records and shortens the computing time considerably [60]. The structures with ATMD using PID controller via modified harmony search algorithm is tested by Kayabekir et al. (2020) to obtain the optimum parameters of PID controller and enhance the effectiveness of the structures with ATMD against the near-fault ground motions. While controlled structural displacements in the case of tuned mass damper (TMD) decreased by 31.22%, these values were reduced by 53.71% in the case of ATMD [61]. In the event of unexpected time delay in the active tendon controlled structures using metaheuristic-tuned PID controller and the reaction of the controller to this situation is examined by Ulusoy et al. (2021) to verify the robustness of structures. Different control limits and time delay values (a case study) should be taken into account to calculate PID control parameters of structures using metaheuristic algorithms and to assess the case [62]. Harmony search algorithm and Flower pollination algorithm have been added separately to PID controller to optimize the objective function and determine the suitable feedback strategies such as displacement, velocity, acceleration in structures with active tendons (on each floor or only the first floor). The performance of optimum tuned PID controller with acceleration feedback on earthquake excited structures is better than the other feedback strategies [63].

4 Conclusion In the present study, the type of active control in structures, active control techniques and a review of the studies on the active structural control are summarized. The conclusions from this study are as follows: • Active structural control performs better in reducing structural reactions than passive structural control. For this reason, it may be more appropriate to use active control systems against the near-fault ground motions. • Some of the issues to be considered in the application of active control are as follows: prediction of unknown ground motion, time delay effect, control force

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capacity, feedback strategy, geometric and material nonlinearity, soil-structure interaction, irregularity of structures, type of ground motions, structural uncertainties, calculation time, installation cost and obtaining the optimum parameters of the controller. • Various control techniques have different problems mentioned above in real structural applications and each of them has a different effect on structural reactions. However, the extended studies attempt to eliminate the problems in structural control.

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Metaheuristics-Based Optimization of TMD Parameters in Time History Domain Melda Yücel, Gebrail Bekda¸s, and Sinan Melih Nigdeli

Abstract In structural engineering, one of the basic engineering problems is to equalize by decreasing and damping vibration movements, which may cause to hurt of living creatures for reasons as collapse, failure, and to damage any structural design (television towers, business centers, hospital buildings, etc.) with the effect of various dynamic forces as powerful water waves and wind, vehicle traffic, high sound together with structural responses (displacement, acceleration, velocity, etc.) that they may be arisen owing to these factors. In this study, for ensuring the most proper namely optimum design of tuned mass dampers (TMDs), which is a passive control system that benefited from damping vibration forces affected to structures, metaheuristic algorithm was benefited that is an optimization technique frequentlypreferred in many scientific areas, and designed based on activities of living creatures like hunting, feeding, communication with congenerous, etc., which provide the surviving in natural life, besides some genetic properties, together with formation mechanisms of various chemical and physical processes, or capabilities based on memory. With this context, optimum TMD designs were obtained by using flower pollination algorithm. While these processes were realized, minimization of critical displacements in the time-history domain was made possible by providing to obtain optimum design parameters of TMDs located on the top story of structures. Keywords Tuned mass dampers (TMDs) · Optimization · Time history domain · Metaheuristic algorithms · Flower pollination algorithm

M. Yücel (B) · G. Bekda¸s · S. M. Nigdeli Department of Civil Engineering, Istanbul University-Cerrahpa¸sa, 34320 Avcılar, Istanbul, Turkey G. Bekda¸s e-mail: [email protected] S. M. Nigdeli e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. Bekda¸s and S. M. Nigdeli (eds.), Optimization of Tuned Mass Dampers, Studies in Systems, Decision and Control 432, https://doi.org/10.1007/978-3-030-98343-7_4

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1 Introduction Many different alternative scenarios can be improved about solving any engineering problems and designing the model. However, besides that the problem can be resolved, there is an important issue, which is how the mentioned problem is analyzed and the obtained benefits in this process. In this respect, design rules determined for the target problem will be guiding sources in the analysis/resolution phase of the problem. The way, method and approaches, which are followed to reach the targeted main design with fulfilling these mentioned rules, make it possible to realize a solution within the most convenient/best namely optimum way. On the other hand, optimum resolution process can be modeled in the way of providing a design with the most proper/low-priced, finding of structural parameters minimizing the weight, fictionalizing of target project in the direction of increasing the profit amount, supplying the safety of livings and also structures by ensuring sufficient strength across to dynamic forces such as earthquake. On the other side, about the problem of modeling any design as optimal, it requires various stages that the issue based on reaching the main purpose determined or goal function by fulfilling some conditions that are considered necessary. But, the complexity of the dealt problem prevents researching each type of possible solution or solution combination, and consequently, finding correct and convenient alternatives turn into the main purpose in an acceptable namely reasonable time concerning the problem. Nevertheless, in designs with multi number members, the mentioned optimization process may be necessitated long times and thus high costs. In this regard, the consideration of randomness and possibility cases in behaviors of metaheuristic methods, which are more advanced and differentiated versions of heuristic approaches that one of the frequently used in optimization applications, can be a solution for various applications and problems. In this regard, it can be recognized that numerous applications have been realized in many different engineering areas. For example, in biomedical engineering, genetic algorithm (GA) [1], cuckoo search (CS) [2, 3], shark smell optimization (SSO) [4], six different metaheuristics including particle swarm optimization (PSO), artificial bee colony (ABC), firefly algorithm (FA), spotted hyena optimization (SHO), etc. [5] were utilized to improve the quality of biomedical images. Also, to detect some diseases/disorders that occurred in different bio-organisms, ant colony optimization (ACO), bacterial foraging optimization (BFO), krill herd (KH) algorithm together with FA and CS [6], flower pollination algorithm (FPA) besides crow search algorithm (CSA) [7] and also five metaheuristic algorithms consisted from grey wolf optimization (GWO), ant lion optimization (ALO), butterfly optimization algorithm (BOA), dragonfly algorithm (DA) with satin-bird optimization (SBO) [8] have been benefited. Furthermore, within the field of electrical and electronics engineering, to optimal adjust the reaction power flow by minimizing the power losses within systems, different versions of PSO and ABC [9], symbiotic organisms search (SOS) [10],

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PSO, FA, ABC, ACO and differential evolution (DE) [11], stochastic fractal search (SFS) algorithm [12] were handled during optimization processes. As to the civil engineering area, different and several optimization applications have been carried out to design the best models in the direction of providing efficient water network systems [13–15], determining the safest soil conditions/parameters [16, 17] or generating the most convenient structures for highways, bridges, buildingtype models (skyscrapers, hostels, television towers, dams, etc.) [18–21]. For example, various optimization studies were handled as optimum modeling of steel trusses by minimizing the structural weight or cost. In the year 2017, Kaveh and Ghazaan used a metaheuristic algorithm to optimize the mentioned truss structures to improve their dynamic performance under frequency constraints [22]. This is the reason why they developed a method called vibrating particles systems (VPS) by inspiring from dynamic behavior of structures. Also, by Javidi et al., a study was achieved to find the optimum results occurred via providing that truss structures with minimum weight by using three variations of CSA [23]. As a different example model to these structures, optimally arranging of bracings on steel frames under seismic loading was conducted by Gholizadeh and Ebadijalal through benefiting from a recently developed metaheuristic algorithm, which is named as center of mass optimization (CMO) [24]. Nevertheless, seismic protection of structures can be provided via optimization approaches by designing the best modeling of different vibration control devices such as base isolation systems, bracings, damper types (tuned mass, tuned liquid, visco-elastic, column dampers), etc. However, it can be seen that the most-applied field of them has been realized for tuned mass dampers (TMDs). As an example of this, Farshidianfar and Soelili aimed to generate an efficient damping mechanism to decrease the maximum displacement and acceleration, which are arisen in stories of a high-rise building where two different historical earthquake excitations have an impact as Tabas and Kobe. In this direction, optimum TMD parameters are provided to determine the usage of ACO by considering soil-structure interaction in the mentioned study [25]. Nigdeli and Bekda¸s also studied multiobjective optimum TMD design for a shear building to decrease the first story displacement and also absolute acceleration of top story by using HS [26]. On the other hand, by Etedali and Mollayi, CS method was used to determine the best parameter values defined as frequency and damping ratio for TMD within a single degree of freedom (sdof) system, where three different dynamic forces including harmonic base acceleration, white noise external force together base acceleration (with white noise) affect, and a prediction process were realized through the usage of obtained data with the least squares support vector machine technique [27]. Also, Yucel et al. realized a combined application to generate an optimized TMD device to design the best vibration system. For this respect, flower pollination algorithm (FPA) and artificial neural networks (ANNs) were employed simultaneously and a prediction model was created to determine TMD parameters [28]. When in some cases, intended for damping of undesired vibrations, there may be no sufficient structural area for the reaction given by TMD, namely lengthening of spring. A study where this case is handled was realized by Liu et al., and the amount of spring lengthening of TMD is tackled as a design constraint

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to provide effective TMD control. In this regard, GA is benefited from the generation of optimum TMD design for a 167 m length tower structure [29]. Kayabekir et al. applied an optimization approach by using an adaptive model of HS to a real-size structure to find the best design of TMD and also the stroke capacity of TMD was controlled [30]. In this chapter, an optimization methodology is given for optimum tuning of TMD positioned on top of structures. The employed algorithm is the flower pollination algorithm (FPA) and the optimum results are presented for passive control of sdof structures as numerical examples.

2 Flower Pollination Algorithm The main foundation where population-based metaheuristic algorithms were created that they are one of the most frequently preferred optimization methods, is vital activities or some properties of any species together with a colony, group, namely community where these species are connected. For example, an ant targets to find and use the shortest one among many alternative ways while foraging; also, a bee aims to direct towards and benefit from the flower where the amount of food (nectar) is the richest. In this regard, as in the mentioned samples, the flower pollination algorithm (FPA), which is one of the population-based algorithms developed by inspiring from many creatures and natural vital behaviors of the community where they live, was developed in 2012 by Xin-She Yang based on pollination activity that is a natural process belonging flowery plants [31]. This property, whose FPA belongs to it, is based on the pollination activity of flowery plants, which can be also realized by wind besides some insect or bird species in nature known as pollinators/pollen carriers. In this process, it can be possible that the mentioned pollinators direct to plants by sourcing from flowery plants, which have nice/aromatic smells together with beautiful, attractive and shiny colors, and thus, pollination arising as a reproduction process can occur. Pollination can realize in two ways as cross and own self. Where, while cross-pollination can arise between two different flowers besides two independent members of a single flower via pollinators, self-pollination occurs between pollination organs, which exist in the self-structure of one species flower. These two pollination types are determined with the help of an algorithm parameter specific to FPA and known as switch probability.

3 Numerical Examples The optimization problem aims to find the best set of TMD parameters that reduces the responses of the structure. In the presented methodology, the objective function is the minimization of the maximum displacement of the top story of the structure where TMD is positioned as seen in Fig. 1. The tuned parameters are related to mass

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Fig. 1 Shear building with TMD

xd kd md

cd

xN

mN cN kN

xi mi ci ki

x1 m1 c1 k1

..

xg

(md ), stiffness (kd ) and damping (cd ) of TMD. Generally, the optimum TMD mass is the maximum. The mass must be limited according to the axial loading capacity of the structure. For better and rapid optimization, the mass is proposed to be taken as a design constant. Related to kd and cd as seen in Eqs. (1) and (2), period (Td ) and damping ratio (ξd ) of TMD are proposed to be taken as design variables.  Td = 2π

md kd

cd 

ξd =

2md

kd md

(1) (2)

The design constants are the structural properties of N-story structure including story masses (mi for i = 1 to N), stiffness (ki for i = 1 to N) and damping coefficient (ci for i = 1 to N). The stiffness and damping properties of SDOF structures can be also defined as period (Ts ) and damping ratio (ξs ) as given in Eqs. (3) and (4).  Ts = 2π

m1 k1

(3)

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c1 

ξs =

2m1

(4)

k1 m1

The design constants also include the ranges of design variable, algorithm parameters and the earthquake data of ground motion (¨xg ). As known, the optimum period of TMDs is close to the period of the structure. For that reason, the range of Td can be 0.5 and 1.5 times of Ts for lower and upper bounds of the range. For ξd , it is defined according to economical conditions. All algorithms have conducted and specific parameters. The population number (p) and iteration number (t) are generally used in all metaheuristic-based algorithms. The employed FPA has a switch probability (sp) to close from one of the optimization types. As the earthquake data, it is useful to select a high number of records for a general optimum solution. For that reason, the numerical examples of this chapter are found by using the set of far-fault around motions given in Table 1. These records are listed in FEMA P695 [32]. Table 1 The earthquake record information [32] Earthquake number Date Name

Component 1

Component 2

1

1994 Northridge

NORTHR/MUL009

NORTHR/MUL279

2

1994 Northridge

NORTHR/LOS000

NORTHR/LOS270

3

1999 Duzce, Turkey

DUZCE/BOL000

DUZCE/BOL090

4

1999 Hector Mine

HECTOR/HEC000

HECTOR/HEC090

5

1979 Imperial Valley

IMPVALL/H-DLT262 IMPVALL/H-DLT352

6

1979 Imperial Valley

IMPVALL/H-E11140

IMPVALL/H-E11230

7

1995 Kobe, Japan

KOBE/NIS000

KOBE/NIS090

8

1995 Kobe, Japan

KOBE/SHI000

KOBE/SHI090

9

1999 Kocaeli, Turkey

KOCAELI/DZC180

KOCAELI/DZC270

10

1999 Kocaeli, Turkey

KOCAELI/ARC000

KOCAELI/ARC090

11

1992 Landers

LANDERS/YER270

LANDERS/YER360

12

1992 Landers

LANDERS/CLW-LN

LANDERS/CLW-TR

13

1989 Loma Prieta

LOMAP/CAP000

LOMAP/CAP090

14

1989 Loma Prieta

LOMAP/G03000

LOMAP/G03090

15

1990 Manjil, Iran

MANJIL/ABBAR–L

MANJIL/ABBAR–T

16

1987 Superstition Hills SUPERST/B-ICC000

SUPERST/B-ICC090

17

1987 Superstition Hills SUPERST/B-POE270 SUPERST/B-POE360

18

1992 Cape Mendocino

CAPEMEND/RIO270 CAPEMEND/RIO360

19

1999 Chi-Chi, Taiwan

CHICHI/CHY101-E

CHICHI/CHY101-N

20

1999 Chi-Chi, Taiwan

CHICHI/TCU045-E

CHICHI/TCU045-N

21

1971 San Fernando

SFERN/PEL090

SFERN/PEL180

22

1976 Friuli, Italy

FRIULI/A-TMZ000

FRIULI/A-TMZ270

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Fig. 2 The Matlab-Simulink block diagram

Since the aim is the reduction of the structural displacement, the maximum amount of displacement of structures without TMD must be previously known. In the methodology, the shear building equation of motion given in Eq. (5) as matrix form is modeled via Matlab and Simulink [33]. It is modeled as Fig. 2 with matrix and vector operations to find structural responses. M¨x(t) + C˙x(t) + Kx(t) = −M{1}¨xg (t)

(5)

In Eq. (5), the matrices of mass (M), damping (C) and stiffness (K) are given as Eqs. (6), (7) and (8), respectively. Also, the displacement vector (x(t)) given as Eq. (9). M = diag[m1 m2 . . . mN ] ⎡

(c1 + c2 ) −c2 ⎢ −c (c2 + c3 ) 2 ⎢ ⎢ . ⎢ ⎢ C=⎢ . ⎢ ⎢ ⎢ ⎣ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ K=⎢ ⎢ ⎢ ⎢ ⎣

(6) ⎤

−c3 . . . . .. .

(k1 + k2 ) −k2 −k2 (k2 + k3 ) −k3 . . . . . . .. .

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ . −cN ⎦ −cN cN ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ . −kN ⎦ −kN kN

(7)

(8)

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x(t) = [x1 x2 . . . xN ]T

(9)

For all earthquake excitations, the simulations of dynamic analysis are done and the responses under the record with the most impact on the displacement are saved. Then, as with all metaheuristic-based optimization processes, an initial solution matrix is generated for design variables. These values are randomly chosen within the solution range. For all sets of solutions, the dynamic analyses are done for all excitations and the maximum one is saved for comparison of candidate solutions. In the dynamic analyses, the matrices and displacement vector are updated as Eqs. (10) to (13). M = diag[m1 m2 . . . mN md ] ⎡

(c1 + c2 ) −c2 ⎢ −c (c2 + c3 ) 2 ⎢ ⎢ . ⎢ ⎢ C=⎢ . ⎢ ⎢ ⎢ ⎣ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ K=⎢ ⎢ ⎢ ⎢ ⎣

(10) ⎤

−c3 . . . . .

(k1 + k2 ) −k2 −k2 (k2 + k3 ) −k3 . . . . . . .

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ . ⎥ −cN (cN + cd ) −cd ⎦ −cd cd ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ . ⎥ −kN (kN + kd ) −kd ⎦ −kd kd

x(t) = [x1 x2 . . . xN xd ]T

(11)

(12)

(13)

Also, the problem is constrained with Eq. (14). It is related to the stroke limitation of TMD. It is limited by using a st_max value and it is taken as 1.5 in the numerical examples. 

max |xN +1 − xN | with TMD ≤ st_ max max[|xN |]without TMD

(14)

After the generation of initial solutions, the iterative optimization stages are started. The solution matrix is updated by a generated new set of solutions and the dynamic analysis is done. Equation (14) is also checked. If Eq. (14) is not provided for both new and existing solutions, the one with the minimum value of Eq. (14) is saved as better. If one of them provides the constraint requirement, it is saved as the

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best one. If both solutions provide the rule of Eq. (14), the one with less maximum displacement is saved. The optimization methodology can be updated to any metaheuristic algorithm. The iterations will be done for several iterations. For example, FPA uses two types of optimization. These are chosen according to sp and the formulations for updating a design variable are given for global and local optimization are given as Eqs. (15) and (16), respectively.

= xti + L xti − g∗ xt+1 i

(15)

  xt+1 = xti + ∈ xtj − xtk i

(16)

Equation (15) uses a Levy distribution (L), while a linear distribution (a random number between 0 and 1 as ε) is used in local pollination. In equations, xi t+1 represents the newly generated solution which is updated by using existing solutions; xi t . xj t and xk t are two randomly chosen exiting solutions.

4 Numerical Examples In order to generate optimal TMD designs with the aim of the usage in SDOF structures, several combinations were created via different structural parameters. All of the optimal values for design variables were provided with FPA in the way of minimization of critical namely maximum displacement (Table 2). Table 2 Optimization results for different structural cases Case

Mass ratio (μ)

Structural Damping period (Ts ) ratio of structure (ξs )

Optimum TMD damping ratio (ξd )

Optimum TMD period (Td )

Maximum displacement with TMD (x1 with TMD )

Maximum displacement without TMD (x1 without TMD )

1

0.05

0.5

0.03

0.1064

0.6107

0.1035

0.1678

2

0.3

2

0.05

0.1189

2.5114

0.2691

0.4842

3

0.01

4

0.2

0.2851

6.0000

0.8142

0.8235

4

0.4

2

0.1

0.1680

2.5513

0.2964

0.3857

5

0.05

3

0.03

0.1676

2.9509

0.7586

1.0673

6

0.35

1.5

0.2

0.0500

1.5505

0.1180

0.2014

7

0.4

5

0.05

0.0731

7.1882

0.9157

2.1884

8

0.25

0.5

0.03

0.0821

0.6052

0.0516

0.1678

9

0.2

4

0.2

0.2885

6.0000

0.7558

0.8235

10

0.01

3

0.1

0.1223

4.1690

0.6106

0.6219

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Fig. 3 The time-domain history for 7th case

In Fig. 3, time-domain history was shown for the 7th case by the consideration both TMD usage and not.

5 Conclusion As a result, it can be seen from Table 2, the optimization approach was found in the direction of generation extremely successful dynamic analyses. The cause of this is to determine minimum displacement values that are smaller than the case without TMD. Also, optimal periods for TMD can be accepted properly according to each structural period in terms of closeness to them. Additionally, it can be recognized from Fig. 3 for a specific case namely 7th, time-domain history, which shows the changing of displacement values along increasing time, reflect the efficiency of optimization applications in the context of that displacement amounts decreased with the usage of TMD in comparison to the initial case of structure (without TMD).

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Robust Design of Different Tuned Mass Damper Techniques to Mitigate Wind-Induced Vibrations Under Uncertain Conditions Javier Fernando Jiménez-Alonso, Jose Manuel Soria, Iván M. Díaz, and Andrés Sáez Abstract Different tuned mass damper techniques have been widely designed and installed to improve the dynamic response of buildings and civil engineering structures when subjected to wind action. The main reason stems from the fact that such vibration absorbers exhibit a good balance between their cost and their effectiveness in mitigating the wind-induced vibrations. Notwithstanding, the appropriate design of these control systems must account not only for the wind loading, but also for the uncertainty associated with the variation of the operational, environmental, degradation and damage conditions of the structure. Multiple approaches have been presented and formulated in order to assist engineering practitioners in the robust design of these control devices. In this chapter, the most recent trends in this research field are presented, whilst a practical method for the robust design of several control devices is further proposed and implemented. To this end, two approaches (namely, probabilistic and fuzzy) are considered to simulate the uncertainty. Additionally, a numerical case-study is included in order to illustrate and analyze in detail the performance of the current proposal.

J. F. Jiménez-Alonso (B) · A. Sáez Department of Continuum Mechanics and Structural Analysis, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, Seville, Spain e-mail: [email protected] A. Sáez e-mail: [email protected] J. M. Soria Department of Signal Theory and Communications, Escuela Politécnica Superior, Universidad de Alcalá, Alcalá de Henares, Spain e-mail: [email protected] I. M. Díaz Department of Continuum Mechanics and Structures, E.T.S. Ingenieros de Caminos, Universidad Politécnica de Madrid, Canales y Puertos, Madrid, Spain e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. Bekda¸s and S. M. Nigdeli (eds.), Optimization of Tuned Mass Dampers, Studies in Systems, Decision and Control 432, https://doi.org/10.1007/978-3-030-98343-7_5

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Keywords Structural control · Structural optimization · Tuned mass dampers · Wind-induced vibrations · Probabilistic approaches · Fuzzy approaches · Uncertain conditions

1 Introduction The dynamic response of current buildings and civil engineering structures is normally sensitive to the wind action together with the uncertainties associated with the modification of their inherent modal properties [1]. Therefore, three alternatives have usually been employed to mitigate the wind-induced vibrations in these constructions [2]: (i) to improve the aerodynamic properties of the construction to reduce the wind force coefficients; (ii) to modify the structural design either increasing construction mass to reduce the air/structure mass ratio or increasing the stiffness or natural frequency of the structure in order to reduce the non-dimensional wind-speed; and (iii) to increase the energy dissipation capacity via the installation of external damping devices. Due to its greater applicability to both existing structures and new construction, the third alternative is being imposed especially for singular constructions [3–6]. In this manner, different damping devices have been widely used to mitigate robustly windinduced vibrations in buildings and civil engineering structures [7, 8]. Among the different control systems, tuned mass dampers (TMD) have been usually employed due to their good balance among performance, cost and easy installation [9]. A TMD is a mechanical device in which the natural frequency, f d [Hz], of the vibration absorber is tuned to the natural frequency, f s [Hz], of the vibration mode of the structure to be controlled [10]. Thus, the effectiveness of this control device is maximized when the tuning is achieved, and therefore, its performance is degraded if any modification of the operational, environmental, degradation and damage conditions originates its detuning [11]. Different techniques have been proposed to guarantee an adequate performance of TMDs during their overall life cycle [11]. Despite the different internal mechanisms, which govern their behavior, all these techniques share the common objective of guaranteeing an adequate performance of the vibration device when the modal properties of the structure vary [12]. According to their constitutive laws, the TMD techniques may be classified in three groups [11]: (i) active TMD (ATMD) [13]; (ii) semi-active TMD (STMD) [14]; and (iii) passive TMD (PTMD) [15]. Each TMD technique uses the following mechanisms to achieve the abovementioned purpose: (i) the driving force generated by an actuator in the case of an ATMD; (ii) the variation of its stiffness and/or damping in the case of a STMD; and (iii) the selection of a robust mechanical parameters [16] or the design of an adaptive device [17] for the case of the PTMD. Although ATMD and STMD have shown a higher performance than PTMD when they are used to mitigate vibrations under uncertain conditions [18], they have a clear limitation when they are employed to control the dynamic behavior of a structure during its overall life cycle

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since their performance is influenced by the reliability of the external power supply [10]. A hybrid strategy [19] is usually considered to overcome this limitation. Thus, the robust performance is achieved by combining a passive and a smart (active or semi-active) technique. During the last fifty years, the scientific community has made a great effort to assist structural and control engineers in the design of control devices, proposing and formulating multiple control algorithms under stochastic conditions [20]. Among these proposals, the optimum design of these damping systems has paid a special attention [21]. This design problem is normally formulated considering two different objectives [22]: (i) to find the optimum placement of the control device; and (ii) to find the optimum sizing of the control systems. Herein an optimum placement of the vibration absorbers is assumed and the study focused on the sizing problem. For this purpose, two types of method can be used: (i) analytical approaches [23] and; (ii) numerical approaches [24]. Among these approaches, the current trend is the use of the numerical approaches, based on the theory of structural optimization [25], for the robust design of these control systems. In spite of the high efficiency of these algorithms [26], all they share a common drawback, they are usually developed for a particular control device, being difficult to set a common framework which simplifies its practical implementation in real-world applications and allow performance comparison among different devices. In order to shed light on this problem, the motion-based design (MBD) method has been recently presented and further implemented to set this common framework [12]. The proposal is a particular case of the more general performance-based design method [27] in which the design requirements are defined in terms of the fulfilment of a performance level of the structure [28]. This chapter is focused on the fulfilment of the fatigue requirements as they are recommended by the European codes [29]. According to this method [10], the design problem may be formulated either as a constrained single-objective optimization problem or a multi-objective optimization problem [30]. Herein the first approach has been considered due to the good balance between its accuracy and its easy implementation when the method is adopted to solve the abovementioned sizing problem. For this purpose, the following aspects must be considered: (i) the design variables are the mechanical parameters which govern the structural behavior of the control systems; (ii) the single-objective function is defined in terms of the cost of the control systems [25]; (iii) the equality constraints are defined in terms of the required form of the frequency response function of the system (structure equipped with the control system); (iv) the inequality constraints are defined in terms of the design requirements (the fulfillment of the performance fatigue damage state of the structure) established by the designer [28]; (v) a search domain is included to guarantee the physical meaning of the solution obtained; and (vi) a natureinspired computational (NIC) algorithm is usually considered as global optimization method due to the non-linear relationship between the inequality constraints and the design variables [31]. Consequently, this design method may be easily adapted to take into account the uncertainties associated with both the variability of the wind action [32] and the

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modification of the modal properties of the structure originated by the abovementioned uncertain conditions [33]. Thus, the numerical quantification of these uncertainties is usually performed by one of these two approaches [34]: (i) a probabilistic approach [35]; and (ii) a fuzzy approach [36]. In order to implement both methods, the design requirements (required damage state) must be re-formulated respectively under a probabilistic or fuzzy approach. Accordingly, this re-formulation allows guaranteeing a robust design for the different control systems [37]. One of the main effects of these uncertain conditions is the detuning between the dumping device and the main structure reducing the performance of the vibration absorber [11]. Two alternatives are usually considered to overcome this limitation [10]: (i) the modification of the mechanical parameters of the control device in order to achieve its adequate adaptation to the new design conditions [38]; and (ii) to design the control system with such robust mechanical parameters that an adequate behavior is guaranteed regardless of the variability of the modal properties of the structure [30]. For our particular case, the second alternative can be easily implemented, since the uncertainty associated with the mentioned phenomena occurs in a different temporalscale, so both phenomena (the variability of the wind action and the variability of the modal properties of the structure) can be simulated independently [20]. Therefore, this chapter focuses on the robust design of different TMD techniques (ATMD, STMD and PTMD) when these vibration absorbers are used to mitigate wind-induced vibrations in buildings and civil engineering structures. For the sake of clarity and conciseness, a numerical case-study has been included to illustrate the implementation of the proposal. Thus, the dynamic response of a steel chimney has been controlled when it is subjected to vortex-shedding-induced vibrations. The mentioned common framework [12] has been implemented for the robust design of these vibration absorbers. The two mentioned approaches (probabilistic and fuzzy) have been taken into account. The results obtained in this study can be easily extrapolated to both other types of control systems and other types of civil engineering structures. Thus, the main purposes of this chapter are: (i) to present the general formulation of the problem about the robust design of different TMD techniques when these vibration absorbers are used to mitigate wind-induced vibrations in buildings and civil engineering structures; (ii) to highlight the influence of the uncertainty on the design of these control devices; and (iii) to show the pros and cons between the approaches considered to simulate the uncertainty. The chapter is organized as follows. In Sect. 2, a TMD–structure interaction model is formulated and further implemented for the numerical assessment of the dynamic response of controlled buildings and civil engineering structures under wind action. Later, the MBD method under uncertain conditions is presented and further adapted for the design of the different TMD techniques in Sect. 3. Subsequently, a numerical case-study, based on a vibration-controlled steel chimney, is illustrated in Sect. 4. The performance of three abovementioned TMD techniques, designed according to the proposed method, is analyzed in detail. Finally, in Sect. 5, some concluding remarks have been included to close the chapter.

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2 Tuned Mass Damper-Structure Interaction Model Subjected to Wind Load An interaction model is needed to be formulated to determine numerically the dynamic behavior of both a building and a civil engineering structure equipped with a control system. For our particular case, a TMD-structure interaction model has been developed. For this purpose, the following assumptions have been considered herein [12]: (i) the behavior of the structure is simulated via a single vibration mode (modal coordinates), hence it is assumed that only a vibration mode is prone to vibrate (however the formulation can be easily generalized to the case in which multiple-vibration modes are excited); (ii) the TMD is modelled via a single degree of freedom system (physical coordinates); (iii) the TMD is modelled in a general way (ATMD), thus the interaction model allows simulating any TMD technique just removing the required element of the equation of motion; (iv) the wind load is simulated by an equivalent harmonic load according to the recommendations of the European codes [39] (only the dynamic effects of vortex-shedding vibrations have been considered herein); and (v) the TMD is located at the antinode of the considered vibration mode. In order to illustrate the configuration of the different TMD-structure interaction models. Figure 1 shows the main mechanical parameters considered to simulate the dynamic behavior of each TMD together with the main structure. In order to represent the structure, a generic building has been considered. In this manner, a TMD, considering its most general configuration (ATMD), is formed by four elements [11]: (i) a sprung mass, m d [kg]; (ii) a viscous damper (dashpot), characterized by its damping coefficient, cd [sN/m]; (iii) a spring, characterized by its stiffness coefficient, kd [N/m]; and (iv) an actuator characterized by its equivalent driving force, f d (t) [N]. As it is also illustrated in Fig. 1, the control force generated by each TMD technique may be decomposed in different terms: (i) a spring, a damper and actuator for the ATMD; (ii) a spring and an actuator (which only applies control forces when energy is dissipated and/or storages by the device) for the STMD; and (iii) a spring and a damper of the PTMD. Although different models have been proposed [40] for the simulation of a STMD (via the modification of its stiffness or damping), herein the semi-active behavior has been simulated via a virtual actuator, f d,mod (t) which models the force generated by a variable damping dashpot (for instance a magnetorheological damper [41]). The equations of motion of the TMD-structure interaction model can be obtained via the application of the second Newton’s law to the two masses, the vibration controller and the equivalent modal mass. The resultant equations may be expressed as follows: m s x¨s (t) + cs x˙s (t) + ks xs (t) + kd (xs (t) − xd (t)) + cd (x˙s (t) − x˙d (t)) = qw∗ (t) − f d (t)

(1)

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Fig. 1 Different TMD techniques in term of their constitutive behavior: a active (ATMD): b semiactive (STMD) and c passive (PTMD)

m d x¨d (t) + cd (x˙d (t) − x˙s (t))+k d (xd (t) − xs (t)) = f d (t)

(2)

where qw∗ (t) = qw (t) · φ T [N] is the projection of the wind load on the considered vibration mode (being qw (t) the wind load [N], φ the considered vibration mode and T the transpose function); x¨s (t) [m/s2 ], x˙s (t) [m/s] and xs (t) [m] are respectively the acceleration, velocity and displacement of the structure; x¨d (t) [m/s2 ], x˙d (t) [m/s] and xd (t) [m] are respectively the acceleration, velocity and displacement of the TMD; and f d (t) [N] is the driving force generated by the actuator of the smart TMD. Equations (1) and (2) can be re-organized in matrix form as follows: 

     ms 0 x¨s (t) x˙s (t) cs + cd −cd + x¨d (t) x˙d (t) −cd cd 0 md        xs (t) 1 ∗ −1 k + kd −kd = + s qw (t) + f d (t) xd (t) −kd kd 0 1

¨ + [C]{x(t)} ˙ + [K ]{x(t)} = {B0 }qw∗ (t) + {Bc } f d (t) [M]{x(t)}

(3) (4)

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   cs +cd −cd ms 0 is the mass matrix; [C] = is the damping where [M] = −cd cd 0 md   k + kd −kd is the stiffness matrix; {B0 } is the input vector assomatrix; [K] = s −k d kd ciated with the pedestrian load; {Bc } is the input vector associated with the driving force; {x(t)} ¨ is the acceleration vector; {x(t)} ˙ is the velocity vector and {x(t)} is the displacement vector. Equation (4) may be converted into a state-space formulation [18] as follows: 

{˙z (t)} = [A]{z(t)} + [B]{u(t)}

(5)

{y(t)} = [E]{z(t)} + [D]{u(t)}

(6)



[0] [I ] [A] = −1 −[M] [K ] −[M]−1 [C]   [0] [B] = −[M]−1 [B0 ]

 (7)

(8)

  where {z(t)} = {x(t)} {x(t)} is the state vector (computed in terms of the relative ˙ displacements, {x(t)}; and the relative velocities, {x(t)}, ˙ of both the structure and the TMD); {u(t)} is the input vector; {y(t)} is the output vector (computed in terms of the displacement experienced by the building); [A] is the state matrix; [B] is the input matrix; [E] is the output matrix; [D] is the feedthrough matrix; and [0] and [I ] are, respectively, zero and identity matrices [18]. Additionally, the definition of the wind load, qw (t) may be performed according to the recommendations of the considered design guidelines (the European code has been considered herein [39]); and the driving force, f d (t), can be computed numerically by a feedback controller [18]. Figure 2 illustrates the general layout of the considered feedback controller. According to this controller, the driving force, f d (t), can be computed in terms of the state vector, {z(t)}, and a gain matrix, [G]. After the implementation of the mentioned controller, the equation system can be transformed as follows: {˙z (t)} = [A]{z(t)} + [B0 ]{qw (t)} − [Bc ]{ f d (t)}

(9)

{˙z (t)} = [A]{z(t)} + [B0 ]{qw (t)} − [Bc ][G]{z(t)}

(10)

{˙z (t)} = ([A] − [Bc ][G]){z(t)} + [B0 ]{qw (t)}

(11)

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Fig. 2 Flowchart for the design of a feedback controller in a state-space formulation

where [B0 ] and [Bc ] are determined from Eq. (8) assuming as pattern load vectors {B0 } and {Bc } respectively. Among the existing algorithms [18] to determine, [G], the linear quadratic regular (LQR) method [18] has been considered herein due to its extensive use for practical engineering applications [12]. According to this algorithm, the following performance-index function, J , is minimized in order to compute the value of the matrix gain, [G]: ts J=

{z(t)}T [Q]{z(t)} + {[G]{z(t)}}T [R]{[G]{z(t)}} dt

(12)

0

where [Q] and [R] are two positive-defined weighting matrices related to the control effectiveness and control effort, respectively; and ts [sec] is the integration time. These matrices may be defined in terms of the mass, [M], and stiffness, [K ], matrices of the proposed interaction model and two weighting factors, αd [-] and βd [-], as follows [18]:  [Q] = αd

 [K ] 0 and [R] = βd [I ] 0 [M]

(13)

Additionally, the driving force, f d (t), determined by the LQR controller, must be adapted for the particular case of a STMD (for instance to simulate the behavior of a magneto-rheological damper). The following relationship, the so-called clipped force method [42], is usually employed for this purpose:

Robust Design of Different Tuned Mass Damper …

⎧ f d (t)x˙r (t) < 0 and | f d (t)| < f max ⎨ sgn( f d (t)) f max f dmod (t) = if f d (t)x˙r (t) < 0 and f min < | f d (t)| < f max fd ⎩ sgn( f d (t)) f min f d (t)x˙r (t) > 0 or | f d (t)| < f min

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(14)

where x˙r (t) [m/s] is the relative velocity between the vibration absorber and the structure, x˙r (t) = x˙d (t) − x˙s (t); sgn() is the sign function; and f min [N] and f max [N] are respectively the minimum and maximum dissipative forces. Thus, Fig. 2 also shows the operating areas of the driving force in terms of both the relative velocity, x˙r (t), and the TMD technique. Finally, the dynamic response of the TMD-structure interaction model under wind action can be obtained by the integration of the above state space system using a Runge–Kutta method as it is implemented in the Matlab suite of software [43].

3 Motion-Based Design Method Under Uncertain Conditions The performance-based method [27] is currently the most widely used approach for the robust design of smart control systems when these damping devices are employed to mitigate the wind-induced vibrations in buildings and civil engineering structures. According to this method, this design problem may be formulated as an optimization problem [25], whose main objective is to find the configuration of the device (design variables) that, minimizing its cost, guarantees the fulfilment of the established design requirements. As, for modern buildings and civil engineering structures under wind action, the most limiting design requirements are usually defined in terms of the required performance level and this level usually depends on the movement of the structure [10], this design process may be denominated as the motion-based design (MBD) method [16]. As required performance level, the fatigue ultimate limit state (FULS) has been considered herein. As one of the possible effect of the wind-induced vibrations on structures is their fatigue damage, the fulfilment of the FULS, as it is defined by the European codes [29], has been considered herein as design requirement. However, additional design requirements can be considered in terms of both the characteristic of the wind action and the modal properties of the structure [32]. The design problem, following the rules of the MBD method, can be formulated either as a constrained single-objective optimization problem or as the combination of two sub-problems (a multi-objective optimization problem together with a decision making problem) [20]. Herein, the first formulation has been considered and described in detail for the sake of simplicity. The method is first presented under a deterministic approach and re-formulated later under a stochastic approach. The re-formulation only implies a new description of the inequality constraints of the problem.

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According to this, the MBD method is based on the general scheme of a constrained single-objective optimization problem [25], which may be expressed as: Minimize f ({θ }) Subjected to

∗ geq, j ({θ }) = geq, j j = 1, 2, . . . , s ∗ g j ({θ }) ≤ g j j = 1, 2, . . . , q

(15)

 l   u θ = {θil } ≤ {θ } = {θ i } ≤ θ u = {θ i } i = 1, 2, . . . , n d ∗ where f ({θ }) is the objective function, geq, j ({θ }) is the jth equality constraint, geq, j is the threshold of the jth equality constraint, s is the number of equality constraints, g j ({θ }) is the jth inequality constraint, g ∗j is the threshold of the jth inequality   constraint, q is the number of inequality constraints, θ l and {θ u } are, respectively, the lower and upper bounds of the vector of design variables, {θ }, and n d is the total number of design variables. In this manner, the MBD problem can be formulated as particular case of a sizing optimization problem in which the mechanical parameters of the control device are considered as design variables [26]. According to the best of authors’ knowledge, a common framework, for the fatigue assessment of steel structures subjected to wind-induced vibrations, is the implementation of the so-called safe life method [44] (as it is recommended by the European code [29]). According to this method, the fatigue assessment of any structural element may be checked following an eight-step procedure [44]: (i) to select the structural element to analyze; (ii) to establish the load event (to define the wind loads on the structure); (iii) to compute the stress history in the structural element for the considered load event; (iv) to perform the cycle counting (the rain-flow counting method has been taken into account herein); (v) to compute the stress range, σ , spectrum from the previous cycle counting; (vi) to compute the number of cycles to failure, Ni ; (vii) to evaluate the cumulated fatigue damage, Ds , according to the Palmgren–Miner’s rule [44]; and (viii) to check the fulfilment of the fatigue limit state. In order to compute either the fatigue strength or the number or cycles to failure, Ni , the S–N curves, as they are defined by the European code [29], have been taken into account herein. The number of cycles to failure, Ni , may be determined as follow, in terms of the corresponding stress range, σi :

⎧  σ 3 C ⎪ ⎪ 6 1.35 D ⎪ 2 · 10 · σ if σi ≥ σ and σ D = 0.737σC ⎪ 1.35 ⎪ i ⎨  σ 5 D Ni = 1.35 L D ⎪ 5 · 106 · σ if σ ≤ σi ≤ σ and σ L = 0.549σ D ⎪ 1.35 1.35 i ⎪ ⎪ ⎪ ⎩ L ∞ if σi ≤ σ 1.35

(16)

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where σC is the reference fatigue strength at 2 · 106 cycles; σ D is the constant amplitude fatigue limit; and σ L is the cut-off limit. Additionally, the cumulated fatigue damage, Ds , may be determined considering the following expression: Ds =

n tot  ni < Dlim Ni i=1

(17)

where n i is the number of cycles corresponding to the stress range, σi ; n tot is the number of stress range classes; and Dlim is the cumulated fatigue damage limit (Dlim = 1 according to the European code [29]). Figure 3 illustrates an example of the implementation of the MBD method for the design of a PTMD when it is employed to control the fatigue damage of a building subjected to a dynamic wind load, qw (t). It is assumed that only the first vibration mode, φ1 (z), is prone to vibrate due to the wind action. Thus, the main objective of this example is to find the optimum parameters of the PTMD (m d , cd and kd ) that, minimizing the cost of the control device (the TMD mass, m d , in this case), guarantees that the maximum cumulated fatigue damage, Dmax , is lower than the allowable cumulated fatigue damage, Dlim . Nevertheless, for the design of the different TMD techniques, the following parameters are normally defined: (i) the mass ratio, μ = m d /m s ; (ii) the frequency ratio, δd = f t / f s (where f t is the natural frequency of the TMD [Hz]; and f s is the natural frequency of the considered vibration mode of the structure [Hz]); (iii) the damping ratio of the TMD, ζd ; [-]; and (iv) the weighting factors, αd [-] and βd [-], which

Fig. 3 Example of the implementation of the motion-based design method for the optimum design of a building equipped with a PTMD

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characterize the driving force, f d (t). The following relationships may be formulated to link these equivalent parameters with the mechanical parameters of a TMD: md = μ · ms

(18)

cd = 4 · m d · π · δd · f s · ζd

(19)

kd = m d · (2 · π · δd · f s )2

(20)

Therefore, four key aspects are considered here to particularize the abovementioned general method: (i) the objective function is defined in terms of the value of the mass ratio, μ, (since the cost of the TMD depends mainly on its mass) [30]; (ii) a NIC optimization algorithm (genetic algorithms) has been selected to solve the optimization problem [31] due to the nonlinear relation between the constraints and the design variables; (iii) the numerical quantification of the uncertainty associated with the variation of the modal properties of the structure is simulated via both a probabilistic and a fuzzy approach [12] (inequality constraints); and (iv) a design criterion is set to constrain the form of the frequency response function of the controlled structure, Hd , (equality constraints). Herein it is assumed that the change of the modal properties of the structure is caused by the modification of the operational, environmental, degradation and damage conditions [33]. On the one hand, according to the probabilistic approach [20], some mechanical parameters which characterize the dynamic behavior of the structure (as the stiffness or the damping) may be considered as uncorrelated random variables which follow a predetermined probabilistic distribution function (normal distributions have considered herein, N (λ, κ), being λ the mean value and κ the standard deviation). In consequence, the response of the structure is equally governed by a joint probabilistic distribution function. In order to estimate this function, the Monte Carlo method may be used [45]. For this purpose, a sample of the possible states of the structure must be generated. The sample size is determined via a convergence study, in order to guarantee that the sample size is large enough to ensure an accurate estimation of the dynamic response of the structure under a preselected significance level. The probabilistic distribution function of the response allows defining the inequality constraints under a probabilistic approach. On the other hand, according to the fuzzy approach [46], the abovementioned mechanical parameters may be considered as uncorrelated fuzzy variables, ξ˜ . Thus, the quantification of the uncertainty (lack of knowledge) about a certain event is computed by considering the possibility that a physical parameter takes a certain value. A membership function, η˜ ξ˜ , may be assigned to each fuzzy variable. The membership function assigns to each fuzzy variable, ξ˜ , a grade of membership ranging between 0 and 1. In this manner, upper grade (grade 1) is assigned to the nominal values, ξn , which means a 100% of belief, and lower grade (grade 0) is assigned to the internal of maximum variability of the considered fuzzy variable,

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which means a 0% of belief. It is usual to considerer as a membership function, ˜ The triangular membership function may be defined as a triangular function, . follows:   ˜ ξn , αl · ξ n , αu · ξn η˜ ξ˜ = 

(21)

where αl is a reduction factor and αu is an incremental factor to establish the variation range of the fuzzy variable, ξ˜ . Based on the associated membership functions, the β-levels, or, β-cuts, may be determined. These levels define the different grades of possibility associated with the fuzzy variables. After certain β-level is set for the different fuzzy variables, the maximum response of the system for this grade of possibility is determined via a combinatorial analysis. A sensitivity analysis must be performed to determine the sampling size for the combinatorial analysis. Thus, the maximum value of the response corresponding to a certain β-cut allows defining the inequality constraints for a fuzzy approach. For the fourth aspect, two types of design criteria may be considered depending on whether equality constraints are included or not in the optimization problem. In the first case, equality constraints are imposed to force the frequency response function of the structure, Hd , to adopt a prescribed shape. As the form of the frequency response function, Hd , is strongly conditioned by the frequency ratio, δd , and damping ratio, ζd , of the TMD, the equality constraints normally act directly on these two parameters, providing thus their values. Among the different proposals [23], the so-called H∞ criterion [16], where frequency response function is adapted to reduce the dynamic response of the structure under a harmonic excitation, has been considered herein. In this way, the formulation of the design optimization algorithm based on the H∞ criterion may be written as: Find{θ } = { μ} Minimizing f (θ ) = μ 1 =0 geq,1 = δd − 1+μ  3μ =0 Subjected to geq,2 = ζd − 8(1 + μ) g1 =

Dmax −1≤0 Dlim

where Dmax =

Ds,α Ds,β

Probabilistic Approach

for Fuzzy Approach

{θ l } ≤ {θ } ≤ {θ u }

(22)

being Dmax the maximum cumulated fatigue damage of the structure defined in terms of the probabilistic or fuzzy approach; Ds,α is the percentile αth of the probability distribution function of the maximum cumulated fatigue damage of the structure;

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f

Fig. 4 Flowchart of the motion-based design method under uncertain conditions

Ds,β is the maximum cumulated fatigue damage of the structure for the β-level; and Dlim the cumulated fatigue damage limit defined in terms of the required performance level (FULS) established by the European code [29]. The main steps of the MBD method under uncertain conditions are illustrated in Fig. 4. Once the final design variables are obtained and after the vibration device is built and installed on the structure, designers must carry out experimental tests to verify that the required performance level (FULS) is fulfilled [16].

4 Application Example In order to illustrate both the practical implementation of the MBD method and the importance of considering the uncertainty for the robust design of different TMD techniques, the following numerical case-study is presented. In this application example, the compliance of an adequate performance level (FULS under wind action) of a numerical steel chimney is guaranteed via the installation of different

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TMD techniques. Additionally, the MBD method under uncertain conditions has been formulated under two approaches (probabilistic and fuzzy).

4.1 Description of the Benchmark Structure and Preliminary Analysis of Its Structural Behavior In this numerical application only the fatigue damage originated by vortex-sheddinginduced vibrations has been considered (cross-wind vibrations). For this purpose, wind loads on the chimney have been computed according to the European code [39]. As benchmark structure, a steel chimney reported in literature has been considered herein [47]. The chimney has a height, H , of 55 m and an outside constant diameter, D, of 1.63 m, as shown in Fig. 5. It is a double-walled chimney composed by an outer tube and inner thermal insulating layer. The chimney shaft is composed of five separated parts, bolted together using socket joints. A finite element model of the chimney was developed using a commercial package [48]. Beams elements, BEAM188 (2 nodes per element, 6 d.o.f. in each node), has been considered to build the finite element model. This numerical model is composed by a mesh of 110 beam elements (Fig. 5). A linear behavior is assumed for the constitutive law of the constitutive material (steel S-235 according to the European code [39]). The following mechanical properties have been adopted [29]:

Fig. 5 Scheme of the benchmark structure (steel chimney) and representation of the first three vibration modes considering the uncertainty associated with the modification of the operational, environmental, degradation and damage conditions

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(i) a Young’s modulus, E s = 210000 MPa: (ii) a Poisson’s ratio, νs = 0.3 [−]; and (iii) a density, ρs = 7850 kg/m3 . A proportional damping for the structure, a ratio, ζs , about 1.2%, is assumed herein. Additionally, the chimney is assumed to be located in a suburban terrain (a flat area with regular cover of buildings), thus the location area is classified as a terrain category III according to the European code (39). In order to check if the effect of vortex shedding must be investigated, the following conditions must be met [39]: H/D > 6

(23)

vcri < 1.25 · v m

(24)

where vcri is the critical wind velocity for the considered vibration mode [m/s] and vm is the characteristic 10 min mean wind velocity at the cross section where vortex shedding occurs [m/s]. The critical wind velocity for the considered vibration mode, vcri , is defined as the wind velocity at which the frequency of vortex shedding equals the natural frequency of the considered vibration mode and can be determined as: vcri =

D · fs St

(25)

where D is the reference width of the cross-section at which resonant vortex shedding occurs (for circular cylinders the reference width is the outer diameter) [m]; f s is the natural frequency of the considered vibration mode of cross-wind vibration mode [Hz]; and St is the Strouhal number (for circular cross-sections may be approximated by 0.18). The characteristic mean wind velocity, vm (z), at a height z above the terrain depends on the terrain roughness and orography and on the basic wind velocity, vb [m/s] (a value of 28 m/s has been taken into account here) and can be determined as follows [39]: vm (z) = cr (z) · c0 (z) · vb

(26)

where cr (z) is the roughness factor and c0 (z) is the orography factor (assumed as 1 in this study). The roughness factor, cr (z), accounts for the variability of the mean wind velocity at the site of the structure due to the height above ground level together with the ground roughness of the terrain upwind of the structure in the considered wind direction. According to the European codes [39], the roughness factor, cr (z), can be expressed as:

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cr (z) =

kr · I n( zz0 ) z ≤ z ≤ z max for min z ≤ z min cr (z min )

83

(27)

where z 0 is the roughness length [m] (which depends on terrain category); kr is the terrain factor depending on the roughness length (which can be calculated by 0.07  where z 0,I I is assumed as 0.05 m); z min is the minimum height 0.19 · z 0 /z 0,I I [m] (which also depends on the terrain category) and z max is the maximum height [m] (which must be taken as 200 m). Due to the location of the chimney, a value of z 0 = 0.3 m and z min = 5 m has been assumed herein. In order to evaluate the Strouhal number, St , a modal analysis has been performed to obtain the numerical natural frequencies and their associated vibration modes. In order to take into account for the modification of the modal properties of the chimney associated with the variation of the operational, environmental, degradation and damage conditions two variation ranges have been considered: (i) ± 10% for each considered natural frequency and associated vibration mode; and (ii) ± 50% for the damping ratio of these vibration modes, ζs . Two approaches have been considered to simulate the uncertain conditions: (i) a probabilistic approach and (ii) a fuzzy approach. For this purpose, the natural frequencies and the damping ratio are assumed as random variables and fuzzy variables respectively. Thus, for the probabilistic approach, the random variables are assumed to follow a normal function; and for the fuzzy approach, the fuzzy variables are assumed to follow a triangular membership function. Figure 6 shows both the probability distribution functions and the membership functions for each considered variable. In this manner, in all analyses to come, the same probabilistic and fuzzy sets have been considered. Figure 5 illustrates the first three lateral vibration modes of the steel chimney and the associated variation range of their natural frequencies. Subsequently, the fulfillment of the vortex-shedding criterion is checked for the first three mentioned vibration modes. As result of this checking, it was verified that only the first vibration mode is prone to vibrate due to vortex-shedding-induced vibrations. The mass mobilized by the first vibration mode, m s , is about 770 kg. Thus, for the fatigue assessment of the chimney, the dynamic response of the chimney under wind action must be computed. For this purpose, the equivalent wind load, qw , which simulates the vortex-shedding action, must be defined. For the definition of the equivalent wind load, the recommendations of the European code [39] has been considered herein. Figure 7 illustrates a schematic diagram for the definition of both the equivalent wind load and the evaluation of the cumulated fatigue damage, Ds . According to this code, the equivalent wind load, qw (t) [N/m], can be defined as follows: qw (t) =

1 ρa · vm (z)2 · D · c f · sin(2π · f s · t) 2

(28)

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Fig. 6 Consideration of the uncertainty for the implementation of the MBD method: a Probabilistic approach and b fuzzy approach

Fig. 7 Vortex-shedding model considered to compute the fatigue assessment of the steel chimney

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85

where ρa is the air density (1.25 kg/m3 ) and c f is the lateral force coefficient (a value of 0.7 is assumed herein). The equivalent wind load is applied on the equivalent height of the chimney, Heq [m], which may be determined as follows (assuming that only the first vibration mode, φ1 (z), is prone to vibrate under vortex-shedding effect) [39]: 6D xmax < 0.1D Heq = 4.8 + 12xmax /D f or 0.1D ≤ xmax ≤ 0.6D 12D xmax ≥ 0.6D

(29)

where xmax is the maximum lateral displacement of the chimney subjected to wind action [m]. Finally, to assess the cumulated fatigue damage, Ds , the following procedure has been considered: (i) to compute the dynamic response of the structure under the equivalent wind action (a simulation time of 300 s and a time increment of 0.005 s has been taken into account in this study); (ii) the stress history of the considered structural element is determined based on the dynamic response; (iii) the cycle counting is computed using the rain-flow counting method; (iv) the stress range spectrum is determined based on the cycle counting; (v) the number of cycles to failure is computed for each stress range of the spectrum; (vi) the fatigue spectrum is normalized in terms of the simulation time; (vii) the number of cycles caused by vortex-shedding, Nv , is computed and (viii) finally, the cumulated damage ratio is computed. The number of cycles, Nv , caused by vortex excited oscillation may be determined as follows [44]:  Nv = 2 · T · f s · ε0 ·

vcri v0

2

    vcri 2 · exp − v0

(30)

where T is the life time (which is equal to 3.2 · 107 multiply by the expected life in years) [sec]; ε0 is the bandwidth factor describing the band of wind √ velocities with vortex-induced vibrations (which may be taken as 0.3); and v0 is 2 times the modal value of the Weibull probability distribution assumed for the wind velocity (which can be taken as 20% of the characteristic mean wind velocity at the height of the cross section where vortex shedding occurs) [m/s]. The fatigue assessment has been performed at the section located at the basis of the chimney. The Eurocode curve 41 has been adopted as category detail for the definition of the S–N curve [29]. As the fulfilment of the FULS is not met, different TMD techniques have been designed, considering the MBD method under uncertain conditions, to control the vortex-shedding-induced vibrations on the steel chimney. As design criterion, a hybrid strategy (considering the H∞ norm) has been taken into account for the different TMD techniques. Thus, for both smart TMD techniques (ATMD and

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STMD), the main parameters which govern the driving force have been established as, αd = 300 and βd = 10−6 , according to the recommendations of some studies reported in literature [18]. As global optimization method, genetic algorithms have been used for solving the optimization problem. The following common parameters have been considered for the optimization algorithm in order to achieve a balanced-equilibrium between the accuracy and the computational cost: (i) as population size, 20; (ii) as maximum number of iterations, 20; (iii) as objective function tolerance, 10−5 (according to this, the algorithm stops if the average relative change of the best value of the objective function is less than or equal to the mentioned value); (iv) as selection ratio, 0.1; (v) as crossover ratio, 0.9, and (vi) as mutation ratio, 0.1. Both sensitivity studies have been carried out to set the sample size for the probabilistic and the fuzzy approaches [49]. In each study, the variation of the response of the system with the size of the sampling has been analyzed. As result of these studies, the following sample sizes have been adopted: (i) 10,000 samples for the probabilistic approach; and (ii) 1000 samples for the fuzzy approach. In order to analyze the influence of the uncertainty on the mechanical parameters of the vibration absorbers, the design of each TMD techniques has been performed for the following uncertainty levels: (i) according to the probabilistic approach, the maximum cumulated fatigue damage of the structure, Dmax , has been determined for five different levels (percentile 50th, 67th, 80th, 95th and 99th); and (ii) according to the fuzzy approach, the maximum cumulated fatigue damage of the structure, Dmax , has been obtained for five different levels (β-level 0, 0.25, 0.5, 0.75 and 1).

4.2 Motion-Based Design of the Benchmark Structure Equipped with ATMD Under Uncertain Conditions Firstly, the MBD method under uncertain conditions has been implemented for the design of the ATMD. The design problem, for the ATMD, can be particularized (based on Eq. (22)) as follows: Find{θ } = {μ} Minimizing f ({θ }) = μ 1 =0 geq,1 = δd − 1+μ  Subjected to geq,2 = ζd − g1 =

3μ =0 8(1 + μ)

Dmax −1≤0 Dlim

where Dmax =

Ds,α Ds,β

Probabilistic Approach

for Fuzzy Approach

(31)

Robust Design of Different Tuned Mass Damper … Table 1 Robust optimum value of the mass ratio, μ, for the three different TMD techniques in terms of the uncertainty level considering the probabilistic approach

Table 2 Robust optimum value of the mass ratio, μ, for the three different TMD techniques in terms of the uncertainty level considering the fuzzy set approach

87

αth

ATMD

STMD

PTMD

50

0.0038

0.0038

0.0045

67

0.0041

0.0041

0.0052

80

0.0043

0.0043

0.0066

95

0.0047

0.0078

0.0103

99

0.0053

0.0123

0.0182

β-level

ATMD

STMD

PTMD

1.00

0.0038

0.0038

0.0038

0.75

0.0044

0.0044

0.0061

0.50

0.0049

0.0052

0.0113

0.25

0.0055

0.0137

0.0177

0.00

0.0060

0.0226

0.0253

The values of the design variable, obtained after the design process, are shown in Table 1 (probabilistic approach) and Table 2 (fuzzy approach). As example of this design process, Fig. 8 illustrates the stress spectrum versus the fatigue strength curve of the steel chimney without and with the ATMD considering the probabilistic approach (percentile 95th) has been considered for the results shown in Fig. 8. Three characteristic situations are illustrated in Fig. 8: (i) f s = 0.58 Hz and ζs = 1.2%; (ii) f s = 0.65 Hz and ζs = 1.2%; and (iii) f s = 0.72 Hz and ζs = 1.2%. Subsequently,

Fig. 8 Stress spectrum versus the fatigue strength curve of the steel chimney without and with the ATMD considering the probabilistic approach (αth = 95%): a f s = 0.58 Hz and ζs = 1.2%; b f s = 0.65 Hz and ζs = 1.2%; and c f s = 0.72 Hz and ζs = 1.2%

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Fig. 9 Stress spectrum versus the fatigue strength curve of the steel chimney without and with the ATMD considering the fuzzy approach (β-level = 1): a f s = 0.58 Hz and ζs = 1.2%; b f s = 0.65 Hz and ζs = 1.2%; and c f s = 0.72 Hz and ζs = 1.2%

Fig. 9 illustrates the stress spectrum versus the fatigue strength curve of the steel chimney without and with the ATMD considering the fuzzy approach (β-level = 1).

4.3 Motion-Based Design of the Benchmark Structure Equipped with STMD Under Uncertain Conditions Subsequently, the MBD method under uncertain conditions has been implemented for the design of the STMD. The design problem, for the STMD, can be particularized (based on Eq. (22)) as follows: Find{θ } = {μ} Minimizing f ({θ }) = μ 1 =0 geq,1 = δd − 1+μ  Subjected to geq,2 = ζd − g1 =

3μ =0 8(1 + μ)

Dmax −1≤0 Dlim

where Dmax =

Ds,α Ds,β

Probabilistic Approach

for Fuzzy Approach

{θ l } = {0.001} ≤ {θ } ≤ {θ u } = {0.05}

(32)

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89

Additionally the value of the driving force has been limited, ( f min = 0 N and f max = 25 N), according to the values recommended by a manufacturer [50]. The values of the design variables, obtained after the design process, are gathered in Table 1 (probabilistic approach) and Table 2 (fuzzy approach). Figure 10 shows the stress spectrum versus the fatigue strength curve of the steel chimney without and with the STMD, considering the probabilistic approach (percentile 95th) for the same three cases shown before. Additionally, Fig. 11 shows the same results for the fuzzy approach (β-level = 1).

Fig. 10 Stress spectrum versus the fatigue strength curve of the steel chimney without and with the STMD considering the probabilistic approach (αth = 95%): a f s = 0.58 Hz and ζs = 1.2%; b f s = 0.65 Hz and ζs = 1.2%; and c f s = 0.72 Hz and ζs = 1.2%

Fig. 11 Stress spectrum versus the fatigue strength curve of the steel chimney without and with the STMD considering the fuzzy approach (β-level = 1): a f s = 0.58 Hz and ζs = 1.2%; b f s = 0.65 Hz and ζs = 1.2%; and c f s = 0.72 Hz and ζs = 1.2%

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4.4 Motion-Based Design of the Benchmark Structure Equipped with PTMD Under Uncertain Conditions Finally, the MBD method under uncertain conditions has been implemented for the design of the PTMD. The design problem, for the PTMD, can be particularized (based on Eq. (22)) as follows: Find{θ } = {μ} Minimizing f ({θ }) = μ 1 =0 geq,1 = δd − 1+μ  Subjected to geq,2 = ζd − g1 =

3μ =0 8(1 + μ)

Dmax −1≤0 Dlim

where Dmax =

Ds,α Ds,β

Probabilistic Approach

for Fuzzy Approach

{θ l } = {0.001} ≤ {θ } ≤ {θ u } = {0.05}

(33)

The values of the design variables, obtained after the design process, are shown in Table 1 (probabilistic approach) and Table 2 (fuzzy approach). Figure 12 illustrates the stress spectrum versus the fatigue strength curve of the steel chimney without and with the PTMD, considering the probabilistic approach (percentile 95th) for the same three cases shown before. Additionally, Fig. 13 illustrates the same results for

Fig. 12 Stress spectrum versus the fatigue strength curve of the steel chimney without and with the PTMD considering the probabilistic approach (αth = 95%): a f s = 0.58 Hz and ζs = 1.2%; b f s = 0.65 Hz and ζs = 1.2%; and c f s = 0.72 Hz and ζs = 1.2%

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Fig. 13 Stress spectrum versus the fatigue strength curve of the steel chimney without and with the PTMD considering the fuzzy approach (β-level = 1): a f s = 0.58 Hz and ζs = 1.2%; b f s = 0.65 Hz and ζs = 1.2%; and c f s = 0.72 Hz and ζs = 1.2%

the fuzzy approach (β-level = 1).

4.5 Discussion of the Results The main design variable, the mass ratio, μ, of each TMD in terms of the uncertainty level is shown in Table 1 (probabilistic approach) and 2 (fuzzy approach). As these results shown, it is possible to find a robust optimum solution for each design problem. In this manner, the proposed method allows designing different TMD techniques considering a common framework, being a potential tool to assist engineering practitioner in the design of control devices to mitigate wind-induced vibrations in buildings and civil engineering structures. According to these results, a clear correlation may be established between the uncertainty level and the robust optimum mass ratio of the different TMD techniques. A similar correlation is shown independently of the method considered to simulate the uncertainty (probabilistic or fuzzy). Additionally, Fig. 14 illustrates the variation of the mass ratio, μ, of the different TMD techniques with the uncertainty level. Concretely, the results considering the probabilistic approach are shown in Fig. 14a and the results associated with the fuzzy approach are shown in Fig. 14b. According to these results, it is clear that an increase in the uncertainty levels implies an increase in the mass ratio. This effect is greater in the case of the PTMD and STMD. It is worth remarking, if we compare the performance among the different TMD techniques, that the ATMD allows reducing the mass of the external control device,

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Fig. 14 Variation of the mass ratio, μ, of the different TMD techniques in terms of the uncertainty level: a Probabilistic approach and b fuzzy approach

especially when the uncertain conditions increase. This fact can make this technique the best option for the retrofitting or controlling of existing building and civil engineering structures. Finally, the two approaches, used to simulate the uncertainty, show a similar performance in relation to the robust value of the mass ratio (and the remaining mechanical parameters) of the different TMD techniques. In this sense, it is necessary to highlight that the fuzzy approach is easier to implement and perhaps the best alternative when the designer has not additional information about the probabilistic distribution function of the considered design variables (an accurate design is obtained with a clear reduction of the computational cost).

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5 Conclusions In this chapter, recent advances have been presented about the robust design of smart vibration absorbers when these damping devices are used to reduce the wind-induced vibrations of buildings and civil engineering structures under uncertain conditions. Among the different control techniques, the chapter focuses on the robust design of different TMD techniques (active, semi-active and passive) since these vibration absorbers have been widely used to mitigate the wind-induced vibrations in multiple buildings and civil engineering structures together with the fact that the effect of other vibration controllers can be simulated via a tuned mass damper. Additionally, the effect of the modification of the structural modal properties (uncertain conditions) due to the variation of the operational, environmental, degradation and damage conditions is taken into account for the design process. Finally, a common framework, proposed recently for the robust design of the abovementioned TMD techniques has been described and further formulated in detail. The framework is based on the implementation of the MBD method under uncertain conditions. According to this method, this design problem may be formulated via a constrained single-objective optimization problem as follows: (i) the design variables are the mechanical parameters of the control system; (ii) the single-objective function is defined in terms of the cost of the control device; (iii) the equality constraints are defined in terms of the required form of the frequency response function of the TMD-structure interaction system; (iv) the inequality constraints are defined, according to either a probabilistic approach or a fuzzy approach, in terms of the design requirements (the fulfillment of the FULS of the structure has been considered herein); (v) a search domain is established to ensure the physical meaning of the solutions obtained; and (vi) NIC algorithms (genetic algorithms herein) are considered as optimization method in order to obtain a global solution due to the non-linear relationship between the constraints and the design variables. As an application example, a numerical case-study has been considered. In this case-study, different TMD techniques have been installed at the top of a steel chimney to mitigate its lateral vortex-shedding-induced vibrations The mechanical parameters of these vibration absorbers have been obtained via the implementation of the MBD method. As result of this study, the performance of the proposed method has been shown up and the influence of the uncertain conditions on the different TMD techniques has been illustrated. Thus, three main conclusions can be obtained from this study: (i) a clear correlation maybe established between the mass ratio and the uncertainty level for the three considered TMD techniques (especially for the STMD and PTMD techniques; (ii) the ATMD is the least sensitive device to the effects of the uncertain conditions; and (iii) a clear reduction of the mass can be achieved for the ATMD which can makes this technique the best option for the retrofitting of existing structures. Additionally, it has been highlighted that although fuzzy theory is less frequently used than probability theory for dealing with the uncertainty in practical engineering

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applications, the latter can be very efficient and simpler for quantifying the uncertainty if there is not enough information to determine the probability distribution function of the uncertain variables. In spite of the goodness of the results obtained, further studies are needed to assess experimentally the performance of structures equipped with intelligent vibration controllers designed considering these recent proposals. Acknowledgements This work was partially funded by two research projects: (i) RTI2018-094945B-C21 (Ministerio de Economía y Competitividad of Spain and the European Regional Development Fund); and (ii) SEED-SD RTI2018-099639-B-I00 (Ministerio de Ciencia, Innovación y Universidades of Spain). Declaration of Conflicting Interests The authors declare no potential conflicts of interest with respect to the research, authorship, and publication of this chapter.

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Optimal Seismic Response Control of Adjacent Buildings Coupled with a Double Mass Tuned Damper Inerter Salah Djerouni, Mahdi Abdeddaim, Said Elias, Dario De Domenico, and Rajesh Rupakhety Abstract Adjacent buildings are exposed to a high risk of pounding against each other during seismic events. In recent strong earthquakes events, the separation gap has been found to be insufficient to prevent structural damage related to pounding phenomena. The inerter-based tuned mass damper has been validated as an effective, lightweight passive control device by incorporating the inerter into a conventional tuned mass damper (TMD). The proposed system’s optimal design is achieved using a constrained optimization problem based on the Grey Wolf Optimizer (GWO) algorithm. This numerical study investigates the capability of reducing the pounding risk of inertially connected tuned mass dampers (TMDs). The presented system connects two high-rise adjacent buildings as a novel seismic protection system. The optimal design of the proposed system is conducted through a constrained optimization problem via a Grey Wolf Optimizer (GWO) algorithm, wherein the pounding gap distance of the two high-rise adjacent buildings is selected as performance index. Optimal results obtained are critically analyzed and compared. For S. Djerouni · M. Abdeddaim LARGHYDE Laboratory, Department of Civil Engineering and Hydraulics, Faculty of Sciences and Technology, Mohamed Khider University, BP 145 RP, 07000 Biskra, Algeria e-mail: [email protected] M. Abdeddaim e-mail: [email protected] S. Djerouni · S. Elias · R. Rupakhety Earthquake Engineering Research Centre, Faculty of Civil and Environmental Engineering, School of Engineering and Natural Sciences, University of Iceland, Austurvegur 2a, 800 Selfoss, Iceland e-mail: [email protected] S. Elias (B) Department of Construction Management and Engineering (CME), Faculty of Engineering Technology (ET), University of Twente (UTWENTE), Enschede, Netherlands e-mail: [email protected] D. De Domenico Department of Engineering, University of Messina, Messina, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. Bekda¸s and S. M. Nigdeli (eds.), Optimization of Tuned Mass Dampers, Studies in Systems, Decision and Control 432, https://doi.org/10.1007/978-3-030-98343-7_6

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comparison purposes, two separate TMDs are mounted on the rooftop of each of the adjacent buildings, and the two systems are optimized independently under the constraint of the same total mass. Performance of these independent TMDs is evaluated to compared to that of inertially connected ones using a large number of ground motions selected from a set of 462 ground motion records. The main response parameters of interest are the minimum seismic gap for pounding mitigation, along with inter-storey drift. When the two buildings have different natural frequencies, the results reveal that the suggested new device outperforms the non-connected TMDs system. Keywords Pounding · Tuned mass damper · Structural optimization · Grey wolf optimizer · Inerter · Tuned mass damper inerter · Adjacent buildings

1 Introduction One of the most catastrophic structural phenomena during recent earthquakes in densely populated metropolitan areas has been pounding of adjacent buildings. Values of ordinary seismic joints recommended by seismic codes were often exceeded by the actual lateral displacements experienced during strong earthquakes [1–4], thus being ineffective in reducing pounding risk. Several post-earthquake reports have revealed that pounding phenomena causes lasting structural damage. After the 1985 Mexico City earthquake, the 1989 Loma Prieta earthquake, the 1995 Kobe earthquake, the 2011 Christchurch earthquake, the 2016 Kumamoto earthquake, and the 2017 Puebla, Mexico earthquake, damages due to pounding were recorded [5–12]. Researchers recommended a variety of control strategies to mitigate pounding risk of adjacent buildings. Most proposed solutions make use of damping devices [13, 14] or devices connecting the adjoining buildings. These devices vary from passive, active and semi-active. Westermo [15] was the first to propose a connection between adjacent buildings through a link and a beam. This solution may lead to a response increase in structures with distinct dynamical characteristics. Other researchers have proposed the use of passive dampers to connect adjacent buildings [16–18]. These devices showed certain limitations, especially under strong ground motions [19]. Henceforth, active and semi-active coupling systems have been reported to be more successful in preventing pounding between adjacent structures [20–25]. Even though active and semi-active based solutions are promising, they are costly: they require an extensive network of sensors and computing stations that are very sensitive to noise and disturbances. Tuned mass dampers have shown excellent potential in reducing seismic response of buildings [26–28]. Motivated by this, coupling of adjacent buildings through TMDs to reduce pounding hazard has been investigated by researchers. Abdullah et al. [29] proposed a shared tuned mass damper to reduce the seismic response and pounding risk of two adjacent buildings. Their proposal consists of a TMD placed

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on one of the buildings and connected to the adjacent building through a spring and a dashpot. Kim and Kang [30] used a similar approach the shared TMD was reported to considerably reduce the minimum separation gap required to prevent pounding. Kim [31] used a TMD linking a building to a second adjacent building through a magnetorheological (MR) damper and a spring. The MR damper was driven using a fuzzy logic control algorithm. The performance of the shared TMD was compared with two independent TMDs installed on each building. It was concluded that the shared TMD can achieve better performance with half of the mass used by the independent TMDs. Guenidi and Abdeddaim [32] connected a semi-active tuned mass damper to an adjacent building via a dashpot. The adjacent building frequencies were varied during the study, and it was observed that the performance of the proposed strategy is not affected by the frequency ratio between the adjacent building in terms of pounding hazard mitigation. Rupakhety et al. [33] showed that shared TMDs as in [29] are actually equivalent to a viscous coupling between adjacent buildings, and not superior to independent TMDs placed on adjacent buildings. It is generally known that the performance of traditional TMDs is proportional to their mass ratios, especially when used to mitigate the seismic response of adjacent buildings, which involves two buildings simultaneously. TMD mass ratios, on the other hand, remain constrained and low due to the limitation on vertical loading on the structure (structural considerations) and the relatively contained space allocated for their positioning (architectural considerations). A new device capable of generating apparent mass ratios through inertia was recently introduced to vibration control applications [34]. This device is referred to as an “inerter” [35]. Inerters were quickly paired with TMDs to improve their performance and overcome their shortcomings by artificially increasing their mass [34]. TMDs and inerters were combined to create so-called tuned mass damper inerters (TMDIs) [36]. TMDIs have been intensively researched throughout the previous decade [37, 38]. They were used to control a broad range of buildings and structures, with a multitude of configurations [38, 39]. The TMDI (in single and multiple configurations) and other inerter-based vibration absorbers, such as the tunable liquid column damper inerter (TLCDI), were found to be more resilient and efficient than conventional TMDs at improving the seismic performance of adjacent high-rise buildings. [40, 41], which represents the motivation to further explore this potential in this research work. A recent study has shown that connecting two TMDs with an inerter can be an efficient strategy for reducing the seismic response of buildings [42–44]. Connecting two tuned mass dampers with an ineter has two benefits (i) it increases the system’s mass ratio and (ii) it prevents inertial forces from being transferred to the structure. This idea is extended in the present study to reduce pounding hazards between adjacent buildings. Hence, TMDs installed on two adjacent buildings are connected to each other through an inerter device to mitigate the pounding phenomena and reduce the seismic response. The best tuning parameters of the devices are identified using a Grey Wolf Optimizer (GWO) algorithm. The optimization is performed in the frequency domain, and it aims at reducing the minimum separation gap while reducing the TMDs mass ratio exploiting the inerter’s mass amplification effect. The resulting optimal parameters are then used to perform time-domain simulation

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under real earthquake ground motions. The performance of the proposed inertially coupled system is compared with the non-coupled tuned mass dampers and the uncontrolled adjacent buildings to demonstrate peculiarities and advantages of the proposed structural control strategy.

2 Equations of Motion for Adjacent Controlled Buildings To examine the control performance of inertially connected tuned mass damper installed on adjacent buildings subjected to earthquake excitation, two 9-story frame buildings shown in Fig. 1a were considered. A tuned mass damper was installed on the roof of the left building, and it was connected to a second tuned mass damper placed on the rooftop of the right building; the connection was achieved through a passive inerter, as shown in Fig. 1c. For comparison purposes, uncoupled (i.e., independent) TMDs were installed on the roof of each building, as shown in Fig. 1b. The properties of the example structures are taken from [45]. The building models were reported as planar frames in [45] which have been reduced to a one-degree-of-freedom per floor model by condensing the unwanted degrees of freedom of the original system. The resulting condensed matrices of the structures are given as Eqs. (A.1)–(A.5) in the Appendix. The natural frequencies of the two buildings are presented in Fig. 2. The building to the right is stiffer. Assuming linear elastic behavior, the seismic response of the two buildings equipped with inertially connected TMDs can be described using the following matrix–vector form of the equation of motion:           M˜ {x(t)} ¨ + C˜ {x(t)} ˙ + K˜ {x(t)} = − M˜ z {l} x¨ g (t) .

(1)

where arrays (vectors) are indicated with curly brackets, while matrices are indicated with square brackets. The 20 × 1 arrays {x(t)},{ ¨ x(t)} ˙ and {x(t)} denote the relative acceleration, velocity, and displacement of each degree of freedom (DOF), respectively (9 DOFs for each floor of the two buildings and one DOF for each TMD),  x¨ g (t) is the ground acceleration during the earthquake  and   l is a column   location ˜ ˜ vector containing ones. The square 20 × 20 matrices M , C and K˜ represent the mass, damping, and stiffness matrices of the adjacent buildings equipped with inter-connected TMDs, respectively. The component of the above-defined matrices and vectors are as follows:   {x(t)} = x1L x2L · · · x1R x2R · · · xnR x˜1 x˜2   {x(t)} = x˙1L x˙2L · · · x˙1R x˙2R · · · x˙nR  ˙ (2) x˙ 1  x˙ 2  L L  R R R {x(t)} = x¨1 x¨2 · · · x¨1 x¨2 · · · x¨n  ¨ x¨ 1  x¨ 2 .

Fig. 1 The n-storey adjacent buildings, a with no control device, b with multiple passive independent TMDs and c with multiple inertially connected TMDs

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Fig. 2 The natural frequencies of left and right building

⎡   ⎢ ⎢ M˜ = ⎢ ⎣

⎤

···

[M L ] [M R ]

.. . −b˜

m˜ 1 + b˜

⎥ ⎥ ⎥. ⎦

m˜ 2 + b˜ ⎡ ⎤ [C L ] {0} {C h1 }T {0}   ⎢ {0} [C ] {0} {C }T ⎥ R h2 ⎥. C˜ = ⎢ ⎣ {C h1 } {0} {0} ⎦ c˜1 {0} {C h2 } {0} c˜2 ⎡ ⎤ [M L ] · · · .. ⎥   ⎢ ⎢ . ⎥ [M R ] M˜ z = ⎢ ⎥. ⎣ ⎦ m˜ 1 SY M

(3)

m˜ 2

SY M ⎡

⎤ [K L ] {0} {K h1 }T {0}   ⎢ {0} [K ] {0} {K h2 }T ⎥ R ⎥. K˜ = ⎢ ⎣ {K h1 } {0} {0} ⎦ k˜1 {0} {K h2 }T {0} k˜2

(4)

where the details of each matrix in the Eq. (3) can be given as follows: ⎡ ⎢ ⎢ ⎢ [M L ] = ⎢ ⎢ ⎢ ⎣

⎤

···

m1 m2

..

. m n−1

SY M

.. ⎥ . ⎥ ⎥ ⎥. ⎥ ⎥ ⎦ mn

(5)

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⎤ c1 + c2 −c2 · · · ··· 0 ⎥ ⎢ c2 + c3 ⎥ ⎢ ⎢ . .. ⎥ [C L ] = ⎢ ⎥. ⎥ ⎢ ⎣ cn−1 + cn −cn ⎦ SY M cn ⎡

(6)

The subscripts L and R denotes the left side and right-side buildings, respectively.

   {C h1 } = 0 1 × n − 1 −c˜1 ,

   {C h2 } = 0 1 × n − 1 −c˜2 .

(7)

The matrices of the right-side buildings [M R ], [C R ], and [K R ] can be formulated similarly. The mass, damping and stiffness matrices of the analysed buildings are presented as Eqs. (A.1)–(A.5) in the Appendix. Here, m˜ 1 , c˜1 , and k˜1 are the mass, external damping coefficient, and stiffness terms corresponding to the TMD placed on the left building, respectively. Similarly, m˜ 2 , c˜2 , and k˜2 are the mass, external damping coefficient, and stiffness terms corresponding to the TMD placed on the right building, respectively. The governing equation of motion (1) can be re-written in state-space form as:  :

 Z˙ (t) = [A]{Z (t)} + [B]{u(t)} {y(t)} = C y {Z (t)}

(8)

with state vector:  {Z (t)} =

x(t) x(t) ˙

 .

(9)

and state matrix [A] in Eq. (8) that can be written as:  [A] =

[I ]20 [0]  −1    20×20 −1   K˜ − M˜ C˜ − M˜  [B] =

[0]20×1 −[1]

 .

(10)

 .

(11)

In order to compute the responses of interest, the output response vector {y(t)} need to be defined as:   {y(t)} = C y {Z (t)} .

(12)

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    where C y and D y are output matrices. When the pounding distance of the adjacent buildings is of interest, they are defined as:        C y 2n×2n = C y L C y R . ⎡  L ⎢ Cy = ⎣

1

⎤ ..

⎡

⎥ ⎦

. 1

.

 R ⎢ Cy = ⎣

−1

(13) ⎤

..

⎥ ⎦

.

n×n

−1

.

(14)

n×n

Given the state space representation in Eq. (8), the transfer function of the controlled adjacent buildings system can be obtained by   [Hs ( )] = C y (i [I ] − [A])−1 [B] .

(15)

3 Grey Wolf Optimizer (GWO) Algorithm Mirjalili suggested a new meta-heuristic technique called the Grey Wolf Optimizer (GWO) algorithm, this later is used to cope with the design optimization of the proposed inertially coupled system [46]. Wolves are apex predators at the top of the food chain. The majority of wolves live in packs, with an average of 5–12 wolves per population. However, because each wolf has a unique role in the population, they follow a fairly tight social hierarchy. The GWO algorithm replicates wolf leadership and predatory behavior before utilizing grey wolf abilities such as search, encirclement, hunting, and other predation-related actions to reach the goal of optimization. Assuming  the number of wolves is s, the location of the jth wolf can  that be expressed as: Q ji = Q j1 , Q j2 , Q j3 , ..., Q js . In the process of mathematically modeling the social hierarchy of wolves, the best (fittest) solution is assigned as the lambda (λ) wolf. Additionally, the second and third-best solutions are defined as gamma (γ ) and epsilon (ε) wolves, respectively. Theta wolves (θ ) are assumed to be the last candidate solutions. The placement of the lambda wolf is related to the position of the prey in the algorithm [47, 48]. Grey wolves’ encircling behavior can be analytically represented as follows:  = |{L}.{Q l (τ )} − {Q(τ )}| .

(16)

{Q(τ + 1)} = {Q l (τ )} − { }.{} .

(17)

where the set τ indicates the current iteration and the set Q l (τ ) represents the location vector of the prey, the set Q(τ ) is the location vector of a grey wolf, the set L is a

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control coefficient, which is determinate by the following formula: {L} = 2 .{e1 }.

(18)

where the set {e1 } is random vector in [0, 1]. In Eq. (17) the set is the convergence factor, which is calculated as follows: { } = 2{a}.{e2 } − {a}.

(19)

where the set {e2 } is the random vector in [0, 1], the set {a} is the control factor, which linearly decreases from 2 to 0 throughout iterations [49]. When the grey wolves catch prey, the leader wolf (λ) initially directs the rest of the wolves to surround the prey. The best solutions from (λ), (γ ) and (ε) are saved in each iteration, while the (ε) wolve update their position based on the best solutions. These steps are mathematically represented as follows: {λ } = |{L 1 }.{Q λ (t)} − {Q(t)}| .

(20)

     γ = {L 2 }. Q γ (t) − {Q(t)} .

(21)

{ε } = |{L 3 }.{Q ε (t)} − {Q(t)}| .

(22)

{Q 1 } = {Q λ } − { 1 }.{λ } .

(23)

    {Q 2 } = Q γ − { 2 }. γ .

(24)

{Q 3 } = {Q ε } − { 3 }.{ε } .

(25)



{Q(τ + 1)} =

{Q 1 } + {Q 2 } + {Q 3 } . 3

(26)

The distance between Q(τ ) and (λ),(γ ),(ε) wolves is evaluated by Eqs. (20)– (25), and the location of the wolves moving to the prey is calculated by the Eq. (26). Figure 9 shows the flowchart of the grey wolf optimizer (GWO) algorithm.

4 Numerical Study To examine the proposed system’s performance, the tuning parameters are optimized using the grey wolf optimization approach presented in Sect. 3.

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The  for the optimum  tuning process are the  frequency   free design variables used ratio f opt , damping ratio ζopt , mass ratio μopt and inertance ratio βopt .  f opt =

k˜1,2 m˜ 1,2 +b˜

ω j,1

μopt =

. ζopt = 

c˜1,2  . 2k˜1,2 m˜ 1,2 + b˜

m˜ 1,2 b˜ . βopt = . Mtot Mtot

(27)

where ω j,1 and Mtot represent, respectively, the first order natural frequency and total mas of building j. The optimization is accomplished in the frequency domain. The H∞ -norm of the transform function of minimum separation distance to avoid pounding is selected as the objective function. The best parameters obtained are then utilized to run a time history analysis on two representative ground motion records. The numerical study is then extended to an extensive ground motion database containing 462 records. The response parameters of interest are the percentage reduction in inter-storey drift in both the buildings and the minimum separation gap to avoid pounding. The variation of the response reduction quantity is shown with respect to the inter-storey drift of NC in both buildings, the pounding distance of NC, spectral displacement, and earthquake’s duration and PGA, respectively. The maximum number of iterations and the number of search agents in the GWO algorithm are considered to be 500 and 30, respectively, for this purpose. These characteristics were specified or selected based on general literature suggestions [50, 51].

4.1 Verification of GWO Algorithm for the Optimal Design of the Proposed Systems (MPS1 and MPS2) In order to illustrate the GWO algorithm’s efficacy and performance in finding the optimum parameters of the MPS1 and MPS2 systems, ground motions from the 1940 Imperial Valley and 1995 Kobe earthquakes are used in simulating the response of the generic frames described in Sect. 2. The optimum parameters of both configuration MPS1 and MPS2 are determined using the GWO algorithm. It should be noted that the equivalent total mass ratio μeq,opt in MPS1 configuration is kept equal to the summation of total mass μopt and inertance ratios βopt in MPS2 configuration. The proposed optimization approach is applied to calculate suitable parameter values for the proposed configurations already shown in Fig. 1, that are: (I) multi passive system denoted MPS1, consisting of two TMDs mounted on each building separately; (II) multi passive system denoted MPS2, which is composed of two TMDs

Optimal Seismic Response Control of Adjacent Buildings … Table 1 Optimization bounds

Design variables

Min value

107 Max value

f opt

0.2

1.2

ζopt (%)

10–3

80

μopt (%)

0.1

0.9

βopt (%)

0

40

Fig. 3 Optimal parameters of MPS1 and MPS2 configurations

placed at the rooftop of each building and connected with an inerter generating an ˜ apparent mass b. The optimization problem  given in Eq. (9) is solved with the GWO algorithm performed in the MATLAB® software. The optimization bounds for design variables as presented in Table 1. In order to make the MPS1 and MPS2 configurations comparable, and considering the fact that TMD in (MPS1) do not have inertance ratio βopt , the parameters defining the TMD (MPS1) performance are: tuning frequency f opt , damping tuning ζopt (%) and equivalent mass ratio μe,opt = μopt + βopt . The optimal design parameters obtained for both configurations are given in Fig. 3.

4.2 Seismic Response of the Proposed Systems In this sub-section, the optimal parameters obtained through the optimization procedure are used to investigate the proposed devices’ efficiency in the time domain under earthquake excitation. For this purpose, two strong ground motion records are used to calculate the seismic response of the two adjacent buildings controlled with MPS1 and MPS2 configurations. As a preliminary study, two ground motions, one far-field and the other near-field are selected. These ground motions are: (i) El Centro; during the Imperial Valley, California earthquake of May 18, 1940, the N-S component was measured at the Imperial Valley Irrigation District substation in El Centro, California, and (ii) Kobe: during the Hyogo-ken Nanbu earthquake on January 17, 1995, the N– S component was reported at the Kobe Japanese Meteorological Agency station.

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The absolute peak acceleration of the ground motions are 3.417, and 8.676 m/s2 , respectively. The ground acceleration time histories are shown in Fig. 4. The maximum inter-storey drift in adjacent structures, as well as the required space to minimize pounding, are shown in Fig. 5. It can be seen that both the configurations reduce the inter-story drift and pounding distance efficiently compared to

Fig. 4 Time histories of 1940 Imperial Valley (left) and 1995 Kobe (right) ground motion accelerations

Fig. 5 Distribution of maximum a inter-storey drift along the floors of the left building, b interstorey drift along the floors in the right building and c minimum pounding gap distance under 1940 Imperial Valley, and 1995 Kobe ground motion accelerations, respectively

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Fig. 6 Inter-storey drift (a) and pounding gap distance (b) frequency response plots corresponding to the uncontrolled and controlled configurations

the uncontrolled case (NC). However, the inertially connected multi-tuned mass dampers (MPS2) surpasses the performance of unconnected multi-tuned mass dampers (MPS1) in terms of separation gap reduction. The reduction is observed in all the floors in both inter-storey drift (except left building) and separation gap despite the devices being placed at rooftop level. The MPS1 is superior to the MPS2 in controlling inter-storey drift of left the building at higher floors (i.e., 6, 7, 8, and rooftop), while MPS2 is superior at lower floors. In case of the Kobe ground motion, where the pounding risk is higher, reduction in pounding distance is very impressive. It’s also worth mentioning that despite using a very small physical mass (see Fig. 3), the coupled solution provides as good, if not better, control performance than the control scheme with independent TMDs. In Fig. 6, the frequency response in terms of inter-storey drift and pounding gap distance is shown for the two studied configurations and compared with the uncontrolled case. Overall, the performances of both the proposed configurations are efficient compared to the uncontrolled scenario; a slight advantage of the proposed device can be observed at frequencies ranging between 1 and 2.2 Hz.

4.3 Performance Under a Wide Range of Ground Motions To assess the performance of the proposed systems in more general terms, the controlled adjacent buildings (MPS1 and MPS2) configurations are submitted to a large database of ground motion records. In particular, a database of 462 ground motion records is used [52]. The duration and elastic response spectra of these ground motions are discussed in detail in this paper [53, 54]. More information about these ground motion records can be found in [55].

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Fig. 7 Variation in effectiveness of MPS1 and MPS2 in reducing the maximum inter-storey drift with uncontrolled maximum inter-storey drift for ground motions with PGA higher than 0.15 g

Due to the large number of records, only those records with PGA ≥ g (g being the acceleration of gravity) are presented here, as ground motions with lower intensities don’t pose significant pounding risk. In Fig. 7, the percentage reductions (with reference to the uncontrolled buildings) in maximum inter-storey drift are presented. The percentage reductions in the left and right buildings are denoted as (J1 ) and (J2 ), respectively. In controlling the maximum inter-storey drift (Fig. 7a) of the left building, both the configurations provide similar performance. The response reduction, in most cases, is not significant, and amplification is observed in some cases. On the right-side building, response reduction by the MPS2 configuration is very impressive, well above that of MPS1, and no case of response amplification. It appears the MPS2 is more effective in controlling the drift of the right building than that of the left building. While this observation is interesting and requires further investigation, our preliminary reasoning is related to the right building being more flexible than the left building. Being more flexible, the right building experiences larger drift demands (see Fig. 7a, b). Since the optimization is based on transfer function related to pounding gap, it appears that the optimal solution in the coupled system corresponds to optimal control of the more flexible system. Since the system is coupled, the parameters of the TMD in the left building is somehow forced by the optimal parameters of the right building (see Fig. 3), which results in lower efficiency in response control. In summary, it appears that optimal control of pounding distance may be indirectly related to optimal control of drift demands of the more flexible building. The dynamical properties of the buildings such as mass, stiffness and damping are primarily responsible for differences in response reduction between adjacent buildings. In Fig. 8, the percentage reduction (with reference to the uncontrolled buildings) in the pounding distance of the controlled structures is presented as a function of that of the uncontrolled structures. as the pounding distance plotted here corresponds to its largest value in all the floors. It can be seen that both the configurations reduce

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Fig. 8 Effectiveness of MPS1 versus MPS2 configuration in lowering the minimum separation gap value to mitigate pounding. The effectiveness variation with different ground motion parameters is presented for ground motions with a PGA higher than 0.15 g

the minimum separation gap, with MPS2 clearly superior to MPS1. At very high demands, i.e., when the pounding distance in the uncontrolled structures is large, the performance of the two configurations is similar. When subjected to less strenuous ground motions, the performance of MPS2 is much better than that of MPS1.

5 Conclusions In this study, the pounding hazard mitigation of two adjacent buildings controlled with an inerter-coupled TMD system is investigated. The parameters of the proposed system are optimized using the grey wolf optimizer algorithm. The proposed system is compared with optimal TMDs placed independently on the roof of the two buildings. The total mass of the control devices in the two configurations are constrained to be equal. Based on numerical simulation and analysis of dynamic response to a large set of ground motion records, it is possible to reach the following conclusions:

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– When the two buildings are controlled independently (i.e., MPS1), inter-storey drift reduction in the stiffer building is more than that in the more flexible building. The opposite is the case when the TMDs at the roofs of the buildings are coupled with an inerter device (i.e., MPS2). – MPS1 configuration seems to be more effective than MPS2 configuration in reducing inter-storey drift demands on the stiffer buildings, while MPS2 configuration is more effecting in reducing inter-storey drift demands on the more flexible building. Since the overall demand in the uncontrolled case is higher in the more flexible building, it can be argued that MPS2 configuration is, in general, more advantageous. – In terms of controlling inter-storey drift demands on the more flexible structure, which faces more demand from the ground motions considered here, MPS2 configuration is far more effective than MPS1 configuration, which provides lower response reduction in most cases, and also amplifies response in many cases. – While both the control configurations are effective and show similar performance in reducing pounding gap when subjected to the most demanding ground motions, MPS2 configuration provides clearly better response reduction than MPS1 configuration in most of the ground motions used in this study. This leads to the important conclusion that MPS2 configuration provides at least the same level of performance as MPS1 configuration, alleviates some dangerous cases of response amplification, and outperforms MPS1 configuration in many cases. – Superiority of MPS2 configuration over MPS1 configuration is valid for a wide range of amplitude, duration, and frequency content parameters represented by the large set of ground motion parameters used in this study. – Despite using much lower physical mass compared to MPS1, the proposed inertercoupled system provides better control performance against a wide range of real ground motions recorded during past earthquakes.

Optimal Seismic Response Control of Adjacent Buildings …

Appendix The mass matrix, stiffness and damping matrices for the two buildings (Fig. 9)

Fig. 9 The complete flowchart of the grey wolf optimizer (GWO) algorithm

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⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ [M L ] = [M R ] = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎤

90.718

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥tons ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

90.718 90.718 90.718 90.718 90.718 90.718 90.718 90.718

(A.1)

⎡

6714.53 ⎢ −4753.16 ⎢ ⎢ ⎢ 1194.81 ⎢ ⎢ −235.21 ⎢ ⎢ [K L ] = ⎢ 39.01 ⎢ ⎢ −6.64 ⎢ ⎢ ⎢ 1.02 ⎢ ⎣ −0.14 0.01

−4753.16 7156.12 −4666.16 1349.70 −223.85 38.10 −5.87 0.78 −0.07

1194.81 −4666.16 7139.68 −4523.12 1176.88 −200.34 30.84 −4.12 0.36

−235.21 1349.70 −4523.12 6480.87 −3998.04 1036.57 −159.55 21.32 −1.84

39.01 −223.85 1176.88 −3998.04 5702.41 −3402.84 810.21 −108.26 9.35

−6.64 38.10 −200.34 1036.57 −3402.84 4640.55 −2598.39 538.66 −46.50

1.02 −5.87 30.84 −159.55 810.21 −2598.39 3322.87 −1624.16 223.16

−0.14 0.78 −4.12 21.32 −108.26 538.66 −1624.16 1764.20 −588.30

⎤ 0.01 −0.07 ⎥ ⎥ ⎥ 0.36 ⎥ ⎥ −1.84 ⎥ ⎥ ⎥ 2 9.35 ⎥ × 10 K N /m ⎥ −46.50 ⎥ ⎥ ⎥ 223.16 ⎥ ⎥ −588.30 ⎦ 403.84

−1793.13 2716.65 −1785.22 521.17 −87.45 15.07 −2.35 0.32 −0.03

455.38 −1785.22 2747.85 −1754.81 462.03 −79.63 12.41 −1.68 0.15

−90.54 521.17 −1754.81 2533.61 −1578.28 414.21 −64.57 8.74 −0.76

15.19 −87.45 462.03 −1578.28 2268.72 −1367.35 329.75 −44.61 3.89

−2.62 15.07 −79.63 414.21 −1367.35 1879.57 −1063.32 223.22 −19.48

0.41 −2.35 12.41 −64.57 329.75 −1063.32 1369.88 −676.12 93.97

−0.06 0.32 −1.68 8.74 −44.61 223.22 −676.12 738.54 −248.35

⎤ 0.00 −0.03 ⎥ ⎥ ⎥ 0.15 ⎥ ⎥ −0.76 ⎥ ⎥ ⎥ 2 3.89 ⎥ × 10 K N /m ⎥ −19.48 ⎥ ⎥ ⎥ 93.97 ⎥ ⎥ −248.35 ⎦ 170.61

(A.2)

⎡

2522.11 ⎢ −1793.13 ⎢ ⎢ ⎢ 455.38 ⎢ ⎢ −90.54 ⎢ ⎢ [K R ] = ⎢ 15.19 ⎢ ⎢ −2.62 ⎢ ⎢ ⎢ 0.41 ⎢ ⎣ −0.06 0.00 ⎡

⎤ 168.82 −116.55 29.30 −5.77 0.96 −0.16 0.03 0.00 0.00 ⎢ −116.55 179.65 −114.41 33.09 −5.49 0.93 −0.14 0.02 ⎥ 0.00 ⎢ ⎥ ⎢ ⎥ ⎢ 29.30 −114.41 179.25 −110.91 28.86 −4.91 0.76 −0.10 0.01 ⎥ ⎢ ⎥ ⎢ −5.77 33.09 −110.91 163.09 −98.03 25.42 −3.91 0.52 −0.05 ⎥ ⎢ ⎥ ⎢ ⎥ 1 [C R ] = ⎢ 0.96 −5.49 28.86 −98.03 144.01 −83.44 19.87 −2.65 0.23 ⎥ × 10 K N .s/m ⎢ ⎥ ⎢ −0.16 0.93 −4.91 25.42 −83.44 117.97 −63.71 13.21 −1.14 ⎥ ⎢ ⎥ ⎢ ⎥ −0.14 0.76 −3.91 19.87 −63.71 85.66 −39.82 5.47 ⎥ ⎢ 0.03 ⎢ ⎥ ⎣ 0.00 0.02 −0.10 0.52 −2.65 13.21 −39.82 47.44 −14.43 ⎦ 0.00 0.00 0.01 −0.05 0.23 −1.14 5.47 −14.43 14.09 ⎡ ⎤ 66.03 −43.97 11.17 −2.22 0.37 −0.06 0.01 0.00 0.00 ⎢ −43.97 70.80 −43.77 12.78 −2.14 0.37 −0.06 0.01 0.00 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 11.17 −43.77 71.56 −43.03 11.33 −1.95 0.30 −0.04 0.00 ⎥ ⎢ ⎥ ⎢ −2.22 12.78 −43.03 66.31 −38.70 10.16 −1.58 0.21 −0.02 ⎥ ⎢ ⎥ ⎢ ⎥ [C L ] = ⎢ 0.37 −2.14 11.33 −38.70 59.81 −33.53 8.09 −1.09 0.10 ⎥ × 101 K N .s/m ⎢ ⎥ ⎢ −0.06 0.37 −1.95 10.16 −33.53 50.27 −26.07 5.47 −0.48 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0.01 −0.06 0.30 −1.58 8.09 −26.07 37.77 −16.58 2.30 ⎥ ⎢ ⎥ ⎣ 0.00 0.01 −0.04 0.21 −1.09 5.47 −16.58 22.29 −6.09 ⎦ 0.00 0.00 0.00 −0.02 0.10 −0.48 2.30 −6.09 8.37

(A.3)

(A.4)

(A.5)

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Optimization of Tuned Liquid Dampers for Structures with Metaheuristic Algorithms Ayla Ocak, Gebrail Bekda¸s, and Sinan Melih Nigdeli

Abstract Tuned liquid dampers (TLD) is a passive control system used to dampen building movement with the help of spring and liquid mass. These devices, which are effective in preventing all kinds of vibrations in the structure, use the geometric properties of the damper and the fluid parameters that provide sloshing in providing the structure control. Using various optimization methods in determining the characteristic features of TLDs plays an important role in increasing the damping performance. For this purpose, Metaheuristic Algorithms inspired by natural events and instinctive behaviors of living things are seen as an advantageous optimization method due to the simple and understandable logic of the mathematical models and the easy selection of design factors for the problem. This study aims to optimize TLD devices by using Jaya (JA) and Teaching-Learning Based Optimization (TLBO) algorithms to determine the design parameters that will minimize the structure movement in a single-story structure exposed to earthquake warnings. Keywords Tuned liquid dampers · Structural control · Optimization · Jaya algorithm (JA) · Teaching-learning based optimization (TLBO) · Metaheuristics algorithm

1 Introduction Tuned liquid dampers are passive control devices that provide damping by making use of the sloshing energy of the liquid they contain. These devices, like other passive control devices, adopt Newton’s principle of conservation of energy. By converting the kinetic energy of any external load on the structure into mechanical energy, it

A. Ocak · G. Bekda¸s (B) · S. M. Nigdeli Department of Civil Engineering, Istanbul University-Cerrahpa¸sa, Avcılar, 34320 Istanbul, Turkey e-mail: [email protected] S. M. Nigdeli e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. Bekda¸s and S. M. Nigdeli (eds.), Optimization of Tuned Mass Dampers, Studies in Systems, Decision and Control 432, https://doi.org/10.1007/978-3-030-98343-7_7

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ensures the sloshing of the liquid in the system. It aims to dampen the movement of the building by providing a cycle without losing the energy coming into the building. We can list some of the advantages of liquid dampers as follows. • Since they have a simple setup and are easy to install, they do not require much cost. Maintenance and repairs are easy. • They can be easily placed not only in new structures but also in old structures. • The frequency of the system can be easily adjusted by factors such as liquid movement and pressure change in the system. Considering the disadvantages. • It may give small errors (like measuring still water level). • It may be insufficient for strong vibrations of larger amplitude (mostly used for wind vibrations). Today, examples can be seen in many buildings. Examples of TLD applications are structures such as Nagasaki Airport Tower in Japan, Tokyo Sofitel Hotel, Yokohoma Marine Tower, Shin Yokohoma Prince Hotel, Sakitama Bridge, Narita Airport Control Tower, and Emley Moor Tower in England. Apart from these, we can show the British Airways i360 structure, the thinnest and tallest tower in the world compared to its diameter, in England, which was opened in 2016. Although it is on the coastline and exposed to a lot of wind load, TLDs placed in the structure ensured that these vibrations are kept under control. TLDs were first used to prevent vibrations in ship machinery and satellites since the 1950s, and their applicability in structures was investigated by Bauer’s studies in the 1980s [1–5]. In 1997, with the work of Modi and Seto, a sloshing damper, also known as a tuned liquid damper, was named [6]. TLDs have been used in vibration control of different types of structures such as towers, wind turbines, buildings, bridges, and offshore platforms since the 1990s [7–11]. Modifying TLDs, multiple tuned liquid column damper (MTLCD), hybrid mass liquid damper (HMLD), tuned liquid column and agitation damper (TLCSD), liquid column vibration absorber (LCVA), tuned vibration damper (TSD), and the pendulum type liquid column damper (PLCD) such as various liquid dampers have been developed [12–18]. Optimization is the process of obtaining the desired minimum or maximum values by the purpose of the problem from the data obtained because of the study of the desired parameters of any problem. Optimization is a process with a wide scope, applicable to all kinds of problems in life. Metaheuristic algorithms are one of the methods used in the implementation of this process. In 2010, Yang wrote a book on the use of metaheuristic algorithms in engineering optimization and mentioned the problems in which metaheuristic algorithms can be applied in optimization [19]. The feature that distinguishes metaheuristic algorithms from other algorithms is that they solve problems by using events found in nature’s cycle. It reflects not only events but also instinctive behaviors in living life to algorithms. For example, when we look at the whale optimization algorithm, behaviors such as the roadmap of the spiral-shaped bubbles formed by humpback whales while hunting [20], the foraging model created by the artificial bee algorithm to search for

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food by honey bees [21], or the bats in the bat algorithm to detect the location of the objects around them using the echolocation called echolocation [22], inspired by nature, it led to the creation of metaheuristic algorithms. Some examples of metaheuristic algorithm types are; Genetic Algorithm (GA) developed by Holland in 1975 [23], Simulating Annealing (SA) proposed by Kirkpatrick et al. in 1983 [24], Ant Colony Optimization (ACO) first used by Dorigo in 1991, in 1995 Particle Swarm Optimization (PSO) proposed by Kennedy et al. [25], Harmony Search Algorithm (HS) proposed by Geem et al. in 2001 [26], Artificial Bee Colony (ABC) introduced by Karabo˘ga in 2005 [21], in 2011 Teaching-Learning Based Optimization (TLBO) [27] introduced by Rao et al. in, Flower Pollination Algorithm (FPA) proposed by Yang in 2012 [28], Jaya Algorithm (JA) proposed by Rao et al. [29] can be shown. The algorithms used for the optimization process in this study are explained below.

2 Methodology 2.1 Optimum Design of Tuned Liquid Dampers via Teaching-Learning Based Optimization Algorithm It is an algorithm inspired by the fact that the same education level, class, or group of students based on the population are obtained by 2 different instructors, with different instructor behaviors, and different results in grade distributions, although the same information is seen as education. In 2011, a model was prepared by using mathematical expressions based on teaching and brought to the literature as an optimization method by Rao et al. It has been found that teachers’ good teaching skills play a major role in the positive outcome of education and students’ understanding [27]. The instructor success level here is evaluated by increasing the grade distribution on average. In this algorithm, in which the same subjects are explained but two different results are obtained, it has been observed that apart from education, students’ interaction with each other and sharing their knowledge contributes to the increase in the average. This variable situation in the results was attributed to the teacher’s teaching ability and the interaction of the students. There is research on this algorithm in many fields in engineering science. In various civil engineering problems, the TLBO algorithm has been applied. During the optimization of passive dampers related to structure control, studies have been done using TLBO [30–32]. Before starting the optimization process, all fixed parameters and design lower and upper limit values should be introduced to the system. Equation 1 shows the design limits. Xjmin ≤ Xj ≤ Xjmax j = 1, 2, 3, . . . , Tn

(1)

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In Eq. 1, Xj represents each variable in the design and j represents the number of variables from 1 to Tn. In the system, random values are assigned within the design constraints and turned into a matrix (Eq. 2). ⎡

⎤ X1,1 · · · X1,T ⎢ ⎥ A = ⎣ ... . . . ... ⎦ XPn,1 · · · XPn,T

(2)

The results obtained by writing the generated random values in the A matrix in the objective function are converted into a solution vector as in Eq. 3. ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ f (X ) = ⎢ ⎢ ⎢ ⎢ ⎣

f (X1 ) f (X2 ) .. . .. . .. .

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(3)

f (XPn ) The mean value of the population is calculated. This value is written inside MT in Eq. 4. MT = [m1 , m2 , . . . , mT ]

(4)

The min value in the vector f (X ) is called the best solution. ˙In Eq. 5 is the value shown as Xteacher . Xteacher = Xmin f (x)

(5)

The best solution determines the new solution in Eq. 6 and is used to find new solutions. MNew,T = Xteacher,T

(6)

As shown in Eq. 7, with the randomly selected r value in the range [0,1], a difference value is obtained from the old mean and the new mean. The teaching factor Tf found here is randomly selected as 1 or 2 as in Eq. 8. DifferenceT = r(MNew,T − Tf MT )

(7)

Tf = round [1 + rand (0, 1){2 − 1}]

(8)

The new solution values in Eq. 9 are updated with DifferenceT from Eq. 7.

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XNew,T = XOld ,T + DifferenceT

123

(9)

Phase 1 is completed with the latest updated solutions. Phase 2 is the situation in which students transfer knowledge by sharing what they have learned with each other. 2 solution values (XP , XT ) ) are randomly selected from the student solutions found in the population. Equations 10 and 11 are selected according to the better solutions from these solutions and the solutions are updated. These operations are repeated as many as the number of iterations. r → [0, 1] XNew= XP+ r(XP − XT )

(10)

XNew= XP+ r(XT − XP )

(11)

2.2 Optimum Design of Tuned Liquid Dampers via Jaya Algorithm Jaya Algorithm (JA) Developed by Rao [18], it is an algorithm that can be expressed with a single equation that does not include a design factor and is derived from the logic of the 2-phase Teaching-Learning Based Optimization algorithm. Thanks to these features, it shortens the optimization process and makes it easier to obtain optimum values. The Jaya algorithm is frequently preferred in solving many engineering problems due to its advantages in practice [33–39]. There are studies on the design optimization of dampers in structure control [40–42]. The algorithm equation of the Jaya algorithm is given in Eq. 12.

 Xi,new = Xi,j + rand Xi,best − Xi,j − rand (Xi,worst − Xi,j )

(12)

In Eq. 12, Xi,new , i the new solution value for the variable, Xi,j , i of the design variable j candidate solution, Xi,best , i of the best solution in the objective function, design variable, Xi,worst , the worst solution in the objective function i represents the design variable.

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2.3 Design Parameters of Tuned Liquid Dampers Tuned liquid dampers use sloshing of the liquid in the tank in the opposite direction of movement to control structure movement. In this sloshing, values such as the geometry of the tank, liquid mass, stiffness, and damping coefficient are among the factors affecting the restriction of movement. In the design of TLD devices, the sloshing liquid must be considered as a part that moves independently of the tank. Therefore, in TLD design, each parameter is calculated separately for liquid, structure, and TLD. In this study, the geometry of the tank was chosen as a cylinder. It is known that in the design of tuned liquid dampers, some of the liquid mass is not involved in sloshing and remains as the passive liquid mass. Based on this, in Eqs. 13 and 14, the mass of liquid involved in sloshing is expressed as ms , the mass of TLD mTLD and mass of empty tank + mass of passive liquid that is not sloshing, as md . tanh( 1.84h ) R 2.2h

ms = mst × R ×

md = mTLD − ms

(13) (14)

For a cylindrical TLD tank, h is the tank height and R is the tank radius. TLD and sloshing stiffness are given in Eqs. 15 and 16, damping coefficients are given in Eqs. 17 and 18. kd = md ×

ks = mst ×

 2π 2

Td

 2 g tanh 1.84h R

1.19h  cd = 2 × ζd × md × kd  cs = ζs × 2 ms Ks

(15)

(16) (17) (18)

The damping ratio of a system with TLD added is calculated as in Eq. 19. ζd =

cd  2md

kd md

(19)

By adding TLD to the structure, the mass, stiffness, and damping coefficients of the system are obtained by using the TLD, the sloshing liquid, and the parameters of the structure (Eqs. 20, 21, and 22). Unindexed terms indicate the parameters of the structure, those with “s” index the sloshing liquid, and a “d” index the parameters of

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the TLD. ⎡

⎤ m0 0 ⎣ 0 md 0 ⎦ 0 0 ms ⎡

K + Kd ⎣ −Kd 0 ⎡ C + Cd ⎣ −Cd 0

⎤ −Kd 0 Kd + Ks −Ks ⎦ −Ks Ks ⎤ −Cd 0 Cd + Cs −Cs ⎦

−Cs

(20)

(21)

(22)

Cs

The basic equation of motion obtained with m, k, and c matrices is as shown in Eq. 23.       [M ] X¨ + [C] X˙ + [K]{X } = −[M ]1 X¨ g

(23)

The equation of motion specified for TLD is solved on Matlab via Simulink [43]. Earthquake ground acceleration records to be applied to the structure were taken from FEMA: Measurement of Building Seismic Performance Factors far-field records [44].

3 The Numerical Example In this study, the TLD device, which contains water as a liquid mass, was placed on a single-storey structure and earthquake warnings were sent to the structure and various design parameters were optimized. For this purpose, a three degree of freedom (3DOF) model consisting of the structure, TLD device, and sloshing liquid mass was created (Fig. 1). The mass of the structure was taken as 100 tons, and a structure mass ratio of 5 tons was taken by determining the 5% for TLD. By choosing the structure period of 1.5 s, the period was optimized so that a value between 0.5 and 1.5 times the value in the structure was chosen for the TLD period. TLD’s period, damping ratio, tank radius, and height are optimized with TLBO and Jaya algorithms. TLD tank radius and height of 0.1 and 10 m were taken as design h > 0.15) constraints and the ratio between effective liquid length and tank height ( 2R made by Fujino et al. was checked [45]. Optimization processes were repeated for 1000 iterations. Optimization results obtained from JA and TLBO algorithms are given in Table 1. The structure displacement and total acceleration values obtained for the critical earthquake warning in the analysis using the optimized values are shown in Table 2.

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Fig. 1 Structure + TLD system model

Table 1 Optimum results for TLD-Water optimized with TLBO and JA Variables

Optimized Values JA

TLBO

Td (s)

1.3142

1.3482

ζ

0.0665

0.0112

R(m)

0.7209

0.7920

h(m)

0.4608

0.3276

Table 2 TLD analysis results for critical earthquake Algorithm

JAYA

Without TLD

With TLD

Displacement (m)

Total acceleration (m2 /s)

Displacement (m)

Total acceleration (m2 /s)

0.2927541

5.2223281

0.2504594

4.3653512

0.2503374

4.3413470

TLBO

The graphs of the displacement and total acceleration values that occurred in the critical earthquake as a result of Jaya and TLBO optimizations are given in Figs. 2 and 3, respectively.

4 Conclusions Mass, stiffness, and damping coefficient are the most important factors in the design of TLDs. With the optimization process, obtaining these values or finding the parameters used in the calculation ensures that the design shows the optimum efficiency.

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Fig. 2 Displacement and total acceleration graphs for JA optimization critical earthquake

Fig. 3 Displacement and total acceleration graphs for TLBO optimization critical earthquake

In this study, using two similar algorithms, single-phase and two-phase, TLD design parameters were optimized, and the performance of the algorithms in reducing the structure motion was investigated. The results obtained from the analysis are as follows. – The critical earthquake displacement value obtained with the TLBO algorithm gave a better result with a difference of 0.1 mm than that obtained with Jaya. – While the TLBO algorithm reduced the critical earthquake displacement value by 14.49%, the Jaya algorithm decreased it by 14.45%. It is seen that both algorithms are successful in reducing the structure motion. – Optimized tank radius and heights, although both algorithms have close values, there are differences. This shows that there are different options for design diversity and that more than one design measure can give the optimum result. – TLBO showed a 16.87% and Jaya a 16.41% reduction effect in total acceleration values. Based on this, it is understood that TLD optimization is effective in reducing the overall acceleration.

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When these results are examined, it has been determined that TLD optimization has remarkable effects in the control of structure motion.

References 1. Hüsem, M., Yozgat, E.: Building Control Systems that Can Be Used in Earthquake Resistant Building Design, vol. 20, p. 121. Chamber of Civil Engineers ˙Izmir Branch (2005) 2. Moiseyev, N.N.: Dynamics of a ship having a liquid load. Izv Akad Nauk SSSR Old Tekhn Naud 7, 27–45 (1952) 3. Jamalabadi, M.Y.A.: Frequency analysis and control of sloshing coupled by elastic walls and foundation with smoothed particle hydrodynamics method. J. Sound Vib. 115310 (2020) 4. Abramson, H.N., Chu, W.H., Garza, L.R.: Liquid sloshing in spherical tanks. AIAA J. I(2), 384–389 (1963) 5. Bauer, H.F.: Oscillations of immiscible liquids in a rectangular container: a new damper for excited structures. J. Sound Vib. 93(1), 117–133 (1984) 6. Modi, V.J., Seto, M.L.: Suppression of flow-induced oscillations using sloshing liquid dampers: analysis and experiments. J. Wind Eng. Ind. Aerodyn. 67, 611–625 (1997) 7. Yamamoto, K., Kawahara, M.: Structural oscillation control using a tuned liquid damper. Comput. Struct. 71(4), 435–446 (1999) 8. Gao, H., Kwok, K.S.C., Samali, B.: Characteristics of multiple tuned liquid column dampers in suppressing structural vibration. Eng. Struct. 21(4), 316–331 (1999) 9. Xue, S.D., Ko, J.M., Xu, Y.L.: Optimum parameters of tuned liquid column damper for suppressing pitching vibration of an undamped structure. J. Sound Vib. 235(4), 639–653 (2000) 10. Lee, H.H., Wong, S.H., Lee, R.S.: Response mitigation on the offshore floating platform system with tuned liquid column damper. Ocean Eng. 33(8–9), 1118–1142 (2006) 11. Zhang, Z., Staino, A., Basu, B., Nielsen, S.R.: Performance evaluation of full-scale tuned liquid dampers (TLDs) for vibration control of large wind turbines using real-time hybrid testing. Eng. Struct. 126, 417–431 (2016) 12. Shum, K., Xu, Y.L.: Multiple tuned liquid column dampers for reducing coupled lateral and torsional vibration of structures. Eng. Struct. 26(6), 745–758 (2004) 13. Banerji, P., Samanta, A.: Earthquake vibration control of structures using a hybrid mass liquid damper. Eng. Struct. 33(4), 1291–1301 (2011) 14. Lee, S.K., Min, K.W., Lee, H.R.: Parameter identification of new bidirectional tuned liquid column and sloshing dampers. J. Sound Vib. 330(7), 1312–1327 (2011) 15. Taflanidis, A.A., Angelides, D.C., Manos, G.C.: Optimal design and performance of liquid column mass dampers for rotational vibration control of structures under white noise excitation. Eng. Struct. 27(4), 524–534 (2005) 16. Taflanidis, A.A., Beck, J.L., Angelides, D.C.: Robust reliability-based design of liquid column mass dampers under earthquake excitation using an analytical reliability approximation. Eng. Struct. 29(12), 3525–3537 (2007) 17. Cavalagli, N., Agresta, A., Biscarini, C., Ubertini, F., Ubertini, S.: Enhanced energy dissipation through 3D printed bottom geometry in Tuned Sloshing Dampers. J. Fluids Struct. 106, 103377 (2021) 18. Sarkar, A., Gudmestad, O.T.: Pendulum-type liquid column damper (PLCD) for controlling vibrations of a structure–theoretical and experimental study. Eng. Struct. 49, 221–233 (2013) 19. Yang, X.S.: Engineering Optimization: An Introduction with Metaheuristic Applications. Wiley (2010) 20. Mirjalili, S., Lewis, A.: The whale optimization algorithm. Adv. Eng. Softw. 95, 51–67 (2016) 21. Karabo˘ga, D.: An Idea Based on Honey Bee Swarm for Numerical Optimization, vol. 200, pp. 1– 10. Technical report-tr06, Erciyes University, Engineering Faculty, Computer Engineering Department (2005)

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22. Yang, X.S.: Bat algorithm for multi-objective optimization. Int. J. Bio-Inspired Comput. 3(5), 267–274 (2011) 23. Holland, J.H.: Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor, MI (1975) 24. Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P.: Optimization by simulated annealing. Science 220(4598), 671–680 (1983) 25. Kennedy, J., Eberhart, R.C.: Particle swarm optimization. In: Proceedings of IEEE International Conference on Neural Networks No. IV, 27 November–1 December. Perth, IEEE Service Center, Piscataway, NJ, 1942–1948 (1995) 26. Geem, Z.W., Kim, J.H., Loganathan, G.V.: A new heuristic optimization algorithm: harmony search. Simulation 76(2), 60–68 (2001) 27. Rao, R.V., Savsani, V.J., Vakharia, D.P.: Teaching-Learning-based optimization: a novel method for constrained mechanical design optimization problems. Comput Aided Des 43, 303–15 (2011) 28. Yang, X.S.: Flower pollination algorithm for global optimization. In: Durand-Lose, J., Jonoska, N. (eds.) Lecture Notes in Computer Science, 27, vol. 7445, pp. 240–249. Springer, London (2012) 29. Rao, R.: Jaya: a simple and new optimization algorithm for solving constrained and unconstrained optimization problems. Int. J. Ind. Eng. Comput. 7(1), 19–34 (2016) 30. Temür, R., Bekda¸s, G.: Teaching learning-based optimization for design of cantilever retaining walls. Struct. Eng. Mech. 57(4), 763–783 (2016) 31. Ni˘gdeli, S.M., Bekda¸s, G.: Teaching-learning-based optimization for estimating tuned mass damper parameters. In: 3rd International Conference on Optimization Techniques in Engineering (OTENG’ 15), 7-9 November. Rome, Italy (2015) 32. Bekda¸s, G., Ni˘gdeli, S.M., Aydın, A.: Optimization of tuned mass damper for multi-storey structures by using impulsive motions. In: 2nd International Conference on Civil and Environmental Engineering (ICOCEE 2017). Cappadocia, Turkey (2017) 33. Öztürk, H.T.: Modeling of concrete compressive strength with Jaya and teaching-learning based optimization (TLBO) algorithms. J. Investig. Eng. Technol. 1(2), 24–29 (2018) 34. De˘gertekin, S.O., Bayar, G.Y., Lamberti, L.: Parameter free Jaya Algorithm for truss sizinglayout optimization under natural frequency constraints. Comput. Struct. 245, 106461 (2021) 35. Kaveh, A., Hosseini, S.M., Zaerreza, A.: Improved Shuffled Jaya algorithm for sizing optimization of skeletal structures with discrete variables. In: Structures, vol. 29, pp. 107–128. Elsevier (2021) 36. Mahjoubi, S., Bao, Y.: Game theory-based metaheuristics for structural design optimization. Comput. Aided Civil Infrastruct. Eng. (2021) 37. Du, D.C., Vinh, H.H., Trung, V.D., Hong Quyen, N.T., Trung, N.T.: Efficiency of Jaya algorithm for solving the optimization-based structural damage identification problem based on a hybrid objective function. Eng. Optim. 50(8), 1233–1251 (2018) 38. Dede, T.: Jaya algorithm to solve single-objective size optimization problem for steel grillage structures. Steel Compos. Struct. 26(2), 163–170 (2018) 39. Rao, R.V.: Jaya: An Advanced Optimization Algorithm and Its Engineering Applications (2019) 40. Bekda¸s, G., Ulusoy, S., Ni˘gdeli, S.M.: Optimum tuning of mass dampers by impulsive motions and several metaheuristic methods. In: 3rd International Conference on Civil and Environmental Engineering (ICOCEE), 24-27 April 2018. ˙Izmir, Turkey (2018) 41. Bekda¸s, G., Kayabekir, A.E., Ni˘gdeli, S.M., Toklu, Y.C.: Transfer function amplitude minimization for structures with tuned mass dampers considering soil-structure interaction. Soil Dyn. Earthq. Eng. 116, 552–562 (2019) 42. Mubuli, A., Nigdeli, S.M., Bekda¸s, G.: Passive control of frame structures by optimum tuned mass dampers. In: International Conference on Harmony Search Algorithm, pp. 111–126. Springer, Singapore (2020) 43. The MathWorks, Matlab R2019a.: Natick, MA (2019)

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Semi-active Tuned Liquid Column Dampers with Variable Natural Frequency Behnam Mehrkian and Okyay Altay

Abstract Semi-active devices offer fail-safe attributes of passive and adaptability of active systems. Variants of tuned mass dampers (TMDs) have been proposed, which can adapt their frequency to an optimal one required by structures. In classical TMDs, an oscillatory solid mass generates restoring forces. The vibration energy of the mass is mitigated by supplementary dissipators. In contrast, tuned liquid column dampers (TLCDs) use Newtonian fluids, such as water, for both restoring and damping forces, which allows simpler manufacturing and versatility. This chapter presents two variants of TLCDs named semi-active TLCDs with movable panels or closable cells (S-TLCDMP , S-TLCDCC ), which can retune their natural frequency by manipulation of cross-sectional areas of vertical tubes. A general mathematical formulation is proposed for TLCDs, which covers the classical uniaxial U-shaped layout as well as the omnidirectional layout with multiple columns. Experimental studies validate the formulation and the frequency adaptation capability of both S-TLCDMP and S-TLCDCC . Keywords Semi-active · TLCD · Movable panels · Closable cells · Vertical cross-sectional area · Frequency adaptation

1 Introduction The field of structural control, which was formally initiated in early 1970s by Yao [1], can be categorized into three major classes of passive, semi-active, and active control systems plus a combination of these classes known as the hybrid control systems [2–7]. In these control classes, passive and semi-active systems may be favored for vibration control of civil engineering structures since, unlike the active B. Mehrkian · O. Altay (B) RWTH Aachen University, Mies-van-der-Rohe-Str. 1, 52074 Aachen, Germany e-mail: [email protected] B. Mehrkian e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. Bekda¸s and S. M. Nigdeli (eds.), Optimization of Tuned Mass Dampers, Studies in Systems, Decision and Control 432, https://doi.org/10.1007/978-3-030-98343-7_8

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counterpart, they are not prone to destabilize the structures and require low energy source to operate. Passive control systems, however, operate with predefined and fixed characteristics and cannot adapt themselves to a new state within a vibration event. This is while the civil engineering structures, which are excited by dynamic stochastic loads like wind and earthquakes, are very likely to require an adjustable control device to reach optimal vibration mitigation. Changing environmental conditions, degradation effects and soil-structure interaction reiterate the importance of this demand even further. Semi-active control systems, in contrast, can introduce a desirable trade-off between active and passive systems since semi-active control devices are reactive but adjustable without considerable need for power. As one important capability, a semi-active device can adapt its frequency within a vibration scenario. Principally, tuning a supplementary device to a natural frequency of an oscillating system leads to significant vibration attenuation [8] at that frequency. Consequently, the possibility to adaptively tune the semi-active control device to structures, which are often oscillating with combination of mode shapes and natural frequencies, is accounted as a pronounced advantage. Semi-active control devices can originate from passive devices. The famous examples of such systems are devices based on tuned mass dampers (TMDs). Nagarajaiah [9] developed the semi-active continuously and independently variable stiffness (SAIVS) device that can switch the stiffness continuously and smoothly. Furthermore, the adaptive length pendulum smart tuned mass damper (ALP STMD) was proposed by Nagarajaiah [10] and the resettable variable stiffness TMD (RVS-TMD) by Lin et al. [11]. In this family, a more versatile passive device has been proposed by Sakai et al. [12, 13] for civil engineering structures entitled a tuned liquid column damper (TLCD). A conventional TLCD is composed of a U-shaped container consisting of a horizontal tube connecting two vertical tubes all with uniform cross-sectional areas. The container can be commonly filled with any Newtonian liquid, such as water. Thanks to its column geometry, TLCDs employ more liquid mass than its predecessor tuned liquid dampers [14] and require simpler manufacturing, implementation and maintenance than TMDs. Moreover, without mechanical elements, TLCDs not only function as a vibration absorber but, when needed, TLCDs can supply the inhabitants of the building with water. Accordingly, passive TLCDs have been extensively studied by structural control community [15–24] and have been successfully employed in several real-world structures [25]. Even though the passive TLCDs are simple and cost-effective, they are particularly useful when suppressing excessive vibrations under a particular frequency and condition is of interest. In this regard, the aim of the present chapter is to introduce semi-active variations of TLCDs. The focus will be on the frequency adaptation methods for TLCDs, where semi-active tuned liquid column dampers with movable panels (S-TLCDMP ) and semi-active tuned liquid column dampers with closable cells (S-TLCDCC ) are mathematically and experimentally presented. Furthermore, the mathematical derivation will not be limited to the conventional uniaxial U-shaped

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layout with two vertical tubes but to a general omnidirectional layout with n number of vertical tubes. This chapter is structured as follows. Sect. 2 describes the mathematical formulation of the tuned liquid column damper with general layout and introduces technologies for semi-active natural frequency tuning. The mathematical and experimental investigations of the S-TLCDMP and the S-TLCDCC are described in Sects. 3 and 4, respectively. Finally, the chapter is concluded in Sect. 5.

2 Tuned Liquid Column Dampers Initially, invented by Frahm [26] to suppress rolling motions of ships, TLCDs are structural control devices, which improve modal response performance of structures. As shown in Fig. 1, a TLCD is assembled by n ≥ 2 vertical tubes, which are distributed symmetrically around an origin and communicating with each other via horizontal tubes. The tube system is partially filled with a Newtonian liquid, such as water, which exhibits out of phase oscillations with the structure generating restoring forces. The oscillation energy is dissipated by the turbulence effects of the liquid flow and the inherent friction effects, which can be deliberately induced by an orifice. In each horizontal tube, from the origin, the length of the liquid streamline is represented as H/2. In Fig. 1b, the liquid portion in the horizontal tube is denoted as ∀ H . Similarly, in each vertical tube, the liquid depth from the horizontal streamline to the hydrostatic liquid level is represented as V . These regions are denoted in the figure as ∀V . The liquid deflection in each vertical tube is discretized by u i , where

Fig. 1 a Plan and b front views of a TLCD with n columns on a structure: (1) vertical tube, (2) horizontal tube, (3) structure, (4) hydrostatic liquid level, (5) orifice

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i ∈ [1, n]. The cross-sectional areas of horizontal and vertical tubes are denoted as A H and A V , respectively. In Fig. 1, the single-degree-of-freedom w discretizes the displacement of the structure in transverse direction. The resulting vibration direction of the structure is inclined with α in the Cartesian coordinate system. The orientation of each horizontal tube of the TLCD is represented with θi = 2(i − 1)π/n. Depending on the number of columns, the TLCDs operate for n = 2 (U shape) uniaxially and for n ≥ 3 (star shape) omnidirectionally [13, 24, 27].

2.1 Mathematical Description The equation of motion (EoM) of a TLCD can be derived from the Euler–Lagrange equation: d dt



∂ Ek ∂ u˙

 −

∂Ep ∂ Ek + = Qu . ∂u ∂u

(1)

Here, the liquid deflection in each vertical tube u i for n ≥ 2 is generalized with [24] u i = ucos(α − θi ).

(2)

The kinetic energy is calculated as Ek =

 n   2 H  2 1 r V u˙ V,i  + u˙ H,i  , ρ AH i=1 2 2

(3)

where ρ is the liquid density and r = A V /A H . Furthermore, u˙ V,i and u˙ H,i represent the velocity vectors of the liquid in i th vertical and horizontal tubes in the Cartesian coordinate system as   u˙  ˙ sin α w˙ cos α uκ ˙ i , V,i = w

(4)

where κi = cos(α − θi ). The potential energy reads Ep =

n 1 ρ AV g (V + uκi )2 , i=1 2

(5)

where g is the gravity. Finally, the non-conservative force is computed as n 1 |u| ˙ uκ ˙ i2 , Q u = − ρλr A V i=1 4

(6)

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which represents the friction and turbulence effects with the loss factor λ [28]. This ˙ u/2 ˙ for n = 2, α = 0, θ1 = 0 and θ2 = π , which equation yields Q u = −ρλr A V |u| corresponds to the well-known representation for TLCDs with two vertical tubes, such as the one in [29]. The EoM of general TLCDs with n columns is determined after introducing Eqs. 2–6 into Eq. 1: ¨ u¨ + δ|u| ˙ u˙ + ωd2 u = −γ1 w,

(7)

using the TLCD parameters δ = r λ/2L 1 , ωd =



2g/L 1 , γ1 = H/L 1 , L 1 = 2V + r H,

(8)

which are the head loss coefficient, the natural angular frequency, the first geometric factor and the first effective length, respectively. In case of earthquake excitation, w¨ is replaced with w¨ t = w¨ + w¨ g . The liquid deflection in each vertical tube u i is obtained by Eq. 2. The corresponding EoM of the controlled structure reads ¨ w¨ + 2Ds ωs w˙ + ωs2 w = f − μ(w¨ + qγ2 u),

(9)

with the coefficient q=

cos2 α n=2 1 n ≥ 3, 2

(10)

where Ds and ωs are its damping ratio and natural angular frequency and f is the mass-normalized excitation force. Here, μ = m d /m s represents the mass ratio of the TLCD and the structure. Finally, γ2 is the second geometric factor of the TLCD, which is computed from the momentum induced by the liquid motion as γ2 = H/L 2 with L 2 = 2V + r −1 H,

(11)

where L 2 is the second effective length with m d = n2 ρ A V L 2 .

2.2 Technologies for Semi-active Natural Frequency Tuning The natural frequency of a TLCD can be controlled by modulating the velocity of the liquid flow. Two applications of this technique are presented in Sects. 3 and 4. Traditionally, the ends of vertical TLCD tubes are open. By sealing them, above the liquid surface, a closed chamber is established, which introduces a supplementary

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air spring to the TLCD. By controlling the air pressure of the chamber, the stiffness of the air spring and consequently the natural frequency of the liquid oscillation can be controlled. Various versions of this technique are described in [30–35]. The TLCD natural frequency can also be controlled by external mechanical spring elements, which are interposed between the TLCD and the structure. By modulating the spring stiffness, the phase shift of the damper restoring force can be adapted semi-actively. Such mechanisms are studied in [36–40]. Apart from natural frequency tuning, the resulting TLCD force can be semiactively modulated by controlling the inherent damping. This damping control is plausible, for example, by closing the orifice [12, 13, 41, 42], by reducing the open ends of vertical tubes [26, 43] and using a controllable magneto-rheological fluid as TLCD liquid [44, 45].

3 Semi-Active Tuned Liquid Column Dampers with Movable Panels The semi-active tuned liquid column damper with movable panels (S-TLCDMP ) can change the cross-sectional area of its vertical tubes by changing the position β of panels, which are attached on the tube walls as shown in Fig. 2 [46]. The panels are mounted perpendicular to the liquid flow direction in the horizontal tubes to keep the streamline length in the horizontal tube independent from β and constant as H/2. Furthermore, the panels consist of two segments, by which transition and deflection regions are created in the tube with the liquid depths V2 and V3 (β), respectively. Here, V3 changes depending on β as the total liquid volume is constant. Correspondingly, the regions in the vertical tube of the S-TLCDMP are denoted as

Fig. 2 a Plan and b side views of L-arms of the semi-active TLCDs with movable panels (STLCDMP ): (1) vertical tube, (2) horizontal tube, (3) movable panels, (4) liquid level at rest

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∀V 1 , ∀V 2 and ∀V 3 . During oscillation, the liquid surface stays ideally in ∀V 3 to avoid any nonlinear amplitude dependency of the tuned frequency.

3.1 Mathematical Description Equation of motion of S-TLCDMP . In fluid dynamics, the local fluid acceleration in a tube can be computed from the time dependent change in the generalized liquid velocity u(s, ˙ t), where s denotes the position along the liquid streamline. As shown in [46], the integral of this change along s of the TLCD corresponds to the liquid acceleration scaled by L 1 :

∂ u˙ ¨ ds = L 1 u. ∂t

(12)

Accordingly, a change in liquid velocity directly affects the first effective length, which in turn changes the natural angular frequency of the TLCD ωd , cf. Eq. 8. If the liquid velocity in the deflection region is assumed to be u, ˙ due to changes in the crosssectional areas, the velocity will change after the transition region to u˙ A V 1 /A V 2 and in the horizontal tube to u˙ A V 2 /A H , where the cross-sections are defined as shown in Fig. 2: Av1 (β) = d1 d3 (β), Av2 = d1 d2 , A H = 2V1 d2 .

(13)

Here, d3 and associated with it also A V 1 depend on the panel position β. From Eq. 12, the first effective length reads L 1 (β) = 2V3 (β) + V2 + (2V1 + V2 )

A V 1 (β) + r (β)H, AV 2

(14)

where r (β) = A V 1 (β)/A H . By substituting this L 1 (β) and r (β) in Eq. 8, the corresponding head loss coefficient δ(β), the natural angular frequency ω(β) and the first geometric factor γ1 (β) of the S-TLCDMP can be calculated. Equation of motion of the coupled S-TLCDMP -structure system. The change in β affects also the EoM of the controlled structure. Corresponding to Eq. 11, the second effective length changes with β and reads L 2 (β) = 2V3 (β) + V2 + (2V1 + V2 )

A V 2 (β) + r (β)−1 H, AV 1

(15)

where, like the classical TLCD for the liquid mass, it holds m d = ρ A V 1 L 2 . Finally, the second geometric factor γ2 (β) can be updated by Eq. 11 depending on the panel position β.

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3.2 Experimental Validation The frequency adaptation capability of the S-TLCDMP is validated experimentally on a model of the damper with two L-arms, which is fabricated by acrylic glass plates. The dimensions of the model are shown in Fig. 3. Each movable panel consists of two acrylic glass segments, which are 60 mm and 385 mm long. The rotation of the panel segments is realized by metal joints. The gap between panels and column walls is insulated by a plastic membrane. For the study, a shaking table is utilized, which can simulate harmonic and random vibrations within 0.1–50 Hz with max. ±50 mm amplitude. In the first part of the study, the S-TLCDMP model is mounted directly on the shaking table to examine its natural frequency for different panel positions. In the second part of the study, the S-TLCDMP model is mounted on a pendulum, which is attached to a frame. Base excitation is applied to the frame by the shaking table. As shown in Fig. 4, the liquid deflection is measured in one of the vertical tubes by a capacitance transducer. The motion of the pendulum is measured by an accelerometer. Both sensors are sampled at 100 Hz. The orifice of the S-TLCDMP is closed (φ = 90◦ ) and opened (φ = 0◦ ) by an actuator. The position of the movable panels is controlled by actuators, which are connected to the upper panel segments via threaded rods. The panels can change their position within β = [45◦ , 72◦ ]. Study 1—Effect of panel position on the natural frequency. In this study, to investigate the change of natural frequency and its calculation, frequency sweep

Fig. 3 a Front and b side drawings of the S-TLCDMP model: (1) vertical tube, (2) horizontal tube, (3) movable panels, (4) orifice

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Fig. 4 Experimental setup of the S-TLCDMP model: (1) shaking table, (2) pendulum, (3) STLCDMP model, (4) movable panels, (5) actuator, (6) capacitance transducer, (7) orifice, (8) accelerometer

tests are conducted on the S-TLCDMP model. For a water of m d = 4.95 kg, the corresponding parameters of the S-TLCDMP are given in Table 1. Furthermore, in Table 2, L 1 , L 2 , γ1 , γ2 and ωd (or f d ) are calculated according Table 1 Parameters of the S-TLCDMP model with m d = 4.95 kg and panel positions β = [45◦ , 72◦ ] β

AV 1

AV 2

AH

V1

V2

V3

H

(°)

(cm2 )

(cm2 )

(cm2 )

(cm)

(cm)

(cm)

(cm)

45

17.6

5.0

100.0

5.0

4.2

18.9

35.0

58

28.2

5.0

100.0

5.0

5.1

9.8

35.0

72

41.5

5.0

100.0

5.0

5.7

5.2

35.0

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Table 2 Further parameters of the S-TLCDMP model with m d = 4.95 kg and panel positions β = [45◦ , 72◦ ] β

L1

L2

γ1

γ2

fd

f d,m

| f d |

(°)

(cm)

(cm)

(−)

(−)

(Hz)

(Hz)

(Hz)

45

53.2

88.5

0.66

0.40

0.97

0.99

0.02

58

43.1

61.3

0.81

0.57

1.07

1.07

0.00

72

43.7

49.5

0.80

0.71

1.07

1.05

0.02

to Eqs. 8, 11, 14, 15. Here, f d,m is the experimentally determined natural frequency and  f d is the difference between f d and f d,m . We observe that the S-TLCDMP modulates its natural frequency by changing its panel positions β. From the low frequency difference  f d , the applicability of the proposed mathematical description is shown. The results are plotted in Fig. 5. Study 2—Vibration control performance. The natural frequency of the pendulum is f s = 1.15 Hz, whereas the natural frequency of the frame is f F = 9.20 Hz. Accordingly, the frame can be considered as a rigid body. The damping ratio of the pendulum is low at Ds = 1.0 %. According to Den Hartog [8], the optimum natural frequency of the damper should be f d,opt = f s /(1 + μ) where the mass ratio of the active liquid mass to and the modal mass of the structure is calculated as μ=

md m d γ1 γ2 . = ms m s + m d (1 − γ1 γ2 )

(16)

Here, m s includes besides structural mass also the dead liquid mass [46]. Accordingly, by changing the panel position, besides natural frequency of the S-TLCDMP , also the required optimum frequency changes. This effect is studied in Fig. 6 on the Fig. 5 Comparison of the measured and calculated natural frequencies of the S-TLCDMP model with the liquid mass m d = 4.95 kg

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Fig. 6 Change of S-TLCDMP performance depending on panel position: (a) natural frequency and optimum frequency vs. panel position and (b) maximum pendulum acceleration versus panel position

S-TLCDMP model with the liquid mass m d = 4.95 kg. The mass of the pendulum corresponds to m s = 10.86 kg. The panel position dependent natural frequencies and the corresponding optimum frequencies are plotted in Fig. 6a. Furthermore, in Fig. 6b, the pendulum acceleration peak values are plotted. The frame is excited harmonically by the shaking table at the resonance frequency of the pendulum.

4 Semi-active Tuned Liquid Column Dampers with Closable Cells Alternatively, the natural frequency of semi-active TLCDs can be adjusted by manipulation of cross-sectional areas at the top of the vertical tubes [47–49]. This approach is rooted in the fact that the liquid in a TLCD with a pair of L-arms will be motionless if we fully and simultaneously prevent the air communication (i) between each L-arm and the ambient air (ii) and between the L-arms themselves. This means that, if we close both vertical tubes at the top in a sealed fashion, a constant amount of air is trapped over the liquid in the tubes, which makes the liquid come to a standstill even if the container of the TLCD is exposed to movement—it is assumed that compression of the trapped air is negligible. The cross-sectional areas of the vertical tubes A V can be divided into smaller individual parts which may be called as cells. Figure 7 illustrates an L-arm of a liquid column damper accommodating closable cells with length d1 and widths d2 in the vertical tubes. The aforementioned closing concept can be applied, not to the whole, but to certain cells at the top of the vertical tubes in order to bring about variable cross-sectional areas in the tubes.

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Fig. 7 a Plan and b side views of an L-arm of a semi-active TLCD with closable cells (S-TLCDCC ): (1) vertical tube, (2) horizontal tube, (3) cells, (4) liquid level at rest, (5) closable valves

In consequence of these variable areas due to the closable (or control) cells, the liquid will be trapped in the closed cells but free to move in the open cells, so that the natural frequency of the damper will be altered. Since this frequency manipulation can be implemented with low source of power and without exerting control force to the liquid, the damper can be known as a semi-active tuned liquid column damper with closable cells (S-TLCDCC ). Considering a semi-active liquid damper with L-arms like the one with closable cells in Fig. 7, as an example, closing the side cells transforms the initial liquid damper with A V = d1 × 3d2 into a new liquid damper with an open cell in the middle and A V = d1 × d2 . Because of this transformation, not only the natural frequency but other characteristics of the liquid damper change, which are investigated in the next subsections for the S-TLCDCC alone as well as coupled with a structure.

4.1 Mathematical Description Equation of motion of S-TLCDCC . Assume a semi-active tuned liquid column damper with closable cells, S-TLCDCC , with n number of L-arms (n ≥ 2), the crosssectional areas of horizontal tubes A H , and the horizontal and vertical liquid lengths H/2 and V , respectively. Each vertical tube has a total cross-sectional area A V , variable open cross-sectional area A V,v , and variable closed cross-sectional area A V˜ ,v . Following Eq. 7 in Sect. 2, the motion equation for the S-TLCDCC reads 2 ˙ u˙ + ωd,v u = −γ1,v w¨ t , u¨ + δv |u|

(17)

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where by defining the variable cross-sectional area ratio rv and the first variable effective length L 1,v , rv =

A V,v , L 1,v = 2V + rv H, AH

(18)

the variable head-loss coefficient δv , the variable natural frequency ωd,v , and the first variable geometric factor γ1,v can be calculated using Eq. 8 where r and L 1 are replaced with rv and L 1,v . The liquid deflection in the i th vertical tube at θi = 2(i − 1)π/n is obtained using Eq. 2. In Eq. 17, w¨ t is the total excitation acceleration on S-TLCDCC ; in case of a structure with the degree of freedom w and under the ground motion wg , w¨ t = w¨ + w¨ g . Equation 17 is valid for a time slot where the desired A V,v is reached and remains time-invariant for that time slot. The equation does not address the transfer time from one time slot to the other (i.e. from one state of A V,v to the other). However, on one hand, each new state of cross-sectional area A V,v can be reached abruptly; on the other hand, depending on the frequency content of the excitation w¨ t , the rate of occurrence of these transfers may be limited, particularly for civil engineering structures. Equation 18 next to Eq. 8 shows that by varying the cross-sectional area of the vertical tubes of an S-TLCDCC ,A V,v , the natural frequency of the damper ωd,v varies since the first effective length L 1,v is affected. By increasing the cross-sectional area A V,v , the effective length increases but the natural frequency ωd,v , although with a lower rate, decreases and vice versa. This influence on ωd,v will be also investigated later experimentally in the next subsection. Furthermore, according to Eq. 8, the first variable effective length also inversely influences the geometric factor γ1,v and the head-loss coefficient δv . The head-loss coefficient δv has additionally a direct relation with the cross-section A V,v as well. Where the variable open vertical cross-sectional area, A V,v , is used to write the motion equation of the S-TLCDCC , the variable closed cross-sectional area A V˜ ,v = A V − A V,v represents the liquid cells which do not participate in the liquid motion. In other words, out of the total liquid mass of the S-TLCDCC m d , A V,v addresses the variable liquid mass, m d,v : m d,v =

n ρ A V,v L 2,v 2

n ≥ 2,

(19)

while A V˜ ,v addresses the variable dead liquid mass m d,v ˜ = m d − m d,v ,

(20)

which transfers to the primary structure. Here, the second variable effective length L 2,v is defined as

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L 2,v = 2V + rv−1 H.

(21)

Equation of motion of the coupled S-TLCDCC -structure system. Assume a host structure with a single floor in plane, which is symmetric along x- and y-axis (Fig. 1). The structure presents equal coefficients of the mass m s , the damping cs and the stiffness ks along the x- and the y-axis. The external force F and the ground motion wg are considered at the angle α with respect to (w.r.t) the first L-arm. In the coupled S-TLCDCC -structure system, the horizontal tubes of each L-arm exerts counteracting control force on the structure. Considering all n L-arms and projecting their control forces on the excitation direction α, the coupled EoM of S-TLCDCC -structure system reads (κi = cos(α−θi ))   m s,v + m d,v w¨ + w¨ g + cs w˙ + ks w+ n  γ2,v i=1

n

m d,v u¨ κi2 = F

n ≥ 2,

(22)

where the variable mass of the structure, which is updated due to the variable dead liquid mass transferred from the damper to the structure (Eq. 20), is m s,v = m s + m dv ˜ ,

(23)

and Eq. 2 has been used to determine the liquid deflection in each L-arm in the summation. Equation 22 can be rewritten based on the variable mass ratio μv = m d,v /m s,v : 2 w = f v − w¨ g (1 + μv ) − qγ2,v μv u, ¨ (1 + μv )w¨ + 2Ds,v ωs,v w˙ + ωs,v

(24)

with the coefficient q defined in Eq. 10; Ds,v and ωs,v are the variable damping ratio and the variable natural angular frequency of the structure, respectively; fv = F/m s,v is the variable mass-normalized external force and the second variable geometric factor is defined as γ2,v =

H L 2,v

(25)

On one hand, it is noticed from Eq. 24 that the mass ratio, the damping ratio and the natural frequency of the structure and the external force are affected by changing the natural frequency of the S-TLCDCC . On the other hand, comparing Eqs. 17 and 24, one recognizes that, unlike the liquid motion equation, the coupled motion equation is dependent to the excitation angle of incidence α, no matter the damper is passive or semi-active. However, this dependence is only limited to the regular liquid column dampers with 2-L-arms. The TLCDs with n ≥ 3 number of L-arms function omnidirectionally and are independent of the excitation angle.

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Calculation of variable vertical cross-section A V,v . The S-TLCDCC gets tuned when the cross-sectional areas of the vertical tubes in all L-arms are optimally adjusted. We may calculate the variable cross-section A V,v by inverting the natural frequency formula in Eq. 8: A V,v

AH = H

g



2 − 2V ,  2 π f opt

(26)

where f opt is the desired or the optimal frequency; f opt may be calculated as presented in literature for tuned dampers (e.g. cf. [8, 19, 50, 51]).

4.2 Experimental Validation This section reports some experimental results on a prototype of a S-TLCDCC , which has been produced for a real-world application. The main aim of this section is to compare the natural frequency obtained from the experiments in laboratory with the ones calculated by the theory (Eqs. 8 and 18) on this prototype excited by a shaking table. Description of the test setup. The prototype S-TLCDCC has been fabricated using 8 and 5 mm aluminum plates based on a 4-L-arm layout specified by C1 -C4 (n = 4). Figure 8 demonstrates the drawings of the prototype S-TLCDCC . The Larms consist of vertical and horizontal tubes denoted by ∀1 -∀4 and ∀1 -∀4 , respectively. Each vertical tube of the L-arms has been divided into two cells by a 5 mm plate, where one is the closable or control cell and the other is the open cell; that is, in total, there are four control and four open cells. The control cells can be automatically closed and opened using cone-shaped silicone stoppers. The stoppers are driven by regulators which are 24 V step motors mounted on holders. The cells can be closed and opened when the stoppers are driven downward and upward through the circular hole at the top of the cells, respectively. The regulators are controlled by a control algorithm programmed in a real-time PC. The control command is either opening the cells or closing the cells. Among the four open cells, two of them, in tubes ∀1 and ∀4 , are occupied by Ultrasonic sensors to track the liquid motion dynamically. It is noted that two sensors are enough to capture the whole information about the liquid state in this 4-L-arm layout. Both the sensors and the regulator-stopper systems are installed outside over the cells of the vertical tubes. The sampling rate of the measurements is 100 Hz. The prototype can be situated at different angles thanks to the two plates at the bottom. The lower rectangular plate with fixed x y coordinate system can be connected to the shaking table and fixed there. The upper curved plate with moving x  y  coordinate system forms the base of the damper and can be rotated about z-axis over the lower plate, so that x  -axis can meet four orientation angles α at 0 to 45◦ spaced at 15◦ w.r.t the x-axis. However, since two Ultrasonic sensors are employed at the

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Fig. 8 The drawings of the prototype of the semi-active tuned liquid column damper with closable cells, S-TLCDCC , with (a) the top, (b) the isometric, (c) the vertical cross-sectional and (d) the horizontal cross-sectional views: (1) regulators, (2) sensors, (3) stoppers, (4) open cells, (5) closable or control cells

perpendicular directions (i.e. along the x  - and the y  -axis), measuring the liquid deflections from 0 to 90◦ is possible. The motion of the shaking table is always along the fixed x-axis. The prototype is filled with tap water up to a certain level. The dimensions and characteristics of the prototype S-TLCDCC are brought in Table 3. In this table, two Table 3 Parameters of the prototype S-TLCDCC for the two control states: (i) when all cells are open rv = 0.96 and (ii) when the control cells are closed rv = 0.36 A V,v

V

H

L 1,v

L 2,v

γ1,v

γ2,v

m d,v

(cm2 )

(cm)

(cm)

(cm)

(cm)

(−)

(−)

(kg)

Cells open rv = 0.96

121.7

16.0

34.2

64.9

67.6

0.53

0.51

16.5

Control cells closed rv = 0.36

45.7

16.0

34.2

44.4

126.8

0.77

0.27

11.6

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states of the damper are reported: (i) when all cells are open meaning the variable open area ratio rv is equal to 0.96 and (ii) when all control cells are closed meaning rv is equal to 0.36. It is noted that, when all cells are open, the cross-section in the vertical tube is slightly less than the horizontal tube due to the 5 mm plate separating the cells. The prototype is tested on a uniaxial shaking table which depending on the weight and the frequency could realize up to ±200 mm stroke on each side. The weight of the empty container of the prototype S-TLCDCC is about 75 kg, so that the total weight on the slider of the shaking table is about 100 kg. Tests and results. The goal of the tests is to find the experimental natural frequencies of the prototype S-TLCDCC with control cells automatically closed and opened. Furthermore, these natural frequencies are investigated at different orientation angles between the x- and the x  -axis. Accordingly, a set of free vibration tests is conducted with the damper situated on the shaking table at different angles. The free vibration is implemented by a sudden 20 mm displacement of the container of the prototype. The liquid motion is recorded till the rest and the measured signal is then post-processed. Figure 9 reports, for example, four time-history plots of the liquid deflections of the S-TLCDCC . The results for 0 and 45◦ in Fig. 9a, b are from the sensor located on tube ∀1 along the x  -axis and for 45 and 90◦ in Fig. 9c, d are from the sensor located on tube ∀4 along the y  -axis. Each plot includes two states of the damper: (i) all cells are open rv = 0.96 and (ii) the control cells are closed rv = 0.36. Furthermore, the spectral amplitudes of the liquid motions for the aforementioned states are computed using the fast Fourier transform (FFT) for all orientation angles and the obtained natural frequencies are depicted in Fig. 10. Firstly, Fig. 10 illustrates that closing the control cells has notably changed the initial natural frequency of the S-TLCDCC . This implies that the regulators could effectively seal the control cells, which causes about 0.16 Hz frequency jump compared to the S-TLCDCC with all cells open. In addition, Fig. 10 indicates that each natural frequency of this 4-L-arm prototype is identical in the L-arms and is independent of the orientation (or excitation) angle α. Secondly, the time history responses of the liquid for the two states in Fig. 9 show that the area ratio has also influenced the damping of the S-TLCDCC , which was noted earlier in the theory (cf. Eqs. 8, 17 and 18). It is seen the liquid motion is dampened remarkably faster for the S-TLCDCC with the area ratio rv = 0.36. Moreover, it is noted that the liquid deflection reduces in L-arm C1 (or C3 ) and increases in L-arm C4 (or C2 ) when the x  -axis (L-arm C1 ) diverges from the excitation axis x. Comparing Fig. 9b and c, however, the response is not identical for the 45◦ case for the L-arms C1 and C4 . This is due to the arrangements of the open cells, two of which carry the sensors; the open cells along x  -axis are basically closer to the excitation axis x while the open cells along y  -axis are farther to the excitation axis. These arrangements make liquid deflection in the open cell of the tube ∀1 (or ∀3 ) higher than the one in ∀4 (or ∀2 ). Although, at 90◦ case, liquid oscillation should be theoretically zero, Fig. 9d reports motions which can be even with different frequency particularly for the area ratio rv = 0.96. As we observed, part of these motions is assumed to be due to the

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Fig. 9 Liquid deflection (a, b) in tube ∀1 of the S-TLCDCC when the L-arm C1 is at 0 and 45◦ and (c, d) in tube ∀4 of the S-TLCDCC when the L-arm C4 is at 45 and 90◦ w.r.t the x-axis for two states of (i) all cells open rv = 0.96 and (ii) the control cells closed rv = 0.36

sloshing effect. The other part, however, comes from an eccentricity effect. When all cells are open, there is an eccentricity in the tubes due to the tube physical dimensions. That is the L-arms are not in reality as center lines but they are distributed over widths. Accordingly, each side of each L-arm has an eccentricity w.r.t the center line of that L-arm. The eccentricity of each side causes a secondary motion in that side, which gradually forms and increases by increasing the orientation angle α. In each L-arm, the secondary motions in the opposite sides of the center line are opposite to each other—note also that the sensor is not centered but at the side cell. Subsequently, in the current 4-L-arm prototype with four cells open, the secondary motion in opposite L-arms are in-phase with each other. This secondary motion as well as the sloshing effect may disturb the main response of the damper and cause a different frequency,

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Fig. 10 Experimental and theoretical natural frequencies of the S-TLCDCC when (a) the L-arm C1 is at 0 to 45 ◦ and (b) the L-arm C4 is at 45 to 90◦ w.r.t the x-axis for two states of (i) all cells open rv = 0.96 and (ii) the control cells closed rv = 0.36

which can be dominant at the most for the ultimate case that is here the C4 at 90◦ orientation. However, here in the present study, only the fundamental frequency of the damper is reported in Fig. 10. For the case with control cells closed, in contrast, the two-sided eccentricity described above is changed into a single sided one in each L-arm. By increasing α, not two opposite motions but a single motion will exist in each vertical tube. Subsequently, in the current 4-L-arm prototype with control cells closed, the secondary motions in opposite L-arms are out-of-phase since the open tubes in ∀1 and ∀3 (or ∀2 and ∀4 ) locate opposite each other w.r.t the center line connecting these tubes. Thus, for this prototype, the disturbance of the main response due to the secondary motion is decreased. It should be pointed out here that the general influence of the cell arrangements is out of the scope of the present study while it demands further investigations. Figure 10 also includes the theoretical natural frequencies of the S-TLCDCC calculated by Eqs. 8 and 18. Table 4 compares the calculated natural frequencies, f cal , with the experimental ones, f ex p , for the two states of the open area ratio rv = 0.96 Table 4 Experimental ( f ex p ) and theoretical ( f cal ) natural frequencies of the prototype S-TLCDCC for the two control states: (i) when all cells are open rv = 0.96 and (ii) when the control cells are closed rv = 0.36; f ex p is for the case of L-arm C1 at 0◦

f ex p

f cal



(Hz)

(Hz)

(%)

Cells open rv = 0.96

0.946

0.875

7.51

Control cells closed rv = 0.36

1.099

1.058

3.70

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and rv = 0.36 when L-arm C1 is situated at 0◦ . Despite the discrepancy, this table shows that the theoretical formulas in Eqs. 8 and 18 could reasonably estimate the natural frequencies of the prototype. One reason for the discrepancy can be that the physical first effective length L 1,v is shorter than the one calculated by the theory in Eq. 18. The theoretical L 1,v takes, for example, the whole H and V while these parameters may be shorter in reality due to the flow at the elbows. It is also observed in Table 4 that, for the higher natural frequency in rv = 0.36 case, the discrepancy has decreased which may imply that the calculation of the first effective length for this S-TLCDCC with the employed open cell arrangements estimates the physical first effective length better. The experimental results in this subsection validate the mathematical derivations so that the natural frequency of the S-TLCDCC is adjusted by adjusting the crosssectional areas of vertical tubes of the damper A V,v using closable cells. In the end, It is worth mentioning that, regardless of the cell size, bigger initial area A V,v results in a wider range of natural frequencies for the S-TLCDCC . Increasing number of cells by decreasing the cell size, on the other hand, results in finer frequency steps for jumping from one natural frequency to the other, which will be useful for fine tuning of this damper.

5 Conclusion This chapter introduced adaptive variants of TLCDs that were semi-active tuned liquid column dampers with (i) movable panels (S-TLCDMP ) and (ii) closable cells (S-TLCDCC ). The core idea behind both the S-TLCDMP and S-TLCDCC was to adapt the natural frequency of the damper by manipulating the cross-sectional areas in vertical tubes A V . Mathematical formulations were presented for the general omnidirectional layout with n number of vertical tubes, which covers also the conventional U-shaped one with two vertical tubes. The formulations were also specified to describe the behavior of S-TLCDMP and S-TLCDCC . In comparison, it was shown that, in addition to A V , the S-TLCDMP changes the vertical liquid height V and S-TLCDCC changes the liquid mass as well. The chapter also presented the outcomes of experimental tests on the S-TLCDMP and S-TLCDCC . Both dampers could effectively alter their natural frequency by manipulation of the variable panel position d3 (β) in S-TLCDMP and the variable cross-section of open cells A V,v in S-TLCDCC both in vertical tubes. The experiments proved that the presented mathematical formulations could predict the natural frequencies of the prototypes reasonably. It was also observed that variation in crosssectional areas in the vertical tubes could influence the damping capacity of the dampers, which was in line with the presented theory. The capability of getting retuned is a prominent feature for novel vibration control devices. Therefore, although capturing a full picture of the attributes of these promising dampers, particularly in interaction with structures, demands further

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studies, the presented results introduce S-TLCDMP and S-TLCDCC as effective and optimal adaptive dampers for varying excitation scenarios. Acknowledgements The authors gratefully acknowledge the financial support of this research by the German Federal Ministry of Education and Research (Bundesministerium für Bildung und Forschung, BMBF) with the grant number: FKZ 03VP04680. The authors would also like to thank the Control Engineering Institute of RWTH Aachen University, and in particular Mr. Markus Zimmer, for their support during the preparation of the experimental setup.

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Optimum Tuning of Active Mass Dampers via Metaheuristics Aylin Ece Kayabekir, Gebrail Bekda¸s, and Sinan Melih Nigdeli

Abstract In the present study, a metaheuristic-based optimization methodology that optimizes both physical parameters of active tuned mass dampers (ATMDs) and the tuning parameters of proportional-integral-derivative (PID) type used in the generation of the control force. As known, ATMDs are positioned on seismic structures to reduce structural responses. As the current target of today’s needs, the methodologies must be suitable for real structures. Due to that, the proposed method considers all limitations in practice like stroke capacity, control force limitation and time delay of the control signal. In addition to that, the process is a time-saving one with automated decision-making for non-effective candidate solutions during dynamic analysis. This method was tested by employing three different algorithms. Keywords Metaheuristics · Optimization · Active tuned mass dampers · Structural control

1 Introduction Structural control with active systems has an important place in current interest, and many studies have been carried out especially since 1990. Saleh and Adeli proposed the use of parallel algorithms in structural control problems [1]. Saleh and Adeli were also used a robust parallel-vector algorithm to solve the complex eigenvalue problem of unsymmetric [2] matrix and Riccati equation [3]. Betti and Panariello proposed an active control tendon concept for structures that are under multiple support excitation [4]. Adeli and Saleh were revealed the idea of using A. E. Kayabekir Department of Civil Engineering, Istanbul Geli¸sim University, 34310 Avcılar, Istanbul, Turkey G. Bekda¸s · S. M. Nigdeli (B) Department of Civil Engineering, Istanbul University-Cerrahpa¸sa, 34320 Avcılar, Istanbul, Turkey e-mail: [email protected] G. Bekda¸s e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. Bekda¸s and S. M. Nigdeli (eds.), Optimization of Tuned Mass Dampers, Studies in Systems, Decision and Control 432, https://doi.org/10.1007/978-3-030-98343-7_9

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a robust parallel-vector algorithm for active tendon-controlled bridges [5]. In two studies conducted in the same year, Chung et al. applied a modified instantaneous control algorithm considering the time delay effect on the structures [6] and Wong and Hart numerically analyzed inelastic responses of active controlled structures during earthquake with the force analogy method [7]. Parallel-vector algorithms used by Saleh and Adeli [1] were also applied to determine closed-loop system sensitivities of multistory space frame structures and steel bridges [8]. Saleh and Adeli employed high-performance parallel algorithms to investigate the effect of the different displacements of controllers for multistory buildings [9]. Chung et al. proposed an acceleration feedback strategy for multi-step control to verify activecontrolled structures subjected to earthquake excitations [10]. Spencer et al. derived an evaluation model that was obtained with the help of experimental data to evaluate the relative effectiveness and feasibility of various structural control algorithms [11]. Lu and Skelton proposed an iterative method for optimum parameters of active and passive control systems that use H2 /H∞ performance requirements [12]. Chung improved a predictive active control algorithm using discrete-time and the method was evaluated on active-controlled single and multiple degrees of freedom systems [13]. Sedarat et al. performed the comparison of single and multiple input in active control systems by investigating multiple settlements of active tendon control systems [14]. In the 2000s, it was seen that the researchers’ interest in the subject increased and many studies were carried out. Under the wind and earthquake excitations, the placement effects of active and passive control systems on the three-dimensional structures were investigated by Arfiadi and Hadi [15]. Bakioglu and Aldemir presented a new algorithm to be used in the sub-optimal solution of the optimal closed-open-loop control based on the prediction of earthquake loads utilizing Taylor series and Kalman filter [16]. Aldemir et al. also proposed an approximate method for optimal closedopen-loop control that predicts near future excitations [17]. The multipoint instantaneous performance index was used for structures with active control systems to reduce earthquake effects [18]. Adeli and Kim also applied a wavelet-hybrid feedback-least mean square algorithm for vibration control of structures [19]. Then, this algorithm was used for control of cable-stayed bridges [20]. In another study, Kim and Adeli proposed combining a passive supplementary and a semi-active tuned liquid column damper [21]. Kim and Adeli have also applied this hybrid system to irregular steel high-rise buildings [22]. In the same year, an eigenvalue assignment algorithm was used by Min et al. [23]. Alivinasab and Moharrami employed a technique based on energy and the effectiveness of this technique was tested on a three-story building controlled by active tendons [24]. Nomura et al. presented an integrated fuzzy active control system developed with particle swarm optimization (PSO) [25]. For this purpose, a semi-active device containing two viscous dampers and an actuator was proposed in the research. Chang and Lin suggested using an optimal H∞ control algorithm to reduce inter-story drift of the building under earthquake effects [26]. As a result of their investigation on control of structures implemented magnetorheological (MR) damper, Kim et al. have revealed a linear matrix inequality (LMI)-based systematic design methodology [27]. In the next research released by the authors

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[28], an advanced version of multi-input, multi-output control system has been also used for vibration control of buildings under seismic excitations. Numerical analyses have been developed for the active control of torsionally irregular structure considering soil-structure interaction by Lin et al. [29]. Bitaraf et al. used a direct adaptive control [30]. Nigdeli and Boduro˘glu developed an active tendon control process for torsionally irregular structures in near-fault regions [31]. Ulusoy et al. optimized PID active tendon-controlled structures by using metaheuristic algorithms [32–34]. In addition to these studies, recently some research on active tuned mass damper (ATMD) is presented in this part. Pourzeynali et al. suggested using a combination of fuzzy logic control (FLC) and genetic algorithm (GA), one of the metaheuristic algorithms, to find optimum parameters of ATMD system [35]. Güçlü and Yazıcı used fuzzy logic and PD controllers to suppress the responses of the multi-degreeof-freedom system resulting from ground accelerations [36]. Then, the optimum design of fuzzy PID controller (FPIDC) was done to control seismic responses of the nonlinear structural systems that include base–structure interaction effects [37]. In addition, self-tuning fuzzy logic controllers (STFLC) were implemented for the active control of structures under earthquake effects [38]. In the next year, Li et al. investigated the optimum design parameters and control performance of ATMD systems attached to asymmetric structures [39]. For the asymmetric structures including the soil-structure interaction (SSI), an investigation was also done by Li on optimum parameters of active multiple-tuned mass dampers [40]. In another study, a control approach employing three algorithms such as discrete wavelet transform (DWT), PSO, and linear quadratic regulator (LQR) algorithms was introduced by Amini et al. to determine control forces [41]. You et al. proposed linear quadratic Gaussian (LQG) controller to the active control of tall buildings with ATMD against along-wind induced excitations [42]. Shariatmadar et al. researched the application of interval type-2 FLC (IT2FLC) to the ATMD system [43]. For a multi-story building subjected to seismic excitations, the combination of the fuzzy logic controller (FLC) and PSO was considered by Shariatmadar and Muscat Razavi to control the action of the ATMD system [44]. To attenuate responses due to wind loads and seismic excitations in tall buildings with ATMD, Soleymani and Khodadad investigated a multi-objective method for adaptive genetic-fuzzy controller [45]. Then, a hybrid active tuned mass damper (HATMD) system was introduced by Li and Cao to reduce oscillations resulting from seismic excitations [46]. Also, Cao and Li added a dashpot between structure and active system to improve the control performance of HATMD in the mitigation of oscillations [47]. In the other study, a hybrid controller using traditional LQR and PID controller was suggested by Heidari et al. [48]. Then, an energy-based predictive (EBP) algorithm was presented by Zelleke and Matsagar to improve the efficiency of vibration control in structures equipped with semi-active tuned mass dampers [49]. Kayabekir et al. employed harmony search for optimum tuning of ATMDs using PID controllers [50]. In addition to the presented studies, the information and detailed review about smart structures exist in several books [51–53]. Fisco and Adeli presented two reviews about active and semi-active methods [54], and hybrid systems [55]. El-Khoury and

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Adeli detailly reviewed vibration control methods that are proposed for dynamic loading [56]. Li and Adeli presented a review for civil and mechanical systems [57]. In the present study, an optimization methodology is presented for Active Tuned Mass Dampers (ATMDs) used in seismic structures. It is aimed to develop a feasible method by considering of time delay of the control signal, the maximum value of control force, the maximum allowed stroke capacity for ATMD. In addition, the optimization of ATMD system was done by metaheuristic algorithms, and three different algorithms are employed as a comparative study to find the best suitable results of optimum physical parameters of ATMD and the control system chosen as Proportional-Integral-Derivative (PID) type controllers. Compared to Kayabekir et al. [50], the advantage of the methodology is the automatic finding of parameters without trapping an undefined response (stability error) during optimization thanks to the integrated comparison and operations done in the dynamic analysis process of the structure that also provides time-saving in the optimization process.

2 Active Control of Structures via ATMDs For seismic structures, tuned mass dampers are mostly installed on the top of the structure since the maximum amplitude of the first mode shape is seen on the top of the structure. In this section, the equation of motion of structure with an ATMD on the top of a shear building is presented. The control algorithm, which is optimized via metaheuristic methods is also summarized in this section. To perform dynamic analyzes, the equations of motion must be solved. A structural system with ATMD attached to the top of an n-story shear building as in Fig. 1 shows n + 1 degrees of freedom behavior. For that reason, the ATMD parameters (mass, damping and stiffness terms) and active control force values are also included in the equations of motion to be written, unlike classical structures. The equations of motion can be written as single matrix equations; M¨x(t) + C˙x(t) + Kx(t) = −M{1}¨xg (t) + F(t)

(1)

in which mass (mi ), stiffness (ki ) and damping coefficient (ci ) of ith story are respectively used in the matrices of mass (M), stiffness (K) and damping (C). These matrices can be seen from Eqs. (2–4). x¨ (t); acceleration, x˙ (t); velocity and x(t); displacement vectors represent time-varying structural responses. x(t) is given in Eq. (5). x¨ (t) and x˙ (t) can be obtained by taking derivatives of the displacement vector. {1} on the right side of the equation is a unit vector with n + 1 elements used to ensure matrix size compatibility in the multiplication and x¨ g symbolize ground acceleration due to earthquake excitations. F(t); control force vector (Eq. 6) includes active force (Fu ) generated by ATMD. M = diag[m1 m2 . . . mN md ]

(2)

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Fig. 1 A shear building with ATMD

⎡

(c1 + c2 ) −c2 ⎢ −c (c2 + c3 ) 2 ⎢ ⎢ . ⎢ ⎢ C=⎢ . ⎢ ⎢ ⎢ ⎣

⎤ −c3 . . . . .

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ . ⎥ −c N (c N + cd ) −cd ⎦ −cd cd

(3)

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⎡

⎤ (k1 + k2 ) −k2 ⎢ −k ⎥ (k2 + k3 ) −k3 2 ⎢ ⎥ ⎢ ⎥ . . ⎢ ⎥ ⎢ ⎥ K=⎢ . . . ⎥ ⎢ ⎥ ⎢ ⎥ . . . ⎢ ⎥ ⎣ −k N (k N + kd ) −kd ⎦ −kd kd ⎧ ⎫ x1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x2 ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎬ . .. x(t) = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ xN ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ xd ⎧ ⎫ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ . .. F(t) = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Fu ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ - Fu

(4)

(5)

(6)

The active control force can be obtained by multiplication of trust constant (Kf ) and current of armature coil (iATMD ) as given in Eq. (7). iATMD can be found via Eq. (8). This equation includes R; the resistance value, Ke , the induced voltage constant of armature coil, the difference of top story x˙ N and ATMD x˙d velocities, and u; the generated control signal. Fu = K f i AT M D

(7)

Ri AT M D + K e (x˙d − x˙ N ) = u

(8)

The control signal expresses the amount of voltage of the control system and it is generated according to a controller algorithm. In the study, for the generation of this signal, a proportional-derivative-integral (PID) controller is employed. PID controller transforms error signals to control signals (u) via feedback control. Equation of the PID controller as follows  u = Kp

1 de(t) e(t) + Td + dt Ti



 e(t)dt

(9)

in which e(t) is the error signal and it is obtained according to the feedback of the response of the system in the time domain. The error signal is transformed to control

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signals via proportional (P), integral (I) and derivative (D) actions. These actions include three controller parameters such as Kp ; proportional gain, Td ; derivative time and Ti ; integral time, and have an effective role in various purposes. The proportional action is effective in increasing the speed of the control response. Integral action provides elimination of steady-state errors and, with the help of derivative action effective damping is provided [58]. There is a relationship directly between velocity and kinetic energy of the structure under the earthquake excitations. For that reason, in the present study, the error signal is specified as the top story velocity. The metaheuristics are chosen as a challenging goal that considers both optimizations of controller and mass damper parameters. Dynamic formulations are solved by a code using Matlab with Simulink [59]. This analysis code is integrated with the iterative metaheuristic-based optimization code. Several metaheuristic algorithms, which are briefly summarized in Sect. 3, are used to check the best suitable algorithm.

3 Metaheuristic Algorithms Metaheuristic algorithms are the iterative methods that iteratively generate new design variables to update a solution matrix that is initially defined by sets of design variables. The examination on choosing the best design variable is a function defined according to a solution of analysis. The modification is done according to special features of metaheuristic algorithms that are inspired by a happening in the world, a production process or a feature of living things. Metaheuristics are effective in the optimization of structural engineering problems including control of structures. The optimization of passive tuned mass dampers (TMDs) is the most investigated problem in this subject. In this process, the mechanical properties such as mass, stiffness and damping coefficient are optimized by using time-domain solutions [60] and frequency domain [61, 62]. The optimum tuning of PID controller has been also considered in several studies [32–34]. A harmony search-based methodology to tune both mass damper properties and PID controller parameters was presented by Kayabekir et al. [50]. To reach the optimum solution, three different metaheuristic algorithms are used in the methodology to find the best suitable optimum solution. These algorithms are proved by the success of optimizing passive [61] and active [32–34, 50] control systems. The results of a single algorithm are not effective in the decision of the best objective function. The algorithms may trap to local optimum results and multiple verifications with different algorithms may prevent it. Also, it may not be possible to find the best algorithm, because one of the algorithms may outperform the other for a specific problem or a case of design constraint, while another one is the best one for a different problem and case. This situation is called as no-free lunch theorem.

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The capacities of the axial force of the structures and the cost-related of the control system can be used to determine the ultimate limits of design variables such as stiffness, mass, damping coefficient. However, a solution range cannot be defined to the PID controller parameters since it depends on different actions. As a result, the same control performance can be obtained for different combinations of these variables [32–34]. The employed metaheuristic algorithms are flower pollination algorithm (FPA) [63], teaching–learning-based optimization (TLBO) [64] and Jaya algorithm (JA) [65]. These algorithms are briefly explained in the following subsections. The unique features of the employed algorithms are the main reason for the selection of the algorithms. FPA is an algorithm that uses Lévy distribution and linear distribution in two different types of optimization. TLBO is free of user-defined algorithm parameters and it uses randomly selected weighting parameters such as teaching factors. JA is a simple algorithm with a single-phase that considers both the best and worst existing solutions in one equation.

3.1 Flower Pollination Algorithm FPA was developed via the pollination process of flowering plants via Yang [63]. The main key of the algorithm is the flower constancy, which is a feature about the tendency of specific pollinators to specific flowers. This relationship is combined with the types of pollination to define two types of the optimization process. These processes are selected due to an algorithm parameter called switch probability (sp) which can be selected between 0 and 1. This parameter is similar to HMCR in HS, and it can be used similarly. The global pollination formulized in Eq. (10) imitates two types called biotic pollination and cross-pollination. The classification of pollination types is pollinatororiented or flower-orientated. As a pollinator-orientated type, biotic pollination involves pollinators as living organisms like insects, bees, animals to carry pollens in long distances. Pollinators fly randomly and it is the main reason to use Lévy distribution concerning Lévy flight. In the mean of flower, cross-pollination is the pollination process between different flowers. The best existing solution is used in Eq. (10) as gi j,* 90% of flowers follow global pollination processes in real life. The other type is local pollination formulated as Eq. (11). j,t+1

= X i + L(X i − gi ) i f sp > r1

j,t+1

= X i + ε(X ia,t − X ib,t ) i f sp ≤ r1

Xi

Xi

j,t

j,t

j,t

j,∗

(10) (11)

In local pollination, abiotic and self-pollinations are used as pollinator and flowerorientated types, respectively. In local pollination, a linear distribution (ε) is used as

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a random number between 0 and 1, because the travel distance of pollen is short in abiotic pollination since the transfer is done via natural events wind or diffusion in water. Two randomly chosen existing solutions (a and b) are chosen in the modification due to self-pollination which involves self-fertilization of the same flower like peach.

3.2 Teaching–Learning-Based Optimization TLBO is a user-defined parameter-free algorithm developed by Rao et al. [64]. It uses two types of education in a classroom. The best solution with the best knowledge is selected as a teacher, and it is used in Eq. (12) with the average value of the existing variables (Xi ave,t ). The aim is to increase the level of the average by the best result. A parameter called teaching factor (TF) is used and TF can randomly be 1 or 2. In that case, it is not a user-defined parameter. j,t+1

Xi

j,t

j,∗

= X i + rand(1)(gi

− T F X iave,t )

(12)

A probability parameter as sp is not used in TLBO. After the teacher phase, the student phase is applied. It is the simulation of a classroom in which the education of students between each other after the education of the teacher. The student phase is formulated as Eq. (13). In this process, random existing solutions are used.  j,t+1 xi

=

j,t

xi + rand(1)(xia,t − xib,t ) i f f (xia,t ) < f (xib,t ) j,t

xi

+ rand(1)(xib,t − xia,t ) i f f (xia,t ) > f (xib,t )

(13)

3.3 Jaya Algorithm Jaya algorithm is another parameter-free algorithm. It is developed by Rao [65] according to the Sanskrit word “Jaya” meaning victory. The unique feature of JA is the usage of both worst (wi j,* ) and best (gi j,* ) solutions in a single phase as seen Eq. (14). Two random numbers between 0 and 1 (r1 and r2 ) are multiplied with these opposite results. j,t+1

Xi

j,t

j,∗

= X i + r1 (gi

j,t

j,∗

− X i ) − r2 (wi

j,t

− Xi )

(14)

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4 Methodology In this section, an optimization method for optimum tuning of ATMD positioned on seismic structures is given. Optimization can be defined as an iterative process in that solution set containing the best combination of the values of the design variables that are searched according to the objective of the optimization such as maximation or minimization of the objective function. The sub-sections of Sect. 4 include the design variables of the problem, the objective function and difficulties on optimum design of ATMDs.

4.1 The Design Variables In the optimization process, the optimum value of 5 design variables including physical parameters of ATMD, and PID controller tuning parameters are searched. For physical parameters of ATMD, kd (Eq. 15) and cd (Eq. 16) are considered as design variables. The optimum mass of ATMD is generally equal to the maximum ultimate limit, so it is defined as a constant value. The other design variables are Kp , Td and Ti used as tuning parameters of PID.  Tatmd = 2π cd 

ξd =

2m d

md kd

kd md

(15) (16)

4.2 The Optimization Objective The optimization has two objectives. In the study, to minimize the maximum top story displacement (xN ), Eq. (17) is considered as the first objective function (f1 ). f 1 = max |x N | + pen

(17)

As seen in Eq. (17), a penalty function (pen) is also added to the objective function. This penalty is used for controlling maximum signal and thereby obtaining costefficiency and feasible design. The pen given in Eq. (18), expresses the maximum amount of the control force in N. Since the value of pen has a very big value compared to displacements in m, elimination of these results is provided. pen = max |Fu |

(18)

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Equation (19) formulates the stroke capacity of ATMD and although the optimization can also be defined as the design constant, this equation is considered as the secondary objective function (f2 ) in the study. In case of the value of the function (f2 ) is greater than the user-defined value stmax, the existing results are compared. In this way, the maximum allowable limit of f2 is taken into account to find the best results as well. f2 =

max(|xd − x N |)with AT M D max(|x N |)without AT M D

(19)

4.3 The Optimization Method The objective functions are found for earthquake records set given in FEMA P:695: Quantification of Building Seismic Performance Factors as far fault records [66]. In general, the solution range of the period of TMDs is determined between 0.8 to 1.2 times of the critical period of the main structures, since the optimum period of TMDs and critical period of main structures are close to each other. In addition, the maximum limit of damping ratio is taken as 30%. For the PID controller parameters, it is hard to define a range. Also, several combinations of PID controller parameters may encounter in NAN solution in MATLAB and the iterative optimization process is interrupted. For that reason, an initial range may be defined. Also, the control force must be limited. If the possibility of a different combination of PID controllers that have very similar performance effect is considered, the optimization process may last too long, and the process may not be feasible. For that reason, several integrations are done. The generated Simulink block diagram for the dynamic analysis of structure with ATMD is shown in Fig. 2. In the block diagram, the analysis of the equations of motion is done via matrices and vectors. On the upper side of the block diagram, the top story displacement is chosen with a “selector” block, and the value of it is compared with the maximum value of uncontrolled structure with the usage of “if” and “else” blocks. If it is bigger than the value of uncontrolled response, the simulation is stopped, and the maximum value of the objective function is taken as the value of the last active step. In that case, time is saved, and NAN results are neglected. Thus, the iterative optimization process is not interrupted. Since the value of the objective function is a big value which is greater than the uncontrolled response, it is easily eliminated in the comparison of candidate solutions. A similar verification is also done for the maximum control force on the right of the block diagram. “if” and “else” blocks are used to check the control force value. If the value of the control force is bigger than the desired value, the simulations are stopped. In that case, the constraint of the control force is considered, and time is saved. Also, the control force value, which is more than the allowed value (Fumax ) is added to the response of the structure as a penalty as defined as “pen”.

Fig. 2 MATLAB block diagram for dynamic analysis of structure with ATMD

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Fig. 3 Flowchart of the dynamic analyses

Another function is the consideration of time-delay by using a transport delay block in Matlab Simulink [59]. The numerical cases also include different cases of time delay values and control force limit. The block diagram used as a module with the optimization code is summarized in the flowchart given in Fig. 3. The flowchart of the optimization process is given in Fig. 4. Firstly, the design constants, ranges of design variables, earthquake excitations and required userdefined limits and algorithm parameters are entered. Structural properties such as mass, stiffness and damping coefficient, control system parameters including R, Kf and Ke , simulation time that determines earthquake excitation time, time delay (td) and mass of ATMD (md ) are the design constants of the optimization. To restrict control force, stmax and maximum top story displacements of structure without active system, it is required to determine limits by a user. After the definition of constant values, an initial solution matrix is generated. This matrix contains sets of design variables that are selected within the desired range, and the number of these sets is equal to the population. After that, the optimization iterations start. New design variables are produced via the features of the algorithms. The objective functions such as f1 and f2 are found. Then, new results are compared with the existing variables. If both new and existing results provide f2 < stmax, set of design variables with a minimum f1 is saved. If not, set of design variables with the minimum stmax is saved. If one of them provides f2 < stmax, the providing result

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Fig. 4 Flowchart of the optimization methodology

is considered and saved. The optimization steps continue for a maximum number of iterations.

5 The Numerical Example In the example, the effects of different stroke limits (stmax) and time delay (td) conditions on the optimum design of the ATMD system are investigated by considering 10-story shear building that has the same structural properties for each story. ATMD system is located to the top story of this structure.

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By using three different algorithms presented in Sect. 3, optimum values are obtained, and the results are presented comparatively. Table 1 shows design variable limits, algorithm parameters, design constants of the structure and the control system. In this section, the optimization results performed with TLBO, FPA and JA algorithms are presented in Table 2. In the tables; f1 , f2 and Nopt are the objective functions and the number of iterations where the optimum result is obtained respectively. Fmax is the maximum control force under the critical earthquake record and it is limited with 10% of the total weight of the structure. Before detailed investigations on optimum results, it should be noted that the inferences do not include two situations (td 20 ms and 30 ms for stmax 2) where the JA algorithm cannot find the optimum result. Considering the optimum results obtained from different algorithms for the certain time delay (td) and stmax values, it is seen that the objective function values are approximately the same although the design variables are different. The reason for this situation, it can be shown that Table 1 The design constants and design variable limits Symbol

Definition

Value

Unit

mi

Mass of the story

360

ton

ki

Rigidity coefficient of the story

650

MN/m

ci

Damping coefficient of the story

6.2

MNs/m

md

Mass of ATMD

180

ton

T atmd

Period of ATMD

0.5–1.5 times of period of structure

s

ξd

Damping ratio of ATMD

1–50

%

Kp

Proportional gain

(−10,000)–(10,000)

Vs/m

Td

Derivative time

(−10,000)–(10,000)

s

Ti

Integral time

(–10,000)-(10,000)

s

stmax

Stroke limit of ATMD

2, 3

–

Td

Time delay

10, 20, 30, 50

ms

R

Resistance value

4.2



Kf

Trust constant

2

N/A

Ke

Induced voltage constant of armature coil

2

V

Pn

Population number

10

–

mt

Maximum iteration number

5000

–

HMCRin

Initial harmony memory considering rate

0.5

–

PARin

Initial pitch adjusting rate

0.05

–

BSCR

Best solution considering rate

0.3

–

sp

Switch probability

0.5

–

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Table 2 The optimum results for ATMD TLBO stmax = 2 Td

10

stmax = 3

20

30

50

10

20

30

50

Tatmd (s) 0.944

0.970

0.953

0.969

1.019

1.012

0.975

0.965

ξd (%)

26.286

27.476

28.707

28.892

11.077

11.522

11.919

11.552

Kp (Ns/m)

483.23

1903.6

−1206.3 −246.96 2243.7

125.845

173.119

878.14

Td (s)

−1107.4 −269.77 460.098

2017.32

Ti (s)

−1895.9 2024.5

8716.61

−8908.1 −2011.9 −9999.45 9953.55

−4943

f2

1.999

1.999

1.999

1.999

2.999

2.999

2.999

2.999

N opt

4991

4147

4919

4655

4713

4106

3076

4976

F max (kN)

3529.4

3521.7

3528.26

3529.46

3528.5

3529.39

3529.97

3530.2

f 1 (m)

0.2498

0.2529

0.2544

0.2613

0.2123

0.2173

0.2200

0.2285

−232.22 −4035.89 −3135.7 −551.3

FPA stmax = 2 Td

10

stmax = 3

20

30

50

10

20

30

50

Tatmd (s) 0.947

0.974

0.951

0.965

1.013

1.008

0.971

0.954

ξd (%)

26.284

27.652

28.638

28.660

10.968

11.433

11.854

11.272

Kp (Ns/m)

53.125

1901.56

1329.62

8935.38

−1549.8 97.979

Td (s)

−10,000 −269.99 −417.22 −55.699 336.27

Ti (s)

−10,000 3889.74

−8026.5 −9734.0 −5012.9 263.328

−2802.0 −10,000

f2

2.000

1.999

1.999

1.999

2.999

2.999

2.999

2.999

N opt

2887

4975

4981

4966

4888

4966

4960

4966

F max (kN)

3498.9

3520.3

3525.37

3530.16

3530.2

3530.16

3530.10

3528.8

f 1 (m)

0.2499

0.2528

0.2544

0.2612

0.2124

0.2173

0.2201

0.2285

−3299.3 1222.1

−5184.83 165.012

−401.1

JA stmax = 2 Td

10

stmax = 3

20

30

50

10

20

30

50

Tatmd (s) 0.943

0.890

0.930

0.958

1.027

1.017

0.978

0.951

ξd (%)

26.213

23.832

27.532

28.281

11.240

11.600

11.991

11.204

Kp (Ns/m)

119.87

260.56

1536.36

369.698

1896.8

50.687

107.450

−2166

Td (s)

−4454.4 −1969.6 −359.21 −1341.0 −274.32 −10,000

Ti (s)

7920.8

616.06

−4067.3 −1835.7 9537.6

−2617.31 2940.09

−2414

f2

1.999

1.998

1.999

2.999

2.999

1.999

2.997

−5037.8 226.89 2.99911

(continued)

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Table 2 (continued) TLBO N opt

4786

1347

4625

3959

4738

2299

1137

4191

F max (kN)

3521.6

3515.8

3494.44

3528.77

3518.8

3518.01

3528.49

3526.1

f 1 (m)

0.2499

0.2556

0.2549

0.2613

0.2125

0.2173

0.2201

0.2285

there are various combinations of interdependent design variables giving the same objective function. According to the optimum results, it is seen that the objective functions tend to increase as the time delay increases, whereas it decreases as the stmax value increases.

6 Conclusions After the general evaluations about the optimum values, it is useful to evaluate the efficiency and performance of the algorithms in finding optimum results in order to suggest the method to be used in the solution of the problem. According to the analysis results, except for two cases where the JA algorithm could not reach the optimum result (td 20 ms and 30 ms for stmax 2), all algorithms are successful in obtaining the optimum value of the objective function. Another criterion used to compare algorithm performances is the speed of reaching the optimum result. According to the analysis results, in each case, a different algorithm has succeeded in finding the result faster. However, JA is better in most cases. It can be also said that other algorithms are approximately similar in terms of iteration numbers. However, considering that TLBO algorithm is a two-phase one, it performs double numerical operations in each iteration. Therefore, FPA algorithm is faster than TLBO algorithm.

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Machine Learning-Based Model for Optimum Design of TMDs by Using Artificial Neural Networks Melda Yücel, Sinan Melih Nigdeli, and Gebrail Bekda¸s

Abstract In this study, it was considered that approach where optimization and machine learning tools are utilized together for inducting possible structural damages to minimum via optimum modeling of tuned mass dampers (TMDs) in frequencydomain. For this respect, in the optimization process realized as the first step, the metaheuristic flower pollination algorithm (FPA) was used for the minimization of the biggest amplitude for the transfer function. In the second step, a rapid-effective prediction tool was developed by training some mechanical parameters belonging to optimum TMD designs via a machine learning-based model. Accordingly, they are possible to directly and quickly determine TMD parameters required for new test models thanks to this tool developed by benefiting from artificial neural networks (ANNs) and obtaining transfer function values at minimum level with the help of parameters. Thus, an equivalent solution was found to optimization analyses taking long times with the development of a system, which can predict quickly, effectively, and without additional operations. Keywords Tuned mass dampers (TMDs) · Frequency domain · Metaheuristic algorithms · Flower pollination algorithm · Machine learning · Artificial neural networks (ANNs)

1 Introduction Machine learning, which is one of the sub-disciplines of artificial intelligence, is a focusing process on methods and algorithms provided to learn by computers. M. Yücel · S. M. Nigdeli · G. Bekda¸s (B) Department of Civil Engineering, Istanbul University-Cerrahpa¸sa, 34320 Istanbul, Turkey e-mail: [email protected] M. Yücel e-mail: [email protected] S. M. Nigdeli e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. Bekda¸s and S. M. Nigdeli (eds.), Optimization of Tuned Mass Dampers, Studies in Systems, Decision and Control 432, https://doi.org/10.1007/978-3-030-98343-7_10

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Here, machine learning can realize learning by producing information about data and connections between data when a sample dataset is given to the system [1]. In this regard, machine learning concept can be defined as the creation of systems, which can learn an issue through providing on realizing of many activities performed by a human, who has a real intelligence and mind, by machines, computers, software and even robots, can sense, and generate reactions to this. Machine learning is one of the technological and scientific sub-branches belonging artificial intelligence area, which can be defined as returning to a human-like formation with transferring of a virtual intelligence to automatic structures containing devices like computers, calculators, robots, etc. by benefiting from many disciplines such as mathematic, engineering, psychology, biology and genetic science, linguistics; and performing some brain activities like data prediction, classification, pattern determination, decision-making, language sense and also speeching by these structures through mimicking of brain mechanism together with human intelligence. For example, any human can learn the meaning of words namely object descriptors from books, articles, dictionaries, or other people, and thus, naturally and speedy, he/she can obtain the definition that belongs to the related object, which is perceived by seeing. Besides, although people don’t know what are the features of a real class/group where objects belong, they can separate them into different groups easily according to visual properties. In this regard, the principal aim of machine learning is to enable the development of intelligent and smart tools through that the mentioned formations can carry out the whole of the expressed tasks and create responses during the desired cases by taking action quickly. On the other side, especially nowadays, it can be seen that different techniques were proposed about the functioning of smart devices in the direction of developments in technology together with diversifying of working principles, and these techniques were differentiated day by day. In the beginning times more generally, in this process based on utilization from learning activities and reactions of a human, then the structure of the brain and nervous system with collaboration mechanism each other; it takes attention in nowadays that solutions become possible intended for more complex, big and various problems like predicting of an image via operation by a system just as a scanner, determination of listened voices belong who, finding of letters in a scanned text possess which language. Nevertheless, they were only possible from the middle of the nineteenth century that these types of developments become noticeable in real life and get applicable. As it was mentioned previously, machine learning, which is regard to the usage of artificial intelligence by various computers, can be performed in many different ways and smart systems, software and tools can be developed in several fields of study thanks to this variety. Due to the realized machine learning processes, time and effort can be achieved saving by pioneering to make activities faster carried out in scientific environments besides plenty many areas of daily life, too. In this regard, it is known that there are different applications performed with especially machine learning methods together with artificial intelligence technologies various areas such as health and pharmaceutical industry, science and education, social media, shopping and e-commerce, security, etc. For example, in the field of healthcare, the

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determination of mental health by evaluating behavior states [2–5]; prediction the possibility of occurring of some diseases and classification of them [6–8]; detection of the cancer/tumor cells in organisms [9–12], etc. have been drawn attention as the most applied sub-areas. Also, especially in recent years, some applications such as prediction demographic properties (gender, age, income status, etc.) and structure of users [13–17], discovering of fake news and contents [18–21], and also detecting the aggressiveness as cyber bullying and [22–26] became more popular and frequentlyused technologies intended for social media applications. Nevertheless, the field of study as an engineering discipline is one of the areas where machine learning applications have already been utilized and become more widespread day by day. They can be expressed that some of the applications such as forecasting water demand for users [27–30], planning of energy by supplying the efficiency [31–34], generation smart road systems with traffic flow modeling [35–37], etc. However, for structural design area within civil engineering that is one of the branches of the mentioned discipline, some applications such as smart energy systems, automatized calculation modules or formulas, rapid management tools for planning of construction areas, etc. started to increase nowadays comparison to past times where classical approaches were more preferred. As several examples to these, they can be given that generation of the cost-planning for structural projects [38–42], determination of the most proper concrete ingredients [43–46], predicting the optimal model properties of structural members such as beam, column, retaining wall, footings, etc. [47–49] and detecting some mechanical properties of both structural materials and members [50–56]. Also, one of the other problems within the structural engineering field is to provide sufficient seismic retrofitting and in this meaning, to generate strengthened and well-designed vibration devices. In this respect, the best mechanical properties of tuned mass dampers (TMD) are forced to estimate by optimizing [57–60], some vibration damping systems (base isolations, PID controllers, etc.) are designed with the quick prediction of mechanical properties [61–63], dynamical behaviors are arisen by estimating of seismic responses [64–67], etc. In this chapter, the expressions provided via the ANN model [57] were presented. The model uses flower pollination algorithm in the optimization and it is given in Sect. 2. Then, ANN is explained in Sect. 3. The ANN model for the estimation of optimum TMD properties is summarized in Sect. 4. The numerical cases are presented in Sect. 5 and the results are concluded in the last section.

2 Optimization Method: Flower Pollination Algorithm (FPA) The process known as optimization is to make possible the most suitable namely handy results intended for any problem under the pre-determined constraints and in the direction of a purpose mainly targeted. In this meaning, some processes are

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considered in the development of metaheuristic algorithms, which are one of the utilized techniques. Also, some physical and chemical operations, the genetic structure of humans inspire designing of these algorithms besides that these processes are more widely some superior abilities of nature and livings taken placed within it, the form of life, vital activities for living. Here, they make it possible that algorithms are used in optimization processes by mathematically expressing the mentioned process and activity, and the most convenient solutions are achieved. For instance, in the flower pollination algorithm (FPA), flight, which is realized by pollinator livings towards the flower, was simulated with a random distribution function. On the other respect, as to harmony search (HS) method, it was symbolized via an algorithm parameter that the case where a musician plays a tune/note from a specific fret by remembering in the way of win the favor of audiences. Flower pollination algorithm (FPA), which is a metaheuristic benefited in this study, was developed by benefiting from a natural process known as pollination realized in the direction of providing the continuity of species of flowery plants and proposed in the year 2012 by Yang [68]. On the other side, it is known that the mentioned pollination event can occur in two different ways in this process: biotic (cross) and abiotic (self) pollination. While biotic one, namely cross-pollination happens between two different flowers or two different members of the same kind of flower, abiotic/self-pollination arises inside of the self-structure of one flower. Also, the formation and realization mechanisms of both types are expressed via Eqs. (1) and (2) for biotic and abiotic pollinations, respectively: X new,i = X old,i + Levy(X best,i − X old,i )

(1)

X new,i = X old,i + rand(0, 1)(X n,i − X m,i )

(2)

where, X new,i and X old,i are old and updated values of the current solution corresponding ith design variable, respectively. Moreover, X best,i symbolizes the most convenient solution among all of the ones that it has the minimum/maximum objective function as the best result for ith design variable. Also, X n,i and X m,i are values of mth and nth solutions selected as random. Here, Levy and rand(0,1) are known as distribution functions calculated as random-flight and linear between [0, 1], respectively.

3 Artificial Neural Networks (ANNs) The structure of artificial neural networks (ANNs) is comprised of the constituents of the central nervous system. The nervous cells in this biologic structure and axons providing the information transfer between the mentioned cells represent the nodes (input, hidden and output) and node connections in ANNs, respectively. They can be seen in Fig. 1 that a real neuron structure, and the working mechanism of ANNs,

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Fig. 1 A real neuron model and a simulation type providing the expression of model as an artificial neuron mechanism regard to ANNs

which is generated in the direction of expression and systematization of this neuron structure as artificial. Each of the input neurons that also can be seen in Fig. 1, represents each definer, namely input attribute, and provides to be operated of sample values belonging to these attributes along the process [69]. Also, input weights symbolized w can be considered as information transfer percent of axons, which play a role in relaying information taken from a real nervous cell. Nevertheless, it requires that the effect of each weight to the relevant input value and combines all of the values eventually occurred to predict of output, and summation function comes into play here. After the combination of each obtained value, this summed result is processed again via a transformation unit named as activation function, and a prediction value is produced

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for the target output. Besides there are many types of the mentioned activation function, some of the most-used are known as linear, sigmoid/logistic, softsign, binary, softplus, exponential, Swish, etc., and expressed in Eqs. (3)–(9), respectively [70, 71]. f (x) = x 1 1 + e−x   x − 1.13 f (x) = |x| + 1   1x >0 f (x) = 0x ≤0 f (x) =

(3) (4) (5)

(6)

f (x) = log(1 + e x ) − 1.21

(7)

f (x) = e−x

(8)

f (x) =

x − 1.29 1 + e−x

(9)

Nevertheless, the connection shape of neurons/nodes each other together with signal flowing namely information transmission determines the kind of network. In this regard, there are two different ANNs structures as feed-forward and backforward (recurrent). In feed-forward ANNs, information flow is uni-directional and this flow realizes from input nodes to output units namely in the way of forward direction. Each node obtaining the mentioned information transmits this information to another node in the forward direction (node where information was not taken). For this respect, a turning/loop or information transfer does not occur intended for back. Some of the kinds of feed-forward ANNs are known as single-layer perceptrons, multilayer perceptrons (MLPs), radial basis function networks. As to back forward ANNs, feedback loops occur for the transmitted information [70, 72].

4 The ANNs Model for TMD In the reference study [57], the minimum value of the acceleration transfer function (TF) for SDOF structure together with design parameters containing the optimum TMD period (Td ) and damping ratio (ξ d,opt ) were predicted via ANNs. Also, mass ratio (μ) and structural period (Ts ) were handled as input parameters. In this scope,

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100 different design combinations for sdof structure were produced during the optimization process, and all of them were utilized for the training of ANNs. Thus, in this process, 10 different structural periods (changing in [0.1–5 s]) and 10 different values for mass ratio (changing in [0.01–0.40]) were combined. At the end of the mentioned applications, 400 different design options were produced within the same ranges of Ts and μ to generate a test model. Here, by using the created ANNs prediction model, optimum parameters of the test model were determined, and then, some formulations were proposed through the provided results to observe the optimum values of design parameters directly. Moreover, these formulations were adjusted via three different versions of curve fitting and compared with well-known literature equations. The ANN-based equations can be seen in Eq. (10) for linear, Eq. (11) for polynomial, and Eq. (12) for exponential in terms of f opt and ξ d,opt , respectively.  Linear =

f opt = −0.6438μ + 0.9966 ξd,opt = 0.5673μ + 0.1235

f opt = −249.91μ5 + 400.09μ4 − 208.03μ3 + 43.801μ2 − 4.1453μ + 1.0675 P.nomial = ξd,opt = −54.673μ4 + 54.639μ3 − 19.274μ2 + 3.2302μ + 0.0237  f opt = 1.0038e−0.747μ E x ponential = ξd,opt = 0.1258e2.8573μ

(10)

(11)

(12)

5 Numerical Investigations In Table 1, optimum damping (ξd,opt ) and frequency ratios (fopt ) of TMDs for different structural cases were presented. Also, the minimum acceleration transfer function values for different cases as test designs with respect to SDOF structure were handled with the obtained parameters and can be seen in Table 2. All of the values were presented and compared according to previously proposed and well-known literature equations together with the generated formulations via curve fitting. Here, the required inherent damping (ξ) value for SDOF structure is handled as 0.05. It can be recognized in Tables 1 and 2, all of the formulations developed by the ANN-based model are extremely effective and useful for decreasing the transfer function values in comparison to the other namely literature equations. Additionally, besides that the best one among the proposed formulations can be accepted as the linear equations concerning the handled cases thanks to that the best minimization performance for TF values was observed with the usage of them.

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Table 1 Comparison of literature equations and proposed formulations for different cases of SDOF structure (Optimum parameters) Case

Mass ratio (μ)

Structural period (Ts )

Den Hartog [73]

Warburton [74]

Sadek et al. [75]

ξd,opt

fopt

ξd,opt

fopt

ξd,opt

fopt

1

0.05

0.5

0.1336

0.9524

0.1098

0.9404

0.2658

0.9420

2

0.1

1

0.1846

0.9091

0.1527

0.8861

0.3470

0.8817

3

0.1

0.5

0.1846

0.9091

0.1527

0.8861

0.3470

0.8817

4

0.05

2

0.1336

0.9524

0.1098

0.9404

0.2658

0.9420

5

0.05

1

0.1336

0.9524

0.1098

0.9404

0.2658

0.9420

6

0.1

1.5

0.1846

0.9091

0.1527

0.8861

0.3470

0.8817

7

0.05

1.5

0.1336

0.9524

0.1098

0.9404

0.2658

0.9420

8

0.1

2

0.1846

0.9091

0.1527

0.8861

0.3470

0.8817

Case

Leung and Zhang [76]

Linear equation

Polynomial equation

Exponential equation

ξd,opt

fopt

ξd,opt

fopt

ξd,opt

fopt

ξd,opt

fopt

1

0.1091

0.8249

0.1519

0.9644

0.1435

0.9462

0.1451

0.9670

2

0.1514

0.7229

0.1802

0.9322

0.2032

0.9205

0.1674

0.9315

3

0.1514

0.7229

0.1802

0.9322

0.2032

0.9205

0.1674

0.9315

4

0.1091

0.8249

0.1519

0.9644

0.1435

0.9462

0.1451

0.9670

5

0.1091

0.8249

0.1519

0.9644

0.1435

0.9462

0.1451

0.9670

6

0.1514

0.7229

0.1802

0.9322

0.2032

0.9205

0.1674

0.9315

7

0.1091

0.8249

0.1519

0.9644

0.1435

0.9462

0.1451

0.9670

8

0.1514

0.7229

0.1802

0.9322

0.2032

0.9205

0.1674

0.9315

6 Results and Conclusion In the result of the performed optimization applications and prediction process via ANNs together with generation effective formulations, the most convenient TMD designs can be created in the scope of controlling the undesired vibrations and decreasing the acceleration transfer function amplitudes. As it can be seen that the worst performance for minimization was found in Leung and Zhang [76] equations that optimum parameters for TMDs were observed similarly for the proposed formulations and Den Hartog [73] equations. However, the best one was determined as linear equations from the proposed three different formulations. Cause of this is that the transfer function amplitudes were minimized effectively for almost all structural cases.

Mass ratio (μ)

0.05

0.1

0.1

0.05

0.05

0.1

0.05

0.1

Case

1

2

3

4

5

6

7

8

2

1.5

1.5

1

2

0.5

1

0.5

Structural period (Ts )

10.8534

12.5871

10.8639

12.6227

12.6227

10.8692

10.8639

12.6227

TF (dB)

Den Hartog [73]

11.6922

13.2003

11.6707

13.2003

13.1830

11.6990

11.6922

13.2203

Warburton [74]

12.6210

14.4102

12.6278

14.4301

14.4301

12.6278

12.6210

14.4301

Sadek et al. [75]

15.2055

16.0810

15.2055

16.0810

16.0810

15.2055

15.2255

16.0810

Leung and Zhang [76]

10.4481

12.3659

10.4481

12.3659

12.3235

10.4568

10.4481

12.3659

Linear equation

Table 2 Comparison of literature equations and proposed formulations for different cases of SDOF structure (TF)

10.5740

12.7447

10.5831

12.7389

12.7389

10.5860

10.5831

12.7447

Polynomial equation

10.4678

12.4275

10.4890

12.4275

12.3207

10.4890

10.4882

12.4275

Exponential equation

Machine Learning-Based Model for Optimum Design … 183

184

M. Yücel et al.

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