Table Of ContentVibration Mitigation
Systems in Structural
Engineering
Vibration Mitigation
Systems in Structural
Engineering
Okyay Altay
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To my family
Contents
Preface xiii
Acknowledgments xv
List of Figures xvii
List of Tables xxxiii
List of Abbreviations xxxv
I Fundamentals 1
1 Theory of Structural Vibration 3
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Vibrations without Damping . . . . . . . . . . . . . . . . . . 3
1.2.1 Free Vibrations without Damping . . . . . . . . . . . 4
1.2.2 Forced Vibrations without Damping . . . . . . . . . . 8
1.3 Vibrations with Damping . . . . . . . . . . . . . . . . . . . . 12
1.3.1 Free Vibrations with Damping . . . . . . . . . . . . . 12
1.3.2 Forced Vibrations with Damping . . . . . . . . . . . . 16
1.4 Frequency-Domain Methods . . . . . . . . . . . . . . . . . . 18
1.4.1 Transfer Function . . . . . . . . . . . . . . . . . . . . 19
1.4.2 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4.3 Deformation Response Factor . . . . . . . . . . . . . . 24
1.4.4 Fourier Transformation . . . . . . . . . . . . . . . . . 27
1.5 Time-Domain Methods . . . . . . . . . . . . . . . . . . . . . 30
1.5.1 Interpolation of Excitation Method . . . . . . . . . . . 31
1.5.2 Newmark’s Method . . . . . . . . . . . . . . . . . . . 33
1.5.3 Central Difference Method. . . . . . . . . . . . . . . . 39
1.5.4 Stability and Computational Error . . . . . . . . . . . 44
1.6 Multi-Degree-of-Freedom Systems . . . . . . . . . . . . . . . 47
1.6.1 Natural Frequencies and Modes . . . . . . . . . . . . . 49
1.6.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . 52
1.6.3 Reduction of Degrees-of-Freedom . . . . . . . . . . . . 52
1.6.3.1 Static Condensation . . . . . . . . . . . . . . 52
1.6.3.2 Kinematic Constraints. . . . . . . . . . . . . 53
vii
viii
1.6.3.3 Rayleigh-Ritz Method . . . . . . . . . . . . . 53
1.7 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 56
1.7.1 Modal Analysis without Damping . . . . . . . . . . . 57
1.7.2 Modal Analysis with Damping . . . . . . . . . . . . . 60
1.8 Damping Models . . . . . . . . . . . . . . . . . . . . . . . . . 62
1.8.1 Viscoelastic Behavior . . . . . . . . . . . . . . . . . . 63
1.8.2 Hysteretic Damping . . . . . . . . . . . . . . . . . . . 66
1.8.3 Coulomb Damping . . . . . . . . . . . . . . . . . . . . 68
1.8.4 Construction of the Damping Matrix . . . . . . . . . . 69
1.9 Nonlinear Vibrations . . . . . . . . . . . . . . . . . . . . . . 72
1.9.1 Newmark’s Method . . . . . . . . . . . . . . . . . . . 72
1.9.2 Central Difference Method. . . . . . . . . . . . . . . . 78
2 Structural Control Systems 81
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.3 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.3.1 Control Devices . . . . . . . . . . . . . . . . . . . . . . 83
2.3.2 Control Strategies . . . . . . . . . . . . . . . . . . . . 84
2.3.3 Materials Incorporated in Control Devices . . . . . . . 84
3 Principles of Structural Control 87
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.2 State-Space Representation . . . . . . . . . . . . . . . . . . . 88
3.3 Structural Control Algorithms . . . . . . . . . . . . . . . . . 93
3.3.1 Operation Modes . . . . . . . . . . . . . . . . . . . . . 93
3.3.2 Controller Algorithms . . . . . . . . . . . . . . . . . . 95
3.3.2.1 On-Off Controller . . . . . . . . . . . . . . . 95
3.3.2.2 Fuzzy Logic Controller . . . . . . . . . . . . 99
II Conventional Damping Systems 103
4 Dissipators 105
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.2 Metallic Dampers . . . . . . . . . . . . . . . . . . . . . . . . 106
4.3 Friction Dampers . . . . . . . . . . . . . . . . . . . . . . . . 109
4.4 Viscoelastic Dampers . . . . . . . . . . . . . . . . . . . . . . 112
4.5 Viscous Fluid Dampers . . . . . . . . . . . . . . . . . . . . . 116
ix
5 Tuned Mass Dampers 119
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.2 Classical Tuned Mass Dampers . . . . . . . . . . . . . . . . . 119
5.2.1 Mathematical Modeling . . . . . . . . . . . . . . . . . 121
5.2.1.1 State-Space Representation . . . . . . . . . . 122
5.2.1.2 Deformation Response Factor. . . . . . . . . 125
5.2.1.3 Parameter Optimization. . . . . . . . . . . . 127
5.2.1.4 Mass Ratio . . . . . . . . . . . . . . . . . . . 128
5.2.2 Pendulum Tuned Mass Dampers . . . . . . . . . . . . 133
5.2.3 Tuned Liquid Dampers . . . . . . . . . . . . . . . . . 136
5.2.4 Tuned Liquid Column Dampers . . . . . . . . . . . . . 140
III Advanced Damping Systems 143
6 Active and Semi-Active Damping Systems 145
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.2 Active Damping Systems . . . . . . . . . . . . . . . . . . . . 145
6.2.1 Description . . . . . . . . . . . . . . . . . . . . . . . . 145
6.2.2 Application Examples . . . . . . . . . . . . . . . . . . 147
6.3 Semi-Active Damping Systems . . . . . . . . . . . . . . . . . 149
6.3.1 Description . . . . . . . . . . . . . . . . . . . . . . . . 149
6.3.2 Application Examples . . . . . . . . . . . . . . . . . . 150
7 Semi-Active Tuned Liquid Column Dampers 155
7.1 Semi-Active Uniaxial Tuned Liquid Column Damper . . . . 155
7.1.1 Mathematical Modeling . . . . . . . . . . . . . . . . . 158
7.1.1.1 Equation of Motion of the S-TLCD . . . . . 159
7.1.1.2 Restoring Force . . . . . . . . . . . . . . . . 164
7.1.1.3 Geometric Factors . . . . . . . . . . . . . . . 165
7.1.1.4 State-Space Representation . . . . . . . . . . 167
7.1.2 Experimental Investigations . . . . . . . . . . . . . . . 169
7.1.2.1 Experimental Setup . . . . . . . . . . . . . . 169
7.1.2.2 Investigations on the Natural Frequency . . . 174
7.1.2.3 Investigations on the Inherent Damping . . . 177
7.1.2.4 Investigations on the Vibration Control Per-
formance . . . . . . . . . . . . . . . . . . . . 180
7.2 Semi-Active Omnidirectional Tuned Liquid Column Damper 188
7.2.1 Mathematical Modeling . . . . . . . . . . . . . . . . . 190
7.2.1.1 Equation of Motion of the Semi-Active O-
TLCD . . . . . . . . . . . . . . . . . . . . . . 190
7.2.1.2 EquationofMotionofanSDoFStructurewith
a Semi-Active O-TLCD . . . . . . . . . . . . 195
