Active Control of Pendulum Tuned Mass Dampers for Tall

ACTIVE CONTROL OF PENDULUM TUNED MASS DAMPERS FOR TALL

BUILDINGS SUBJECT TO WIND LOAD

Dissertation

Submitted to

The School of Engineering of the

UNIVERSITY OF DAYTON

In Partial Fulfillment of the Requirements for

The Degree of

Doctor of Philosophy in Engineering

Mohamed A. Eltaeb, M.S.

UNIVERSITY OF DAYTON

Dayton, Ohio

December, 2017

ACTIVE CONTROL OF PENDULUM TUNED MASS DAMPERS FOR TALL

BUILDINGS SUBJECT TO WIND LOAD

Name: Eltaeb, Mohamed Ali

APPROVED BY:

------Reza Kashani, Ph.D. Dave Myszka, Ph.D. Committee Chairperson Committee Member Professor Associate Professor Department of Mechanical Department of Mechanical and Aerospace Engineering and Aerospace Engineering

------Elias Toubia, Ph.D. Muhammad Islam, Ph.D. Committee Member Committee Member Assistant Professor Professor Department of Civil and Department of Mathematics Environmental Engineering

------Robert J. Wilkens, Ph.D., P.E. Eddy M. Rojas, Ph.D., M.A., P.E. Associate Dean for Research and Innovation Dean, School of Engineering Professor School of Engineering

Mohamed A. Eltaeb

2017

ABSTRACT

ACTIVE CONTROL OF PENDULUM TUNED MASS DAMPERS FOR TALL

BUILDINGS SUBJECT TO WIND LOAD

Name: Eltaeb, Mohamed A. University of Dayton

Advisor: Dr. Reza Kashani

Wind induced vibration in tall structures is an important problem that needs effective and practical solutions. TMDs in either passive, active or semi-active form are the most common devices used to reduce wind-induced vibration. The objective of this research is to investigate and develop an effective model of a single multi degree of freedom (MDOF) active pendulum tuned mass damper (APTMD) controlled by hydraulic system in order to mitigate the dynamic response of the proposed tall building perturbed by wind loads in different directions. The proposed APTMD can be tuned to the first three dominant frequencies of the targeted structure in three directions (X, Y, γ) simultaneously and add damping to the corresponding modes. Another design requirement of the APTMD is the capability of adjusting its properties (stiffness and damping) to compensate for the detuning occurred due to the varying operating conditions such as an environment, or imposed loading. These supplemental damping devices offer attractive means to protect

iii tall buildings against natural hazards and make a genuine contribution to the building sway, which has such a great economic and social effects.

The targeted structure for the proposed approach in this work is a MDOF model representing a full scale concrete tall building. This building is a modern high-rise building designed as a flexible and slender structure, asymmetric geometry, excited by wind loads in multiple directions. The building has 41 stories above the ground where each floor has three degrees of freedom, two in 푥, 푦 directions (planar) and one around the axis perpendicular to 푥 − 푦 plane γ (rotational). The first 15 modes of the building will be included in this study, five modes in each direction.

The innovative idea of this work is involving the Stewart Platform, was originally designed in 1965 as a flight simulator, and it is still commonly used for that purpose. It is controlled by hydraulic system that is used for motion control (position control) of the pendulum TMD relative to the building. The pendulum itself is a passive device but as it is comprised with active-controlled hydraulic actuators, the legs of the Stewart Platform in our case, it becomes an active system. The electrohydraulic servo valve is used to control the hydraulic system of the proposed active PTMD because it can offer more responsive and accurate control tasks in a timely manner. By combining the muscle of the hydraulic power and the accuracy of electrical control, electrohydraulic control valves can control hydraulic systems precisely and efficiently.

The desired control force is calculated from the acceleration, velocity, and displacement feedbacks of the MDOF system and active PTMD in order to achieve the different tuning frequencies and damping effects. The proposed tasks for the conduct of

‘Multi Tuning-frequency Passive/on demand Active Pendulum Tuned Mass Damper’ research are:

a) To synthesize the control scheme for active Pendulum TMD that can be tuned

simultaneously to multiple directions replacing multiple more massive PTMDs.

Such attributes lowers the cost, weight and space requirement associated with

dampening multiple modes using multiple TMDs.

b) To increase the effectiveness of the proposed active PTMD, which leads to

lowering its weight (50% less) without degrading its performance. With its small

size and multi-frequency tuning capacity, the proposed APTMD is as effective as a

passive TMD many times more massive.

c) To obtain a high fidelity model of the structure targeted for damping

The synthesis and analysis of the proposed passive/on-demand active PTMD is presented. The effectiveness of the proposed tuned mass damper is numerically demonstrated, by interfacing its model with that of a high-rise building.

I dedicate my dissertation work to my family. A special feeling of gratitude to my loving parents, Al-Haj Ali Eltaeb and Fatima Eblaiblu whose words of encouragement and support have provided me with strength and patience.

I also dedicate this dissertation to my darling wife, Amna who has supported me throughout my study journey far from home. I will always appreciate all she has done.

ACKNOWLEDGMENTS

My special thanks are in order to my committee chair Professor Dr. Reza Kashani, my advisor, for providing the time and resources necessary for the work contained herein, and for directing this dissertation and bringing it to its conclusion with patient guidance and expertise. It has been an honor to be his Ph.D. student. I would also like to express my appreciation to my committee members, Dr. Mohammad Islam, Dr. Dave Myszka, and Dr.

Elias Toubia.

Also, I would like to acknowledge friends and family members who supported me during my work on this research. First and foremost, I would like to thank my father and mother for their constant love and pray for me to get success. And most of all for my loving, supportive, and patient wife Amna whose faithful support during the stages of this work is so appreciated. Also, I thank my darling children for their constant support and encouragement throughout this work.

vii

TABLE OF CONTENTS

ABSTRACT ...... iii

DEDICATION ...... vi

ACKNOWLEDGMENTS ...... vii

LIST OF FIGURES ...... xi

LIST OF TABLES ...... xiv

I. INTRODUCTION ...... 1

1.1 Problem Statement ...... 1 1.2 Research Objectives ...... 3 1.3 Approach ...... 5

II. LITERATURE REVIEW AND BACKGROUND ...... 6

2.1 Tall Buildings...... 6 2.1.1 Lateral Loads Affecting Tall Buildings ...... 6 2.1.1.1 Wind Loads ...... 7 2.1.1.2 Seismic loads ...... 8 2.1.2 Wind Induced Building Motion ...... 8 2.1.3 Design Approaches Against Wind Excitation ...... 11 2.1.3.1 Mechanical Design Approach ...... 12 2.1.3.2 Comfort Criteria (human response to building motion) ...... 13 2.1.4 Design Approaches Against Seismic Excitation ...... 13 2.2 Structural Vibration ...... 14 2.2.1 Control of Vibration ...... 15 2.2.1.1 Structure Modification and/ or Damping ...... 16 2.2.2 Vibration Control Systems ...... 17 2.2.2.1 Passive Control System...... 17 2.2.2.2 Active Control System ...... 18 2.2.2.3 Semi Active Control System ...... 20 2.3 Tuned Mass Damper (TMD) ...... 21 2.3.1 Passive Tuned Mass Damper (PTMD) ...... 22 2.3.1.1 Translational TMD...... 24 2.3.1.2 Pendulum TMD ...... 25 viii

2.3.2 Active Tuned Mass Damper (ATMD) ...... 26 2.3.3 Semi-Active Tuned Mass Damper (SATMD) ...... 33 2.4 Active Pendulum Tuned Mass Damper and Structure Model ...... 34 2.4.1 Structure Model ...... 35 2.4.2 Active PTMD (APTMD) ...... 37 2.4.2.1 The Mathematical Model of the PTMD ...... 38 2.4.2.2 Auxiliary Damping and Stiffness...... 42 2.4.2.3 Pendulum Dynamics Coupled with MDOF Structure ...... 44

III. MULTI TUNING-FREQUENCY PASSIVE/ON-DEMAND ACTIVE PENDULUM TUNED MASS DAMPER ...... 48

3.1 Multiple Frequency Tuning PTMD ...... 49 3.2 Passive/On-Demand Active Tuned Mass Damper ...... 50 3.3 Control Strategy ...... 51 3.3.1 Active Control ...... 54 3.3.2 The Control Scheme ...... 56 3.4 Illustrative Example ...... 59 3.5 Modeling of the Structure ...... 60 3.6 Application of the Proposed Passive/On-demand Active PTMD ...... 62

IV. EFFECTIVENESS ENHANCEMENT OF TMD ACTIVE CONTROL ...... 66

4.1 The Impact of Mass Ratio on TMD Effectiveness ...... 67 4.2 Active Tuned Mass Damper (ATMD) ...... 69 4.2.1 Control Strategies for Active Tuned Mass Dampers ...... 71 4.3 The Proposed Active Pendulum Tuned Mass Damper ...... 72 4.4 Active Control Strategy of the Proposed PTMD ...... 75 4.4.1 Effectiveness Enhancement of Active Damping ...... 75 4.4.2 Active Tuning ...... 77 4.5 Illustrative Example ...... 80 4.5.1 The Predicted Peak Accelerations ...... 80 4.6 Application of the Active PTMD ...... 81

V. HYDRAULIC ACTUATION SYSTEM ...... 86

5.1 Stewart Platform ...... 86 5.2 Fluid Power ...... 88 5.3 Hydraulic Systems ...... 88 5.3.1 Electrohydraulic Servo Valve ...... 89 5.4 Sizing of the Hydraulic Actuation System...... 89 5.4.1 Maximum Force and Stroke of the Hydraulic Actuator ...... 90 5.4.2 The Selection of the Hydraulic Cylinder ...... 94 5.4.3 The Selection of the Electro-Hydraulic Servovalve ...... 96 5.4.4 Servovalve D661 Highresponse Series ...... 97 5.5 The Hydraulic Circuit of the Actuation System ...... 99

VI. CONCLUSION AND FUTURE WORK ...... 101

6.1 Conclusion ...... 101 6.2 Future Work ...... 103

REFERENCES ...... 104

APPENDIX A Electrohydraulic Servovalve Selection Guide ...... 113

APPENDIX B Inverse Kinematics of the Stewart Platform ...... 114

LIST OF FIGURES

Figure 1: Pendulum tuned mass damper ...... 4

Figure 2: Generation of eddies ...... 8

Figure 3: Wind induced motion on tall buildings ...... 10

Figure 4: Block diagram of structure passively controlled ...... 18

Figure 5: Schematic diagram of the structural control problem ...... 19

Figure 6: Concept of the AMD control system ...... 20

Figure 7: Schematic of various TMD utilizing inertial effects...... 22

Figure 8: Schematic of a unidirectional translational TMD ...... 25

Figure 9: Schematic of a PTMD ...... 25

Figure 10: Active tuned mass damper (ATMD) ...... 27

Figure 11: The configuration of pendulum tuned mass damper (PTMD) ...... 35

Figure 12: Colocated FRFs mapping perturbations to accelerations in all 3 directions on floor 41 ...... 36

Figure 13: Modes shapes of vibration in Y-direction ...... 36

Figure 14: Schematic geometry of the PTMD mass with auxiliary damper and spring ....38

Figure 15: Schematic of a MDOF flexible main structure equipped with a PTMD ...... 46

Figure 16: The passive/on-demand active PTMD ...... 51

Figure 17: Block diagram of the structure + PTMD ...... 52

Figure 18: Feedback interaction of an APTMD with its target tall structure ...... 53

Figure 19: Block diagram of the active PTMD ...... 58

Figure 20: FRFs and the resonant time traces of the structure’s acceleration along Y direction, measured at the top floor ...... 63

Figure 21: FRFs and the resonant time traces of the structure’s acceleration along X direction, measured at the top floor ...... 64

Figure 22: FRFs and the resonant time traces of the structure’s angular acceleration around Z axis, measured at the top floor...... 65

Figure 23: Frequency response functions of an underdamped structure without and with two TMDs with 1% and 5% mass ratios ...... 68

Figure 24: The schematic presents an ATMD with the active element U, appended to a structure ...... 69

Figure 25: The active PTMD ...... 74

Figure 26: Block diagram of the structure + APTMD ...... 79

Figure 27: Excursion, FRFs, and the resonant time traces of the structure’s acceleration along Y direction, measured at the top floor

(Large mass and No control) ...... 82

Figure 28: Excursion, FRFs, and the resonant time traces of the structure’s acceleration along Y direction, measured at the top floor

(Small mass and No control) ...... 83

Figure 29: FRFs and the resonant time traces of the structure’s angular acceleration around Z axis, measured at the top floor (Small mass with control) ...... 84

Figure 30: Specifications of Stewart Platform...... 87

Figure 31: Moog series G77XK electrohydraulic Servovalve ...... 89

xii

Figure 32: Structure frequency response function in Y direction ...... 91

Figure 33: Control forces for the six hydraulic actuators (Mode 2) ...... 92

Figure 34: Strokes of the six hydraulic actuators ...... 93

Figure 35: Servovalve D661 Highresponse Series with servo-jet pilot stage ...... 98

Figure 36: The hydraulic circuit for one cylinder (leg) ...... 100

Figure 37: Stewart Platform ...... 115

xiii

LIST OF TABLES

Table 1: Human perception levels ...... 15

Table 2: The requirements for the hydraulic system ...... 93

Table 3: Key input parameters for the hydraulic actuation system ...... 98

xiv

CHAPTER I

INTRODUCTION

1.1 Problem Statement

Modern design and development of construction technologies have resulted in tall, slender, and flexible structures such as high-rise buildings (skyscrapers) and airport towers.

These tall structures are susceptible to excessive vibration when perturbed by environmental dynamic loads such as wind, and could experience large amplitude motion especially at the top of the structure. Wind loading occurs rather frequently and is a major source of large lateral building vibration. Low-frequency vibration (range of 0 – 1 Hz) which commonly occurs in high-rise buildings causes discomfort to the occupants and adversely affects the intend use of buildings known as serviceability issue (1).

A common technique for abating the vibration of tall buildings under wind loads is using Tuned Mass Dampers (TMDs) (1). TMD was first developed by Herman Frahm in

1909 to reduce the vibration at the hull of ships (2) (3). TMDs have been implemented in tall buildings and towers to reduce wind induced vibrations. Examples include the Citicorp

Center Office Building in New York City, the Chiba Port Tower in Japan, the Centrepoint

Tower in Sydney, the John Hancock Tower in Boston, and the Taipei 101 Tower in Taiwan

(4) (1).

A TMD type is a passive system made up of a mass, a restoring mechanism (e.g. a spring), and a dissipative mechanism (e.g., a viscous damper). The natural frequency of the

TMD is tuned to a particular structural natural frequency (resonant frequency) so that when the structure is excited at the tuned frequency, the damper resonates out-of-phase with the structure and hence mitigates the vibration of the structure (3) (5).

Despite the widespread use of conventional (passive) TMDs, they have some major drawbacks in their performance. One drawback is their sensitivity to changes in their operating environment such as structural deterioration or imposed loads which result in detuning. Failure to account for detuning may cause the performance of TMDs to degrade over time (6) (3) (7).

The ability of the TMD to actively change (tune) its natural frequency to match the resonant frequency of the structure which represents the dominant mode would mitigate vibrations associated with that mode. Such a vibration absorption device is called active tuned mass damper (ATMD). By introducing an active element to act between the structure and the TMD, active TMD (ATMD), the effectiveness of the TMD can be enhanced (1).

The objective of this research is to investigate and develop an effective model of a single multi degree of freedom (MDOF) active pendulum tuned mass damper (APTMD) controlled by a hydraulic system in order to mitigate the vibrations of tall buildings perturbed by wind loads in multiple directions.

The innovative idea of this work is using the Stewart Platform (SP), which was originally designed in 1965 as a flight simulator, to actuate the PTMD in multiple directions and provide the tuned damping corresponding to each direction. SP is still commonly used for that purpose (flight simulator) and is a classic example of a mechanical design,

2 controlled by hydraulic system, used for motion control (position control) of the TMD relative to the building. SP has a remarkable range of motion and can be positioned and oriented accurately and easily. Due to the ability to provide an enormous amount of rigidity and stiffness for a given structure mass, a Stewart platform can provide significant positioning and orientation (8) (9).

The pendulum itself is a passive device, but as it is comprised of active-controlled hydraulic actuators (hydraulic actuators acted as the Stewart Platform legs), it becomes an active system. The desired control force is achieved from the feedback of the velocity and displacement of the active PTMD in order to achieve the different tuning frequencies and damping effects in various directions. In addition, the effectiveness of the proposed

APTMD is enhanced by feeding back the structural acceleration in different directions to the proposed APTMD.

1.2 Research Objectives

Pendulum tuned mass dampers (PTMDs) as shown in Figure 1 are commonly used in adding damping to tall structures and quiet their vibration. These multi-directional

TMDs are made up of viscous dampers and a mass block suspended by steel cables. The objective of this research is:

1. One Active PTMD that can be tuned simultaneously to multiple modes, eliminating

the need for having multiple PTMDs tuned to multiple modes.

2. Many tall structures neither have the room to accommodate nor the load carrying

capacity to take the weight of an optimally sized PTMD (which can be many

hundreds of tons). A small active PTMD can be as effective as a large passive

PTMD.

3. An APTMD can constantly (continuously) tune itself to the frequency of the target

mode of the structure (which can change over time) and thus always be optimally

tuned and provide the best damping performance.

(a) PTMD installed on the top of Taipei 101 (b) PTMD configuration

Figure 1: Pendulum tuned mass damper

1.3 Approach

In this work, a novel PTMD is presented which can, on-demand, turn into an active

PTMD (APTMD) (10), (11) with multiple, adjustable tuning frequencies and damping ratios in multiple directions. This device is optimally designed as a passive PTMD tuned to the first mode of the structure. It also has the look of a passive PTMD as it appears from

Figure 1, but instead of using shock absorber type viscous dampers as its dissipative elements it uses hydraulic cylinders which along with flow control valves act as passive viscous dampers. Upon switching the active feature on, the flow control valves will be placed outside of the hydraulic circuits and the fluid automatically is re-routed through servovalves instead, turning the device into an active PTMD. In the active mode, the

APTMD extends the extent and bandwidth of its energy dissipation effectiveness to that of a more massive PTMD tuned to a single-mode. In addition, the APTMD can be tuned to more than one mode in more than one direction, replacing multiple PTMDs and in turn lowering the cost, weight and space requirement associated with dampening multiple modes using multiple TMDs. Another advantage of the proposed APTMD is having a high effectiveness which made it possible to lower its mass even more (50 %) than regular active

TMDs using modern control method.

CHAPTER II

LITERATURE REVIEW AND BACKGROUND

2.1 Tall Buildings

Modern design and development of construction technologies, have resulted in taller and lighter structures. As an example, the world’s tallest man-made structure, the

Burj Khalifa tower, stands a remarkable 828 m from its base with an estimated weight in excess of 110,000 tons (6). With every step that the high-rise buildings take toward the sky, more challenges and difficulties face architects and engineers. As the height of skyscrapers increases due to the improvement in the field of structural system design and the use of advanced and high strength materials, the choice of structural systems becomes constrained due to limitations imposed by the height of buildings. Modern structures, particularly those that are tall, slender and flexible are sensitive to wind loads. They are susceptible to large lateral motion and respond dynamically to the effects of wind (12) (13).

2.1.1 Lateral Loads Affecting Tall Buildings

Tall buildings (skyscrapers), are more sensitive to wind and earthquake induced lateral loads than low rise buildings. Wind loads increase with the building height and earthquake loads increase with the building mass. For this reason, wind loads are an important factor and have a significant effect on the design of tall buildings while they are

6 generally not an issue in the design of the structure for low rise buildings. Wind loads occur rather frequently and are a cause of large lateral building sway (12).

2.1.1.1 Wind Loads

Due to the various flow conditions arising from the interference of wind with buildings, wind becomes a phenomenon of great complexity. Wind is composed of a very large number of eddies of varying sizes and rotational features carried along in a general stream of air moving relative to the earth’s surface. These eddies give wind its stormy or turbulent character. Dynamic loading on tall buildings resulting from wind, depends on the size of eddies. Vortices (large eddies), whose dimensions are equivalent to the structure size, increase the well correlated pressures as they surround the structure. In contrast, small eddies result in pressures on various parts of a structure that become practically uncorrelated with distance of separation. Eddies generated around a typical structure are shown in Figure 2.

The dynamic response of buildings to wind loads occurs as a result of various phenomena include buffeting, vortex shedding, galloping and flutter. Slender structures are more likely to be sensitive to a long-wind loading, in line with the wind direction, as a consequence of turbulent buffeting where wind hits the building with excessive force.

Cross-wind response is expected to be created from vortex shedding or galloping but may also result from excitation by turbulence buffeting. Flutter is a coupled motion, often being

a combination of bending and torsion, and can result in instability. For tall building structures, flutter and galloping are generally not an issue (13).

(a) Elevation (b) Plan

Figure 2: Generation of eddies (13)

2.1.1.2 Seismic loads

The propagation of energy waves, formed because of seismic motion in the earth’s crust acts upon a building foundation becoming the earthquake loads on that building. The lateral inertia forces on a structure created by an earth quake are functions of (12)

 The magnitude and duration of the earthquake

 The distance of the structure from the center of the earthquake

 The mass of the structure, the structure system, and the soil structure interaction

2.1.2 Wind Induced Building Motion

Not only the wind approaching a building is a complicated phenomenon, but the flow pattern created around a building is similarly complex. This is due to the distortion of the flow, flow separation, the formation of vortices, and development of the wake. As a

result, large wind pressure fluctuations happen on the building façade, and consequently, large aerodynamic forces are imposed on the structural system and intense local forces act on the surface of such a structure (13).

Wind induced building motion can essentially be divided into three types as shown in

Figure 3 – b

 Along wind motion

 Across wind motion

 Torsional motion

As the wind speed on the surface area of the building façade increases, the effect of the wind, which can affect more than one face of the building, increases. The building sway is considered highest when the wind loads are perpendicular to the building face. Along wind motion occurs when the dynamic response of the building (sway) is parallel to the direction of the wind as shown in Figure 3-b. This motion is induced by instabilities in wind speed, and the difference in wind pressure between the windward (upstream) and leeward (downstream) faces (faces perpendicular to the wind direction) of the building.

Across wind motion occurs when the dynamic response of the building (sway) is perpendicular to the direction of the wind as shown in Figure 3-b. When tall buildings block the motion of the air mass stream, the wind (air stream) splits into two, passing both sides and the back face of the building. The air mass stream will circulate on the sides and rear facades of the building, creating vortices depending on the velocity of the wind, size and aspect ratio of the building. Turbulent air flow around tall buildings forms vortices by accelerating wind speeds in the shear layers which occurs because of the compression of the streamlines around the sides of the building as shown in Figure 3 - a. When vortices, 9

spiral flow formations created by turbulence, shed (break away) from the surface of the building, they create negative pressure in the cross wind direction. They are shed alternately from either side (along the wind direction) of the building, following each other on opposite faces and interacting sequentially.

(a) The formation of turbulent air flow

(b) Tall building motion under the effect of wind

Figure 3: Wind induced motion on tall buildings (12)

The vortices shed rate is a function of the building shape, dimensions, and the wind speed. Consequently, in addition to along wind motion, across wind motion which is considered more significant and critical, occurs because of the forces devolved on the sides of the building cannot counteract each other. Therefore, while the windward face of the

building is subject to positive wind pressure and the leeward face is subject to negative wind pressure (suction), the cross wind faces of the building are subject to alternating positive/negative wind pressure.

Tall buildings are not only exposed to along wind and cross wind aerodynamic forces, but they may also experience torsional forces as shown in Figure 3-b. These may arise if the building shape is asymmetric, or if the building is subjected to asymmetric wind loads. The wind force acting perpendicular to the building surface passes through the geometric center of the building. On the other hand, the reaction of this force passes through the stiffness center of the building. If these two centers do not pass through the same axis, the resulted eccentricity leads to torsional moments, thus, torsional motion occurs about the vertical axis of the building.

Generally, a large number of wind loading codes consider estimating the along wind forces for regular structures and comparatively few of them for across wind and torsional building responses, which by their nature are much harder to codify with precision than along building responses. For the case of high-rise buildings, all of the three types of wind induced building motions, along wind, across wind, and torsional building responses together with the wind loads must be taken into consideration. Moreover, wind tunnel tests of a scaled model and/or numerical analysis are suggested for estimating wind loads in such cases (12) (13).

2.1.3 Design Approaches Against Wind Excitation

Design approaches for controlling wind induced vibration in tall buildings and protecting serviceability can be divided into three main groups

 Architectural design

 Structural design

 Mechanical design

2.1.3.1 Mechanical Design Approach

In this research, we will discuss the mechanical design approach where auxiliary damping systems are used to mitigate the effects of wind induced vibration. Auxiliary damping systems can be divided into four groups: passive systems, active systems, semi- active systems, and hybrid systems (12).

In structural and mechanical systems, the damping is the rate at which the motion energy is dissipated to the environment as heat. Damping is helpful in tall buildings affected by wind loads, as it subdues building movement, decreasing the occupant’s perception to the building motion. Increasing the effective damping to control structure vibrations can be a cost-effective solution and sometimes it is the only feasible and practical solution available to abate the resonant vibration of a structure.

Some examples of passive dampers are: Tuned Mass Damper (TMD), Distributed

Viscous Dampers, Tuned Liquid Column Dampers (TLCD), also known as Liquid Column

Vibration Absorbers (LVCA), Tuned Sloshing Water Dampers (TSWD), Impact Type

Dampers, Visco-Elastic Dampers, and Friction Dampers. Examples of active dampers include: Active Tuned Mass Damper (ATMD) and Active Mass Driver (AMD). Examples of semi-active dampers include: Variable Stiffness Dampers, Magneto-Rheological (MR)

Dampers, Electro-Rheological (ER) Dampers, and Variable Friction Dampers. On one hand, passive damping systems due to their lower cost, robustness, and not relying on an external power supply are preferred by vibration control industry. On the other hand, active or semi-active dampers may be the ideal solution for certain vibration problems because of

their ability to control wide range of vibration. More details about passive, active, and semi- active systems to control vibrations will be covered later on in sections (2.2.2.1, 2.2.2.2,

2.2.2.3) (13).

2.1.3.2 Comfort Criteria (human response to building motion)

Major innovations and the use of advanced technology in structural systems have led to the control of wind induced motion on tall buildings limiting its effects. However, the dynamic response of the building is still a concern, causing discomfort to the occupants and posing serious serviceability issues (14). Comfort criteria in tall buildings design do not have ideal international standards. However, there are a significant number of studies into the importance of physiological and psychological parameters that may affect human perception to structural vibration in the low frequency range of 0 -1 Hz commonly occurring in tall buildings. These parameters include the occupant’s expectancy and experience, their activity, body posture and orientation, visual and acoustic cues, and the amplitude, frequency, and accelerations for both the translational and rotational motions to which the occupant is subjected. Table 1 shows some guidelines on general human perception levels (13), (14).

2.1.4 Design Approaches Against Seismic Excitation

Viscous and viscos-elastic (VE) dampers (energy dissipation devices) are known as effective solutions to reduce the excessive vibration induced by seismic loads but are not attractive solutions to wind-induced vibration of tall structures. On the other hand,

TMDs (reactive energy dissipation devices) are not as efficient as viscous and VE dampers in mitigating seismic vibration but they are the common solution for reducing wind induced vibrations on tall structures (15) (16). However, despite the TMDs not being the best

solution to seismic vibration of a structure but sometimes they are the only available solution such as in historical buildings that cannot be retrofitted to accept viscous or VE dampers.

The reason for vibration due seismic loads and vibration due to wind loads requiring different damping devices is that the former vibration has a broadband and the latter has a narrowband spectra. Most of the time, there is one dominant mode of the structure that couples with the wind load causing vibration that has a dominant single frequency and requires a tuned vibration damper capable of dampening that specific mode. In contrast,

Seismic induced vibrations occur randomly at different frequencies and require a broadband vibration damper such as viscous or VE dampers.

2.2 Structural Vibration

Structures nowadays, due to modern design and construction technologies and the use of light weight and strong materials continue to soar skyward, where they become exposed to complex wind drifts. As a result, this race toward new heights accompanied by increased flexibility and a lack of adequate inherent damping, increase the susceptibility of the structure to the action of wind (14). Structures have multiple degrees of freedom with an infinite number of modes of vibration (17).

For each mode of vibration, there is a corresponding natural frequency, mode shape, and damping ratio (17), (18) . In real physical systems, however, the first few vibration modes are important for systems designers because they (the first few vibration modes) carry the highest vibration energy of the system. The dominant modes, in the case of tall buildings, are the first few modes which normally occur in the frequency range of less than 1 Hz. The

first mode of the structure called the fundamental mode, which is also the most important mode, has the lowest natural frequency and the largest amplitude of motion (17).

Table 1: Human perception levels (13) Level Acceleration m/sec2 Effect

1 <0.05 Human cannot perceive motion

2 0.05 – 0.1 Sensitive people can perceive motion

3 0.1 - 0.25 Majority of people will perceive motion

4 0.25 – 0.4 Desk work becomes difficult or almost

impossible

5 0.4 – 0.5 a. People strongly perceive motion

b. Difficult to walk naturally

6 0.5 – 0.6 Most people cannot walk naturally

7 0.6 – 0.7 People cannot walk or tolerate motion

8 > 0.85 Objects begin to fall and people may get

injured

2.2.1 Control of Vibration

There are several codes and benchmarks which have been made by civil engineers for estimating vibration range on structures and decreasing it to the satisfactory vibration level before building the structure. Despite this, excessive vibration levels can still be found in high-rise buildings after the construction is finished. Annoying vibrations could be reduced by modifying the structure, and/ or introducing auxiliary damping into it (19).

2.2.1.1 Structure Modification and/ or Damping

1) Alternation of natural frequency: the resonance, which occurs when the frequency

of the perturbation matches the resonant frequency of the structure, can lead to

severe vibration in the structure. When the frequency of the perturbation source is

almost constant, the vibration can be alleviated by changing the resonant frequency

of the structure mis-matching the perturbation frequency. This can be done by either

changing the stiffness or the mass of the structure. For example, the vibration of a

fan blade can be reduced by modifying the stiffness or the mass of the blade and

thereby changing its natural frequency and avoiding resonance which can cause

fatigue on the blade (19).

2) Force relocation: The simplest and lowest cost method to reduce the vibration in

structure is rearrangement of the source of vibration. For instance, relocating a gym

in a building can play a significant role on floor vibration reduction such as using a

ground story to be a gymnasium instead of upper or middle levels. Also, mounting

equipment close to walls or columns rather than installing it in the middle of the floor

can help mitigate the floor vibration resulting from this equipment (17).

3) Energy dissipation: This method can be applied to reduce the effect of resonance,

by adding broadband or tuned dampers to the structure (19).

4) Auxiliary mass: adding mass to a structure is another way to lower its vibration.

The content of a building such as furniture, machines, and partition, as well as the

occupants of a floor, reduces the motion on that floor by adding mass to it.

2.2.2 Vibration Control Systems

Irrespective of the remedial methods (methods of vibration control) considered, the techniques for suppressing the undesirable vibration could be categorized into three classes of passive, active, and semi-active.

2.2.2.1 Passive Control System

A passive control system can be defined as a system that does not require an external power source to function, and the structure motion is employed to produce the relative motion of the passive devices which, in turn, opposes the motion of the structure and abates its vibration. The response of the structure to the external loads excites the passive system which, in turn, develops the required inertial force to mitigate the induced motion at the location of the passive control system (17) (20) (21).

Soong and Constantinou (22) presented supplemental energy dissipation systems, which are passive control systems that can be used to increase the energy dissipation capacity of a structure through localized, discrete energy dissipation devices.

Passive control has the advantages of being a) relatively inexpensive, b) not needing external energy to operate, c) being inherently stable, and d) begin robust (working in the face of even severe events). Note that the passive control system and the structure interact with each other rather than behave independently (23). The dashed line in Figure 4 shows the interaction between the control and the structure. Passive supplemental damping strategies, including viscous dampers, sliding friction, and tuned mass dampers are well understood and are widely accepted by the engineering community as a means for mitigating structure vibrations. However, these passive systems are frequently too large.

Moreover, they are unable to adapt to structural changes and to varying usage patterns and loading conditions (17) (24).

Figure 4: Block diagram of structure passively controlled (20)

2.2.2.2 Active Control System

An active control system is one that requires an external power supply to control the actuators which apply controlled forces to the structure in a desired manner. The signals received by the control actuators in an active control system are a function of the structural response measured with suitable physical sensors such as optical, mechanical, electrical, and chemical sensors (20) (21). Active control is utilized to govern the structural response to either internal excitations such as machinery and human activities, or external excitations such as wind and earthquake loads, where the safety and comfort level of occupants are important. This type of control systems, including active mass dampers, hybrid mass dampers, tendon controls, which may employ electro-hydraulic, pneumatic, electro- magnetic, or motor driven ball-screw actuation, involves a wide range of different actuators depending on the structure size and geometry to produce the control force (21). The actuation force within an active control is typically generated based on feedback information from the measured response of the structure and the external excitation. The 18

response information recorded from the structure motion is monitored by a controller

(computer) which decides for the right control signal to be sent to the actuator based on prescribed control algorithm. The control forces generated by electrohydraulic actuators, which is the actuator type that will be used in this research, requires large power supplies, which are on the order of tens of kilowatts for small structures and may reach several megawatts for large structures (20) (25).

Schematic diagram of structure control shown in Figure 5 presents an algorithm for structure active control. This diagram illustrates the mentioned control strategy that utilizes the measured structural responses to determine an appropriate control signal, which is generated based on pre-defined algorithm, and send it to the actuator that will counteract the structure responses to the excitation and thereby enhance structure safety and serviceability. The example depicted in Figure 6 presents control of a tall building using an active mass damper system (AMD).

Figure 5: Schematic diagram of the structural control problem (5)

In this type of control, a relatively small auxiliary mass, which is usually 1% of the modal mass, is installed on the top floor, and an actuator, which typically is hydraulic actuator for such structures because it can supply high forces, is installed between the auxiliary mass and the structure. An AMD system measures the responses of the building at its location, which is the highest, and responds in the desired manner (5).

Figure 6: Concept of the AMD control system (5)

The significant feature of an active control is using external power supply that produces the control action. This is considered the main negative aspect of active control systems because such a system will be vulnerable to power failure especially during a strong natural hazards (21).

2.2.2.3 Semi Active Control System

Semi-active control system is really a passive control system which has been adapted to allow for the adjustment of its mechanical properties (17) (20). These systems adjust their mechanical properties based on the feedback from the excitation and /or from 20

the measured response of the structure. Semi-active control systems combine the two techniques of active and passive systems together. On one hand, like in active control systems, they use a controller to monitor the feedback measurements and to generate the control forces which are developed through an appropriate (based on a prescribed control algorithm) adjustment of the mechanical properties of the semi-active system. On the other hand, and as in passive control systems, they develop the control forces as a reaction to the structural motion (20).

Examples of such devices include, variable-orifice fluid dampers, variable-stiffness devices, and controllable friction devices, smart tuned mass dampers and tuned liquid dampers, controllable fluid dampers, and controllable impact dampers (24).

2.3 Tuned Mass Damper (TMD)

The majority of tall buildings and towers around the world are equipped with different types of vibration absorbers to mitigate their dynamic responses caused by the environmental loads such as winds and earthquakes. Several factors affect the selection of the vibration absorber including efficiency, capital cost, operating cost, maintenance requirements, and safety. For high rise structures, TMD, in either passive, active or semi- active form, are the most common devices used to reduce wind-induced motion (26).

This section presents a thorough description of tuned mass damper (TMD), including its theory. Also, it talks about recent advances in TMD design and their practical applications in tall structures.

Figure 7: Schematic of various TMD utilizing inertial effects (14).

(Con: controller, a: actuator, Ex: excitation, S: sensor)

2.3.1 Passive Tuned Mass Damper (PTMD)

Whenever the term TMD is mentioned in the field of vibration control of structures, it refers to the passive type which can have several different configurations. Typically, a

TMD is a structural control device that consists of an inertial mass attached to the structure at the location of higher frequency, usually near the top, via spring and damping mechanisms, normally viscous or viscoelastic dampers, as shown in Figure 7. Tuning is the action of substantially matching the natural frequency of the attached TMD to that of the structure by selecting the appropriate TMD parameters (stiffness, damping, and mass).

The TMD will resonate out-of-phase when it is properly tuned to the fundamental (first) mode of the structure (the lowest natural frequency), which most of the time carries the highest vibrational energy of the structure, and the resulted energy will be dissipated to the environment as heat by the damper (6).

The concept of the tuned mass damper (TMD) was first applied by Frahm in 1909 to reduce the rolling motion of ships as well as ship hull vibration. Later, Ormondroyd and Den

Hartog (1928) presented the theory of the TMD, followed by a detailed discussion of optimal tuning and damping parameters in Den Hartog’s book on mechanical vibrations

(1940). Since then, tuned mass damping has become one of the most investigated fields in structural vibration control (2).

There are some significant factors (constraints) that need to be considered when designing a TMD. The major and chief of which is the size and amount of mass that can be practically placed on the top of a building. Space restriction due to the relative motion of the TMD to the building is another design factor as well as low friction bearing surface which enables the mass to respond to the building motion at low levels of excitations.

Friction is a major engineering issue that is associated with a sliding mass arrangement of traditional TMD configurations which become more important when the TMD is utilized as an additional damping to increase the serviceability of the building (21). Another limitation is the lack of adaptation to detuning conditions, and as a consequence, passive

TMD efficiency depends on the accuracy of its primary tuning (3). TMDs are typically effective through the use of multiple TMD configurations, which consist of a collection of several mass dampers with distributed natural frequencies under random loading, leading to more effective and dynamic responses of the structure. However, in many cases, space constraints will not allow for the use of traditional TMD configurations which occupy large areas of the building rather than requiring the installation of an alternative configuration, such as a pendulum TMD which occupies less space and can have a bigger mass that moves freely in the space, to accommodate the TMD in the structure (14) (27).

Currently, there are several TMD configurations installed in tall buildings, bridges, towers, and smoke stakes for dynamic response control primarily against wind-induced motion. As of today, most TMD applications have been made to mitigation wind-induced motion (21). Despite the restrictions that have been mentioned previously, passive TMDs are still the favorable vibration absorber for structure engineering because they are relatively inexpensive systems that perform effectively when properly tuned to the specified mode (14). The two common types of passive TMDs are translational TMDs and pendulum TMDs (2).

2.3.1.1 Translational TMD

The two typical types of translational TMD systems are unidirectional systems and bidirectional systems (2). The motion of the TMD mass in a unidirectional system is constrained to one direction by putting the mass on roller bearings or a set of rails that function as a roller, permitting the mass to slide laterally, out-of-phase, and relative to the floor as depicted in Figure 8. On the other hand, the mass can move in two orthogonal directions which helps control the motion of a structure in two cross coordinate axes. This system is called a bidirectional system. In both systems, a set of springs and dampers are located between the TMD mass and vertical support structure, which transfer the lateral damping force to the structural frame (2).

Some existing examples of early installed versions of translational TMD systems include the Washington National Airport Tower, the John Hancock Tower in Boston, the Chiba

Port Tower in Japan, Citicorp Center in Manhattan, New York City, and the Canadian

National Tower in Toronto (2) (14).

Figure 8: Schematic of a unidirectional translational TMD (6)

2.3.1.2 Pendulum TMD

In pendulum TMDs, the mass is hung by cables to the floor, as shown in Figure 9, instead of sitting on roll bearings or rails, enabling the system to act as a pendulum, which hence eliminates the problems associated with bearings in translational TMDs as depicted in Figure 8. When the pendulum gets excited by the structure motion, it generates a horizontal force that counteracts the floor motion and reduces the structure’s vibration. The behavior of both the pendulum and the translational TMDs can be modeled by an equivalent single degree of freedom (SDOF) system attached to the structure as shown in

Figure 8 and 9 (2).

Figure 9: Schematic of a PTMD (6)

There are several parameters that make pendulum TMDs favorable over translational

TMDs. They include the absence of bearings, they can use larger mass es, multi- directionality, they occupy less space, and have longer life. Utilizing bearings for translational TMD as a support is expensive, because it involves many mechanical systems such as rail and brake systems, and can lead to wear over time. Examples of early versions of PTMD applied to structures include Crystal Tower in Osaka, Japan, Higashiyama Sky

Tower in Nagoya, Japan, and Taipei 101 in Taipei, Taiwan (2).

2.3.2 Active Tuned Mass Damper (ATMD)

Passive TMDs are favored for their simplicity and reliability, but these devices are incapable of adjusting to the variation in system parameters, due to the change in the operational environment. As a result, mistuning problems occur, which is the sensitivity to the fluctuation in tuning the frequency of TMDs to the controlled frequency of the structure, and significantly reduces the effectiveness of the TMD. Clearly, more efficient and swifter TMD systems could be reached with the ability to respond actively to changes in structure parameters and, thereby, would allow absorption of broadband vibrations. Such a vibration absorbers is called active tuned mass damper (ATMD) which is really a traditional TMD with an external hydraulic or electro-mechanical actuator inserted between the primary structure and the auxiliary mass as shown in Figure 10.

The appropriate control force provided by actuator systems counteracts the dynamic response of the structure. The actuators are driven by an appropriate algorithm such as a closed loop (feedback), in which the control forces are determined by the feedback response of the structure (1) (28) (14) (29) (30).

Figure 10: Active tuned mass damper (ATMD) (1)

The selection of the vibration absorber type for high rise buildings and structures depends on several major factors including efficiency, size and compactness of the structure, capital cost, operation cost, maintenance, and safety. The use of Active TMDs as vibration absorbers has several major advantages in mitigation of structural vibration over passive TMDs. They include the ability to reduce vibration in multiple modes (even in different directions) with one TMD, requiring less space (by using one rather than multiple

TMDs), increasing the effectiveness by utilizing the control force, self-tuning to the changing structure’s frequency, using a smaller mass than an equivalent passive TMD, and higher efficiency. Whereas the regular TMD can add 3% to 4% additional damping of the critical damping, the active TMD can add 10% or more (26). Also, ATMDs can reduce the dynamic response of the structure by 40 % to 50% or more (31). ATMDs, unlike some other active control devices, can be installed in different types of structures such as buildings, towers, bridges, and stacks (31).

Increasing the effectiveness is the major feature of utilizing active control in TMD, in which a relatively small mass with control force can be used to subdue the dynamic response of the structure. To reach the highest possible effectiveness of ATMD, it is advised that an optimal active control force be determined using an appropriate control strategy. The issue associated with the selection of a control technique is how to operate the active TMD most effectively in order to mitigate the response of the main structure

(32). Control performance depends on the properties of the controller, the targeted structure to be damped, and the operation environment. Therefore, control system design strategies and control performances of different active control techniques should be studied and investigated to find out the appropriate control method for structural control.

In an effort to enhance the effectiveness of a TMD system, Chang and Soong (1980)

(33), and Isao (1992) (32), introduced an active control force to act between the structure and TMD system, thereby, increasing the effectiveness of the TMD. It has been proven by

Chang and Soong that the effectiveness of TMDs can be considerably increased by the addition of an active actuation. The appropriate control forces were calculated using an optimization process. Numerical results using realistic parameter values show that significant reduction in building displacement and acceleration can be reached when the

TMD system is actively operated. Moreover, reduction in TMD strokes or the mass ratio can also be achieved. Another important finding based on the numerical results is that the

TMD can become ineffective when high frequency excitations are encountered; however, by adding active control force, this situation can be better improved. Further study for designing an effective active TMD to control a tall building subjected to stationary random wind forces was proposed by Abdel-Rohman (1984) using the pole assignment method.

The numerical results of an example indicate that the design of an optimal ATMD required at least a parametric study to select the ATMD parameters.

Chang and Yang (34) proposed a form of a closed-loop complete feedback control algorithm to control a building (MDOF system) using an ATMD. The building is assumed to be under stationary Gaussian white noise ground excitation. The control force is computed from classical feedback control of the acceleration, velocity, and displacement of the building. The passive properties and the gain coefficients of the actuator were derived by minimizing the displacement variance of the building. After the simulation has been performed to evaluate the ATMD, the results show that the control efficiency of the

ATMD, based on velocity feedback, depends on the properties assumed for the passive device. Also, for the same level of reduction in structural displacement, the control force required is smaller using complete feedback. Another similar study was presented by

Seshasayee and Yang (30) for the design of ATMD to suppress the first mode of a tall building subject to wind loads, which was assumed as white noise excitation. In their study, the control force was generated based on complete feedback (of displacement, velocity, and acceleration). Two examples are studied, one is a 162 m tall planar frame and the other is 400 m tall, to evaluate the control procedure and effectiveness of ATMD. The same examples are then studied for MDOF system subject to nonwhite excitation which simulates wind loads better. The results show that the ATMD design introduced in this study is effective in mitigating the rooftop displacement in comparison to the passive version of it. Though, the scholars found that the stroke is longer when active control is applied, which is an undesirable aspect.

In addition to the traditional classical feedback control, many other control techniques are used to drive ATMDs. these techniques include the linear quadratic regulator (LQR), the linear quadratic Gaussian (LQG), positive position feedback (PPF), sliding-mode control, and the use of H2 and Hinf to obtain the optimal control force for minimizing the wind induced vibration of a tall building (35). Ki-Pyo (2014) (35) computed the ATMD control force based on the LQG technique for obtaining the reduced along-wind responses, resulted from the fluctuating along-wind load of a tall building using numerical simulation. This study concluded that the ATMD system using a LQG controller is effective and useful for mitigating wind-induced vibration on tall buildings.

Another control method is the classical LQR algorithm which can calculate the optimal active control forces of the linear system by minimizing the cost function. The control force is obtained from feedback of state variables and accelerations. The gain matrix in this control method is obtained from the discrete algebraic Riccati equation (36).

Further study about LQR control technique has been presented by Alavinasab and

Moharrami (37) where a new energy-based technique is proposed to eliminate trial and error in finding appropriate gain matrices in LQR controllers used in the active control.

They calculate the control force based on displacement and velocity of the structure. With minimizing the performance index by tuning some weighting matrices, the optimum control is reached in this method. It was concluded from this work that an increase in the magnitude of weighting matrices in the LQR method does not always guarantee achieving the best results and optimum external power.

ATMDs used for vibration control of tall buildings under cross wind excitation where the control action is achieved by fuzzy logic controller (FLC) was presented by Bijan

(2004) (38). The main advantage of the fuzzy controller is its inherent robustness and ability to handle any non-linear behavior of the structure. In fact, the feedback control theory can also deal with certain levels of nonlinearities inside the system. However, designing a proper feedback gain usually requires the deep understanding of the system’s dynamics. Even with some estimation, key parameters have to be known ahead of time.

However the fuzzy controller allows the user to apply control effort much easier without worrying about the robustness and that its implementation does not require a mathematical model of the structure. The model building used in this study is the 76-story, 306-m tall reinforced concrete office tower proposed for the city of Melbourne, Australia. The results show that the fuzzy controller and LQG controller are similar in terms of the building response, the active control force required, and the stroke of the actuator. However, the fuzzy controller performs much better in terms of required control power, and requires a fewer number of sensors for the structure. An ATMD driven by a fuzzy controller is shown to be more robust than an ATMD driven by a LQG controller in reducing acceleration response of the structure (1). Great potential for active control has been earned by adopting the fuzzy algorithm to drive the ATMD.

Like FLC, if the system cannot be accurately modeled, then the neural network controller can be used to drive the ATMD. Ghaboussi and Joghataie (39) developed and tested a new neural-network-based control algorithm in the computer simulation of active control of a three-story frame structure subjected to ground excitations. A controller neural network has been developed to adjust control signals based on the immediate response of the structure and actuator. Results of this initial study indicate that a potentially powerful adaptive controller in structural control problems can be obtained utilizing neural network-

based control algorithms after additional research on this control method is done.

Neurocontrollers are also effective control devices for nonlinear control problems.

Another method among the various control strategies and theories for ATMD control applied on structural vibration is the non-linear control strategy known as sliding mode control. Adhikari and Yamaguchi (40) discussed the application of sliding mode control (SMC) for controlling the vibration of tall buildings with ATMD installed at the top floor.

In the civil engineering field, non-linear control theories, in comparison with linear feedback control theories, have certain positive aspects, such as adaptivness and robustness making them a superior choice for the control of vibration problems and favor them over linear feedback control schemes. It is shown in this study that the direct application of the conventional SMC theory may cause large responses in the building due to the interaction effect between ATMD and the building which results due to the large responses of ATMD.

This may lead to excessive control force to be acted on the building and hence exciting the structure instead of controlling it. This paper therefore proposed to use a compensator

(filter) in order to rectify this problem and exclude unwanted effects of the ATMD response on the control scheme. It was proved by this study that the application of the compensator in such a problem helps reduce the interaction effect of ATMD on the structure.

Although the simpler classical feedback control theory has the ability to subdue the structural response due to the external loads in an acceptable level of human comfort, it, however, requires thorough understanding of the system dynamics and even with some approximations, key design parameters have to be known in advance, which are not always available. Sufficient damping against wind and earthquake excitations can be provided by

the LQR control method. The sliding mode control (variable structure controller) outperforms their counterpart LQR control strategy due to the non-linearity properties of the system. Due to the uncertainties and non-linear behavior of the structure and when the structure cannot be accurately modeled, then neural network controller and fuzzy controller are the better options to drive the ATMD. These observations and control classifications have been made by

Nerves and Krishnan (1995) (29) based on the results obtained from the simulation study for application in active control scheme for tall civil structures. In this study, the authors compared different control schemes to reach the best option for the controller.

Considering that the inertia force generated by TMD mass is what adds damping to the structure, increasing such inertia force by active means will result in increase in the effectiveness of the TMD. Nishimura et. al. (41) suggested that the inertia force of a one degree of freedom ATMD appended to a structure, as shown in Figure 10, can be increased by having the active element actuate the system proportional to the acceleration of the structure. As it can be expected, introduction of such actuation also increase the excursion of the ATMD (the relative motion of the TMD mass with respect to the structure). To lower such excursion, Nishimura et. al. (41) augmented the actuation by feeding back the excursion velocity. Such feedback makes the actuator to act as an active viscous damper.

2.3.3 Semi-Active Tuned Mass Damper (SATMD)

Due to some shortcomings of active and passive TMDs, many attempts have been made to develop new designs of TMDs called semi-active TMDs (SATMD) to improve their performance. SATMDs consume small amounts of power to control either component of TMD (stiffness or damping) through the use of variable orifice dampers, variable friction dampers, and controllable fluid dampers (magnetorheological) and / or modifying the

stiffness of the TMD (17). The power used to control stiffness and/or damping in SATMD is much lower than the power required to drive the mass in ATMD (6). Several studies such as (Symans and Constantinou, 1999), (Housner, 1997), and (Spencer and Nagarajaiah,

2003) have been made to improve TMD performance and to talk about its new versions, semi-active and hybrid TMDs, by combining passive and active TMDs. Also, also they present a full review of active and semi-active TMDs (20) (21) (24).

A good comparison of the three types of TMDs are shown clearly in Figure 7 where the configuration of each type can be recognized easily. The classical design of a TMD is shown first in the figure where no modification is added to the TMD, and then a TMD with adjustability built into it which changes the stiffness and damping is shown next. Next to the right is the final improvement added to TMD by adding control force to TMD to increase its effectiveness.

2.4 Active Pendulum Tuned Mass Damper and Structure Model

Suspended mass, as shown in Figure 11, is possibly the simplest form of TMDs.

By involving viscous damping elements in their design, they can suppress structural vibrations effectively and increase the serviceability of high-rise buildings. Such TMDs are called Pendulum TMDs, or PTMDs. The typical schematic geometry is shown in Figure

11. PTMDs are commonly suggested for tall structures with large masses and low natural frequencies (42). The motivation in this work is to develop and investigate a single active

PTMD that a) can be simultaneously tuned to multiple modes of high rise structure and mitigate their corresponding vibrations and b) have a smaller mass than an equivalent passive PTMD, by improving its effectiveness using a modern control theory.

Figure 11: The configuration of pendulum tuned mass damper (PTMD)

2.4.1 Structure Model

The targeted structure for the proposed approach in this work is a MDOF model representing a full scale tall building. This building is a modern high-rise building designed as a flexible and slender structure, with asymmetric geometry, subject to multi-directional wind loads. The building has 41 stories above the ground where each floor has three degrees of freedom, two in x, y directions (planar) and one around the axis perpendicular to x-y plane (rotational). The first 15 modes of the building will be included in this study, with five dominant modes in each direction as it appears in Figure 12 below. Figure 12 illustrates the acceleration of the floor 41 (top floor) of the targeted building in response to

the wind excitation in random direction and Figure 13 illustrates the five dominant modes in y-direction.

Figure 12: Collocated FRFs mapping perturbations to accelerations in all 3 directions on floor 41

Figure 13: Modes shapes of vibration in Y-direction

2.4.2 Active PTMD (APTMD)

Viscous dampers in passive PTMDs, which are connected to the mass from one end and to the vibrating structure at the other, as it appears in Figure 11 are responsible for the energy dissipation from the damped structure. Several reasons make active PTMDs of interest to civil engineers in the case of vibration control of tall buildings because they:

 Eliminate the need for using multiple traditional translational TMDs tuned to multiple

frequencies

 Provide higher damping effectiveness than an equivalent passive PTMD of an equal

size

 Provide as much damping, using a smaller mass, as an optimally designed passive

TMD with substantially larger mass. This is of interest when the vibrating structure,

e.g., the high-rise, cannot support the weight of a massive passive PTMD.

 Occupy less space than a passive type.

An example of well-known high-rise buildings that are equipped with PTMD with some activeness built into it is The Chifley Tower in Sydney, Australia. It is 52 story, 209 m tall, office building constructed in 1992 and equipped with a 400 ton PTMD with a tuning frame hanging below the upper suspension point of the mass. The pivot point of the suspension cables which in turn changes their length can be changed by the tuning frame, and hence changes the natural frequency of the pendulum. Utilizing this vibration absorber added 3% damping to the structure depending on the level of vibration. Another example where the tuning frame was involved is Park Tower in Chicago. The damper consists of a

272 tons mass with a pendulum length up to 7.46 m (42). The behavior of spatial PTMD

has been fully studied by Lourenco (7), and an adaptive PTMD has been developed and tested by Lourenco (6).

This work aims to develop a single multi-degree of freedom for APTMDs to mitigate the wind-induced vibration on a tall building of 41 stories described in the previous section. The proposed APTMD, configured as a spatial pendulum with six hydraulic actuators as its active elements, targets the first three modes of the building. The final target of the proposed APTMD is the ability to change damping and stiffness of the first three modes (푋, 푌, 휃) of the targeted building simultaneously through the active control of the six hydraulic actuators configured as the legs of a classical Stewart Platform.

2.4.2.1 The Mathematical Model of the PTMD

(a) Spatial pendulum (b) x – direction (c) y – direction

Figure 14: Schematic geometry of the PTMD mass with auxiliary damper and spring

In this derivation of equations of motion for the MDOF structure coupled with a PTMD, a five – DOF system is considered first as presented in Figure 14, then extended to the

MDOF. The origin of the system is set up to coincide with the suspension point of the pendulum mass. The vectors 푢, 푣, 푎푛푑 푤 are the displacements of the suspension point of the PTMD in x, y, and z directions respectively. 훩 is the angle of swing away of the auxiliary mass from the vertical line, also known as planar angle. 휑 is the angle of the auxiliary mass rotating about the vertical line, also known as the spherical angle. 퐿푎 is the pendulum length and 푚푎 is the pendulum mass. Also, two sets of linear spring and dampers are connected to the pendulum link at the distances ℎ푥 and ℎ푦 away from the suspension point in both horizontal directions (푥 푎푛푑 푦).

The position vector, in the space where 푢, 푣, 푎푛푑 푤, are corresponding to 푥, 푦, 푎푛푑 푧 directions respectively, of the auxiliary mass relative to the origin is:

푢 + 퐿푎푠𝑖푛휃푐표푠 휙 푟푎 = {푣 + 퐿푎 sin 휃 sin 휙} (2.1) 푤 − 퐿푎 cos 휃

By taking the first derivative of the position, the velocity vector of the auxiliary mass is found to be:

푢̇ + 퐿푎 푐표푠 휃 푐표푠 휙 휃̇ − 퐿푎 푠𝑖푛 휃 푠𝑖푛 휑휑̇ 푣푎 = {푣̇ + 퐿푎 푐표푠 휃 푠𝑖푛 휑̇ 휃 + 퐿푎 푠𝑖푛 휃 푐표푠 휑휑̇ } (2.2) 푤̇ + 퐿푎 푠𝑖푛 휃 ̇휃

 Kinetic energy, potential energy, and dissipation functions

Calculating the kinetic energy of the auxiliary mass from the previous equations

1 훵 = 푚 푣2 푎 2 푎 푎

1 2 = 푚 [(푢̇ + 퐿 푐표푠 휃 푠𝑖푛 휑̇ 휃 − 퐿 푠𝑖푛 휃 푠𝑖푛 휑 휑̇) 2 푎 푎 푎

2 +(푣̇ + 퐿푎 푐표푠 휃 푠𝑖푛 휑̇ 휃 + 퐿푎 푠𝑖푛 휃 푐표푠 휑 휑̇)

2 +(푤 +̇ 퐿푎 푠𝑖푛 휃̇ 휃) ]

1 푇 = 푚 [푢̇ 2 + 푣̇ 2 + 푤̇ 2 + 퐿2 휃̇ 2 + 퐿2 휑̇ 2푠𝑖푛2휃 + 2푢̇퐿 푐표푠 휃 푐표푠 휑휃̇ 푎 2 푎 푎 푎 푎

−2푢̇퐿푎 푠𝑖푛 휃 푠𝑖푛 휑휑̇ + 2푣̇퐿푎 푐표푠 휃 푠𝑖푛 휑휃̇ + 2푣̇퐿푎 푠𝑖푛 휃 푐표푠 휑휑̇

+2푤̇퐿푎휃̇ 푠𝑖푛 휃] (2.3)

The potential energy of the auxiliary mass relative to the origin, is

혝푎 = 푚푎𝑔(푤 − 퐿푎 푐표푠 휃) (2.4)

Where ( 𝑔 ) is the gravitational acceleration. The kinetic energy of the main structure is

푇 푢̇ 푚푢푢 푚푣푢 푚푢푤 푢̇ 1 2 1 훵 = 푀∆̇ = {푣̇ } [푚푣푢 푚푣푣 푚푣푤 ]{푣̇ } 푚 2 2 (2.5) 푤̇ 푚푤푢 푚푤푣 푚푤푤 푤̇

Where M is the 3 × 3 mass matrix corresponding to the pendulum suspension point, in x, y, and z directions. The potential energy of the main mass is

푇 푢 퐾푢푢 퐾푣푢 퐾푢푤 푢 1 푇 1 혝 = ∆ 퐾∆= {푣} [퐾푣푢 퐾푣푣 퐾푣푤 ]{푣} 푚 2 2 (2.6) 푤 퐾푤푢 퐾푤푣 퐾푤푤 푤

Where K is the 3 × 3 stiffness matrix. The Raleigh dissipation function for the main mass

푇 푢̇ 푐푢푢 푐푣푢 푐푢푤 푢̇ 1 푇 1 퐹 = ∆̇ 퐶∆̇ = {푣̇ } [푐푣푢 푐푣푣 푐푣푤 ]{푣̇ } is 푚 2 2 (2.7) 푤̇ 푐푤푢 푐푤푣 푐푤푤 푤̇

where C is the 3 × 3 damping matrix.

 Lagrange’s Equation

The kinetic energy, potential energy, and dissipation function for the combined main structure and PTMD are: (43)

푇 = 푇푚 + 푇푎 (2.8a)

혝 = 혝푚+ 혝푎 (2.8b)

퐹 = 퐹푚 (2.8c)

The lagrangian of the system is L = T – V. The Lagrange’s equation of the system in case of free vibration is (43)

푑 휕푇 휕푇 휕푉 휕퐹 − + + = 0 (2.9) 푑푡 휕푞푟̇ 휕푞푟 휕푞푟 휕푞푟̇ where 푞푟 and 푞푟̇ are the general coordinates and velocities of the system. For the total system (main structure and PTMD), the general coordinates are 푢,푣,푤,휃,푎푛푑 휑 and the general velocities are 푢̇, 푣̇, 푤̇, 휃̇, 푎푛푑 휑̇

 Equations of motion

Substituting 퐸푞. 8 𝑖푛푡표 퐸푞. 9 produces the following system of equations for a 5-DOF system, including the motion of a suspended PTMD from the top floor, corresponding to the first three DOF in the mass, damping, and stiffness matrices for free vibration:

푚푎 0 0 푢̈ 푢̇ 푢 [푀 + [ 0 푚푎 0 ]] {푣̈ } + 퐶 {푣̇ } + 퐾 {푣} = 푚푎퐿푎 0 0 푚푎 푤̈ 푤̇ 푤

− cos 휃 cos 휑 휃̈ + sin 휃 cos 휑 휃2̇ + 2 cos 휃 sin 휑 휃̇휑̇ + sin 휃 sin 휑 휑̈ + sin 휃 cos 휑 휑2̇ ̈ ̇ 2̇ × − cos 휃 푠𝑖푛 휑 휃 + sin 휃 sin 휑 − 2 cos 휃 cos 휑 휃휑̇ − sin 휃 cos 휑 휑̈ + sin 휃 sin 휑 휑 𝑔 − sin 휃 휃̈ − cos 휃 휃2̇ − { 퐿푎 }

(2.10)

The equation of motion corresponding to the planar pendulum motion (휃 − 퐷푂퐹) is

2 퐿푎휃̈ − 퐿푎 sin 휃 cos 휃 휑̇ + cos 휃 cos 휑 푢̈ + cos 휃 sin 휑 푣̈ + sin 휃 푤̈ + 𝑔 sin 휃 =0 (2.11)

The equation of motion corresponding to spherical pendulum motion (휑 − 퐷푂퐹) is

퐿푎 sin 휃 휑̈ + 2 퐿휃푎 cos 휃 ̇휑̇ − sin 휑 푢̈ + cos 휑푣̈ = 0 (2.12)

2.4.2.2 Auxiliary Damping and Stiffness

A linear auxiliary damper and spring are introduced, as shown in Figure 14. The linear spring constant is 푘푥 in the x-direction and 푘푦 in the y-direction. Similarly, the damping coefficient is 푐푥 in the x – direction and 푐푦 in the y – direction. The spring and damper, in both horizontal directions, connected to the pendulum length at a distance ℎ푥 and ℎ푦 away from the suspension point of the pendulum. Ignoring the vertical motion of the attached point of the spring and damper, the position of the attachment point relative to the vertical line passing through the moving suspension point, in x and y directions, is

푟푝,푥 = ℎ푥 sin 휃 cos 휑 (2.13a)

푟푝,푦 = ℎ푦 sin 휃 cos 휑 (2.13b)

By taking the first derivative of the position, velocity is found 42

푣푝,푥 = ℎ푥 cos 휃 cos 휑 휃̇ − ℎ푥 sin 휃 sin 휑 휑̇ (2.14a)

푣푝,푦 = ℎ푦 cos 휃 sin 휑 휃̇ + ℎ푦 sin 휃 cos 휑 휑̇ (2.14b)

The kinetic energy of the auxiliary mass remains the same, as in Eq. 3. The potential energy becomes

1 1 혝 = 푘 푟2 + 푘 푟2 + 푚 𝑔(푤 − 퐿 푐표푠 휃) 푎 2 푥 푝,푥 2 푦 푝,푦 푎 푎

1 1 = 푘 ℎ2 sin2 휃 cos2 휑 + 푘 ℎ2 sin2 휃 sin2 휑 + 푚 𝑔(푤 − 퐿 푐표푠 휃) 2 푥 푥 2 푦 푦 푎 푎 (2.15)

The Raleigh dissipation function for the auxiliary system is introduced. The auxiliary dissipation function is as follows:

1 1 퐹 = 푐 푣2 + 푐 푣2 푎 2 푥 푝,푥 2 푦 푝,푦

1 = 푐 (ℎ2 cos2 휃 cos2 휑 휃2̇ − 2ℎ2 cos 휃 cos 휑 휃̇sin 휃 푠𝑖푛 휑 휑̇ + ℎ2 sin2 휃 sin2 휑 휑2̇ 2 푥 푥 푥 푥

1 + 푐 (ℎ2 cos2 휃 sin2 휑 휃2̇ + 2ℎ2 cos 휃 cos 휑 휃̇sin 휃 푠𝑖푛 휑 휑̇ + 2 푦 푦 푦

2 2 2 2̇ ℎ푦 sin 휃 sin 휑 휑 (2.16)

The kinetic energy, potential energy, and dissipation function for the main structure stay the same. The dissipation function for the total system (main and auxiliary systems) is

퐹 = 퐹푚 + 퐹푎 (2.17)

These results are then used in Lagrange’s equation (Eq. 9) and the system of equations are found for the simple 5 – DOF system including a linear damper and spring connected to the auxiliary system. Eq. 10 corresponding to the translational DOF of the suspended

moving point remains unchanged. The equation of motion corresponding to the planar pendulum motion (휃 − 퐷푂퐹)(Eq. 11) becomes

2 퐿푎휃̈ − 퐿푎 sin 휃 cos 휃 휑̇ + cos 휃 cos 휑 푢̈ + cos 휃 sin 휑 푣̈ + sin 휃 푤̈ + 𝑔 sin 휃

2 2 푘푥ℎ푥 푘푦ℎ푦 + 푠𝑖푛 휃 cos 휃 푐표푠2 휑 + sin 휃 cos 휃 푠𝑖푛2 휑 푚푎퐿푎 푚푎퐿푎

2 푐푥ℎ푥 + (cos2 휃 푐표푠2 휑휃 − cos 휃 cos 휑 sin 휃 sin 휑휑̇ ) 푚푎퐿푎

푐 ℎ2 + 푦 푦 cos2 휃 푠𝑖푛2 휑휃̇ + cos 휃 cos 휑 sin 휃 sin 휑휑̇ = 0 (2.18) 푚푎퐿푎

The equation of motion corresponding to the spherical pendulum motion (휑 − 퐷푂퐹) (Eq.

12) becomes

2 2 푘푥ℎ푥 − 푘푦ℎ푦 퐿푎 sin 휃 휑̈ + 2 퐿휃푎 cos 휃 ̇휑̇ − sin 휑 푢̈ + cos 휑푣̈ − (sin 휃 sin 휑 cos 휑) 푚푎퐿푎

2 푐푥ℎ푥 + (− cos 휃 cos 휑 sin 휑 휃̇ + sin 휃 sin2 휑휑̇) 푚푎퐿푎

푐 ℎ2 + 푦 푦 (cos 휃 cos 휑 sin 휑 휃̇ + sin 휃 cos2 휑휑̇) = 0 (2.19) 푚푎퐿푎

2.4.2.3 Pendulum Dynamics Coupled with MDOF Structure

Considering the flexible MDOF main structure shown in Figure 15 with three translational degrees of freedom in x, y, and z directions, the Kinetic energy of the main mass is: (42)

푇 푢̇ 푚푢푢 푚푢푣 푚푢푤 M푢푟 푢̇ 1 푇 1 푣̇ 푚푣푢 푚푣푣 푚푣푤 M푣푟 푣̇ 훵푚 = ∆̇ 푀∆̇ = { } [ ]{ } (2.20) 2 2 푤̇ 푚푤푢 푚푤푣 푚푤푤 M푤푟 푤̇ ∆푟̇ M푟푢 M푟푣 M푟푤 M푟푟 ∆푟̇

where [M] is the mass matrix for the entire main structure. The first three rows and columns correspond to the three degrees of freedom (DOF) of the main structure at the suspension

̇ 푇 point from which the PTMD is suspended. ∆= {푢̇ 푣̇ 푤̇ ∆푟̇ } is the velocity vector of the main system, where 푢̇, 푣̇, 푎푛푑 푤̇ are the nodal velocities of the suspension point, and ∆푟̇ are the velocities of the remainder of the DOFs for the main system.

Likewise, the subscript r in the mass matrix of main system represents the remainder of the rows and columns of the main mass matrix.

The potential (Strain) energy of the main mass is: (42)

푇 푢 푘푢푢 푘푢푣 푘푢푤 K푢푟 푢 1 푇 1 푣 푘푣푢 푘푣푣 푘푣푤 K푣푟 푣 혝푚 = ∆ 퐾∆= { } [ ]{ } (2.21) 2 2 푤 푘푤푢 푘푤푣 푘푤푤 K푤푟 푤 ∆ ∆ 푟 K푟푢 K푟푣 K푟푤 K푟푟 푟

푇 where [K] is the stiffness matrix of the main structure. ∆= {푢 푣 푤 ∆푟} Is the displacement vector of the main system, where 푢, 푣, 푎푛푑 푤 are the nodal displacements of the suspension point, and ∆푟 are the displacements of the remainder of the DOFs for the main system. Likewise, the subscript r in the stiffness matrix of main system represents the remainder of the rows and columns of the main stiffness matrix.

Figure 15: Schematic of a MDOF flexible main structure equipped with a PTMD

The Raleigh dissipation function of the main system is: (42)

푇 푢̇ 푐푢푢 푐푢푣 푐푢푤 C푢푟 푢̇ 1 푇 1 푣̇ 푐푣푢 푐푣푣 푐푣푤 C푣푟 푣̇ 퐹푚 = ∆̇ 퐶∆̇ = { } [ ]{ } (2.22) 2 2 푤̇ 푐푤푢 푐푤푣 푐푤푤 C푤푟 푤̇ ̇ ̇ ∆푟 C푟푢 C푟푣 C푟푤 C푟푟 ∆푟 where [C] is the damping matrix of the main structure.

Lagrange’s equation including the generalized forces (43) is given by

푑 휕푇 휕푇 휕푉 휕퐹 − + + = 푄푟 (2.23) 푑푡 휕푞푟̇ 휕푞푟 휕푞푟 휕푞푟̇ where 푄푟 is the generalized force 푄 = [푃푢 푃푣 푃푤 푃푟 0 0 ] , where 푃푢, 푃푣, 푎푛푑 푃푤 are the arbitrary forces in the x, y, and z directions respectively applied to the main structure at the suspension point of the PTMD, typically in the top floor. (푃푟) is the arbitrary force applied to the remainder of the DOFs of the main system.

Substituting equations (20) through (22) into (23) produces the following equations of motion corresponding to the first three (translational) DOFs of the MDOF system with

PTMD suspended from the top floor for forced vibration.

푚푎 0 0 0 푢̈ 푢̇ 푢 0 푚푎 0 0 푣̈ 푣̇ 푣 푀 + [ ] { } + 퐶 { } + 퐾 { } = 푚푎퐿푎 0 0 푚푎 0 푤̈ 푤̇ 푤 ̈ ̇ ∆ [ 0 0 0 0 ] ∆푟 ∆푟 푟

̈ 2̇ ̇ 2̇ − cos 휃 cos 휑 휃 + sin 휃 cos 휑 휃 + 2 cos 휃 sin 휑 휃휑̇ + sin 휃 sin 휑 휑̈ + sin 휃 cos 휑 휑 − cos 휃 푠𝑖푛 휑̈ 휃 + sin 휃 sin 휑 − 2 cos 휃 cos 휑 휃̇휑̇ − sin 휃 cos 휑 휑̈ + sin 휃 sin 휑 휑2̇ 𝑔 + − sin 휃 휃̈ − cos 휃 휃2̇ − 퐿푎 { 0 }

푃푢 푃 { 푣 } (2.24) 푃푤 푃푟

The equations of motion corresponding to the planar pendulum motion (휃 − 퐷푂퐹) and to the spherical pendulum motion (휑 − 퐷푂퐹) are the same as given in Eqs. 18 and 19.

Equations 18, 19, and 24 represent the integrated finite element of MDOF flexible structures coupled with spatial PTMD for forced vibration (7) (42) (44).

CHAPTER III

MULTI TUNING-FREQUENCY PASSIVE/ON-DEMAND ACTIVE PENDULUM

TUNED MASS DAMPER

The use of tuned mass dampers (TMDs) in tall buildings and towers for mitigating wind induced vibration have resulted in significant improvements in serviceability of such structures. In applications where more than one mode of vibration with distinctly different natural frequencies are in need of damping, multiple passive TMDs each tuned to one of the natural frequencies are needed. In many such applications, the excessive cost, weight, and space requirement of multiple passive TMDs are not acceptable. This chapter presents the unique trait of the proposed passive pendulum tuned mass damper (PTMD), i.e. the ability to on-demand switch to an active PTMD. In its default passive state, the proposed device is passively tuned to the first mode of the structure and acts as a traditional passive

PTMD. In its active state, while staying tuned to the first mode passively, the PTMD simultaneously tunes itself to multiple higher order modes with corresponding optimal damping ratios.

In its passive state, the PTMD which is tuned to the first mode of the structure adds tuned damping to that mode. In its active mode, the device tunes itself to 3 frequencies adding tuned damping to the first 3 modes (with three distinctly different natural

48 frequencies) of the structure, attenuating the resonant vibration of the high-rise building in three directions, associated with the first three modes of the structure, simultaneously.

3.1 Multiple Frequency Tuning PTMD

Considering that the first translational (sway) mode of a tall structure plays a dominant role in its dynamic response, a PTMD is normally tuned to the first natural frequency of the structure (6) (45) (46) (47). In certain applications multiple modes with distinctly different natural frequencies are in need of being damped, necessitating the use of multiple TMDs with different dynamic characteristics, each tuned to one of the target natural frequencies.

The extent of vibration energy that a TMD dissipates as well as its frequency range

(bandwidth) increase with increase in the mass ratio of the TMD and the corresponding size of the viscous dampers used in its make-up (6) (48). The TMDs used in tall buildings have a very small mass ratio (less than 1%) and correspondingly small damping. Such

TMDs a) have a rather narrow bandwidth making their damping effectiveness highly sensitive to their tuning accuracy and 2) experience large excursions (relative motion of the TMD mass with respect to the structure) even when the structure is subject to moderate perturbations. Due to the change, over time, in the resonant frequencies of a structure caused by deterioration, change in the operating conditions of the structure, remodeling, etc. such TMDs are most likely in need of frequent re-tuning (6) (49). Moreover, the damping ratio of the TMD optimized at the design stage will not remain optimal during the life of the TMD.

The proposed PTMD can, on-demand, turn into an active PTMD (APTMD) with multiple, adjustable tuning frequencies and damping ratios in multiple directions. This

device is optimally designed as a passive PTMD tuned to the first natural frequency of the structure. It also has the look of a passive PTMD, see Figure 16, but instead of using shock absorber type viscous dampers as its dissipative elements it uses hydraulic cylinders which along with flow control valves act as passive viscous dampers. Upon switching the active feature on, the flow control valves will be bypassed and the fluid is re-routed thru servovalves, turning the device into an active PTMD. In the active mode. The APTMD is automatically tuned to more than one mode in more than one direction, replacing multiple

PTMDs lowering the cost, weight and space requirement associated with dampening multiple modes using multiple TMDs. Moreover, such APTMD with a smaller mass can possess the bandwidth and energy dissipation effectiveness of a more massive passive

PTMD.

3.2 Passive/On-Demand Active Tuned Mass Damper

The passive/on-demand active PTMD uses 6 hydraulic cylinders in place of the viscous dampers commonly used in passive PTMDs. Figure 16 depicts the proposed passive/on-demand active PTMD with the mass suspended by 6 steel wire ropes and actuated by 6 hydraulic cylinders. The hydraulic cylinders along with the moving mass form a 6-legged (hexapod) closed-chain mechanism with 6 degrees of freedom, commonly known as Stewart platform, capable of moving in any direction and orientation, generating controllable dynamic motions (50) (8) (51).

Except for the use of hydraulic cylinders in place of purely passive viscous dampers, the proposed PTMD resembles a passive PTMD commonly used in many modern high-rises and towers. In its default state, the hydraulic cylinders are configured to act as passive viscous dampers turning the device to a passive PTMD. When the need arises

(depending on the extent and frequency content of the vibration) the hydraulic cylinders switch to active actuators and turn the device into an active PTMD with multi degree-of- freedom tuning capacity (capability). The hydraulic circuit and the switching logic are discussed in section 5.3 in chapter 5. Figure 18 illustrates the interface between the proposed passive/on demand active PTMD and the targeted tall structure in a

MATLAB/SimMechanics platform.

Figure 16: The passive/on-demand active PTMD

3.3 Control Strategy

A block diagram presentation of the structure and the passive/on-demand active

PTMD is shown in Figure 17 and its corresponding model is shown in Figure 18. The structure is subject to the perturbation force (the term ‘force’ is used for force/moment) u1.

As shown in the block diagram, the PTMD interacts with the structure in a feedback manner by providing a reactive force vector R in response to the vibratory motion of the structure,

at the PTMD installation location. Note that the reactive force vector R is the inertia force vector of the PTMD.

The supervisory controller with continuous access to the vibration attributes of the structure decides whether to run the device as a passive or an active PTMD. When the perturbation in such a way that mainly the first mode of the structure is excited with low to moderate severity, the PTMD will be configured as passive by disengaging the cylinders

(the legs) from the hydraulic power supply and routing the hydraulic fluid in each cylinder from one side of the piston to the other side, thru its corresponding flow control valve.

Figure 17: Block diagram of the structure + PTMD

When the perturbation becomes a) multi-directional and sets higher modes of the structure in motion and/or b) large in severity, the PTMD will revert to active by bypassing the flow control valves from the hydraulic circuit and connecting the cylinders to the power supply, 52

through their corresponding servo-valves. Proper adjustment of the servo-valves by the combination of a centralized and 6 distributed controllers enables the PTMD to simultaneously tune itself to multiple frequencies and thus adding tuned damping to multiple modes.

Figure 18: Feedback interaction of an APTMD with its target tall structure 53

In the unlikely event of losing power, the system revert to passive and hydraulic cylinders act like passive viscous dampers with the system still tuned to the dominant translational mode with the lowest natural frequency; in other words, the proposed damping system at the minimum level of effectiveness is similar to that of a passive PTMD of the same size.

3.3.1 Active Control

The underlying principle (the basis) for the active control algorithm of the PTMD is the on-going adjustment of the PTMDs’ frequency-dependent stiffness and damping.

This is achieved by active, simultaneous actuation of the hydraulic cylinders subjects the

PTMD mass to the combined stiffness plus damping force vector of U2 shown in Equation

(3.1).

푈2 = 퐾 ∗ 푃 + 퐶 ∗ Ṗ (3.1)

In this equation K is the 6x6 matrix providing the PTMD’s additional (beyond what the pendulum provides) stiffness and C is the 6x6 matrix providing damping coefficients required for multi degree of freedom tuning, and P and P are the position and velocity vectors of the center of mass of the PTMD mass measured in the global Cartesian coordinate system installed on the structure.

With its spatial force and position control ability Stewart platform closed-chain mechanism with hydraulic cylinders as legs, is selected as the actuation of choice for the proposed passive/on-demand active PTMD.

In the absence of any constraints, the 6-dof nature of a Stewart platform mechanism enables it to adjust its stiffness and damping in up to 6 directions (3 translational and 3 rotational).

But considering that in PTMDs, a) the rotation around X and Y coordinates (roll and pitch) 54

are restrained and b) the motion in the vertical direction is negligible, the number of directions in which the resilience and damping are adjusted reduces to 3, i.e., X, Y, and γ directions. With nearly decoupled 3 degrees of freedom, the actuation U2 formulated in global coordinate system is shown in Equation (3.2).

 K 0 0 XC  0 0 X  XX    UKYCY2  0YY 0  0 0  (3.2)      0 0 KC  0 0  where Ki and Ci are the additional stiffness and damping coefficients needed in i=X, Y, and

γ directions over what the pendulum itself is providing. If the natural modes were not nearly decoupled then the desired stiffness and damping coefficient matrices shown in

Equation (3.2) would not be symmetric.

To meet the dimensional compatibility of the matrices in conducting matrix algebra required for the computation of the forces of the 6 legs, the 3x3 stiffness and damping coefficient matrices in Equation (3.2) are padded with enough zeros to 6x6 matrices as shown in Equation (3.3)

  K XX0 0 0 0 0 XC   0 0 0 0 0  X       0KYY 0 0 0 0YC   0 0 0 0 0 Y  000000Z   000000 Z U2       000000   000000    000000   000000              0 0 0 0 0 KC   0 0 0 0 0  P KCP

(3.3) where Z is the negligible vertical motion and α and β are the constrained roll and pitch angular motions of the mass. 55

3.3.2 The Control Scheme

Equation (3.3) is the basis for the feedback control algorithm of the proposed passive/on-demand active PTMD. Implementing such algorithm requires collocated arrangement of motion sensors and actuators in the global coordinate systems. Although not impossible, it is rather impractical to implement. The more practical and convenient alternative is transforming the control force U2 of Equation (3.3) from the global coordinate system to the control force defined in the local coordinate systems of the legs. The local,

leg space, control force u2 is shown in Equation (3.4).

푢2 = 푘 ∗ 푝 + 푐 ∗ 푝̇ (3.4) where p and p are the displacement and the velocity vectors of the legs and k and c are the stiffness and damping coefficient matrices of the legs. The transformation is done using a) the principle of virtual work relating the actuation in the local and global coordinate systems, and b) the relationship between the motion in the local and global coordinate systems, as described below:

T   Let UUUUU2 2XYZ 2 2 2 represent the vector of force and moment that the legs collectively impart on the TMD mass in the global coordinate system and

T u2  u 21 u 22 u 23 u26  represent the vector of actuated leg forces (in the local leg

T coordinate systems). Moreover, let p   l1  l 2  l 3  l6  represent the vector of

T virtual legs displacement and PXY    000  represent the vector of virtual displacement associated with the mass.

Note that the displacement in Z direction is negligible and the rotation around X and Y axes

(yaw and pitch) are constrained by the parallel pendulum links. Assuming that the friction

forces in the joints and the gravitational effects of the legs are negligible and considering that the main gravitational effect due to the weight of the TMD mass is taken up by the pendulum (not the legs of the mechanism), the virtual work contributed by all the active forces is

푇 푇 푢2 훿푝 − 푈2 훿푃 = 0 (3.5.1)

Taking into account the relationship between the virtual displacements δp and δP, via the

6x6 Jacobian matrix of the closed-chain mechanism J, i.e.,

p J P (3.5.2) and substituting it in Equation (3.5.1) results in

TT u22 J U  P 0

Considering that the above equation holds for any virtual displacement, one can conclude that

푇 푇 −푇 푢2 퐽 − 푈2 = 0 → 푢2 = 퐽 푈2 (3.5.3)

Combining Equations 3.3, 3.4, 3.5.2 and 3.5.3 results in Equations 3.6.1 and 3.6.2 relating the stiffness and damping coefficients in the legs space to that in the global space.

k JT KJ 1 (3.6.1) T 1 c J CJ (3.6.2) where superscript -T signifies the ‘inverse of transpose’.

The inverse kinematics of the Stewart platform mechanism, the derivation of its Jacobian matrix as well as the stiffness and damping coefficient matrices of Equations 3.6.1 and

3.6.2 are presented in Appendix B.

Having stiffness and damping coefficient matrices in the local coordinate systems of the legs (hydraulic cylinders) i.e., k and c, allows for the evaluation of the instantaneous desired

T force of each leg (the elements of u2  u 21 u 22 u 23 u26  ) by a centralized

Proportional + Derivative (PD) controller being fed by the displacement vector of the legs

(hydraulic cylinders); k and c are the corresponding proportional and derivative feedback gain matrices, correspondingly. The evaluated u2, will in turn be used as the reference input vector to 6 parallel Proportional (P) distributed controllers generating the control signals to the servo-valves. The feedback signals to the distributed P controllers are the measured force in each leg. The active control strategy is depicted in the block diagram of

Figure 17.

Figure 19: Block diagram of the active PTMD

With the desired force of all the legs u2 realized, the pendulum mass will experience the desired force U2. This in turn will result in the PTMD experiencing its desired instantaneous global stiffness and damping as it appears in Figure 19.

3.4 Illustrative Example

The applicability and multi-mode damping effectiveness of the proposed passive/on-demand active PTMD is numerically demonstrated by incorporating it in the model of a tall, nearly uniform high-rise building perturbed by wind in multiple directions.

The building has 41 stories above ground with each floor having three degrees of freedom, two translational in X and Y directions and one rotational, denoted by γ, around the Z axis

(perpendicular to the X-Y plane). Due to the uniformity in geometry and mass distribution in each floor, vibration in those three directions is nearly decoupled from each other.

Tuning the pendulum to the first mode (in Y direction with the natural frequency of 0.185 Hz) requires a wire-rope length of 6.8 meters. This length provides the same swinging natural frequency in both Y and X directions. Although passively tuning a TMD to the first mode of a structure suits the tuning requirement of that mode, but the TMD may or may not be optimally tuned for mode 2, 3…, depending on how close their natural frequencies are to that of mode 1. For example in the building used in our numerical demonstration, modes 2 and 3 with vibration mainly in X and γ directions, respectively, have the natural frequencies of 0.29 and 0.56 Hz which are 60% and 300% higher than the natural frequency of mode 1. With the natural frequencies this far apart from each other, the only effective way of adding tuned damping to the first 3 modes would be using three

TMDs, each one tuned to one of the modes. In addition to being economically unattractive such solution would have a large weight and space requirement penalty. A single passive- on-demand active PTMD proposed in this work can optimally be tuned to all 3 modes, simultaneously.

3.5 Modeling of the Structure

The numerical model of the building, in which the first 15 modes of vibration are included, is developed. The nearly decoupled modes of vibration of the building enable one to associate each mode with the vibration in one direction only. For example the 1st,

4th, 7th, 10th and 13th modes with the corresponding frequencies of 0.18, 0.80, 1.93, 3.23,

4.63 Hz are the modes with vibration mainly in Y direction. The 2st, 5th, 8th, 11th, 14th modes with the corresponding frequencies of 0.29, 0.95, 2.02, 3.46, 5.10 Hz are the modes with vibration mainly in X direction. And the 3st, 6th, 9th, 12th, 15th modes with the corresponding frequencies of 0.56, 1.45, 2.77, 4.2, 5.96 Hz are the modes with vibration mainly in γ direction (rotation around Z axis).

Assuming the structure vibrates in the linear region and using the modal data evaluated by finite element modal analysis, the state space model of the building is formulated. Equation

(3.7) presents the structure of such model

0 I 0 z 2 z u1 nn2    y 0  z  D  u 1 (3.7)

where n is the number of modes considered in the model, 0 and I are the n x n zero and

T identity matrices, z [ q q ] is 2nx1 the state vector in which q [ q12 q qn ] is the nx1 modal displacement and q is the nx1 modal velocities. y is the output vector of

displacements. Moreover, n  diag([ n12  n  nn ]) and  diag([ 12  n ]) are the natural frequency and modal damping ratio matrices, respectively. Also,

1(x 1 )  2 (x 1 )  n (x1 )      1(x 2 ) 2 (x 2 )n (x2 )         12(xn ) (xn )n (x r )     1(y 1 )  2 (y 1 )  n (y1 )  (y )  (y )  (y )    1 2 2 2 n 2    (y )  (y )  (y )   12n n n r    ()()()        1 1 2 1 n 1  1()()()  2  2  2 n  2  

12()()() n  n n  r  (3.8)

T is the 3rxn modal matrix, u1 [ u 1xy u 1 M 1 ] is the 3rx1 vector of the perturbation input in which u1x and u1y are the two rx1 vectors of disturbances acting in the X and Y directions, and M1γ is the rx1 vector of torsional disturbance, and r is the number of floors. And lastly,

D is the direct through matrix which is a 0 matrix when any of the states or their linear combination is the output.

The modal (eigenvector) matrix for the building is given as a 123 x 15 matrix. Note that the number of floors is 41 and each floor has 3 planar degrees of freedom (the flexibility of each floor is ignored) with significant motion. Equation (3.8) shows the structure of the matrix made up of eigenvectors.

The FRF magnitudes of the 41th floor acceleration in Y, X, and γ directions, over the frequency range of 0.1-1 Hz are shown in Figure (12). Harmonic perturbations, with spatially varying amplitudes along the height of the structure (zero at the first and

maximum at the top floors), mimicking the effect of vortex shedding, are used to excite all the floors simultaneously. The modal damping ratio of 1.5% is used for all the modes.

As stated earlier and shown in Figure 12, the modes in 3 directions are for the most part decoupled in this building; note that at any natural frequency the corresponding mode has most of its vibration in one direction, only. In addition, the natural frequencies of the first

3 modes corresponding to vibration in Y, X, and γ directions are far enough apart (in percentage terms) from each other such that a single passive PTMD cannot effectively add damping to all three modes.

3.6 Application of the Proposed Passive/On-demand Active PTMD

The proposed passive/on-demand active PTMD is synthesized to add damping to the first three nearly-decoupled modes of the aforementioned tall building, simultaneously.

The natural frequencies of modes 1-3 are 0.18, 0.29, 0.56 Hz, respectively. The pendulum length is selected so that the PTMD, with its hydraulic cylinders configured as passive viscous dampers, is optimally tuned to the lowest natural frequency of the structure targeting the first mode. The PTMD defaults to its passive configuration when the structure is perturbed mainly along the Y direction and vibrates mostly in its first mode.

With the parameters of the proposed PTMD on hand, the mechanical, hydraulics, and controls aspects of the device are modeled. With the model of the PTMD interfaced with the model of the structure, a number of frequency and time-domain simulations are conducted the results of which are presented in the remaining of this section.

Figure 20 shows the frequency response functions as well as the resonant time traces of the structure’s acceleration along the Y direction, measured at the top floor. Harmonic perturbation with spatially varying amplitude along the height of the structure is used to

perturb all the floors simultaneously, vibrating the floors in Y direction. The severity of the harmonic perturbations is selected corresponding to 10 year return period wind (where the building is located) inducing 32 milli-g peak accelerations in Y direction.

Figure 20: FRFs and the resonant time traces of the structure’s acceleration along Y

direction, measured at the top floor

When modes 2 and 3 of the structure are also perturbed, the hydraulic cylinders which are arranged as the ‘legs’ of a Stewart platform mechanism are automatically reconfigured to act as the active elements turning the PTMD to an active PTMD (APTMD).

They are actively controlled to provide damping and additional direction-dependent stiffness for the APTMD to optimally tune the pendulum TMD to the second and third

modes of the structure as well. Thus in this way the APTMD can add damping to the first three modes, simultaneously.

As in Figure 20, Figures 21 and 22 show the frequency response functions as well as the resonant time traces of the structure’s acceleration along the X and around the Z axes, measured at the top floor. Harmonic forces with spatially varying amplitude along the height of the structure, are used to perturb all the floors, inducing peak accelerations of around 18 milli-g in X direction and 4.5 milli-radian/sec2 (torsional acceleration) in γ direction at the top floor.

Figure 21: FRFs and the resonant time traces of the structure’s acceleration along X

direction, measured at the top floor

Figure 22: FRFs and the resonant time traces of the structure’s angular acceleration

around Z axis, measured at the top floor

Clear from Figures 20 thru 22 one multi tuning-frequency passive/on-demand active Pendulum Tuned Mass Damper can be tuned to the first 3 modes of the structure, adding sizeable damping to all 3, simultaneously. A single proposed PTMD can effectively replace multiple passive tuned mass dampers with different tuning frequencies, in applications where more than one mode are in need of damping. Moreover, when operating in passive mode the proposed PTMD performs as robustly and as effectively as any optimally designed passive PTMD of the same size adding tuned damping to its target mode (normally the first mode) of the structure, with no need for external power.

CHAPTER IV

EFFECTIVENESS ENHANCEMENT OF TMD ACTIVE CONTROL

A multi-layer control strategy along with a novel active pendulum tuned mass damper (PTMD) configuration, for mitigating the vibration of tall structures, are presented and discussed in this chapter. The proposed control scheme a) increases the damping effectiveness of the APTMD and b) enables the APTMD to simultaneously tune itself to multiple frequencies, with corresponding optimal damping ratios, and thus add tuned damping to the corresponding modes of those frequencies. The enhanced damping effectiveness component of the proposed control scheme allowing for the use of a small

TMD and the multi-frequency tuning component of the proposed control strategy eliminating the need for using multiple TMDs tuned to different frequencies, result in the control of structural vibration using a relatively small mass.

The superior damping effectiveness and multi-frequency tuning capacity of the proposed active PTMD are numerically demonstrated by introducing it into the model of a multi-degree of freedom asymmetrical tall building. As will be shown in the subsequent sections of this Chapter, the APTMD increased the damping by 100% over a passive TMD of the same size, while tuned to the first three modes (along X, and Y axes and around Z axis) of the target structure.

4.1 The Impact of Mass Ratio on TMD Effectiveness

Tuned mass dampers have been used for adding tuned damping to various structures, successfully. Considering that the first vibrational mode of a structure plays a dominant role in its dynamic response, a TMD is normally tuned (and must remain tuned) to the first natural frequency of the structure. The damping (energy dissipation) effectiveness of a TMD depends on a) the size of its mass compared to the modal mass of its target mode, i.e., its mass ratio, and b) the accuracy of its tuning.

Due to the tight restriction on their weight and space requirements, the mass ratio of a TMD used in a large and massive structure such as a tall building, is very small. The small mass ratio necessitates the use of low internal damping in the TMD, itself. The highly underdamped nature of such TMD results in

o a tuned mass damper with a very narrow frequency range making the effectiveness

of the TMD highly sensitive to its tuning accuracy (a slight change in dynamics of

the structure will render such TMD detuned and hence less effective),

o a rather long delay in reaching steady state motion (and thus becoming fully

effective) when the disturbance starts perturbing the structure and also a long delay

time to stop when the disturbance has ceased,

o excessive excursion, and

o low damping effectiveness even when a TMD is well-tuned.

The above-listed issues could have been addressed if the mass ratio of the TMD

was large, e.g. around 5%, but such large mass ratio would make the TMD

considered for a high-rise application very massive/large and thus impractical.

The frequency response functions of an underdamped structure without and with two different TMDs are shown in Figure 23. One TMD has a small (0.125%) and the other has a large (0.5%) mass ratio. The optimal value of damping ratio in the larger TMD is more than twice as much (5%) of the smaller TMD (2%), i.e., the smaller TMD is in excess of 2 times more underdamped than the larger TMD.

Figure 23: Frequency response functions of an underdamped structure without and with

two TMDs with 1% and 5% mass ratios

Clear from Figure 23, increase in mass ratio and the corresponding damping ratio of a TMD increases the energy dissipation effectiveness as well as the bandwidth of that

TMD. The problem is that in many applications, there is neither enough space to

accommodate nor the target structure can take the weight of a TMD with large enough mass ratio.

4.2 Active Tuned Mass Damper (ATMD)

The effectiveness of a tuned mass damper can be increased by introducing a controllable element e.g., a linear actuator, into its make-up, resulting in an active tuned mass damper

(ATMD) Figure 24 (52), (53), (41), (54).

Figure 24: The schematic presents an ATMD with the active element U, appended to a

structure

The use of an active tuned mass damper makes it possible to overcome the drawbacks associated with a TMD without increasing its mass ratio. A properly controlled active TMD tuned to a single mode a) has a wider frequency range and thus be more robust to tuning inaccuracies, b) can self-tune itself, and c) possesses higher effectiveness in adding damping to its target mode. Moreover, an ATMD can be used to add damping to more than one mode of vibration when multiple modes are in need of being damped. This

eliminates the need for using multiple passive TMDs with different dynamic characteristics, each tuned to one of the target modes.

But considering that a properly controlled active TMD is a) by far more effective than an equally sized passive TMD and b) can be tuned (add damping) to more than one mode of the structure (In case of pendulum TMD), it could be an economically attractive solution in certain applications. This is true despite the perceived higher initial and running costs of ATMDs compared to their passive counterparts. With cost comparison based on effectiveness, an ATMD is no more expensive than a passive TMD of many times larger in size or multiple passive TMDs tuned to different frequencies. As to the running cost, with proper design and control strategy an ATMD can switch to a passive system under typical loading conditions consuming no energy and revert to active only when the structure reaches a certain vibration threshold which requires more effectiveness. Also in the unlikely event of losing power, the active TMD reverts to a passive TMD.

In this work, a novel active pendulum tuned mass damper is presented which a) with a smaller mass possesses the bandwidth and energy dissipation effectiveness of a more massive passive PTMD and b) has multiple, adjustable tuning frequencies and damping ratios in multiple directions. With its small size and multi-frequency tuning capacity, the proposed APTMD is as effective as a passive TMD many times more massive.

Moreover, it can add damping to more than one mode in more than one direction, replacing multiple more massive PTMDs. As stated earlier, such attributes lowers the cost, weight and space requirement associated with dampening multiple modes using multiple TMDs.

Following a short review of existing active tuned damping techniques, the proposed active pendulum tuned mass damper along with its corresponding control strategy is introduced.

In subsequent sections, the chapter continues with numerical demonstration of the enhanced damping effectiveness as well as multi-mode damping capacity of the proposed

PTMD by incorporating it in the numerical model of a high-rise building perturbed in multiple directions.

4.2.1 Control Strategies for Active Tuned Mass Dampers

To increase the effectiveness, the required force produced by the active element should be determined using an appropriate control strategy. Over the last two decades, a number of researchers have suggested different control strategies for a one degree of freedom tuned mass dampers.

Chang and Soong (1980) (33), and Isao (1992) (32) calculated the required active force using an optimization process and showed significant improvement in the damping effectiveness of the ATMD without increasing its excursion, compared to a passive TMD of equal size. Abdel-Rahman (1984) (55) used pole assignment method to evaluate the active force. In separate studies, Chang and Yang (1995) (34) and Seshasayee and Yang

(1996) (30) computed the active force by feeding back the acceleration, velocity, and displacement of the structure.

In addition to the use of classical feedback control (feeding back displacement, velocity, and acceleration of the structure), modern control technique including linear quadratic regulator (LQR) Alavinasab and Moharrami (2006) (37), linear quadratic

Gaussian (LQG) Ki-Pyo (2014) (35), H2 and Hinf control as well as nonlinear control techniques such as sliding-mode control Adhikari and Yamaguchi (1997) (40), fuzzy logic

Bijan (2004) (38), and even neural network control Ghaboussi and Joghataie (1995) (39) have been suggested to obtain the most ideal control force for mitigating the vibration of

the structure. Despite the effectiveness of all of these control schemes, most of these controllers are either model based and require a reasonably accurate model of the structure and TMD which is rarely available especially applications of tuned damping to very large structures, and/or are in need to feeding back all the states of structure and TMD, or are somewhat complex to implement. These observations and control classifications have been made by Nerves and Krishnan (1995) (29) based on the results obtained from the simulation study of applying various active control schemes to large structures.

As will be described in the subsequent sections, the authors have expanded the non-model- based classical feedback control proposed by Nishimura et. al (32), by extending it to multi directions and cascading it with a novel multi frequency-tuning strategy.

4.3 The Proposed Active Pendulum Tuned Mass Damper

Figure 16 depicts the proposed active PTMD with the mass suspended by steel wire ropes and actuated by 6 hydraulic cylinders. The hydraulic cylinders (actuators) along with the moving mass form a 6-legged (hexapod) closed-chain mechanism with 6 degrees of freedom. This mechanism, commonly known as Stewart platform, is capable of moving in any direction and orientation, generating controllable dynamic motions (50) (8) (51).

The hydraulic cylinders (legs) are the active elements of the APTMD responsible for realizing the spatial motion of the pendulum mass decided by the control algorithm.

Except for the use of hydraulic cylinders in place of passive viscous dampers, the proposed active PTMD resembles a passive PTMD commonly used in many modern high- rise buildings and towers. In its default state, the hydraulic cylinders are configured to act as passive viscous dampers turning the device to a passive PTMD. When the need arises

(depending on the extent and frequency content of the structural vibration) the hydraulic

cylinders switch to active actuators and turn the device into an active PTMD with enhanced damping effectiveness as well as multi degree-of-freedom tuning capacity (capability).

The hydraulic actuation system of the proposed passive/on-demand active PTMD consists of six double acting single ended cylinders. Each hydraulic cylinder is equipped with a temperature-compensated flow control valve and a 4-way electrohydraulic servovalve; the former is used when the device is in its passive state and the latter along with the hydraulic power supply are used when the device is in active state. Refer to section 3.3 in the previous chapter for more on the hydraulic circuit of the APTMD.

The PTMD installed on a structure and interacts with the structure in a feedback manner by providing a reactive force vector in response to the acceleration of the structure at the PTMD installation location. Note that the reactive force vector is the inertia force vector of the PTMD. Figure 25 illustrates the model of the proposed passive/on demand active PTMD attached to the targeted tall building in a MATLAB/SimMechanics platform which is an upgraded version of the previous model of the passive/on demand active PTMD shown in Figure 18. In this APTMD model the structure acceleration is fedback to the

APTMD to enhance its effectivness. As a result, the weight of the PTMD is decreasesd by

50% without degrading its performance.

Figure 25: The active PTMD

4.4 Active Control Strategy of the Proposed PTMD

The control scheme proposed for the multi-degree of freedom PTMD is made up of two parts of a) active effectiveness enhancement and b) active multi-frequency tuning.

In addition, a high-level supervisory logical oversees the vibration circumstances of the structure both in terms of severity as well as frequency and decides to bring none, one or both of these controllers. Since the hydraulic cylinders (legs) are to actively actuate the

PTMD, the control force vector which is formulated in the Cartesian coordinate system

(global) need to be transformed to the leg space (local) prior to realization.

4.4.1 Effectiveness Enhancement of Active Damping

Considering that the inertia force generated by TMD mass is what adds damping to the structure, increasing such inertia force by active means will result in increase in the effectiveness of the TMD. As suggested by Nishimura et. al. (41), the inertia force of a one degree of freedom ATMD appended to a structure, as shown in Figure 25, can be increased by having the active element actuate the system proportional to the acceleration of the structure. As it can be expected, introduction of such actuation also increase the excursion of the ATMD (the relative motion of the TMD mass with respect to the structure). To lower such excursion, Nishimura et. al. (41) augmented the actuation by feeding back the excursion velocity. Such feedback makes the actuator to act as an active viscous damper.

As stated earlier, in this work the control scheme proposed by Nishimura et. al.,

(41) for a one degree of freedom ATMD is a) extended to a multi-degree of freedom ATMD and b) is cascaded by yet another feedback scheme to tune the ATMD to multiple frequencies in multiple directions, simultaneously.

G 00X s X    UGY2 00Ys  00G  s  (4.1)

 where U2 is the vector of ATMD’s active force, X s , Ys , and  s are the absolute

acceleration of the structure along x and y and around z axes, and GX , GY , and G are the corresponding active gains.

As referred to at the beginning of this section, the control force vector of Equation (4.1) formulated in the Cartesian coordinate system need to be transformed to the leg space using the principle of virtual work. Assuming that the friction forces in the joints and the gravitational effects of the legs are negligible and considering that the main gravitational effect due to the weight of the TMD mass is taken up by the pendulum (not the legs of the mechanism), the virtual work contributed by all the active forces is

uTT p U P 0 22 (4.1.1) where  p and  P are the virtual displacements in the leg space and the Cartesian

 coordinate system installed on the structure. Moreover, u2 is the active force vector corresponding to , but defined in the leg space. Clear from Equation 4.1.1, the principle of virtual work indicates that the work done by the global force vector is the same as the work done by the local force vector defined in the leg space.

Using the Jacobian matrix of the closed-chain mechanism J, relating small motion of the

TMD mass to the corresponding small motion of the legs , i.e.,

p J P (4.1.2) 76

and substituting  p from Equation (4.1.2) in Equation (4.1.1) results in

TT u22 J U P 0 

Considering that the above equation holds for any virtual displacement, one can conclude that

uTT J U  0  u   JT U  2 2 2 2 (4.1.3) It should be noted that this active scheme can be turned on and off, depending on the severity of perturbation on the structure.

4.4.2 Active Tuning

To tune the PTMD to different frequencies in different directions and add the desired damping levels to the PTMD corresponding to those tuning frequencies the

PTMDs’ frequency/direction dependent stiffness and damping coefficient need to be realized actively by the actuators. This is achieved by active, simultaneous actuation of the actuators (hydraulic cylinders) subjecting the PTMD mass as well as the structure to the

 combined stiffness plus damping force vector of U2 shown in Equation (4.2)

 K 0 0 XC  0 0 X  XX        U2 KP CP 0 KYY 0 Y  0 C 0  Y      0 0 KC  0 0   (4.2) where Ki and Ci are the additional stiffness and damping coefficients needed in i=X, Y, and

T γ directions over what the pendulum itself is providing. And the vectors PXY  

T and PXY  are the position and velocity vectors of the center of mass of the

PTMD mass measured in the Cartesian coordinate system installed on the structure

measuring the relative motion of the PTMD mass with respect to the structure. Note that the formulation of Equation (4.2) is based on the assumption of the 3 target modes of the structure are nearly decoupled; if the natural modes were not nearly decoupled then the desired stiffness and damping coefficient matrices shown in Equation (4.2) would not be symmetric.

T Considering the impracticality of measuring XY  , and yet the convenience and ease of measuring the displacement of the actuators (legs of the Stewart platform), we

 transform, via the use of the principle of virtual work, the control force U2 of Equation

(4.2) from the Cartesian coordinate system installed on the structure to the control force defined in the local coordinate systems of the legs results in the leg space, control force

vector u2 shown by Equation (4.3)

" 푢2 = 푘 ∗ 푝 + 푐 ∗ 푝̇ (4.3) where p and p are the displacement and the velocity vectors of the legs and k and c are the stiffness and damping coefficient matrices of the legs defined by Equations 4.3.1 and

4.3.2

T 1 k J KJ (4.3.1) T 1 c J CJ (4.3.2) where J is the Jacobian matrix of the PTMD behaving as a closed-chain Stewart platform mechanism and superscript -T signifies the ‘inverse of transpose’.

As stated earlier the supervisory logical constantly assessing the vibration conditions of the structure, both in terms of severity as well as frequency, decides to bring none, one (either

  u2 or u2 ) or both of the control scheme discussed above. If the supervisory logical

decides to run both controllers, the control force vector defined in the local leg coordinate system will be made up of the summation of Equation 2 and 3, presented by Equation (4.4).

′ " −푇 ′ 푢2 = 푢2 + 푢2 = 퐽⏟ 푈 2 +⏟ 푘 ∗ 푝 + 푐 ∗ 푝̇ (4.4) ′ " 푢2 푢2

The block diagram presentation of the structure and the passive/on-demand active PTMD is shown in Figure 26.

Figure 26: Block diagram of the structure + APTMD

4.5 Illustrative Example

The enhanced effectiveness and multi-frequency tuning capacity of the proposed active PTMD (APTMD) are numerically demonstrated by introducing it into the model of a multi-degree of freedom asymmetrical high-rise building. The building has 41 stories above ground with each floor having three degrees of freedom, two translational in X and

Y directions and one rotational, denoted by γ, around the z axis. Due to the uniformity in geometry and mass distribution in each floor, vibration in those three directions is nearly decoupled from each other.

4.5.1 The Predicted Peak Accelerations

The highest predicted wind-induced accelerations at the top occupied floor of the targeted building, taken at structural level “38th Floor” (171 m above the ground), are found to be 25.5, 31.9, and 4.7 mill-g in the directions X, Y, and γ respectively, including an allowance for the uncertainties in the wind climate. The peak accelerations were determined as a function of the highest return period for the provided frequencies, and an overall damping ratio of 1.5% of critical damping ratio. The torsional component, which was included in the total acceleration predictions, was calculated at a representative distance of 9.6 m, based on the radius of gyration, from the reference z-axis (the perpendicular axis on the x-y plane).

4.6 Application of the Active PTMD

The proposed PTMD equipped with six hydraulic cylinders (actuators) is synthesized to add damping to the first three nearly-decoupled modes of the aforementioned tall building with the natural frequencies of 0.18, 0.29, 0.56 Hz, simultaneously. Figures 27, 28, and 29 show the PTMD excursion, the frequency response functions as well as the resonant time traces of the structure’s acceleration along the Y direction (1st mode), measured at the top floor. Harmonic perturbation with spatially varying amplitude along the height of the structure is used to perturb all the floors simultaneously, vibrating the building in Y direction.

The effectiveness of the active PTMD is increased by feeding back the acceleration of the main structure to the PTMD system in the three directions (X, Y, and γ). Usually active control increases the response motion of the auxiliary system, in our case, (from

0.015 to 0.025 m) which is slightly higher with active control. However, the acceleration feedback method increases the effectiveness of the PTMD, where it makes possible to keep the efficiency of the PTMD while lowering its mass by 50 % which is a major accomplishment and that can be observed clearly from the three Figures (27, 28, and 29) that show the excursion and effectiveness of the PTMD in Y-direction (mode one). The new control theory makes the current active PTMD twice as effective as the passive/on demand active PTMD of the same size which was developed (in the previous chapter).

Effectiveness

Excursion

Figure 27: Excursion, FRFs, and the resonant time traces of the structure’s acceleration along Y direction, measured at the top floor (Large mass and No control)

Effectiveness

Excursion

Figure 28: Excursion, FRFs, and the resonant time traces of the structure’s acceleration

along Y direction, measured at the top floor (Small mass and No control)

Effectiveness

Excursion

Figure 29: FRFs and the resonant time traces of the structure’s angular acceleration

around Z axis, measured at the top floor (Small mass with control)

The presented modern control method increases the effectiveness of the PTMD in the other two directions (X and γ) as well. In addition to the capability of replacing multiple passive

TMDs with different frequencies, in applications where more than one mode are in need of damping, the proposed PTMD, using the modern control method, becomes highly efficient and performs as robustly and as effectively as that of double in size optimally designed passive/on demand active PTMD adding tuned damping to its target modes (the first three modes) of the structure.

CHAPTER V

HYDRAULIC ACTUATION SYSTEM

In this chapter, the hydraulic system used in the Stewart platform to provide the desired control force for actuating the PTMD is presented. In order to design the appropriate hydraulic system for the proposed PTMD, several factors are needed to be considered including, but not limited to, enough actuation force and range of motion. The hydraulic cylinders which serve as the legs of the Stewart platform behave as synthetic spring-damper systems tuning the PTMD to different frequencies in different directions (x, y, and γ).

5.1 Stewart Platform

The Stewart platform (SP) is the most widely used parallel manipulator for motion control. It was originally designed in 1965 by Stewart and Gough as a flight simulator and is still commonly used for the same purpose. Some industries using the Stewart platform design include aerospace, automotive, transportation and machine tool technology. The SP has a remarkable range of motion and can be positioned and oriented accurately and easily.

Due to their high rigidity Stewart platforms can provide significant positioning.

SP is a parallel mechanism that consists of a rigid top mobile plate connected to a fixed base plate through six independent moving legs. Usually, the legs are connected to both the mobile platform and the fixed base by universal joints as shown in Figure 30. Each 86

leg has two parts, an upper and a lower, connected by a cylindrical joint allowing the legs to be length adjustable. The top plate can be positioned and oriented based on the length to which the legs are adjusted. The SP can place the platform in six DOFs, three rotational and three translational, and can support a large load because of its design where the six legs are evenly spaced and distributed around the center and share the load on the top plate.

Linear actuation of the Stewart platform is accomplished by varying the lengths of the legs; hydraulic cylinders can be used for this purpose (9).

Figure 30: Specifications of Stewart Platform (9).

5.2 Fluid Power

The technology that utilizes the generation, control, and transmission of power, using confined fluids is called fluid power. Several applications in the industry use fluid power systems. They include, but not limited to, fluid power steering and brakes in automobiles. Fluid power is called hydraulics when the fluid is liquid and is called pneumatic when the fluid is gas. Thus, fluid power is the common name used for both hydraulics and pneumatics.

Fluid power systems are used mainly to perform work which is accomplished by applying pressurized fluid directly on an operating cylinder or motor. A linear motion is resulted from a force applied by cylinder whereas rotary motion is resulted from a torque applied by a motor. Hence, the cylinder and the motor also called actuators provide the desired work done by a fluid power system. Obviously, to make sure that the work delivered by the fluid power system is performed smoothly, accurately, efficiently, and safely, flow control components such as servo control valve are needed to fulfill the task

(56).

5.3 Hydraulic Systems

The science of transferring force and/ or motion through the medium of pressurized liquid is called Hydraulics. Hydraulic systems have several advantages as power sources for operating various industrial applications. They combine operational advantages such as high power/ mass ratio, fast response, and high stiffness. Technical advantages such as ease of installation, simplification of inspection, and minimum maintenance requirements are other reasons that make the hydraulic systems desirable power sources (57).

5.3.1 Electrohydraulic Servo Valve

A servovalve is a directional control valve that can control both the flow rate and the direction of fluid flow. Servovalves have very accurate control of position, velocity, and acceleration of an actuator because of the feedback – sensing devices coupled with them.

Larger electrohydraulic servovalves use an electrical torque motor (servo), a double nozzle pilot stage, and a sliding spool second stage. Figure 31 shows external and cross section views of an electrohydraulic servovalve (56). By combining the muscle of the hydraulic power and the accuracy of electrical control, electrohydraulic control valves can control hydraulic systems efficiently (56).

Figure 31: Moog series G77XK electrohydraulic Servovalve

5.4 Sizing of the Hydraulic Actuation System

There is an optimum size for the piping in each line of a hydraulic system. It must be large enough to carry the oil flow with an acceptable pressure loss, but should be no

larger than the necessary size. Undersize piping creates excessive loss and will contribute to overheating of the oil. Oversize piping will reduce losses but will be more expensive to purchase and install the hydraulic components for. A well-designed system may use several pipe sizes in various parts of the circuit. By knowing the circumstances and the loads that the hydraulic system will experience, the right size of it can be determined.

5.4.1 Maximum Force and Stroke of the Hydraulic Actuator

From the given data of the targeted structure Table 2, the highest predicted peak acceleration of the structure is 31.9 milli-g which is then multiplied by safety factor of 1.5 and used as the peak structure acceleration.

The predicted peak acceleration:

푎푐푐 = 31.9 ∗ 9.81 ∗ 1.5/1000 = 0.46935 푚/푠푒푐2 acc = 0.46935 ∗ 1000 = 469.35 mm/sec2 (5.1)

Where: acc: The acceleration of the structure experienced at the top floor

From Figure 32, the peak magnitude of the acceleration frequency response of the structure in y direction is used to calculate the disturbance force amplitude. Note that a) frequency response function is the ratio between the acceleration of the building at floor 41 as output and the disturbance force on the entire building as input and b) the first mode which has the highest motion is mainly in the Y direction.

acc = 32.3 dist_amp , in dB scale (5.2)

acc = 43.301 dist_amp , in linear scale (5.3)

The disturbance force: 90

acc 469.35 dist_amp = = = 10.839 KN mag 43.301 (5.4)

Where dist_amp: the amplitude of the disturbance force

Figure 32: Structure frequency response function in Y direction

As the Stewart platform contains six hydraulic actuators (legs) and each actuator has its own controller, six time dependent actuator forces and strokes can be obtained by running the model of the PTMD. Using the disturbance force of Eq. (5.4) as perturbation input to the model of the structure treated with the APTMD and measuring the highest force among the six force traces from mode 2 (in x direction), the maximum control force

(to be used for sizing the actuators) can be obtained. Also, in a similar way, the highest stroke among the six stroke traces from mode 1 (y direction) will be chosen. Six traces of control force and stroke can be seen in Figure 33 and Figure 34.

Figure 33: Control forces for the six hydraulic actuators (Mode 2)

1) Maximum control force:

AF = 7.614e04 N

AF = 76.140 KN

Where

AF: Actuation force

2) The stroke of the hydraulic cylinder:

The stroke multiplied by safety factor of 2 is: disp = 0.139 ∗ 2 = 0.278 m (5.5)

Figure 34: Strokes of the six hydraulic actuators

Table 2: The requirements for the hydraulic system

Item Value Unit Desired control force 76.140 KN Stroke of the hydraulic cylinder 0.278 m

In the following sections, the calculations for the sizing of the hydraulic system (the double acting cylinder and the Electrohydraulic Servovalve) are presented. The calculations are based on the design criteria listed in Table 2.

5.4.2 The Selection of the Hydraulic Cylinder

Cylinders are used in the majority of fluid power applications to convert fluid energy into straight line motion. For this reason, they are often called linear actuators. A double acting cylinder with a single end is chosen for this system. The rod size is chosen to have a 2 in diameter based on the CAD configurator provided by EATON company. The size of the actuator is determined by calculating the area of the cylinder that can provide the required control force and having the proper length that can handle the displacement of the piston. Medium duty cylinders are chosen for this hydraulic system. The system pressure is chosen to be rather low at 100 bar. The cylinders are sized to produce a stall force 30% greater than the maximum desired force, as shown in Equations (5.6) and (5.7)

1.3∗AF 1.3∗7.614e04 (N) A = = = 0.0099m2 (5.6) cylinder Ps 100e05 (N/m2)

A 0.0099 D = 2 ∗ √ cylinder = 2 ∗ √ = 0.112 m (5.7) cylinder π π where

Acylinder: actuator area

Dcylinder: actuator diameter

Ps: supply pressure

Referring to the NFPA (National Fluid Power Association) Standards, which allow for the easy interchange of cylinders from one manufacturer to another, the standard size hydraulic cylinders with the closest area to the result of the above calculations are selected.

The maximum speed of the cylinders is calculated considering the largest product of the amplitude of motion (stroke) and the corresponding frequency, which in our case happens the frequency of the 3rd mode of the structure: 94

Vpiston = stroke ∗ frequpper = 0.278 ∗ 0.559 = 0.155 m/sec (5.8)

where

Vpiston: the maximum speed of the cylinder

Stroke : maximum stroke of the cylinder

Frequpper: upper bound frequency of the targeted modes of the structure

Using the maximum required load velocity and the actuator area from the above calculations, the valve loaded flow and the load pressure drop can be calculated:

3 QL = Acylinder ∗ Vpiston = 0.0099 ∗ 0.155 = 0.0015 m /sec (5.9)

AF 7.614e04 Loaded pressure drop (PL) = = = 7.691e + 06 (5.10) Acylinder 0.0099 where

PL: the loaded pressure drop (Pa)

The no load flow:

Ps 100e5 3 QNL = QL ∗ √ = 0.0015 ∗ √ = 0.0031 m / sec (5.11) Ps−PL 100e5−7.691e6

Valve rated flow will be determined at 1000 psi valve drop, which is the typical rating for servovalves, with a 10 % margin increase

3 푄푟 = 1.1 ∗ (푄푁퐿) = 1.1 ∗ 0.0031 = 0.0034 푚 / sec

푄푟 = 0.0034 ∗ 15850.37 = 54.415 푈푆 𝑔푝푚 (5.12) where

Qr: servovalve flow rate 95

5.4.3 The Selection of the Electro-Hydraulic Servovalve

Proper valve selection in a hydraulic system is of high importance because the hydraulic valve is the core of any hydraulic system. For closed loop control of hydraulic systems utilizing electrical feedback, the optimum performance will be achieved if the operation frequency of the servovalve that corresponds to 90° phase lag 3 times or more larger than the load resonant frequency. This operation frequency defines the effective bandwidth fv of the servovalve. A critical (zero lapped) valve (no mechanical deadband), i.e., a spool valve in which the width of the lands on the spool are the same as the width of the flow port in the body of the valve is selected for this application. fvalve ≥ frequpper ∗ 3 = 0.559 ∗ 3 = 1.68 Hz (5.13)

Based on the the calculated flow rate for one electro-hydraulic servovalve in Eq.

5.12 which is 55.8 gpm at the highest demand (the oil consumption of one cylinder with the highest demand) and refering to the valve selection chart provided by MOOG, See appendix A, any Servovalve that has equal or higher range of flow capacity will be an acceptable chioce as long as the sevovalve is not highly oversized. Although the calculation used to come up with the flow rate was based on a single servovalve, the power supply needs to run 6 servovalves at the same time. Therefore, the power supply source needs to supply 55.8 gpm to each servo valve (one for each leg) to enable each hydraulic actuator to provide the desired control force.

Also, sizing of the power source depends on how conservative the design engineer decides to make the hydraulic system. At the highest conservative design of the hydraulic

system, the power source needs to provide 6*55.8 gpm, assuming all of the cylinders with highest demand at the same time (a lot of oil consumption). However, chances are that not all cylinders will consume the same amount of oil at the same time because the mass moves laterally either in one direction at a time (x or y) or in a combined direction (xy). In this case, some actuators (cylinders) will be in an extraction situation and others will be in a retraction situation.

Based on the chart provided by MOOG, shown in Appendix A, it can be seen that the standard model of Electrohydraulic Servovalve D661, Figure 35, will satisfy the system requirments. As presented in MOOG’s catalog of the selected Servovalve that main spool stroke is 3 mm and the spool drive area is 2.0 cm2. By assuming that the spool is circular, the diameter of the spool will be:

A d = 2 ∗ √ spool = 1.6 cm (5.14) spool π

5.4.4 Servovalve D661 Highresponse Series

The flow control servovalves D661 High response series are throttle valves for 2,

3, and 4 way two stage servovalves with servo-jet high response pilot stage; see Figure 35.

These valves are suitable electrohydraulic position, velocity, pressure or force control systems including those with high dynamic response requirements. The spool of the main stage is driven by a jet pipe pilot stage in an electrically closed loop. The torque motor of the valve requires 24 VDC power supply. For more details about D661 highresponse series, two stage servovalves, see appendix (B) (58).

The significant input parameters for the hydraulic actuation system are listed in Table 3.

Having the parameters of the hydraulic system and using Simulink/SimHydraulics toolbox from MATLAB, the hydraulic system can be simulated. 97

a) Cross section view b) External view

Figure 35: Servovalve D661 Highresponse Series with servo-jet pilot stage

Table 3: Key input parameters for the hydraulic actuation system

Item Value Unit

Supply Pressure 1e07 Pa

Cylinder area 0.0099 m2

Cylinder bore 11.2 cm

Stroke 0.28 m

Valve rated flow 55.83 gpm

Valve frequency >5 Hz

Spool drive area 2.0 cm2

Spool diameter 1.6 cm

Spool stroke 3 mm

5.5 The Hydraulic Circuit of the Actuation System

The hydraulic actuation system of the proposed passive/on-demand active PTMD consists of six double acting single ended cylinders. Each hydraulic cylinder is equipped with a temperature-compensated flow control valve and a 4-way electrohydraulic servovalve; the former is used when the device is in its passive state and the latter along with the hydraulic power supply are used when the device is in active state. The control scheme of the hydraulic system shown in Figure 36 is available in each leg of the 6 legs of the Stewart platform, but there is only one power source and reservoir (Pump and tank) for the entire hydraulic actuation system of the proposed passive/on-demand active PTMD.

Two 3-way solenoid valves installed on both ports of each cylinder are used to switch between the passive and active states of the PTMD; see Figure 36. In one of their positions, the 3-way valves route the hydraulic fluid through a flow control valve allowing the fluid to communicate between the two sides of the piston in each cylinder, making the hydraulic cylinders act like viscous damping devices. In this position, the entire power hydraulic system is bypassed and not used. In their other position, the 3-way valves connect the hydraulic power supply through the servovalves to the cylinder making the hydraulic cylinders act as actuators (active elements).

As shown in Figure 36, the hydraulic actuation system is force-controlled using a distributed (local), closed loop control scheme. The reference input to this distributed controller is the desired force vector of all the legs evaluated in the centralized controller discussed in the body of this paper. The force vector measured by the force sensors installed on each cylinder is used as the feedback signal. With the error signal vector as the input, he local PID controllers generate the control signal vector driving the Servovalves and

actuating the hydraulic cylinders (legs of the PTMD). In addition to providing effective damping and simultaneous tuning to multiple modes, the hydraulic system can be used to: a) move the mass for maintenance purposes and b) perturb the mass during the fine-tuning process following the installation and c) block the flow of hydraulic fluid between the two sides of the piston and thus act as a hydraulic snubber slowing down the mass and bringing it to a graceful stop, in the unlikely event of the mass exceeding its allowable motion. It should be mentioned that this feature is not a replacement for a separate stand-alone snubber system that should be part of such design, but just an additional enhancement.

Figure 36: The hydraulic circuit for one cylinder (leg)

100

CHAPTER VI

CONCLUSION AND FUTURE WORK

6.1 Conclusion

As tall structures such as high-rise buildings become taller, lighter and more slender, they become more susceptible to excessive resonant vibration during wind events

(14), (6) , (47). The same is true for air traffic control towers, steel stacks in power plants, and other tall structures subject to wind induced vibration. Tuned mass dampers (TMDs) are commonly used in adding damping to such structures and quiet their vibration. The use of tuned mass dampers (TMDs) in tall buildings and towers for mitigating wind induced vibration have resulted in significant improvements in serviceability of such structures.

In this dissertation, a novel passive pendulum tuned mass damper (PTMD) which on-demand can switch to an active PTMD is proposed. In its default state, the proposed device is passively tuned to the first mode of the structure and acts as a traditional passive

PTMD. In its active mode, while staying tuned to the first mode passively, the device simultaneously tunes itself to 3 frequencies adding tuned damping to the first 3 modes (with three distinctly different natural frequencies) of the structure with corresponding optimal damping ratios, attenuating the resonant vibration of the high-rise building in three directions, associated with the first three modes of the structure.

101

Moreover, the acceleration feedback method applied to the multi-frequency tuning active PTMD was discussed in this dissertation. The APTMD is tuned to the first three modes (one in each direction x, y, and γ) of the targeted structure and its performance is found to be optimum. The acceleration feedback method increases the effectiveness of the

PTMD, where it makes it possible to keep the efficiency of the PTMD while lowering its mass by 50 % (from 450 ton to 225 ton) without degrading its control performance and that is a major accomplishment. The new control theory makes the current active PTMD twice as efficient as the passive/on demand active PTMD of the same size which was previously developed in the chapter 3 from this dissertation.

With its small size and multi-frequency tuning capacity, the proposed APTMD is as effective as a passive PTMD many times more massive and can add damping to more than one mode in more than one direction, replacing multiple more massive PTMDs. The enhanced effectiveness and multi-frequency tuning capacity as well as validity, and feasibility of the proposed active PTMD (APTMD) are numerically demonstrated by introducing it into the model of a multi-degree of freedom MDOF asymmetrical high-rise building perturbed by wind in multiple directions. The building has 41 stories above ground with each floor having three degrees of freedom, two translational in X and Y directions and one rotational, denoted by γ, around the z axis. The numerical model of the building, in which the first 15 modes of vibration are included, is developed.

102

The significance of using the active PTMD is summarized below:

 Provide higher damping effectiveness than an equivalent passive PTMD of the

equal size

 Provide as much damping, using a smaller mass, as an optimally designed passive

TMD with substantially larger mass. This is of interest when the vibrating structure,

e.g., the high-rise, cannot support the weight of a massive passive PTMD.

 Occupy less space than passive type.

In addition, several reasons make the proposed active PTMD (APTMD) an

advancement beyond the previously developed active TMDs including:

 Eliminate the need for using multiple traditional translational TMDs tuned to

multiple frequencies.

 Occupy less space than passive and regular active TMDs by eliminating the need

for multiple TMDs

6.2 Future Work

1. Interface the proposed APTMD to another building that has strong coupling

between modes

2. Try using optimal control for hydraulic system to achieve better

performance

3. Introduce self-tuning to the controller

103

REFERENCES

1. Aly AM. Proposed robust tuned mass damper for response mitigation in buildings

exposed to multidirectional wind. The structural design of tall and special Buildings.

2014 Jun 25; 23(9):664-91.

2. Connor JJ. Introduction to structural motion control: Prentice Hall; 2003.

3. Setareh M, Ritchey JK, Baxter AJ, Murray TM. Pendulum tuned mass dampers for

floor vibration control. Journal of performance of constructed facilities. 2006

Feb;20(1):64-73.

4. Ghorbani‐Tanha AK, Noorzad A, Rahimian M. Mitigation of wind‐induced motion of

Milad Tower by tuned mass damper. The structural design of tall and special

buildings. 2009 Jun 1;18(4):371-85.

5. Spencer BF, Sain MK. Controlling buildings: a new frontier in feedback. IEEE

Control Systems Magazine. 1997 Dec 1;17(6):19-35.

6. Lourenco R. Design, construction and testing of an adaptive pendulum tuned mass

damper Waterloo: The University of Waterloo; 2011.

7. Lourenco R, Roffel AJ, Narasimhan S. Adaptive pendulum mass damper for control

of structural vibration. InProceedings of the CANSMART 2009 Workshop 2009.

104

8. Luh CM, Adkins FA, Haug EJ, Qiu CC. Working capability analysis of Stewart

platforms. Journal of Mechanical Design. 1996 Jun 1;118(2):220-7.

9. Smith N, Wendlandt J. Creating a Stewart Platform Model Using SimMechanics. The

Matheworks—MATLAB Digest. 2002 Sep;10(5).

10. Nerves AC, Krishnan R. Active control strategies for tall civil structures. InIndustrial

Electronics, Control, and Instrumentation, 1995., Proceedings of the 1995 IEEE

IECON 21st International Conference on 1995 Nov 6 (Vol. 2, pp. 962-967). IEEE.

11. Ladipo I, Muthalif AG, Rozaime A, Asyraf A. Design and development of an active

mass damper for broadband vibration control. InMechatronics (ICOM), 2011 4th

International Conference On 2011 May 17 (pp. 1-4). IEEE.

12. Gunel, Mehmet Halis and Husevin Emre Ilgin. Tall Buildings: Structural Systems and

Aerodynamic Form: Routledge; 2014.

13. Mendis P, Ngo TD, Haritos N, Hira A, Samali B, Cheung J. Wind loading on tall

buildings. Electronic Journal of Structural Engineering. 2007.

14. Kareem A, Kijewski T, Tamura Y. Mitigation of motions of tall buildings with

specific examples of recent applications. Wind and structures. 1999;2(3):201-51.

15. Moutinho C. An alternative methodology for designing tuned mass dampers to reduce

seismic vibrations in building structures. Earthquake Engineering \& Structural

Dynamics. 2012; 41(14): p. 2059--2073.

105

16. Shen KL, Soong TT. Modeling of viscoelastic dampers for structural applications.

Journal of Engineering Mechanics. 1995 Jun;121(6):694-701.

17. Alhujaili FA. Semi-Active Control of Air-Suspended Tuned Mass Dampers. Dayton:

University of Dayton; 2012.

18. Murray TM, Allen DE, Ungar EE. Floor vibrations due to human activity.

19. Harris CM, Piersol AG. Harris' shock and vibration handbook. New York: McGraw-

Hill; 2002.

20. Symans MD, Constantinou MC. Semi-active control systems for seismic protection of

structures: a state-of-the-art review. Engineering structures. 1999 Jun 30;21(6):469-

87.

21. Housner G, Bergman LA, Caughey TK, Chassiakos AG, Claus RO, Masri SF, Skelton

RE, Soong TT, Spencer BF, Yao JT. Structural control: past, present, and future.

Journal of engineering mechanics. 1997 Sep;123(9):897-971.

22. Soong TT, Costantinou MC, editors. Passive and active structural vibration control in

civil engineering. Springer; 2014 May 4.

23. Dyke SJ, Spencer Jr BF, Quast P, Sain MK. Role of control-structure interaction in

protective system design. Journal of Engineering Mechanics. 1995 Feb;121(2):322-

38.

106

24. Spencer Jr BF, Nagarajaiah S. State of the art of structural control. Journal of structural

engineering. 2003 Jul;129(7):845-56.

25. Soong TT, Reinhorn AM, Wang YP, Lin RC. Full-scale implementation of active

control. I: Design and simulation. Journal of Structural Engineering. 1991

Nov;117(11):3516-36.

26. Kwok KC, Samali B. Performance of tuned mass dampers under wind loads.

Engineering Structures. 1995 Nov 1;17(9):655-67.

27. Kareem A, Kline S. Performance of multiple mass dampers under random loading.

Journal of structural engineering. 1995 Feb;121(2):348-61.

28. Ladipo I, Muthalif AG, Rozaime A, Asyraf A. Design and development of an active

mass damper for broadband vibration control. InMechatronics (ICOM), 2011 4th

International Conference On 2011 May 17 (pp. 1-4). IEEE.

29. Nerves AC, Krishnan R. Active control strategies for tall civil structures. InIndustrial

Electronics, Control, and Instrumentation, 1995., Proceedings of the 1995 IEEE

IECON 21st International Conference on 1995 Nov 6 (Vol. 2, pp. 962-967). IEEE.

30. Ankireddi S, Y. Yang HT. Simple ATMD control methodology for tall buildings

subject to wind loads. Journal of Structural Engineering. 1996 Jan 19;122(1):83-91.

31. Cao H, Li QS. New control strategies for active tuned mass damper systems.

Computers & structures. 2004 Oct 31;82(27):2341-50.

107

32. Nishimura I, Kobori T, Sakamoto M, Koshika N, Sasaki K, Ohrui S. Active tuned

mass damper. Smart Materials and Structures. 1992 Dec;1(4):306.

33. Chang JC, Soong TT. Structural control using active tuned mass dampers. Journal of

the Engineering Mechanics Division. 1980 Dec;106(6):1091-8.

34. Chang CC, Yang HT. Control of buildings using active tuned mass dampers. Journal

of Engineering Mechanics. 1995 Mar;121(3):355-66.

35. You KP, You JY, Kim YM. LQG control of along-wind response of a tall building

with an ATMD. Mathematical Problems in Engineering. 2014 Sep 14;2014.

36. Amini F, Hazaveh NK, Rad AA. Wavelet PSO‐Based LQR Algorithm for Optimal

Structural Control Using Active Tuned Mass Dampers. Computer‐Aided Civil and

Infrastructure Engineering. 2013 Aug 1;28(7):542-57.

37. Alavinasab A, Moharrami H, Khajepour A. Active Control of Structures Using

Energy‐Based LQR Method. Computer‐Aided Civil and Infrastructure Engineering.

2006 Nov 1;21(8):605-11.

38. Samali B, Al-Dawod M, Kwok KC, Naghdy F. Active control of cross wind response

of 76-story tall building using a fuzzy controller. Journal of engineering mechanics.

2004 Apr;130(4):492-8.

39. Ghaboussi J, Joghataie A. Active control of structures using neural networks. Journal

of Engineering Mechanics. 1995 Apr;121(4):555-67.

108

40. Adhikari R, Yamaguchi H. Sliding mode control of buildings with ATMD.

Earthquake engineering & structural dynamics. 1997 Apr 1;26(4):409-22.

41. Nishimura I. Vibration control of building structures by active tuned mass

damper (Doctoral dissertation).

42. Roffel AJ. Condition assessment of in-service pendulum tuned mass dampers

Waterloo: Diss. University of Waterloo; 2012.

43. Meirovitch L. Methods of analytical dynamics: Courier Corporation; 2010.

44. Roffel AJ, Lourenco R, Narasimhan S, Yarusevych S. Adaptive compensation for

detuning in pendulum tuned mass dampers. Journal of Structural Engineering. 2010

Aug 2;137(2):242-51.

45. Chung LL, Wu LY, Lien KH, Chen HH, Huang HH. Optimal design of friction

pendulum tuned mass damper with varying friction coefficient. Structural control and

health monitoring. 2013 Apr 1;20(4):544-59.

46. Roffel AJ, Lourenco R, Narasimhan S, Yarusevych S. Adaptive compensation for

detuning in pendulum tuned mass dampers. Journal of Structural Engineering. 2010

Aug 2;137(2):242-51.

47. Housner G, Bergman LA, Caughey TK, Chassiakos AG, Claus RO, Masri SF, Skelton

RE, Soong TT, Spencer BF, Yao JT. Structural control: past, present, and future.

Journal of engineering mechanics. 1997 Sep;123(9):897-971.

109

48. Roffel AJ, Narasimhan S, Haskett T. Performance of pendulum tuned mass dampers

in reducing the responses of flexible structures. Journal of Structural Engineering.

2012 Dec 29;139(12):04013019.

49. Aly AM. Proposed robust tuned mass damper for response mitigation in buildings

exposed to multidirectional wind. The Structural Design of Tall and Special Buildings.

2014; 23(9): p. 664--691.

50. Tosatti LM, Bianchi G, Fassi I, Boer CR, Jovane F. An integrated methodology for

the design of parallel kinematic machines (PKM). CIRP Annals-Manufacturing

Technology. 1998 Jan 1;47(1):341-5.

51. Wen F, Liang C. Displacement analysis of the 6-6 Stewart platform mechanisms.

Mechanism and Machine Theory. 1994 May 1;29(4):547-57.

52. Kwok KC, Samali B. Performance of tuned mass dampers under wind loads.

Engineering Structures. 1995 Nov 1;17(9):655-67.

53. Nishimura I, Kobori T, Sakamoto M, Koshika N, Sasaki K, Ohrui S. Active tuned

mass damper. Smart Materials and Structures. 1992 Dec;1(4):306.

54. Kumar A, Poonama BS, Sehgalc VK. Active vibration control of structures against

earthquakes using modern control theory. Asian Journal of Civil Engineering

(Building and Housing). 2007 Jan 1;8(3):283-99.

55. Abdel-Rohman M. Optimal design of active TMD for buildings control. Building and

Environment. 1984; 19(3): p. 191-195.

110

56. Esposito A. Fluid power with applications: Prentice-Hall International; 2000.

57. Zhang Q. Basics of hydraulic systems: CRC Press; 2008.

58. Moog Jr William C, inventor; Patents G, assignee. Electrohydraulic servo valve. US

patent US Patent 2,767,689. 1956 oct 23.

59. Gu M, Peng F. An experimental study of active control of wind-induced vibration of

super-tall buildings. Journal of Wind Engineering and Industrial Aerodynamics. 2002

Dec 31;90(12):1919-31.

60. Zasso Aea. Wind induced dynamics of a prismatic slender building with 1: 3

rectangular section. In 2008.Proceedings of the BBAA VI International Colloquium

on Bluff Bodies Aerodynamics & Applications; 2008.

61. Baker WF, Korista DS, Novak LC. Burj Dubai: Engineering the world's tallest

building. The structural design of tall and special buildings. 2007 Dec 1;16(4):361-75.

62. Ho PH. Economics planning of super tall buildings in Asia Pacific cities. Strategic

Integration of Surveying Services. 2007; 17.

63. Armstrong PJ. Architecture of tall buildings. McGraw-Hill Companies; 1995.

64. Reaveley LD, Shapiro D, Moehle J, Atkinson T, Rojahn C, Holmes W. Guidelines

and Commentary for the Seismic Rehabilitation of Buildings (ATC-33). InBuilding

an International Community of Structural Engineers (pp. 1123-1130). ASCE.

111

65. Ankireddi S, Y. Yang HT. Simple ATMD control methodology for tall buildings

subject to wind loads. Journal of Structural Engineering. 1996 Jan 19;122(1):83-91.

66. Ladipo I. Design and development of an Active Mass Damper for broadband vibration

control. 2011.

67. Roffel AJ, Narasimhan S, Haskett T. Condition assessment of in-service pendulum

tuned mass dampers Waterloo: Diss. University of Waterloo; 2012.

68. Sacks MP, Swallow JC. Tuned mass dampers for towers and buildings. Structural

Engineering in Natural Hazards Mitigation. 1993;: p. 640--645.

69. Gerges RR, Vickery BJ. Optimum design of pendulum-type tuned mass dampers. The

Structural Design of Tall and Special Buildings. 2005; 14(4): p. 353--368.

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APPENDIX A

Electrohydraulic Servovalve Selection Guide

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APPENDIX B

Inverse Kinematics of the Stewart Platform

The inverse kinematics for the Stewart Platform is relatively straightforward and can be stated as: given the position and orientation (poise), obtain the leg lengths (51) (8).

th Referring to Figure 37, the i leg length li is given by:

m T m li( bdAq i  ( i )) ( bdAq i  ( i ))  gAd i ( , ) (B1)

ri where

m  rii d Aq is the vector connecting the origin of the world coordinate system with the

joints on the platform

 bi represents the vector from the origin of the base (world) coordinate frame B to the

joint that connects the i th leg with the base.

m  qi is the vector that connects the origin of the coordinate system of the moving

platform M with the leg joint on the platform in the platform coordinate system.

 d[ X , Y , Z ], is the position vector of the platform coordinate system’s (M) origin in

the base (global) coordinate system, B.

 li is the length of the leg. 114

Figure 37: Stewart Platform

   sinsincossincoscossinsinsincoscoscos     A    sincoscossinsincoscossinsinsincossin   (B2)  sin sincos coscos   is the rotation matrix relating the platform’s coordinate system, M, to the base co-ordinate system, B. Here A is constructed using Roll-Pitch-Yaw (RPY) angle rotations, where

R(roll) = γ around the Z axis.

P(pitch) =  around the Y axis.

Y(yaw) = Ψ around the X axis.

Equation B1 represents the inverse kinematic solution. For some R, and d, the i th leg length

li  can be easily calculated.

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If Equation B1 is expanded using a Taylor series expansion, and considering the

first order term only, the change in leg length,  li , is obtained as a row vector Ji multiplied by the incremental displacement column vector

T PXYZ         as given in Equation (B3)

l J P ii (B3) where

gggggg J  iiiiii,,,,, i       x y z (4) is the ith row of the Jacobian matrix. Assembling the equations for all the legs of the mechanism,

p = J P

T where p   l1,,,,,  l 2  l 3  l 4  l 5  l 6  is the change in the length of the legs. In the limit, this equation relates the speed of the joints of the mechanism to the speed at the end effector.

For the mechanism of Figure 37, the Jacobian matrix can be written as:

T ee12.... e6 J   q e q e .... qe 1 1 2 2 66

th  rb ii where ei is a unit vector in the direction of the i leg, that is .  rb ii

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