Submitted to the Graduate School ... Department of Civil Engineering and Geological Sciences ... 2.4 Multi-Degree-of-Freedom Coupled Building System .
SEMIACTIVE CONTROL OF CIVIL STRUCTURES FOR NATURAL HAZARD MITIGATION: ANALYTICAL AND EXPERIMENTAL STUDIES
A Dissertation
Submitted to the Graduate School of the University of Notre Dame in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
by
Richard E. Christenson, B.S.
B.F. Spencer, Jr., Director
Department of Civil Engineering and Geological Sciences Notre Dame, Indiana December 2001
SEMIACTIVE CONTROL OF CIVIL STRUCTURES FOR NATURAL HAZARD MITIGATION: ANALYTICAL AND EXPERIMENTAL STUDIES
Abstract by Richard E. Christenson
The research detailed within this dissertation will investigate innovative smart structures, including the seismic protection of buildings and the mitigation of wind vibrations in cable structures. The focus is on understanding the dynamic characteristics of these smart structures, identifying viable semiactive control strategies, assessing the merits of the control strategies relative to passive and active control alternatives, and demonstrating the structural control concepts. Analytical, numerical and experimental methods are employed in this research. Coupled building control is shown to be a viable method to protect tall buildings from seismic excitation. Various coupled building configurations are examined and coupled building design guidelines identified. Constraints on the maximum control force are enforced. A semiactive control strategy applied to a coupled building pair provides performance bounded by passive and active control strategies. Active coupled building control, employing acceleration feedback, is experimentally verified. The semiactive control of cable structures is examined, studying the vibration reduction of long cables. The effect of cable sag, axial stiffness, angle of inclination, and damper location on the control performance is examined. Specific levels of sag, axial stiffness, angle of inclination, and damper location resulting in poor performance are identified. A semiactive control strategy is shown analytically to achieve similar performance to
Richard E. Christenson
active control, with performance well beyond that achieved with passive control. A semiactive control strategy is verified experimentally on a 12.65 meter cable experiment employing a smart shear mode magnetorheological fluid damper. The experimentally achieved performance levels are explained by including control-structure interaction. Structural control is shown analytically and experimentally to be a viable method of protecting civil structures from natural hazards, such as seismic and rain-wind induced vibration. Semiactive control strategies, when applied to civil structures, can provide increased performance over passive control without the concerns of energy and stability associated with active control.
To my wife, Kimberly. Your love and support have been continuous. For that I am grateful.
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CONTENTS
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Structural Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Semiactive Control Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Overview of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 COUPLED BUILDING CONTROL: BACKGROUND . . . . . . . . . . . . . . . . . . . . . . 16 2.1 Coupled Building Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Two-Degree-of-Freedom Coupled Building System . . . . . . . . . . . . . . . . . . . 21 2.3 2DOF Coupled Building Optimal Passive Control Strategy . . . . . . . . . . . . . 32 2.4 Multi-Degree-of-Freedom Coupled Building System . . . . . . . . . . . . . . . . . . 41 2.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3 COUPLED BUILDING CONTROL: ANALYTICAL STUDIES . . . . . . . . . . . . . . . 51 3.1 Coupled Building Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2 Effects of Building Configuration on RMS Response . . . . . . . . . . . . . . . . . . 58 3.3 Efficacy of Semiactive Coupled Building Control . . . . . . . . . . . . . . . . . . . . . 66 3.4 Constraint on Maximum Allowable Control Force . . . . . . . . . . . . . . . . . . . . 70 3.5 Low-Rise Coupled Building System Analysis . . . . . . . . . . . . . . . . . . . . . . . . 74 3.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 iii
4 COUPLED BUILDING CONTROL: EXPERIMENTAL VERIFICATION . . . . . . 83 4.1 Coupled Building Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.2 Experimental Coupled Building Control-Oriented Design Model . . . . . . . . . 88 4.3 Experimental Active Coupled Building Control Strategy . . . . . . . . . . . . . . . 91 4.4 Experimental Active Coupled Building Results . . . . . . . . . . . . . . . . . . . . . . . 93 4.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5 CABLE DAMPING CONTROL: BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . 105 5.1 Cable Damping Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.2 In-Plane Motion of Cable with Sag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.3 Cable Damping Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6 CABLE DAMPING CONTROL: EFFECTS OF CABLE SAG . . . . . . . . . . . . . . . 120 6.1 Effects of Sag on Damping Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.2 Effects of Sag on RMS Cable Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.3 Effects of Sag on Damper Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.4 Effects of Sag on Cable Modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7 CABLE DAMPING CONTROL: EXPERIMENTAL VERIFICATION . . . . . . . . 137 7.1 Cable Damping Experimental Setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.2 System Identification of Cable Damping Model . . . . . . . . . . . . . . . . . . . . . 142 7.3 Passively-Operated Smart Damping Control Strategy . . . . . . . . . . . . . . . . . 152 7.4 Experimental Semiactive Cable Damping Control Strategy . . . . . . . . . . . . 155 7.5 Experimental Semiactive Cable Damping Results . . . . . . . . . . . . . . . . . . . . 158 7.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
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8 INVESTIGATING EXPERIMENTAL AND SIMULATION CABLE DAMPING CONTROL PERFORMANCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 8.1 Investigating Cable Bending Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 8.2 Investigating Semiactive Cable Damper. . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 8.3 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 9 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 9.1 Coupled Building Control Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 9.2 Cable Damping Control Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 9.3 Future Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
APPENDIX A: Root Mean Square Responses of a First Order Linear System using the Solution to the Lyapunov Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
APPENDIX B: Modeling Tall Adjacent Buildings using the Galerkin Method . . . . 193
APPENDIX C: Modeling Tall Adjacent Buildings using the Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
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LIST OF TABLES
Table 1.1: Loss of Life and Property Damage for Recent Earthquakes Disasters . . . . . . . 2 Table 2.1: Details of 2DOF Coupled Building System . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Table 2.2: Transfer Function Results of Passive Control Strategy for the 2DOF Coupled Building System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Table 2.3: RMS Response Results of Passive Control Strategy for the 2DOF Coupled Building System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Table 3.1: Performance of Passive, Active and Semiactive Control Strategies . . . . . . . 69 Table 3.2: Performance of Passive, Active and Semiactive Control Strategies for Various Levels of Ground Acceleration with a Constraint on the Maximum Allowable Control Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Table 3.3: Summary of Full-Scale Structural Frame ModelS . . . . . . . . . . . . . . . . . . . . . 75 Table 3.4: Comparison of Passive and Active Control Strategies for the Low-Rise Coupled Building System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Table 4.1: Peak Magnitude of Coupled Building System Transfer Functions. . . . . . . . . 96 Table 4.2: RMS Performance of Coupled Building System to Simulated Earthquakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Table 6.1: Comparison of peak modal damping ratios with a linear passive viscous damper at xd = 0.02 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Table 7.1: Control Performance for Cable Damper Experiment . . . . . . . . . . . . . . . . . . 159 Table 7.2: Control Strategy Cost Function and Shaping Filter Combinations . . . . . . . 162 RMS
Table 7.3: Control Performance, w e
, for Additional Control Strategies . . . . . . . . . 164
RMS
Table 7.4: Control Performance, ( w e ), for 1st Antisymmetric and 2nd Symmetric Mode Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
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LIST OF FIGURES
Figure 1.1: Collapse of the original Tacoma Narrows bridge, November 7, 1940. . . . . . . 2 Figure 1.2: Structural failures during recent strong motion earthquakes. . . . . . . . . . . . . . 3 Figure 1.3: Control strategies and associated supplemental damping devices. . . . . . . . . . 5 Figure 1.4: Examples of passive control strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Figure 1.5: Examples of active control strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Figure 1.6: Actively controlled Kyobashi Seiwa building in Tokyo, Japan. . . . . . . . . . . . 9 Figure 2.1: Examples of full-scale coupled building implementations. . . . . . . . . . . . . . 20 Figure 2.2: 2DOF coupled building system undergoing ground excitation and the resulting 2-DOF model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Figure 2.3: Plot of positive complex pole of SDOF system. . . . . . . . . . . . . . . . . . . . . . . 26 Figure 2.4: Root locus plot of the 2DOF coupled building system as connector stiffness and connector damping is varied. . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Figure 2.5: Transfer function from the ground acceleration to displacement, velocity and absolute acceleration as connector stiffness and connector damping is varied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Figure 2.6: RMS responses over a range of connector stiffness and connector damping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Figure 2.7: Optimal transfer functions from ground acceleration to displacement for the 2DOF undamped coupled building system. . . . . . . . . . . . . . . . . . . . 35 Figure 2.8: Optimal poles for the 2DOF coupled building system. . . . . . . . . . . . . . . . . . 38 Figure 2.9: Optimal transfer functions of ground acceleration to absolute accelerations for the 2DOF coupled building system. . . . . . . . . . . . . . . . . . 39 Figure 2.10: Optimal RMS of 2DOF coupled building system. . . . . . . . . . . . . . . . . . . . 39 vii
Figure 2.11: High-rise MDOF coupled building system. . . . . . . . . . . . . . . . . . . . . . . . . 41 Figure 2.12: Convergence of undamped natural frequencies for Galerkin and Finite Element methods of the first three modes of each building. . . . . 45 Figure 2.13: Power spectral density of ground excitation. . . . . . . . . . . . . . . . . . . . . . . . 47 Figure 2.14: Estimating RMS ground motions from historical records, where the bold section defines the portion of the earthquake used for the RMS calculation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Figure 3.1: High-rise MDOF coupled building system for analytical studies. . . . . . . . . 52 Figure 3.2: Semiactive damper dissipative forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Figure 3.3: Frequency analysis of uncoupled 50-, 30- and 20-story building responses, H ye w(ω) , due to a filtered ground excitation. . . . . . . . . . . . . . . . 60 Figure 3.4: Effect of building height and coupling link location on coupled building performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Figure 3.5: Effect of mass density and stiffness on coupled building performance. . . . . 65 Figure 3.6: High-rise MDOF coupled building system for semiactive control. . . . . . . . 66 Figure 3.7: Semiactive coupled building control RMS responses over range of control forces as compared to passive and active control strategies. . . . . . . 68 Figure 3.8: RMS response profiles of absolute story acceleration and interstory drift ratio over the height of both buildings for uncoupled and optimal passive, active, and semiactive control strategies. . . . . . . . . . . . . . . . . . . . . 68 Figure 3.9: Semiactive performance with identified maximum allowable control force for three levels of excitation as compared to passive and active control strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Figure 3.10: Beam element, 5- and 3-story building models, and building deflection for the low-rise coupled building system. . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Figure 4.1: Schematic of coupled building experiment. . . . . . . . . . . . . . . . . . . . . . . . . . 84 Figure 4.2: Two-story coupled building model for experimental verification. . . . . . . . . 85 Figure 4.3: Control actuator, consisting of a servo-motor with ball-screw mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Figure 4.4: Comparison of the experimental and curve-fit transfer functions. . . . . . . . . 89
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Figure 4.5: Experimental transfer functions of ground acceleration to absolute story accelerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Figure 4.6: Time history response to El Centro simulated ground acceleration. . . . . . . . 98 Figure 4.7: Time history response to Hachinohe simulated ground acceleration. . . . . . . 99 Figure 4.8: Time history response to Northridge simulated ground acceleration. . . . . . 100 Figure 4.9: Time history response to Kobe simulated ground acceleration. . . . . . . . . . 101 Figure 5.1: In-plane static profile z(x) and dynamic loading f(x,t) of inclined cable with sag and transverse damper force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Figure 5.2: Typical static sag profiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Figure 5.3: Ideal semiactive damper dissipative forces. . . . . . . . . . . . . . . . . . . . . . . . . 118 Figure 6.1: Natural frequency and damping ratio in the first two modes for the linear designs for xd = 0.02. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Figure 6.2: Modal frequency and damping ratios over a range of sag with a damper at xd = 0.02. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Figure 6.3: Frequency and damping ratios of first symmetric mode as a function of damper location xd for several sag levels. . . . . . . . . . . . . . . . . . . . . . . . . . 126 Figure 6.4: Frequency and damping ratios of first antisymmetric mode as a function of damper location xd for several sag levels. . . . . . . . . . . . . . . . . 126 Figure 6.5: RMS displacement for a semiactive, passive viscous, or active dampers at xd = 0.02 as a function of the RMS force. . . . . . . . . . . . . . . . . 128 Figure 6.6: Minimum RMS displacement for a semiactive, passive viscous, or active dampers at xd = 0.02. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Figure 6.7: Minimum RMS displacement expanded views near three pairs of peaks (xd = 0.02). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Figure 6.8: RMS velocity for minimum displacement with a semiactive, passive viscous, or active damper xd = 0.02. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Figure 6.9: RMS displacement with a semiactive, passive viscous, or active damper at various damper locations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Figure 6.10: RMS displacement, relative to the optimal passive linear damper, with an active or semiactive damper at various damper locations. . . . . . . 132
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Figure 6.11: Natural frequencies as a function of the independent parameter λ2 for sag cables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Figure 6.12: Cable mode shapes at various sag levels. The antisymmetric modes are shown in gray. The natural frequencies (in nondimensional rads/sec) are given for the symmetric modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Figure 6.13: Expanded view of some cable modeshapes. . . . . . . . . . . . . . . . . . . . . . . . 134 Figure 7.1: Schematic of smart cable damping experiment. . . . . . . . . . . . . . . . . . . . . . 138 Figure 7.2: Flat-sag cable experimental setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Figure 7.3: Brass weights to insure dynamic similitude. . . . . . . . . . . . . . . . . . . . . . . . . 139 Figure 7.4: Smart shear mode magnetorheological fluid damper. . . . . . . . . . . . . . . . . . 140 Figure 7.5: Permanent magnet shaker. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Figure 7.6: In-plane static profile z(x) and dynamic loading f(x,t) of inclined cable with sag and transverse damper force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Figure 7.7: Transfer functions comparing flat-sag cable model (black) to experimental data (grey). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Figure 7.8: Phenomenological model of shear mode magnetorheological damper. . . . 146 Figure 7.9: Comparison of shear mode MR damper analytical model (black) and experimental data (grey). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Figure 7.10: Schematic of control signal to permanent magnet shaker. . . . . . . . . . . . . 148 Figure 7.11: Comparison of frequency content of actual (experimental) shaker force to target (analytical). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Figure 7.12: Comparison of frequency content of analytical (solid) and experimental (grey) shaker force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Figure 7.13: Schematic of process to calculate experimental performance measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Figure 7.14: Passively-operated smart damper cable response versus damper voltage for various levels of excitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Figure 7.15: Optimal passively-operated smart damper voltage versus excitation level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Figure 7.16: Passively-operated smart damper cable response versus damper voltage for various modes excited. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 x
Figure 7.17: Control design filter to weight the spectral content of the shaker excitation in the H2/LQG control design. . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Figure 7.18: Actual (grey) damper displacement and zero-mean (black) damper displacement used by control strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Figure 7.19: Controller performance at evaluation point and over length of cable. . . . 160 Figure 7.20: Additional control design filters to weight the spectral content of the shaker excitation in the H2/LQG control design. . . . . . . . . . . . . . . . . . 161 Figure 7.21: Controller performance at evaluation point for additional controllers. . . . 163 Figure 7.22: Controller performance at evaluation point for additional cable excitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Figure 8.1: Profile of cable at different instances in time for smart cable damping control strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Figure 8.2: Effect of bending stiffness () on optimal damping coefficient of passive cable damper for various damper locations. . . . . . . . . . . . . . . . . . . . . . . . . 173 Figure 8.3: Effect of bending stiffness () on achievable modal damping for passive and active optimal control strategies, and various damper locations. . . . . 174 Figure 8.4: Effect of bending stiffness () on the reduction of RMS response for various damper locations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Figure 8.5: Force for active, ideal semiactive, and smart dampers. . . . . . . . . . . . . . . . . 176 Figure 8.6: Cable damping performance versus damper location including damper dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Figure 8.7: Cable damping performance versus arctangent slope parameter . . . . . . . . 179 Figure 8.8: Comparison of ideal semiactive arctangent damper model to experimental data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Figure 8.9: Schematic of semiactive arctangent damper model with compliance and corresponding force of each element. . . . . . . . . . . . . . . . . . . . . . . . . . 180 Figure 8.10: Comparison of semiactive arctangent damper model with compliance to experimental data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Figure 8.11: Performance of semiactive arctangent damper model with compliance compared to previous damper models and experimental results. . . . . . . . . 182 Figure C.1: Degrees-of-freedom for beam element. . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 xi
ACKNOWLEDGEMENTS
I would like to thank my advisor, Prof. B.F. Spencer, Jr., for his excellent guidance and support throughout the course of this research. I very much appreciate the support and contribution of Prof. E.A. Johnson at the University of Southern California and Prof. K. Seto at Nihon University, Tokyo, Japan. I gratefully acknowledge the partial support of this research by the National Science Foundation under grant CMS 99-00234 (Dr. S.C. Liu, Program Director), the National Science Foundation Graduate Research Traineeship Fellowship, and the National Science Foundation Summer Institute in Japan Program. I also acknowledge support from industry in the form of equipment and information from the LORD Corporation, Ishikawajima-Harima Heavy Industries Co., LTD., and Quanser Consulting. Lastly, I want to express my appreciation for the assistance in setting up and conducting my experiments from undergraduates Joseph Winkels, Kimberly Rubeis, Chad DeBolt, and David Preissler under the National Science Foundation, Research Experiences for Undergraduates (REU) program, and to all my fellow students and researchers who have helped me in conducting this research.
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CHAPTER 1: INTRODUCTION
Civil structures, such as buildings and bridges, are an integral part of modern society. Traditionally, these structures were designed to resist static loads. Civil structures are, however, subjected to a variety of dynamic loadings, including winds, waves, earthquakes, and traffic. These dynamic loads can cause severe and/or sustained vibratory motion, both of which can be detrimental to the structure and its material contents and human occupants. The extent of protection required for these structures may range from reliable operation and occupancy comfort to human and structural survivability. An example of a civil structure that required protection for reliable operation and occupancy comfort is the 60-story John Hancock Tower, in Boston, Massachusetts. The wind-induced lateral and torsional vibration of the building resulted in acceleration levels too large for occupancy comfort on the upper floors. Additionally, glass panes from the over 10,000 windows of the John Hancock Tower began to fail and fall to the ground. Two 300 ton tuned mass dampers were installed, in 1977, on the 58th floor to increase the damping ratio of the building and reduce accelerations. A recent example of a civil structure undergoing vibration is the Trans-Tokyo Bay Crossing bridge located in Tokyo, Japan. This steel box-girder bridge was opened to the public in December 1997. However, during construction the two longest spans of the bridge, measuring 240 m, experienced significant wind induced vibration of the bridge deck due to vortex shedding. The vortex-induced vibration of the first vertical mode resulted in a maximum vibration amplitude of more than 0.5 m. The vibration raised issues of serviceability, fatigue and yielding failure for the structure. The bridge was aug-
1
mented with passive tuned mass dampers to mitigate the vertical motion of the bridge deck (Fujino and Yoshida, 2001). An historic example of a civil structure that did not survive its dynamic loading is the wind induced torsional vibration of the original Tacoma Narrows bridge in Tacoma, Washington. The vibration of this bridge was so severe that it led to the collapse of the bridge on November 7, 1940, as shown in Figure 1.1.
Figure 1.1: Collapse of the original Tacoma Narrows bridge, November 7, 1940. Civil structures also fail during large seismic events, often resulting in loss of human life and property damage. In recent years, tens of thousands of people have died and billions of dollars in property damage have been lost as a result of earthquakes. Figure 1.2 shows the structural damage of civil structures during recent seismic events. Some of the most significant earthquakes, in terms of loss of life and loss of property, in the past 10 years are listed in Table 1.1. TABLE 1.1: LOSS OF LIFE AND PROPERTY DAMAGE FOR RECENT EARTHQUAKES DISASTERS Location Northridge, California Kobe, Japan Kocaeli, Turkey Chi-Chi, Taiwan Bhuj, India
Date 01/17/94 01/17/95 08/17/99 09/28/99 01/26/01
Magnitude 6.8 6.8 7.8 7.7 8.0
Loss of Life 60 5,502 15,637 2,400 20,005
Property Damage $20 billion $147 billion $6.5 billion $14 billion $4.5 billion
data obtained from the NESDIS National Geophysical Data Center, Significant Earthquake Database (http://www.ngdc.noaa.gov/seg/hazard/sig_srch.shtml)
2
1994 Northridge Earthquake
1995 Kobe Earthquake
1999 Kocaeli, Turkey Earthquake
2001 Bhuj, India Earthquake
Figure 1.2: Structural failures during recent strong motion earthquakes. These recent events remind us of the vulnerability of our society to natural hazards. The protection of civil structures, including material content and human occupants, is, without doubt, a world-wide priority. The challenge of structural engineers is to develop safer civil structures to better withstand these natural hazards.
1.1 Structural Control Structural control for civil structures was born out of a need to provide safer and more efficient designs with the reality of limited resources. The purpose of structural control is to absorb and to reflect the energy introduced by dynamic loads such as winds, waves, earthquakes, and traffic. Today, the protection of civil structures from severe 3
dynamic loading is typically achieved by allowing the structures to be damaged. Consider the conservation of energy relationship proposed by Uang and Bertero (1988) E = Ek + Es + Eh + Ed
(1.1)
where E is the total energy input to the structure from the excitation, E k is the kinetic energy of the structure, E s is the elastic strain energy of the structure, E h is the energy of the structure dissipated due to inelastic deformation (e.g., allowing damage to the structure), and E d is the energy dissipated by supplemental damping devices. For traditional structures, the right hand side of Equation (1.1) includes only E k , E s , and E h . By including the energy term E d through structural control, the energy dissipated by supplemental damping devices, the kinetic, elastic, and, most importantly, the inelastic deformation energy can be reduced, preserving the primary structure. There are three primary classes of supplemental damping devices, categorized into three corresponding control strategies. The first class of supplemental damping devices is passive. Passive devices are non-controllable and require no power. The second class of supplemental damping devices is active. Active devices are controllable, but, require significant power to operate. The third class of supplemental damping devices is semiactive. Semiactive devices combine the positive aspects of passive and active control devices in that they are controllable (like the active devices) but require little power to operate. Figure 1.3 shows graphically how these control devices and their control strategies are related. In 1994, the First World Conference on Structural Control was held in Pasadena, California (Housner, et al., 1994a). The success of this conference led to the Second World Conference on Structural Control held in 1998 in Kyoto, Japan (Kobori, et al., 1998). This next year, in 2002, the Third World Conference on Structural Control will be held in Como, Italy. These conferences are indicators of the continued research and support in the area of structural control for civil structures. Indeed, within the field of struc4
Passive Control Strategies
Active Control Strategies
PASSIVE DEVICES
ACTIVE DEVICES
non-controllable no power required
controllable significant power required
Semiactive Control Strategies SEMIACTIVE DEVICES controllable little power required
Figure 1.3: Control strategies and associated supplemental damping devices.
tural control for civil applications, significant research has been conducted, experimental studies performed, and full-scale applications brought to fruition.
Passive Control Strategies Passive control strategies dissipate and isolate structures from the energy of dynamic loadings (Housner, et al., 1997). In a passive control strategy, a passive energy dissipation device is attached to the civil structure. Passive energy dissipation devices include metallic, friction, viscoelastic, and viscous fluid dampers, tuned mass dampers and tuned liquid dampers (Soong and Dargush, 1997). Passive devices are characterized by the dissipative nature of their control forces and the fixed characteristics of the devices (e.g., damping coefficient). Passive devices are often optimally tuned to protect the structure from a particular dynamic loading, and thus the performance of these devices is suboptimal for other loading scenarios and configurations. For example, a passive damper opti5
mally designed to reduce cable responses in the first mode may not be optimal to reduce the cable responding in the second and higher modes. Base isolation is one of the more successful passive control strategies. In a base isolation system, the structure sits on top of rubber bearings that isolate the structure from the moving ground. Another passive energy device is the tuned mass damper (TMD). A TMD transfers energy from the primary structure to the TMD, and provides energy dissipation. Additionally, passive energy dissipation devices can be placed between story levels in a passive bracing system. Schematics of these passive control strategies are shown in Figure 1.4. civil structure
passive energy dissipation device
Passive Bracing System civil structure tuned mass damper
base isolators
Base Isolation
Tuned Mass Damper
Figure 1.4: Examples of passive control strategies. 6
Passive control strategies are popular and have been widely implemented. Passive devices are inherently stable, require no external energy to operate and are relatively simple to design and build. However, the performance of optimal passive control is sometimes limited, in that they are typically designed protect the structure from one particular dynamic loading.
Active Control Strategies At the other extreme of structural control are active control devices. Yao (1972) first proposed the active structural control of civil structures. These control strategies deliver force into the structure to counteract the energy of the dynamic loading and have the ability to control different vibration modes and to accommodate different loading conditions (Housner, et al., 1997). Active devices can provide increased performance over passive strategies, using global response information to determine appropriate control forces, in contrast to being limited, as passive devices are, to the local responses. For example, a passive tuned mass damper must provide control forces based on the response of the floor where it is located. In contrast, an active control strategy can measure and estimate the response over the entire building to determine appropriate control forces. As a result, active control strategies are more complex than passive strategies, requiring sensors and evaluator/controller equipment. Active control devices typically require significant energy to develop the magnitude of forces required for civil infrastructure applications. The uninterrupted supply of energy from external sources, especially during natural hazards when the control strategy is most expected to operate, is of concern. Active and hybrid control strategies reduce unwanted responses by appropriately adding energy to or removing energy from the system. However, given a shift in the dynamics of the structure, the performance of the active strategy may be less than expected and may even result in an unstable condition, whereby unbounded energy is specified by the controller. 7
Some examples of active control strategies include active base isolation, active bracing and an active mass driver (Spencer and Soong, 1999, and Soong and Spencer, 2001). These are natural extensions of passive control strategies. The main differences are the sensors that measure the building responses and the control computer that sends out a control signal to the actuator to provide appropriate force to the structure. Examples of active control are shown in Figure 1.5. Active control strategies been proposed and implemented in a number of civil structures (Spencer and Sain, 1997). In 1989, the Kajima Corporation installed the first full-scale application of active control to a building (Kobori, 1994). Two active mass drivcivil structure
control actuator
Active Bracing System control actuator civil structure
mass
control computer sensors
base isolators
control actuator
Active Base Isolation
Active Mass Driver
Figure 1.5: Examples of active control strategies. 8
ers were installed on the roof of the 11-story Kyobashi Seiwa building in Tokyo, Japan, to reduce building vibration under strong winds and moderate seismic events. Sensors were placed at the roof, 6th floor, and basement levels and the control computer is located on the 11th floor, as illustrated in Figure 1.6. There are currently nearly 40 buildings and towers implemented with active control strategies. Additionally, 15 bridge towers have been implemented with active and hybrid control devices during bridge erection. Spencer and Soong (1999); and Soong and Spencer (2001) provide detailed lists of these full-scale applications. These full-scale active control strategies are located in Japan, China, Taiwan and Korea. Despite numerous success stories, engineers have yet to fully embrace active control. Some reasons include, the capital cost and maintenance, the reliance on external power, system reliability and stability, and acceptance by the profession (Spencer and Sain, 1997).
wind vane
AMD-1 Sensor AMD-2 11th floor
6th floor
Control Computer
Sensor
Observation System
Basement
Sensor
Figure 1.6: Actively controlled Kyobashi Seiwa building in Tokyo, Japan. 9
Semiactive Control Strategies Semiactive control devices, also called “smart” control devices, assume the positive aspects of both the passive and active control devices. A semiactive control strategy is similar to the active control strategy. Only here, the control actuator does not directly apply force to the structure, but instead it is used to control the properties of a passive energy device, a controllable passive damper. Semiactive control strategies can be used in many of the same civil applications as passive and active control. Semiactive control strategies are dissipative in nature, inherently stable, and require a little energy to operate (Spencer and Sain, 1997). Semiactive control strategies appear to be particularly promising in addressing a number of the challenges facing active control strategies, in that the devices are low power, fail-safe, and reliable. Semiactive control performance is bounded by passive and active control. Numerous studies indicate that semiactive control can potentially achieve the majority of the performance of fully active systems. A detailed description of semiactive control devices is presented in Section 1.2.
Hybrid Control Strategies The three primary classes of supplemental damping devices can be combined in various combinations, resulting in hybrid control strategies. Hybrid strategies typically require less, though still significant, energy. These strategies provide performance bounded by passive and active control strategies. The most common hybrid control strategy employs the hybrid mass damper (HMD). The HMD combines a passive tuned mass damper augmented with an active control actuator. The HMD is the most common control device for full-scale civil applications.
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1.2 Semiactive Control Devices Semiactive devices are different from active devices in that semiactive devices can only produce dissipative forces. Semiactive devices include variable orifice dampers, variable friction dampers, controllable tuned liquid dampers, controllable fluid dampers, etc. These devices can be viewed as controllable passive devices, in that the characteristics of the passive devices can be changed in real time. In this manner, semiactive devices can produce the desired dissipative control forces. This section provides a sampling of the extent of research conducted and wealth of literature available on designing and applying various semiactive control devices to civil structural applications.
Variable Orifice Damper The variable orifice damper uses a controllable, electromechanical, variable-orifice valve to vary the flow of hydraulic fluid through a conventional hydraulic fluid damper. Variable orifice dampers have been applied to full-scale building (Kobori, et al., 1993; Kurata, et al., 1999, 2000) and bridge (Sack and Patten, 1994; Patten, 1998, 1999) structures.
Variable Friction Damper Variable friction dampers generate control forces through surface friction and controlling the slippage of the device. To date, only analytical studies have been conducted for these devices as applied to civil structural control. These devices have, however, been proposed to reduce interstory drifts of seismically excited buildings (Dowdell and Cherry, 1994; Inaudi, 1997).
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Controllable Tuned Liquid Dampers Controllable tuned liquid dampers use the motion of a column of fluid, varied with a controllable orifice, to reduce structural responses. These dampers are similar in concept to tuned mass dampers (TMDs), to absorb the energy of the structure by vibrating themselves, however, where TMDs are typically designed for one loading condition, the controllable tuned liquid damper can remain effective for a variety of loading conditions (Kareem, 1994, Lou, et al., 1994, Yalla and Kareem, 2000).
Controllable Fluid Dampers Controllable fluid dampers are similar to the variable orifice dampers; however, they use controllable fluids, such as electrorheological (ER) and magnetorheological (MR) fluids, that do not require a mechanical valve. These ER and MR fluids are able to change between free flowing Newtonian fluid and a semi-solid with controllable yield strength within milliseconds when exposed to electric or magnetic fields, respectively. These fluids date back to the late 1940’s (Winslow, 1947, 1949; and Rabinow, 1948). Only recently have controllable fluid dampers been proposed for civil applications. Some examples of literature proposing ER fluids for the application to civil structural control include Burton, et al. (1996), Gavin, et al. (1996a, b), and Makris, et al. (1996). Some examples of literature proposing MR fluids for the application to civil structural control include Dyke, et al., (1996a, b, 1998), Spencer, et al., (1997), Jansen and Dyke, (2000), Johnson, et al., (2001a, b), Ramallo, et al., (2001), Spencer, et al., (2000), Yi and Dyke, (2000), and Yoshioka, et al., (2001).
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1.3 Overview of Dissertation This dissertation investigates two innovative semiactive systems. The first is the seismic protection of adjacent buildings. The second is the mitigation of wind induced vibration of cable structures. The semiactive control of coupled buildings is investigated, where the potential for coupling adjacent buildings with semiactive dampers for seismic protection is examined with respect to active and passive control strategies (Chapters 2-4). Chapter 2 contains a literature review of the history and current status of coupled building control. Following this is an examination of a simplified two-degree-of-freedom (2DOF) coupled building model to understand the effect of coupling on the eigenvalues, transfer functions, and root mean square (RMS) responses of the system. Optimal passive control strategies are identified for both undamped and damped 2DOF coupled building systems. Since higher modes can contribute to the vibration of tall and flexible buildings, multi-degree-of-freedom (MDOF) building models are also considered. An accurate, low-order, model is developed for the MDOF coupled building system and a passive control strategy presented. Chapter 3 details the analytical studies on coupled building control problem. Two coupled building control strategies are proposed in this chapter: an active coupled building control strategy employing H2/LQG control and absolute acceleration and actuator displacement feedback; and a semiactive control strategy employing a clipped optimal H2/ LQG control strategy. Next, the effect of building configuration on the coupled building system is examined. Building configurations such as relative building heights, building mass, and building stiffness, as well as the connector location are studied. The efficacy of semiactive control for the coupled building problem is presented for an example that is similar in configuration to the Triton Square office complex, a set of three high-rise buildings in Tokyo, Japan, that were coupled in March 2001. The effect of constraining the maximum allowable control force on system performance is studied.
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In Chapter 4, active coupled building control employing absolute story acceleration and actuator displacement feedback is experimentally verified. The experimental setup for the active coupled building control experiment is described, a control oriented model designed, active control strategy identified, and experimental results presented. The second of the complementary research efforts in this dissertation is the mitigation of wind induced vibratory responses of cables (Chapters 5-8). Semiactive dampers are examined to provide transverse control of cables. Chapter 5 contains a literature review of the history and current status of cable damping control. A model for the cable damping system with sag is developed. In Chapter 6, analytical studies on cable damping control are performed. Passive, active and semiactive control strategies are examined. Effects of cable sag on the performance of the control strategies are investigated. Regions of cable sag that result in reduced levels of performance are identified and explained. In Chapter 7, semiactive cable damping, employing a smart shear mode magnetorheological fluid damper attached to a 12.6 meter cable, is experimentally verified. The experimental setup for the smart cable damping control experiment is described, a control oriented design model developed, a semiactive control strategy identified, and experimental results presented. Various levels and modes of excitation are considered. Also, the performance of the smart damper operated in a purely passive mode, where constant levels of current are supplied to the damper, is examined. Chapter 8 investigates the experimental and simulation cable damping control explaining the difference in performance. Two factors are considered to have a possible effect. First, the bending stiffness of the cable, neglected in the simulation studies, is examined. Next, the properties of the semiactive damper are examined. This investigation offers an explanation to the difference in cable damping performance and suggests a solution to experimentally regain this performance.
14
Chapter 9 provides conclusions for the coupled building control and cable damping control. Additionally, this chapter proposes a number of research areas for future studies.
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CHAPTER 2: COUPLED BUILDING CONTROL: BACKGROUND
Seismic events such as the 1994 Northridge and 1995 Kobe (Hyogo-ken Nanbu) earthquakes are recent reminders of the vulnerability of our cities’ infrastructures to strong motion earthquakes. Strong seismic events can cause severe inelastic behavior in civil structures, threatening the safety of occupants and resulting in potential human and material losses. Civil structures are traditionally protected from large seismic events through redundancies. In recent years, medium- and high-rise structures have begun employing control techniques such as active mass drivers (AMDs) to help mitigate responses. Ultrahigh-rise buildings, such as recent trends are producing, are relatively flexible and difficult to control with AMDs, due to long actuator strokes and large energy requirements. Coupling buildings has been shown to be a viable alternative for the protection of adjacent flexible structures (Seto, 1994a). Coupled building control uses dissimilar adjacent structures to impart forces upon one another in such a manner that critical responses are mitigated. This concept was first introduced by R.E. Klein nearly three decades ago (Klein, et al., 1972). Recently, coupled building control has received much attention in Japan and the U.S. as a number of researchers are studying various control strategies, and full-scale applications are beginning to appear. Over the past three decades, coupled building research has steadily gained momentum from proposed research concepts to actual implementation. Numerous passive and active control strategies have been considered for low- to high-rise buildings. In this research, an active control strategy employing acceleration feedback, and, further, semi16
active “smart” dampers, are proposed to connect and control adjacent flexible high-rise structures. This chapter contains a literature review of coupled building control, examination of a simplified two-degree-of-freedom coupled building model to understand the effect of coupling on the dynamic characteristics of the building models, and formulation of the multi-degree-of-freedom coupled building model used for the analytical studies of the research.
2.1 Coupled Building Literature Review In 1972, Klein, et al. (1972) first proposed the concept of coupling two tall buildings in the U.S. In 1976, Kunieda (1976) proposed coupling multiple structures in Japan. In the mid 1980’s, Klein and Healy (1987) suggested a rudimentary semiactive approach, coupling two buildings with cables that could be released and tightened (when slack is available) to provide specified dissipative control forces. They observed that the structures being coupled with a single link must have different primary natural frequencies to insure controllability. They also proposed that the buildings be connected near the top as this is a region where the vibratory modes will have non-zero amplitudes. In the 1990’s, interest in coupling civil structures was renewed due to advancements in structural control and the apparent limits of existing technology (e.g., base isolators, AMDs, etc.). Graham (1994) coupled single-degree-of-freedom building models for both passive and active control strategies and concluded that, in addition to a passive control strategy, an active LQR control approach can effectively reduce the response of the two coupled buildings. Further studies would continue to show the effectiveness of passive and active control strategies for the coupled building problem. Passive control strategies have been studied for both high- and low-rise buildings. Gurley, et al. (1994), Kamagata, et al. (1996), Fukuda, et al. (1996) and Sakai, et al. 17
(1999) have each studied the case of coupling tall flexible structures with passive devices, while Luco, et al. (1994, 1998), Xu, et al. (1999a) and Ko, et al. (1999) have studied connecting low- to medium-rise structures with passive devices. Each of these papers reports positive results in mitigating the responses due to wind and seismic excitations. Additionally, Fukuda, et al. noted, as Klein and Healy had implied, that when a coupling link is placed at a node of a vibratory mode, that mode cannot be controlled by the link, reiterating the importance of the location of the coupling link along the height of the buildings. Active control strategies have been studied extensively for flexible structures. Seto, et al. (1994a, 1994b, 1995, 1996, 1998), Haramoto, et al. (1999, 2000), Matsumoto, et al. (1999), Mitsuta and Seto (1992), Hori and Seto (1999) and Yamada, et al. (1994) have studied connecting tall flexible structures using active control techniques to control the long period motion, as well as the higher modes, with encouraging results. The higher modes of flexible structures may be more susceptible to seismic excitations and are a concern for this class of buildings. Seto, et al. have successfully controlled the first two modes of two and three adjacent flexible building models in simulation and experimentally. They intentionally placed coupling links at the vibrational nodes of the first neglected mode, making it uncontrollable, to prevent spillover of the controller into this higher mode. In addition to the numerous analytical studies actively coupling adjacent buildings for response mitigation, there has been significant experimental work. Mitsuta, et al. (1992) performed experimental tests on two adjacent single-degree-of-freedom (SDOF) building models and adjacent single- and 2-DOF building models. The building masses were coupled with an active control actuator, using absolute displacement sensors for the feedback measurement. Yamada, et al. (1994) coupled a pair of 2-story and 3-story building models at the second story with a negative stiffness active control device and was able to effectively reduce the displacements of these low-rise building models. A number of experiments have been conducted on coupling two continuous plates, representing flexible high-rise structures (Fukuda, et al. 1996, Hori and Seto, 1999, Kamagata, et al. 1996, 18
Seto, 1996, 1998, Seto, et al. 1994a, 1994b, 1995). These active control experiments have used one and two control actuators. The active control strategies for these experimental tests employ displacement measurements for feedback. The direct measurement of displacement on large-scale structures is difficult to achieve. Additionally, nearly all of the experimental tests performed to date have produced active control forces using electromagnetic actuators. The exception is Yamada, et al. (1994) who used a spring in series with a stepping motor of rack and pinion mechanism to realize their negative stiffness control strategy. The idealized actuators have little device dynamics, and thus control-structure interaction is not significant in the resulting experiments. Since control-structure interaction can have a significant effect on the ability of the control actuator to produce desired forces at the structures resonant frequencies, the inclusion of this phenomenon for actuators models more representative of full-scale devices is important (Dyke, et al. 1995). Numerous papers have been published in Japanese concerning the coupled control of adjacent structures (Ezure, et al. 1993, Ezure, et al. 1994, Ikawa, et al. 1996, Iwanami, et al. 1986, Iwanami, et al. 1993, Kageyama, et al. 1994, Maeda, et al. 1997, Mitsuta, et al. 1992, Okawa, et al. 1990, Seto 1998, Seto, et al. 1994c, Sugino, et al. 1997, Toba, et al. 1994, 1995). This research has focused on the passive and active control of two and three adjacent structures, studying roughly the same concepts as the English publications. In addition to these research activities, full-scale tests are being performed and full-scale applications are being realized. Three coupled building control applications, all located in Japan, are pictured in Figure 2.1. In 1989, the KI (Kajima Intelligent) Building complex was constructed in Tokyo, Japan. This complex coupled the 5-story and 9-story structures in a low-rise office complex with passive yielding elements connected at the 5th floor.1 The general contracting firm, Konoike, has implemented four substructure coupling projects in recent years and, in 1998, coupled four of their headquarter buildings, one 12story and three 9-story buildings, in Osaka, Japan, with passive visco-elastic dampers.2 1. Kajima Corporation: Technical pamphlet 91-62E
19
Iemura, et al. (1998) has studied passive and active control of two low-rise structures and is preparing full-scale tests to verify the concept at the Disaster Prevention Research Institute (DPRI) in Kyoto, Japan. Here they will connect 3- and 5-story building frames at the 3rd floor. The Triton Square office complex, located on the Tokyo waterfront on Harumi Island, completed construction in March 2001. The complex is a cluster of three buildings, 195 m, 175 m, and 155 m tall. The 195 m and 175 m tall buildings are coupled at a height of 160 m. The 175 m and 155 m tall buildings are coupled at a height of 136 m. The three buildings are coupled with two 35-ton active control actuators for wind and seismic protection.
Kajima Intelligent Building Complex
Konoike Headquarter Buildings
Triton Square Office Complex Figure 2.1: Examples of full-scale coupled building implementations. 2. http://www.konoike.co.jp/
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Experimental studies to verify active coupled building control have traditionally employed displacement feedback. The direct measurement of displacements on larger scale structures is difficult to achieve, thus acceleration feedback, as considered in this dissertation, is an appealing control strategy for coupled building control. Active control strategies employing acceleration feedback have been shown in previous experiments to be effective for other civil structure applications, including an active bracing system (Spencer, et al. 1993), an active tendon system (Dyke, et al. 1994a, 1994b) and active mass driver systems (Dyke, et al. 1996b, Battaini, et al. 2000). In Chapters 3 and 4, acceleration feedback is shown, through simulation and experiment, respectively, to be an effective method of response reduction for the active coupled building problems (Christenson, et al. 1999b, Hori, et al. 2000). In Chapter 3, semiactive coupled building control is proposed (Christenson, et al. 1999a, 1999b, 2000a, 2000b, 2000c). Recently Zhu, et al. (2001) have also proposed semiactive coupled building control. Zhu, et al. consider coupling two single-degree-offreedom masses with a semiactive connector with positive results. The single-degree-offreedom building models in Zhu, et al. do not allow for coupling link position interference with vibratory nodes to be considered nor for higher mode participation and matching to be examined. These are important features of the coupled building system, as they can significantly effect system performance, and are examined in Chapter 3, using the multidegree-of-freedom (MDOF) building model developed in this chapter.
2.2 Two-Degree-of-Freedom Coupled Building System The most basic representation of the coupled building problem is the two-degreeof-freedom (2DOF) coupled building system. Here, two buildings, each modeled as a single-degree-of-freedom (SDOF) structure, are connected with a passive coupling link, and the resulting 2DOF system is examined. This simplified coupled building system is stud21
ied in order to gain valuable insight into the effect of coupling on the dynamics of the system. The passive control strategy involves placing stiffness and damping elements between the two masses. The selection of an optimal stiffness and damping coefficient for the connector link is critical to the performance of the coupled passive system. When considering structures that are both internally damped, a closed-form solution for the optimal connector stiffness and connector damping is not readily available. The determination of the optimal values is accomplished here through an iterative search process.
2DOF Coupled Building Evaluation Model The evaluation model for the 2DOF coupled building system is developed. The coupled building model presented in this section is comprised of two SDOF structures, with mass (m1 and m2), stiffness (k1 and k2) and damping (c1 and c2) associated with each structure, and a spring and damper (k3 and c3) located in the coupling link between the two masses. This system, and the 2DOF model representing it, are depicted in Figure 2.2. The system parameters are assigned such that the 2DOF system represents a first mode analysis of two typical tall buildings. The stiffness and damping for the ith building are related to the natural frequency ω oi and damping ratio ζ i by 2
k i = ω oi m i
(2.1)
c i = 2ζ i ω oi m i
(2.2)
The stiffness and damping of the connector link, k3 and c3, are set by the designer. Building 1 is intended to represent a 50-story building and building 2 represents a 45-story building. The buildings are considered to be connected at the top floor of the 45story building. The lumped masses are determined using the eigenvector method (Seto, et al. 1987), so that the displacement of the lumped masses have physical meaning as the displacement at the coupling link of 50- and 45-story high-rise buildings bending in flexure. 22
x1
x2 k3 m2
m1 c3 k2
k1 c1
c2
..
xg(t) Building 1
Building 2
..
xg(t)
x1
k1
x2
k3
m1
k2
m2
c1
c3
c2
Building 1
Building 2
Figure 2.2: 2DOF coupled building system undergoing ground excitation and the resulting 2-DOF model.
The first natural frequencies of building 1 and building 2 ( ω oi ) are set to 0.200 and 0.247 Hz, respectively. The SDOF building model stiffness is then determined from Equation (2.1). Both buildings have damping ratios 2% of critical damping ( ζ i = 0.02 ). The physical parameters of the 2DOF coupled building system are given in Table 2.1. The equations of motion for the 2DOF system shown in Figure 2.2 are Mx˙˙ + Cx˙ + Kx = – G x˙˙g m1 0
where M =
0 m2 x = x1 x2
T
,C =
c1 + c3
–c3
–c3
c2 + c3
,K =
.
23
(2.3) k1 + k3 –k 3
–k 3
, G = M 1 , and k2 + k3 1
TABLE 2.1: DETAILS OF 2DOF COUPLED BUILDING SYSTEM Building 1 (50-STORY BUILDING) (i = 1 )
Building 2 (45-STORY BUILDING) (i = 2 )
Mass ( m i )
2.7612x107 kg
1.8401x107 kg
Stiffness ( k i )
4.3629x107 N/m
4.4315x107 N/m
Damping Ratio ( ζ )
2%
2%
0.200 Hz
0.247 Hz
Natural Freqs.
1st mode
The second order differential equation of Equation (2.3) can be written as a first T
order linear time-invariant system with state vector z = x x˙
T T
as
z˙ = Az + Bx˙˙g
where A =
0 –1
I –1
–M C –M K
0
and B =
–1
(2.4)
.
–M G
The input to this system is a ground acceleration. For the 2DOF analysis a white noise will be used as the ground excitation. This excitation possesses the spectral energy content to excite both buildings, which we might expect from a seismic excitation. The evaluation responses are the displacement and the velocity relative to the ground, and the absolute acceleration of mass 1 and mass 2, given by y e = Cz + Dx˙˙g I 0
(2.5)
0 I
0 where C = and D = 0 . –1 –1 0 –M C –M K The solution of the state vector for the first order linear Equation (2.4) is t
z ( t ) = Φ ( t, 0 )z ( 0 ) + ∫ Φ ( t, τ )Bx˙˙g ( t ) dτ 0
24
(2.6)
where the state transition (or principal) matrix for time invariant systems is Φ ( t, τ ) = e
A(t – τ)
and the initial conditions are given by z ( 0 ) .
2DOF Coupled Building Root Locus Analysis A root locus analysis of the 2DOF coupled building system is performed using the eigenvalues of the state space A matrix defined in Equation (2.4) and connector damping and stiffness are varied. By observing the shift of the coupled system poles as functions of the coupling stiffness and damping, the physical transformation that occurs as the two structures become increasingly coupled can be examined. As the stiffness and damping of the coupling member increases, the two uncoupled SDOF structures become a coupled 2DOF system. As the stiffness or damping continue to increase and become significantly large, the two buildings become effectively rigidly connected and behave as a single SDOF oscillator. The rigidly connected SDOF system will contain a single natural frequency located between the two natural frequencies of the two uncoupled SDOF structures. The poles of the rigidly connected coupled building system are the same for large connector stiffness or large connector damping. The difference between using a spring or damper in the coupling link is how (the path) and which poles (building 1 or building 2’s) move from the uncoupled poles to the rigidly connected poles. To observe this phenomenon the poles of the 2DOF coupled building system are examined for both an increase in connector stiffness and an increase in connector damping. As part of this analysis, first consider the characteristic equation of a SDOF sys2
2
tem, s + 2ζω o s + ω o = 0 , and the corresponding poles s = – ζ ω o ± jω o 1 – ζ 2 . Sketching the positive complex pole, as in Figure 2.3, the angle θ is observed to be related to the damping of the SDOF system by cos θ = ζ . For more critical values of damping ( ζ → 1 ), the angle θ approaches zero. Thus, the smaller the angle θ , the more damping is present in the SDOF system. The angle θ is a good measure of system damping. 25
positive complex pole
x
ωo 1 – ζ
ωo
2
θ ζω o
real
imaginary
Figure 2.3: Plot of positive complex pole of SDOF system.
Figure 2.4 is a root locus plot of the 2DOF coupled building system, where the four poles of the system are examined as connector stiffness and connector damping are increased. Both methods, increasing stiffness and increasing damping, have the same beginnings and ends, but the means by which they achieve these are very different. The difference is the path, and thus the angle θ , that the poles follow as the stiffness and damping in the coupling link are increased. As the stiffness increases, the poles of building 1, the taller, more flexible, structure (denoted “1” in Figure 2.4), are shifted predominantly away from the imaginary axis to become the poles of the rigidly connected system and the angle θ is slightly increased. The poles of building 2 (denoted “2” in Figure 2.4) are shifted away from the real axis and approach ± i∞ at large values of stiffness, whereby the angle θ is always increased. As the damping in the coupling link increases, the poles of building 1 are shifted and become the poles of the rigidly connected system. For a portion of this shift, the angle θ decreases; however, at some point θ begins to increase. The poles of building 2 move to the real axis and then approach ( – ∞, 0 ) on the real axis such that the angle θ is decreased to a value of zero. This analysis considers the effects of stiffness and damping independently on the poles of the coupled building system. When stiffness and damping are used jointly, in 26
enlarged view 1.8
1.7
k3 ^
imaginary
1.6
2 k3=c3=0
1.5
c3 ^
k3=c3=inf
1.4
c3 ^
1.3
1.2 −0.2
1
k3 ^ k3=c3=0
−0.18
−0.16
−0.14
−0.12
−0.1
−0.08
−0.06
−0.04
−0.02
0
real 2
1.5
1
imaginary
0.5
θ2
0
θ1
−0.5
−1
−1.5
−2 −2
−1.8
−1.6
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
real
Figure 2.4: Root locus plot of the 2DOF coupled building system as connector stiffness and connector damping is varied. some instances, adding stiffness can be beneficial in reducing unwanted responses of the system. This will be discussed in Section 2.3 in the context of designing the optimal connector stiffness and damping. A similar root locus analysis, observing the damping via root locus plots, has been reported for the actively controlled 2DOF system (Mitsuta, et al. 1994). For the active system it is determined that an optimal level of control force can be specified, that unlimited control force it not necessarily beneficial to increasing the damping of the coupled building system. This is similar to the optimal finite level of damping observed in this 2DOF passive coupled building analysis.
27
2DOF Coupled Building Transfer Function Analysis A transfer function analysis can provide further insight into the modal responses of each structure as the parameters of the coupling link are varied. From the worst case in an H ∞ sense, minimizing the resonant peaks of the transfer function are of interest. From an H 2 sense, the area under the transfer function is of interest. The transfer functions of the ground acceleration to the displacement, velocity and absolute acceleration of the two buildings as the connector stiffness and damping is varied are presented in Figure 2.5. As the stiffness in the coupling link is increased the natural frequencies of both buildings increase. The difference between the two frequencies does not increase monotonically for all values of the stiffness. In particular, for small stiffness increases in the coupling member, the first natural frequency increases faster than the second. However, as the stiffness becomes larger, the difference does increase monotonically. The increasing frequencies were observed in the root locus analysis with the poles moving away from the origin. The first natural frequency tends to a value between the two uncoupled natural frequencies (the natural frequency of the rigidly connected system). The second natural frequency continues to get very large, eventually above the significant frequency content of the ground excitation. As the damping in the coupling link is increased, the natural frequencies are initially observed to increase. As the buildings become more coupled, one of the resonant peaks dampens out, until only one resonant peak is observed for the rigidly connected system. The increase in damping was examined in the root locus analysis with one pole shifting towards the real axis and becoming critically damped. Increasing the stiffness has the effect of increasing the natural frequencies of the system. While this may be a valid technique to avoid inputs with a narrow energy content, this type of control may not be valid for seismic inputs with wide energy spectrums seen historically, and especially not for the white noise excitation assumed here.
28
H x 1 x˙˙g(ω)
H x 1 x˙˙g(ω)
30
30
vary stiffness 10
10
magnitude (dB)
20
magnitude (dB)
20
0
−10
0
−10
Building 1 Building 2 Rigid Connection
−20
−30
vary damping
0
0.05
0.1
0.15
0.2
Building 1 Building 2 Rigid Connection
−20
0.25
0.3
0.35
0.4
0.45
−30
0.5
0
0.05
0.1
0.15
frequency (Hz)
0.2
0.25
H x˙1 x˙˙g(ω)
10
10
magnitude (dB)
20
magnitude (dB)
0.45
0.5
vary damping
20
0
0
−10
−10
−20
−20
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
−30
0.5
0
0.05
0.1
0.15
frequency (Hz)
H
a
x˙˙1 x˙˙g
0.2
0.25
0.3
0.35
0.4
0.45
0.5
frequency (Hz)
(ω)
H
30
a
(ω)
x˙˙2 x˙˙g
30
vary stiffness
20
vary damping
20
10
10
magnitude (dB)
magnitude (dB)
0.4
30
vary stiffness
0
0
−10
−10
−20
−20
−30
0.35
H x˙2 x˙˙g(ω)
30
−30
0.3
frequency (Hz)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
−30
0.5
frequency (Hz)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
frequency (Hz)
Figure 2.5: Transfer function from the ground acceleration to displacement, velocity and absolute acceleration as connector stiffness and connector damping is varied. 29
0.5
2DOF Coupled Building RMS Analysis Insight is also gained through observing the time-domain responses of the buildings, including displacements, velocities and absolute accelerations. Root mean square (RMS) responses are useful in determining the effectiveness of various control system parameters on the overall response of the system to a random excitation. In Appendix A, a differential equation for the covariance of the states of the first order linear state space system is derived in the form of the Lyapunov equation, where the mean square value of the states is determined as the solution to this equation. In this section, the mean square values of the output responses of the 2DOF coupled building system are examined as the connector stiffness and damping are varied. The RMS displacements, velocities and absolute accelerations of the two buildings as functions of connector stiffness and damping are shown in the Figure 2.6. These plots show definite minimums for particular values of damping and zero stiffness (when considering the respective maximum responses over both buildings). For the response of building 1, the taller building, it is observed that zero stiffness and finite damping results in the minimum of the evaluation responses. When considering building 2, the shorter stiffer structure with responses less than that of building 1, a finite amount of stiffness and damping is required to reduce the responses to minimum values. Also, the minimum RMS responses for each of the three responses considered are not achieved at the same value of stiffness or damping. The optimal stiffness and damping values are indicated on Figure 2.6 to reduce the response in each plot, to illustrate this point. The designer must select values of stiffness and damping that reduce building RMS responses to meet the prescribed evaluation criteria.
30
RMS displacement of building 1 x 10
5.5
10
4.5 9
5
9
RMS displacement of building 2
6
x 10
6
10
6
8
8
min. response
7
7
6
6
4
5
5
4
4
2
4 7
6.5
3
7
7
6
7.5 8
7 7.5 8 0
8.5 2
1
2 3.5
3
1
8
4
5
6
7
8
9
0
10
4
5
5.5
0
4.5
1
2
3
4
6
5
4.5 7
6
8
9
10
stiffness
x 10
RMS velocity of building 1
6
4
4
4.5
8.5
8.5
stiffness
x 10
min. response
3.5
7.5
7
6.5 1
10
5
6.5 6
5.5
3
0
4
damping
5. 5
damping
4 6
RMS velocity of building 2
6
7
10
9
6
x 10
x 10
9
6 6
8
8
8
7.5
7
5
8
9.5 10
9
9.5 10
10.5
0
1
11 3
2
10.5 6
11.5 8
7
9
0
10 6
6 8. 9 5 1
2
10.5
9.5
6
7
6.
6.5 7
8
9
10 6
x 10
11
8.5
8 8
4
5
8
4 10
9
5
11.
3
1
11
11.512 12.5 13 1
11.5 12 12.5 13
11
13.5
2
14 5 14. 3
4
3
12
9.5 10.5
12
12.5 13
13.5
5
stiffness
14 14.5 15
6
2 9.5 13.5
8
9
9 8.5
8
10
14 14.5 15
10.5 12 11.5
1
15 5
7
min. response
8
12
8.5
.5
11
damping
10
.5
6 11
0.5
8.5
8.5
7
9.5
5
9
8
10
min. response
6
x 10
9
9
7
0
5
6
8
1
4
RMS absolute acceleration of building 2 10
9
10
3
stiffness
x 10
2
6
6.5 7 75
8
0
x 10
6
damping
5.5
17
11
11
11 5 5
4
5.5
6.5
10.5
RMS absolute acceleration of building 1
0
6
10
stiffness
10
6
2
5
9
8.5
8
3
10
9.5
7.5
1
9.5
9
8.5
5.
2
4
8
7
min. response 5
3
0
9
7.
5.5
5.5 5
5.
8.5
5
4
damping
6
7
6.5
damping
7
min. response
6
6
0
10
0
11
12.5 13
13.5 14 1
2
3
9.5 10
8
10.5 12 11.5 4
8.5
8.5
9
5
11 6
9.5 10
9
10.5 7
9.5 10
8
9
stiffness
6
x 10
Figure 2.6: RMS responses over a range of connector stiffness and connector damping.
31
10 6
x 10
2.3 2DOF Coupled Building Optimal Passive Control Strategy Previously, the effects of varying the connector stiffness and damping on the eigenvalues, transfer functions, and RMS responses were considered. In this section, the optimal passive design will be discussed. Now that the effect of the coupling stiffness and damping on the coupled building system is known, consider the design of the optimal passive system for the two SDOF building models. First consider an undamped coupled building system and the P and Q theory to design the optimal connector link. Next, the more complex damped coupled building optimum coupling link parameters are examined.
Undamped Coupled Building System
For the undamped coupled building system, P, Q theory is used to design the optimal passive controller to reduce the resonant peaks (Seto, 1998). For three undamped SDOF structures, the method of P, Q theory has been extended to P, T, Q theory (Iwanami, et al. 1986). Neither the zero connector stiffness and damping, the maximum (infinite) connector stiffness or the maximum (infinite) connector damping produce optimal systems. Neither result in systems with any damping. The responses are unbounded at the natural frequencies, which is highly undesirable. The optimal values for the connector stiffness and damping are somewhere between zero and infinity. This trend was seen in previous root locus, transfer function, and RMS response analyses. As before, the connector stiffness is used to vary the natural frequencies and the connector damping is used to vary the magnitude of the transfer function. When the connector stiffness is increased, the natural frequencies of the system are also increased, beginning at the uncoupled natural frequencies and approaching the higher fused natural frequencies. The bounds for the natural frequencies of the system as the con-
32
nector
stiffness
is
varied,
and
damping
set
to
zero,
are
ω o1 ≤ ω 1 ( k 3, 0 ) ≤ ( k 1 + k 2 ) ⁄ ( m 1 + m 2 ) and ω o2 ≤ ω 2 ( k 3, 0 ) ≤ ∞ . When the connector damping is increased, the magnitude of the transfer function at the resonant peaks, frequencies ω 1 ( k 3, c 3 ) and ω 2 ( k 3, c 3 ) , decreases. The bounds for the magnitude of the transfer function as the connector damping is varied lie between the magnitude of the undamped and the fused transfer functions. The method of the theory of P, Q requires observing two points, called P and Q, that are the intersection of the transfer functions of the two uncoupled buildings with the transfer function of the rigidly connected coupled building system. As the stiffness of the connector is changed, the locations of P and Q are varied and thus the magnitude of P and Q are varied. The points P and Q, however, remain invariant to the connector damping (for the undamped buildings). When the stiffness is zero, point P is higher than point Q ( magP ( 0 ) > magQ ( 0 ) ) for the displacement responses. As the stiffness is increased, the magnitude of point P is increased and the magnitude of point Q is decreased. So point P, when considering displacements, is the larger magnitude point and can not be reduced by increasing stiffness. When considering absolute accelerations, point Q is of larger magnitude, thus increasing the connector stiffness can bring down Q, while raising P, to equal magnitudes. However, P, Q theory is defined with respect to displacements, so the discussion regarding absolute accelerations will be given in subsequent sections. The magnitude of the transfer function at the frequencies of points P and Q cannot be reduced by varying the damping. As was previously stated, points P and Q are invariant to connector damping. The magnitude at all other points can be significantly reduced by varying the connector damping such that point P, or point Q, is a maximum on the transfer function curve. Seto (1998) has developed the analytical solutions for the optimal damping ratio of the connector to reduce the resonant peaks of the transfer functions of displacement to ground excitation. The P, Q theory developed for the coupled building problem is very 33
specifically defined and somewhat restrictive in that it is limited to reducing displacement responses, reducing the H ∞ norm of the transfer function, and requiring a specific mass and frequency ratio of the two buildings. The P, Q theory requires first, an optimal mass and stiffness relationship must be determined to insure that points P and Q are of equal heights. Here, the natural frequencies should be fixed, and the masses will be varied. The optimal mass ratio is thus defined, in terms of the natural frequencies, as m ω o1 µ = ------2 = -------m1 ω o2
(2.7)
Seto’s closed form optimal damping ratio, to reduce the resonant peak of the displacement transfer function of building 1 or of building 2, are as follows: 1
ζ opt =
2
3
4
2 – 4 + 4µ – 3µ – 2µ – µ -------------------------------------------------------------- or ζ opt = 2 3 8µ ( 1 + 3µ + 3µ + µ )
2
3
4
1 + 2µ – 3µ – 4µ + 4µ ------------------------------------------------------------2 3 8µ ( 1 + 3µ + 3µ + µ )
(2.8)
i
where ζ opt = c 3 ⁄ ( 2 k 2 m 2 ) is the optimal damping ratio of the coupling link to reduce the peak of the displacement transfer function of the ith building to ground acceleration.
Undamped Coupled Building System Numerical Example
Consider the mass and stiffness of the building 1, as defined previously in Table 2.1. Assume also, the frequency of building 2 remains the same as in Table 2.1, 0.247 Hz. The new mass and stiffness of building 2, from Equation (2.7), are optimally set 7
7
to m 2 = 2.2366 ×10 kg and k 2 = 7.1460 ×10 N/m. For the 2DOF building system considered in this dissertation, with no associated structural damping, the optimal connector stiffness is zero and the optimal damping, from Equation (2.8), to reduce the peak of the displacement transfer functions are
34
k3 = 0
(2.9) 6
c 3 = 2ζ opt k 2 m 2 = 5.5746 ×10 N-sec/m
(2.10)
Applying this solution, the optimal coupled building system from ground acceleration to displacement for the 2DOF undamped coupled building system is shown in Figure 2.7. Although the closed-form solutions of the P, Q point theory for coupled building is very attractive, in that it is a closed-form solution, the method is restrictive in requiring particular building properties and in the responses optimized. A more general approach will be examined for the damped coupled building system. 30
20
P
Q
magnitude (dB)
10
0
−10
−20
−30
Bldg 1:Optimal Bldg 2:Optimal Bldg 1:Uncpld Bldg 2:Uncpld Rigid Connection 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
frequency (Hz)
Figure 2.7: Optimal transfer functions from ground acceleration to displacement for the 2DOF undamped coupled building system.
Damped Coupled Building System
Now consider two SDOF buildings that are inherently damped, as previously studied. Together the buildings form a 2-DOF system. For a given building configuration (given structural mass, stiffness and damping), the natural frequencies of the system ( ω 1 ( k 3, c 3 ) and ω 2 ( k 3, c 3 ) ) are functions of the connector stiffness and damping. 35
When the connector stiffness is increased, the natural frequencies of the system are also increased, beginning at the uncoupled natural frequencies and approaching the higher rigidly connected natural frequencies. The bounds for the natural frequencies of the system as the connector stiffness is 2
2
ω o1 ( 1 – ζ 1 ) ≤ ω 1 ( k 3, c 3 ) ≤ [ ( k 1 + k 2 ) ⁄ ( m 1 + m 2 ) ] ( 1 – ζ 1 ) 2
ω o2 ( 1 – ζ 2 ) ≤ ω 2 ( k 3, c 3 ) ≤ ∞ where the undamped natural frequency of building 1 is ω o1 = undamped natural frequency of building 2 is ω o2 =
(2.11) k 1 ⁄ m 1 and the
k 2 ⁄ m2 .
When the connector damping is increased the magnitude of the transfer function (in this case the transfer function of the absolute acceleration of the building to a ground excitation) at the resonant peaks, frequencies ω 1 ( k 3, c 3 ) and ω 2 ( k 3, c 3 ) , decreases. The bounds for the magnitude of the transfer function as the connector damping is varied lie between the magnitude of the undamped and the rigidly connected transfer functions. Connector damping does affect the natural frequencies of the system, but the effect is not so dramatic, and thus not focused on in attempting to understand this process. Again points P and Q can be observed, as in the undamped coupled building system. As the stiffness of the connector is changed, the locations of P and Q are varied and thus the magnitude of P and Q are varied. Important to note is that the points P and Q no longer remain invariant to the connector damping for the case of the damped building. The magnitude of points P and Q are functions of both the connector stiffness and the connector damping. Again, damping does not have a dramatic effect on the placement of P and Q, however, it does effect the magnitude of the transfer functions. The method of theory of P, Q can not be applied here, as P and Q are no longer invariant to connector damping. For damped structures other methods to determine optimal connector stiffness and damping values have been considered including Genetic Algo-
36
rithms (GAs) (Sakai, et al. 1999) and search techniques that consider a range of stiffness and damping values. This study will employ the latter method. The optimal connector stiffness and connector damping for this study are defined such that the system results in minimum absolute acceleration of the building to a random base excitation of constant spectral density S o . This objective is a different objective, an H 2 sense, from the undamped coupled building design. The optimal coupled building system is realized when the magnitude of transfer function of the building absolute accelerations to ground acceleration is minimized over all frequencies. A MATLAB program is developed that computes the absolute RMS accelerations of buildings 1 and 2 and after a numerical search determines the values of stiffness and damping that result in the smallest maximum absolute RMS accelerations. In other words, the coupling stiffness k3 and damping c3 that give min max σ a x˙˙i k 3, c 3 i are determined, where σ
a
x˙˙i
(2.12)
is the absolute RMS acceleration of the ith building.
Damped Coupled Building System Numerical Example
Consider again the mass and stiffness of the building 1, as defined previously in Table 2.1, and the mass and stiffness of building 2 as defined previously in the Undamped Coupled Building System Numerical Example. The optimal stiffness and damping values, to reduce the maximum absolute RMS acceleration of the system, are found using the above mentioned algorithm to be 5
k 3 = 3.1429 ×10 N/m 6
c 3 = 4.7959 ×10 N-sec/m
37
(2.13) (2.14)
Applying these values, the optimal poles of the 2DOF damped coupled building system, are shown in Figure 2.8. The optimal transfer functions of ground acceleration to absolute accelerations for the 2DOF coupled building system are given in Figure 2.9. The optimal RMS responses of the 2DOF coupled building system are shown in Figure 2.10. Although adding connector damping appears from the root locus in Figure 2.8 to increase the damping in both structures, recall that for this system the optimal stiffness and damping levels are such that the maximum absolute RMS acceleration is minimized. If the damping were increased beyond the level identified in Equation 2.14, the maximum absolute RMS acceleration (building 2) would actually increase, as shown on the contour plot in Figure 2.10.
1.8
1.7
k3 ^ optimal point
imaginary
1.6
2 k3=c3=0
1.5
c3 ^
optimal point
1.4
c3 ^
1.3
1.2 −0.2
k3=c3=inf 1
k3=c3=0 −0.18
−0.16
−0.14
−0.12
−0.1
−0.08
−0.06
−0.04
real 2
enlarged view
1.5
1
optimal points
imaginary
0.5
0
−0.5
−1
−1.5
−2 −2
−1.8
−1.6
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
k3 ^
0
real
Figure 2.8: Optimal poles for the 2DOF coupled building system. 38
−0.02
0
30
20
magnitude (dB)
10
0
−10
−20
−30
Bldg 1:Optimal Bldg 2:Optimal Bldg 1:Uncpld Bldg 2:Uncpld Rigid Connection 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
frequency (Hz)
Figure 2.9: Optimal transfer functions of ground acceleration to absolute accelerations for the 2DOF coupled building system. Max. absolute RMS accelerations of building 1
Max. absolute RMS accelerations of building 2 6
6
10
x 10
10
x 10
10.5
10
9
9.5
9
11
9
8
8.5
9
8
7
7
6
6
8.5
8.5
.5
11
11
9.5
12
4
3
2
1
11.5 12 12.5 13
11
10.5
11.5 12 12.5 13.5 0 13 0 1
14 2
13.5
3
4
14 15 5
8
stiffness
8.5
2 13.5
14
8
1
12 9
0
10
8
9.5
15 15 5 7
8
10
14.5
6
8
8
9
13
13
14.5
optimal point
3
12 12.5
9.5
5
4
11.5
11
.5
10
10
8.5
damping
10
.5
10
optimal point
5
9
damping
8
13 14 13.5 0
11 11.5 12.5 1
9 10.5
8.5
10
2
3
8.5
4
10.5 5
9
9 10
9.5 11 11 5
12
6
9.5 7
stiffness
6
x 10
8
10.5 9
10 10 6
x 10
Figure 2.10: Optimal RMS of 2DOF coupled building system. The frequency and RMS results of the passive control strategy are summarized in Tables 2.2 and 2.3. The resonant peaks of all transfer functions are reduced significantly, from 10.6 to 18.5 dB, a reduction, in a linear scale, to 22%-30% of the uncoupled resonant peaks. The RMS responses are also attenuated significantly for the 2DOF system considered. RMS responses of displacements, velocities and absolute accelerations, are reduced to 60%-67% of the uncoupled responses. 39
TABLE 2.2: TRANSFER FUNCTION RESULTS OF PASSIVE CONTROL STRATEGY FOR THE 2DOF COUPLED BUILDING SYSTEM RESPONSE
UNCOUPLED
PASSIVE STRATEGY (% REDUCED)
mag ( x i ) (dB)
building 1 building 2
24.0 20.3
13.7 (10.3%) 10.6 (9.7%)
mag ( x˙ i ) (dB)
building 1 building 2
26.0 24.1
16.1 (9.9%) 13.5 (10.6%)
mag ( ˙˙ x i ) (dB)
building 1 building 2
28.0 28.0
18.5 (9.5%) 16.9 (11.1%)
TABLE 2.3: RMS RESPONSE RESULTS OF PASSIVE CONTROL STRATEGY FOR THE 2DOF COUPLED BUILDING SYSTEM RESPONSE
UNCOUPLED
PASSIVE STRATEGY (% REDUCED)
building 1 building 2
8.2035 5.9803
4.9569 (40%) 3.9931 (33%)
rms max ( x˙ i ) (m/sec)
building 1 building 2
10.3118 9.2806
6.5187 (37%) 5.8455 (37%)
rms max ( ˙˙ x i ) (m/sec2)
building 1 building 2
12.9723 14.4137
8.6292 (33%) 8.6289 (40%)
--
--
38,255
rms
max ( x i
f
rms
) (m)
(kN)
Active control for SDOF coupled building models has been shown to provide no significant further reductions in responses (Graham, 1994). For one reason, the additional information that the active system can utilize, the absolute responses as opposed to the relative responses of the passive strategy, is not adequate to result in significantly increased performance. To see the interesting and beneficial effects of active control, multi-degreeof-freedom (MDOF) building models are considered.
40
2.4 Multi-Degree-of-Freedom Coupled Building System The 2DOF system was sufficient to introduce the concept and mechanics of the coupled building system. However, a more detailed analysis is required to effectively compare active and semiactive damping control strategies to passive control. When considering the control of flexible high-rise buildings, higher mode participation comes into effect. To capture this participation, multi-degree-of-freedom (MDOF) models must be examined for the building models.
MDOF Coupled Building Evaluation Model The coupled building system consists of two dissimilar buildings with given height, mass, stiffness and damping properties. The buildings are connected with a single coupling, which may be either a passive or an active control device. The coupled building system is subjected to ground excitation to simulate a seismic event. The coupled building system considered in this section represents a flexible highrise coupled building system as shown in Figure 2.11. An evaluation model must be developed to reproduce the salient features of the coupled building system. High-rise buildings
f(t)
1 h1
m1
2
hc
m2
EI1
EI2
ζ1
ζ2
h2
.. xg(t)
y x
Figure 2.11: High-rise MDOF coupled building system.
41
are commonly modeled as cantilevered beams, and cantilevered beams are often modeled using the Galerkin (Cook, 1989) (see Appendix B) and finite element (Clough and Penzien, 1993) (see Appendix C) methods. The Galerkin method here uses mode shapes of the uncoupled beam as the trial functions to represent the behavior of the structure in the coupled building system. The finite element approach places a series of beam elements on top of one another, each beam element representing a story level. (The finite element method using weighted residuals does use the Galerkin method; here, however, the stiffness and mass matrices for the finite element method are computed directly from EulerBernoulli beam theory, not a weighted residual method.) The Galerkin and finite element methods are compared below for accuracy and efficiency. The equations of motion for the coupled building system, modeled using either the Galerkin or the finite element methods, can be written in terms of mass, stiffness, damping matrices and the generalized or physical coordinates Mq˙˙( t ) + Cq˙ ( t ) + Kq ( t ) = – Gx˙˙g ( t ) + Pf ( t )
where
M=
M1 0
, C =
0 M2
q(t ) =
q1 ( t ) q2 ( t )
C1 0
,
K=
0 C2
K1 0 0 K2
, G =
(2.15) G1 G2
, P =
P1
, and
P2
, and where M k , C k , and K k are the mass, damping, and stiffness matri-
ces of the kth building respectively, and G k and Pk are the loading matrices for the ground acceleration and coupling force for the kth building. These matrices are defined for the Galerkin method in Appendix B and for the finite element method in Appendix C. A linear time-invariant state space equation for the coupled buildings can be written, as the 2DOF system was, as z˙(t) = Az(t) + Bx˙˙g(t) + Ef ( t )
42
(2.16)
T
where the state is z(t) = q T ( t ) q˙ T ( t )
A=
0
I
–1
–1
–M K –M C
0
,B=
–1
and the coefficient matrices are defined as
0
, and E =
.
–1
–M G
M P
Three outputs are identified for the system described in Equation (2.16): evaluation outputs, measured outputs, and connector outputs. Evaluation outputs y e(t) , includes the absolute acceleration and interstory drift ratio for each story of both buildings, are used to evaluate the performance of the system. The measured outputs y m(t) , consisting of the absolute acceleration of each building at the location of the coupling link and the relative displacement across the coupling link, are used as input for the active and semiactive control strategies. The connector output y c(t) , consisting of the relative velocity of the two buildings at the connector link, is used by the passive control strategy to determine the control force and by the semiactive control strategy to determine the dissipative nature of the control force. These output are y e(t) = x˙˙a (h ) x˙˙a (h ) d ⁄( ∆h ) d ⁄( ∆h ) 1 1 2 2 1 2 y m(t) = x˙˙a (h ) x˙˙a (h ) ∆x(h ) 1 c 2 c 3
T
T
= C e z( t ) + F e f ( t )
= C m z(t) + F m f ( t )
y c(t) = ∆x˙(h c) = C c z(t) + F c f ( t )
(2.17)
(2.18) (2.19)
where ∆h is the building story height, h k = ∆h 2 ( ∆h ) ... n ( ∆h ) is a vector of the story heights for the kth building with n stories, d k = x k(h k) – x k(h k – ∆h) is the interstory drift vector for the kth building, ∆x(h c) = x 2(h c) – x 1(h c) is the relative displacement of two buildings at the height of the coupling link, ∆x˙(h c) = x˙2(h c) – x˙1(h c) is the relative velocity of the two buildings at the height of the coupling link, and C e, C m, C c, F e, F m , and F c are defined appropriately for the Galerkin and finite element methods.
43
Passive Control Strategy Numerous studies have been completed employing passive coupling strategies to tall flexible buildings with positive results. The passive control strategy is implemented here by placing a linear viscous damping element between the two buildings at the roof of the shorter buildings. The passive control force in the coupling link, f (t) , is given by f (t) = c y c(t)
(2.20)
where y c(t) = ∆x˙(h c) = x˙2(h c) – x˙1(h c) as defined in Equation (2.19). The damping coefficient, c is varied to find the optimal coupling link damping values that minimize the measure of performance defined later in this section.
Comparison of Galerkin and Finite Element Methods Frequently tall buildings are modeled as cantilevered beams. The Galerkin method (see Appendix B) and the finite element method (see Appendix C) are used to model the MDOF coupled building model, consisting of two cantilevered beams connected by a coupling link to each other at some point along the height of the beams. The two models are compared to determine the lowest order model that effectively models the coupled building system. The convergence of the models’ prediction of the undamped natural frequencies is observed as the number of shape functions and elements increases. The damping of 6
the connector is selected as c = 2.0462 ×10 , which provides a reasonable level of coupled building interaction. Figure 2.12 shows the convergence of natural frequencies to a final value as the number of modes (for the assumed mode method) and the number of elements (for the finite elements method) are increased. The Galerkin method can provide reasonably accurate estimates of the first three natural frequencies requiring significantly fewer degrees-of-freedom to obtain this accuracy than the FE model. Thus, the Galerkin method will be employed to model flexible buildings for the remainder of this study. Additionally, modeling each building with 5 44
Building 1
Building 2 0.3
0.25 Assumed Modes Galerkin Finite Element
0.29
Mode 1
0.23
0.28
0.22
0.27
natural frequency (Hz)
natural frequency (Hz)
0.24
0.21
0.2
0.19
0.18
0.26
0.25
0.24
0.23
0.17
0.22
0.16
0.21
0.15
0.2
2
4
6
8
10
12
14
16
18
Mode 1
20
2
4
6
8
1.3
1.6
1.29
1.59
Mode 2
1.27
1.26
1.25
1.24
1.23
10
12
14
16
18
1.53
1.5
20
4
6
8
10
number of modes/elements
3.6
14
16
18
20
4.4
4.39
Mode 3
3.58
3.57
3.56
3.55
3.54
3.53
4.37
4.36
4.35
4.34
4.33
3.52
4.32
3.51
4.31
6
8
10
12
14
Mode 3
4.38
natural frequency (Hz)
natural frequency (Hz)
12
number of modes/elements
3.59
3.5
20
1.54
1.51
8
18
1.55
1.52
6
16
1.56
1.21
4
14
1.57
1.22
1.2
12
Mode 2
1.58
natural frequency (Hz)
natural frequency (Hz)
1.28
10
number of modes/elements
number of modes/elements
16
18
4.3
20
number of modes/elements
6
8
10
12
14
16
18
number of modes/elements
Figure 2.12: Convergence of undamped natural frequencies for Galerkin and Finite Element methods of the first three modes of each building.
45
20
shape functions is sufficient to insure that the maximum absolute RMS acceleration of the system to the filtered white noise has converged to less than 1% error for each the uncoupled, optimal passive, and optimal active (as defined in Chapter 3) coupled building systems. Employing 5 shape functions for each building results in a 10 degree-of-freedom model for the coupled building system.
Coupled Building Ground Excitation The ground excitation is modeled as a filtered white noise corresponding to the Kanai-Tajimi spectrum with local ground conditions given by ω g = 12 rad/sec and ζ g = 0.6 (Soong and Grigoriu, 1993). The transfer function representation of the KanaiTajimi filter in the Fourier domain is 2
2ζ g ω g jω + ω g - . H x˙˙g v(ω) = --------------------------------------------------2 2 – ω + 2ζ g ω g jω + ω g
(2.21)
Approaching zero frequency, historical earthquakes have low energy content. In order to better represent the frequency content of seismic excitations at low frequencies, a filter is prepended to the Kanai-Tajimi filter. The parameters of this second filter are selected as ω p = 2.2 rad/sec and ζ p = 0.6 (Clough and Penzien, 1993). The transfer function representation of this filter in the Fourier domain is 2
–ω - . H vw(ω) = --------------------------------------------------2 2 – ω + 2ζ p ω p jω + ω p
(2.22)
The white noise w(t) is a zero-mean ( E [ w(t) ] = 0 ) Gaussian white noise process with autocorrelation E [ w(u)w(t) ] = 2πS o δ ( u – t ) . Figure 2.13 shows that the frequency characteristics of the Kanai-Tajimi filter defined above captures the pertinent frequency content of four major seismic events: El Centro (the N-S component recorded at the Imperial Valley Irrigation District substation in El Centro, California, during the Imperial Valley, California earthquake of May, 18, 46
1940), Hachinohe (the N-S component recorded at Hachinohe City during the Takochioki earthquake of May, 16, 1968), Northridge (the N-S component recorded at Sylmar County Hospital parking lot in Sylmar, California, during the Northridge, California earthquake of January 17, 1994), and Kobe (the N-S component recorded at the Kobe Japanese Meteorological Agency (JMA) station during the Hyogo-ken Nanbu earthquake of January 17, 1995). 1
Power Spectral Density
10
0
10
−1
10
ElCentro Hachinohe Northridge Kobe −2
10
Kanai−Tajimi 0
1
10
10
Frequency [Hz]
Figure 2.13: Power spectral density of ground excitation.
In addition to the frequency content, consider the intensity of the ground acceleration. The ground excitation is a zero mean Gaussian process and can be characterized by its root mean square (RMS), or standard deviation. The RMS of the ground acceleration is determined for the larger magnitude portions of the historical earthquakes. Figure 4 shows the time histories of each of the historical earthquakes and identifies the portion of the time history used to calculate the RMS ground acceleration. The RMS ground accelerations for the El Centro and Hachinohe earthquake records are determined to be 0.65 and 0.41 m/sec2, respectively. The RMS ground accelerations for the Northridge and Kobe earthquakes are determined to be 1.69 and 1.80 m/sec2, respectively. 47
From the analysis of RMS ground accelerations of historical earthquakes, three levels of intensity of the ground acceleration are chosen. The first level represents far-field El Centro and Hachinohe type earthquakes and is set to 0.5 m/sec2. The second level represents near-field Northridge and Kobe type earthquakes and is set to 1.75 m/sec2. The third level of RMS ground acceleration is considered as an upper bound and is set to 3.27 m/sec2. El Centro
10
0
−10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
Hachinohe
10
0
−10
Northridge
10
0
−10
Kobe
10
0
−10
Figure 2.14: Estimating RMS ground motions from historical records, where the bold section defines the portion of the earthquake used for the RMS calculation.
Measure of Performance Root mean square (RMS) responses are useful in determining the effectiveness of various system parameters on the overall response of the system to a random excitation. 48
The covariance of the state vector and output of the first order linear state space system defined in Equations (2.16)-(2.17) can be found using the Lyapunov equation (as shown in Appendix A). An optimal semiactive damping control strategy is applied to this system. For the uncoupled, passive and active cases, RMS values for the linear systems can be determined analytically via the Lyapunov equation. For the nonlinear semiactive damping case, the RMS values must be determined via simulation. As stated earlier, the responses of the system are stationary, and, in fact, they are also ergodic. Ergodicity demands that the ensemble average (mean) is equal to the time average. The RMS responses are determined as the square root of the mean squares of a discrete series of time values n
rms ye
=
1 2 --- ∑ y e ( t i ) n
(2.23)
i=1
where y e is the evaluation output defined in Equation (2.17), t i = i ( ∆t ) , where ∆t is the time step of the simulation, and n is sufficiently large.
2.5 Chapter Summary In this chapter, a 2DOF coupled building model is examined to provide insight into the effect of passively coupling two structures on the dynamics of the coupled system. It is observed from the 2DOF model, as has been noted in previous research (Seto, 1998), that to add damping to the coupled building system, the viscous damping of the coupling link should be increased to some optimal, yet finite, value. Adding stiffness to the coupling link only serves to shift the frequencies of the coupled building system. Additionally, a MDOF coupled building model is developed. The buildings are modeled using the Galerkin method, where the mode shapes of the uncoupled cantilevered beam are used to represent the behavior of the tall coupled building. A ground excitation is modeled as a filtered white noise process corresponding to the Kanai-Tajimi spectrum. 49
Maximum absolute RMS story acceleration and interstory drift ratio are identified as measure of performance for the coupled building system. A MDOF coupled building model, as developed in the latter part of this chapter, can provide further insight into coupled building considerations. The coupled building configuration, including the relative height, mass, and stiffness of the two buildings, and the height of the coupling link, is important in the design of a coupled building control strategy, as it can significantly effect the performance of the system. Chapter 3 proposes some new control strategies and examines analytical studies on the performance of coupled building control using the MDOF coupled building model.
50
CHAPTER 3: COUPLED BUILDING CONTROL: ANALYTICAL STUDIES
This chapter details the analytical studies on the coupled building control problem. Two coupled building control strategies are proposed in this chapter: an active control strategy employing H2/LQG control with absolute building acceleration and actuator displacement feedback, and a semiactive control strategy employing clipped optimal H2/ LQG control. The effect of building configuration on coupled building control is examined, where building heights, connector location, building mass, and building stiffness are varied. The efficacy of semiactive control for the coupled building problem is examined. A low-rise coupled building system is also considered to ensure the results shown for highrise structures are similar for their low-rise counterparts.
3.1 Coupled Building Control Strategies The coupled building control strategies are applied to the high-rise multi-degreeof-freedom (MDOF) coupled building system developed in Section 2.4, and shown again here in Figure 3.1. The evaluation model for the coupled building system is obtained using the Galerkin method, where the response of the kth building is assumed to be represented by the finite series T
x k ( y, t ) = f k (y)q k ( t )
(3.1)
where f k is a vector of trial functions for the kth building, as defined in Appendix B as the mode shapes of a cantilevered beam, and q k is a vector of generalized coordinates of the kth building. 51
1
2 f(t)
h1
m1
m2 hc
(EI)1
h2
(EI)2
ζ1
ζ2 .. xg(t)
y x
Figure 3.1: High-rise MDOF coupled building system for analytical studies.
The combined equations of motion for the coupled building system, as previously defined in Equation (2.15), are Mq˙˙( t ) + Cq˙ ( t ) + Kq ( t ) = – Gx˙˙g ( t ) + P f (t)
(3.2)
where the mass, damping and stiffness matrices are defined as
M=
M1 0
,
C =
0 M2
q(t ) =
q1 ( t ) q2 ( t )
C1 0 0 C2
,
K=
K1 0
,
G =
0 K2
G1 G2
,
P =
P1
,
P2
,
hk
where M k = m k ∫ f k ( y )f k ( y ) dy , C k = M k F k C k F k , T
–1
0 hk
hk
∂2 T ∂2 T K k = ( EI ) k ∫ 2 f k ( y ) 2 f k ( y ) dy , G k = m k ∫ f k ( y ) dy , and P k = f k(h c) , ∂x 0 ∂x 0
52
and
2ζ 1, k ω 1, k 0
Ck =
0
0
0
2ζ 2, k ω 2, k 0
0
0 0
0 0
, and ζ i, k and ω i, k are the modal damping
... 0 0 2ζ n, k ω n, k
ratio and the undamped natural frequency, respectively, for the ith mode of the kth building. For control purposes, the equations of motion are written as the linear time-invariant state-space equation, as previously defined in Equations (2.16) through (2.19), z˙(t) = Az(t) + Bx˙˙g(t) + E f (t)
(3.3)
y e(t) = C e z(t) + D e x˙˙g(t) + F e f (t)
(3.4)
y m(t) = C m z(t) + D m x˙˙g(t) + F m f (t) + v
(3.5)
y c(t) = C c z(t)
(3.6) T
where the states are z(t) = q T ( t ) q˙ T ( t )
. The evaluation output y e(t) is the absolute
acceleration and interstory drift over the height of both buildings, the measured output y m(t) is the absolute acceleration of both buildings and the relative displacement of the buildings at the location of the coupling link, the connector output y c(t) is the relative velocity at the location of the coupling link, and the coefficient matrices are defined as
A =
0
I
–1
–1
–M K
B =
–M C –1
Ce =
,
– F stories M K
–1
,
–M G –1
– F stories M C
∆ stories F stories
–1
0
F e = F stories M P , C m = 0
0 –1
,
M P
–1
D e = 1 – F stories M G , 0
,
0
–1
– F hc M K ∆ hc F h
E =
–1
– F hc M C 0
c
53
–1
, D = 1 – Fhc M G , m 0
–1
T
F m = F hc M P , 0
∆ stories =
and
∆ stories,1
0
0
∆ stories,2
∆hc Fh
Cc = 0
,
∆ stories,i
,
where
F stories =
c
1 0 = –1 1
0 0 , ... 0 0 0 –1 1
f 1 (h 1)
,
T f 2 (h 2)
T
F hc =
f 1 (h c)
,
and
T f 2 (h c)
∆ hc = – 1 1 , and where ∆h is the building story height, h k = ∆h 2 ( ∆h ) ... n k ( ∆h )
T
is a vector of the story heights for the kth building with nk stories, and h c is the height of T
the coupling link. Note here the notation f k (h k) = f k(1∆h) f k(2∆h) ... f k(n k ∆h)
T
.
The coupled building control strategies define the second input, f (t) , the force of the coupling link, defined by the passive, active, or semiactive control. The passive control strategy, as defined in Section 2.4, is to serve as a baseline against which the semiactive damping control strategy is compared. Additionally, comparison with the fully active control strategy is useful as it bounds the achievable performance. The proposed active and semiactive control strategies are identified subsequently.
Active Control Strategy Employing Acceleration Feedback The active control forces are realized by a control actuator connecting the buildings at the height h c , as shown in Figure 3.1. H2/LQG control theory is used. A filter is augmented to the model of the structural system to shape the spectral content of the excitation in the H2/LQG design. The same Kanai-Tajimi filter used to shape the ground acceleration for evaluation purposes is used here in the control design. The evaluation outputs, y e(t) , as defined in Equation (3.4) are minimized using the cost function
54
J = lim
τ→∞
1 --- E τ
τ
∫ ( ye
T
2
( t )Qy e ( t ) + f ( t ) ) dt
(3.7)
0
where Q is a weighting matrix for the evaluation outputs. The active control force is proportional to the state estimate f (t) = – Kzˆ
(3.8)
where zˆ is an estimate of the state and K = EP , where E is defined in Equation (3.3) and P satisfies the algebraic Riccati equation A T P + PA – PEE T P + Q = 0
(3.9)
By varying the weighting matrix Q , a family of controllers that use varying force levels can be designed. The absolute accelerations and interstory drift ratios are weighted in this study through a Q matrix of the following form
Q =
α accel Ψ
0
0
α drift Ψ
(3.10)
where Ψ is a diagonal matrix to weight the story responses over the two buildings and α accel and α drift are coefficients to weight the relative importance of absolute acceleration and interstory drift responses. It is desired to place larger weight on the stories that experienced the larger responses. The diagonal elements of the matrix Ψ are set to the uncontrolled absolute root mean square (RMS) story accelerations of the two buildings. Other forms of Ψ considered were: a vector of ones; a vector of zeros with 1 at the story location of maximum absolute RMS acceleration; uncontrolled interstory drift responses of the two buildings; and vectors of each of the first few uncoupled mode shapes of both buildings. The matrix Ψ equal to the uncontrolled absolute RMS story accelerations was found to be most effective in reducing the maximum absolute RMS acceleration of the coupled building system. The weights α accel and α drift are varied such that the maximum absolute RMS accelerations are minimized while ensuring that the maximum interstory drift ratio does 55
not get larger than the maximum uncoupled interstory drift ratio. This assumes that the maximum interstory drift ratio is small enough that the buildings remain undamaged and the critical response is then the acceleration. A maximum drift ratio of 0.005 is used, as specified by the Structural Engineers Association of California’s (SEAOC) “operational performance” level (Vision 2000 Committee, 1995). This assumption will be examined for the coupled building example presented in Section 3.3. The optimal α accel and α drift are determined from a systematic search over a range of weights. A standard Kalman filter observer is used to estimate the states of the system zˆ˙ = ( A – K KF C m )zˆ + K KF y m + ( E – K KF F m ) f (t) T –1
T + BQ D T ) ( R where K KF = ( P˜ C m KF m KF + D m Q KF D m )
(3.11)
is the estimator gain and P˜ is
computed from the Riccati equation ˜ T + BQ D T ) ( R T –1 T AP˜ + P˜ A T – ( P˜ C m KF m KF + D m Q KF D m ) ( C m P + D m Q KF B ) = – BQ KF B T
(3.12)
where Q KF is the magnitude of the excitation spectral density S x˙˙g x˙˙ (ω ) , R KF the magnig
tude of noise spectral density S vv(ω ) , E [ x˙˙g ] = 0 , E [ v ] = 0 , where E [ · ] is the expectation operator, and excitation x˙˙g and sensor noise v are uncorrelated. For this study the measurement noise for the two accelerometers is assumed to have a variance of –6
6 ×10
(m/sec2)2 and the measurement noise for the displacement is assumed to have a –3
variance of 4 ×10
m2. The measurement noise corresponds to about 0.1% of the uncon-
trolled RMS responses, respectively. The noise was not seen to significantly affect the performance of the controller. The H2/LQG controller, designed using the Control Toolbox in MATLAB, is employed to determine K and K KF . The optimal active control force in the coupling link, f
actv
( t ) , as determined from Equations (3.8) and (3.11), is given by
56
zˆ˙( t ) = A c zˆ ( t ) + B c y m ( t ) f (t) = f
actv
(3.13)
( t ) = C c zˆ ( t )
where y m ( t ) , defined in Equation (3.3), is the measured responses of the system, consisting of the absolute acceleration and relative displacement of the two buildings at the height of the coupling link.
Semiactive Control Strategy The semiactive control strategy employs semiactive control devices that can change their dynamic characteristics in real time to provide a range of dissipative control forces. In Figure 3.2, the achievable forces of a passive viscous damper are indicated by a straight line. An active control strategy could produce forces in any of the four quadrants. A semiactive device produces forces in the first and third quadrants. Note, that Klein’s 1987 semiactive control strategy, using cables in tension to provide dissipative forces, limits the semiactive control forces to only the fourth quadrant of Figure 3.2. Previous studies of such semiactive dampers have shown a clipped-optimal control strategy to achieve good performance (Dyke, et al. 1996a, 1996c). Clipped-optimal control is implemented by determining desired control forces as if the system were fully active and employing a bang-bang approach to make the semiactive device replicate the
f(t) semiactive device nondissipative
r mpe
a us d visco dissipative
dissipative yc(t) nondissipative
Figure 3.2: Semiactive damper dissipative forces. 57
desired forces. The ideal semiactive device can only produce dissipative control forces given as actv f (t) f (t) = 0
f f
actv
actv
( t ) ⋅ y c(t) < 0
( t ) ⋅ y c(t) ≥ 0
(3.14)
where y c(t) = ∆x˙(h c) = x˙2(h c) – x˙1(h c) is the relative velocity across the coupling link, as defined for Equation (3.3).
3.2 Effects of Building Configuration on RMS Response The building configuration is defined by the height of the buildings, the location of the coupling link, and the mass density and stiffness of the buildings. In this section, various configurations are considered, along with their effect on the coupling capabilities of actively and passively controlled coupled building systems. Active and passive control are examined as they provide an upper and lower bound to semiactive control, and the responses to these linear systems is more readily available. The parameters h 1 , m 1 and ( EI ) 1 are fixed for the taller 50-story building, and the following ratios are examined for their effects on the response reduction capabilities of the coupled building system: (i) height ratio h 2 ⁄ h 1 ; (ii) normalized coupling height h c ⁄ h 2 ; and (iii) mass density and stiffness ratio m 2 ⁄ m 1 = ( EI ) 2 ⁄ ( EI ) 1 . The assumption is made that increasing the mass density is accomplished by increasing the size of the floor space and is directly related to an increase in the stiffness. Keeping the mass and stiffness ratios equal has the benefit of leaving the natural frequencies of the system constant to study the effect of varying the mass density and stiffness independently of changes in the natural frequencies. The examination is done in two parts. First, the height ratio and the coupling link location are varied while holding the mass density and stiffness ratio con-
58
stant. Next, the link location is fixed and the height ratio and the mass density and stiffness ratio are varied. The parameters of building 1 are fixed to that of a 50-story high-rise building with height h 1 = 200 m. The story height for both buildings is ∆h = 4.0 m. The mass den5
sity (mass per unit height) of building 1 is m 1 = 4 ×10 kg/m. The stiffness parameter for 13
building 1 is ( EI ) 1 = 8.18 ×10
N-m2, where E is Young’s Modulus and I is the
moment of inertia of the building. EI is, then, a beam bending stiffness comparable to the composite stiffness of the building. An in-plane model is developed for the coupled building system. The buildings are each modeled as flexural (Euler-Bernoulli) beams using the Galerkin method. The first five modes of each building are included in the analysis to capture the significant dynamics of the system. The first five natural frequencies of building 1 are: 0.20, 1.25, 3.51, 6.89, and 11.39 Hz. The natural frequencies of building 2 will vary depending on the height of the building. Classical viscous damping is assumed for each building, with 1% of critical damping in each mode. Consider the transfer function H ye x˙˙g(ω) from the ground acceleration to the evaluation/regulated outputs of a single building model and the Laplace domain representation of the Kanai-Tajimi filter H x˙˙g w(ω) from Equations (2.21) and (2.22). While observing the transfer function H ye x˙˙g(ω) provides insight into the frequency characteristics of the building model, the quantity H ye w(ω) = H ye x˙˙g(ω) ⋅ H x˙˙g w(ω) provides insight into the response of the building model to the filtered ground excitation. This insight is beneficial in understanding the behavior of the coupled building system. The magnitude of H ye w(ω) for the absolute acceleration of the roof and interstory drift ratio of the top floor are shown in Figure 3.3 for the 50-story building model described previously and for 30-story (120 m) and 20-story (80 m) buildings with the same mass density and stiffness parameters.
59
These plots illustrate the modal participation in the absolute acceleration and drift responses. The interstory drift ratio is dominated by the first mode. The peaks of the second mode for the 50-, 30- and 20-story building models is an order of magnitude less than that of the first mode. The absolute acceleration plots show that higher mode participation is more significant than for the drift ratio. The higher mode participation is attenuated, at a level around the second mode, by the Kanai-Tajimi filter. For the 50- and 30-story buildings the second mode plays a significant role in the response, for the 20-story building it is less.
Varying Building and Connector Heights Holding the mass and stiffness ratio constant at unity, the height of building 2 is varied over the range 0.2h 1 ≤ h 2 < h 1 and the height of the coupling link is varied over the
Acceleration Magnitude
range 0.2h 2 ≤ h c < h 2 . Figure 3.4 illustrates the maximum RMS responses of any story of
50-story
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Figure103.3: Frequency analysis of uncoupled 50-, 30- and 20-story building responses, H ye w(ω) , due to a filtered ground excitation. 2
60
either building, as a percentage of the same for the uncoupled system, using the passive and active control strategies described previously. When the natural frequency of a dominant mode of one building is nearly the same as a natural frequency of the other building, the ability of a controller to affect that mode is significantly degraded. As indicated in the discussion of Figure 3.3, maximum absolute accelerations are dominated by the first few modes of building 2. Consequently, when any of the first few modes of either building match those of the other building, limited improvement should be expected, in particular for absolute acceleration responses. This Passive Control Strategy absolute acceleration % of uncoupled 80
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Figure 3.4: Effect of building height and coupling link location on coupled building performance. 61
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phenomenon may be observed particularly for three cases. The first case is when the two buildings are nearly the same height ( h 2 ⁄ h 1 ≅ 1.0 ), where the natural frequencies of one building nearly match those of the other building (i.e., ω i, 1 ≅ ω i, 2 for all i ). As seen in Figure 3.4, passive and active control are able to achieve only a minimal effect when the buildings are nearly the same height (near the top edge of the graphs). A second case is when the second mode of building 2 has the same natural frequency as the third mode of building 1 (i.e., ω 2, 2 = ω 3, 1 ). This match, which occurs when the height ratio h 2 ⁄ h 1 is 0.60, does not affect the interstory drifts (because drifts are dominated by mode 1 of building 1), but has a significant affect on the absolute accelerations. The dashed horizontal line on Figure 3.4 shows where this height ratio occurs. The third case is when the fundamental natural frequency of building 2 matches the second natural frequency of building 1 (i.e., ω 1, 2 = ω 2, 1 ). For this case, which occurs with h 2 ⁄ h 1 = 0.40 , the second natural frequency of building 2 is above the dominant excitation range, so the first mode of building 2 has the largest effect on the absolute accelerations. This is seen by the degradation in improvements in absolute acceleration on Figure 3.4 near the horizontal dash-dot line. Another consideration is the location of the coupling link in relation to nodes of the dominant modes of the two buildings. When the link is located near the node of a vibratory mode, that mode is nearly uncontrollable. Additionally, since the sensor measurements used herein include absolute accelerations at the coupling link, placing the coupling link and the sensors at a node makes that mode unobservable as well as uncontrollable. The node of the first mode is at ground level; it goes without saying that all responses are uncontrollable for this configuration and the control has no effect. The second mode, however, has a node at 0.783h . As seen previously, if the second mode of building 2 is uncontrollable, the absolute accelerations can be expected to suffer for h 2 > 0.4h 1 (for shorter buildings, the absolute accelerations are dominated by the first mode, so the node of the second mode has little effect). The vertical dashed line on Figure 3.4 marks where the coupling link and sensors are at the node of the second mode of build62
ing 2; near this dotted line, the absolute accelerations are only minimally reduced. When the node of the second mode of building 1 coincides with the coupling link, the absolute accelerations will suffer to a lesser extent, however this effect can be noticed on the performance of the absolute acceleration response reduction. The diagonal dashed line on Figure 3.4 marks where the coupling link and sensors are at the node of the second mode of building 1. The nodes of the third mode, at 0.504h and 0.868h , have similar, albeit less dramatic, effects on the performance of passive and active control strategies. The dash-dot vertical and diagonal lines on Figure 3.4 mark where the coupling link and sensors are at the nodes of the third mode of building 2 and building 1, respectively. The performance of the passive and active control strategies, as shown on Figure 3.4, follow the similar trends with respect to regions of degraded performance. The best configuration is one where the building height ratio is such that the dominant vibratory modes do not coincide with each other and the coupling link is placed away from any dominant vibratory modes of the two buildings. When the building frequencies do coincide and/or the coupling link is located near a vibratory node, the performance is degraded. In addition to an overall slightly better performance, active control is able to restrict the regions of reduced performance more so than passive control, providing the greatest benefit to performance near these regions of concern. For coupled building configurations where vibratory modes of the two structure may nearly coincide, or where the coupling link must be placed near a vibratory mode, active coupled building control can provide significantly increased performance. As an example, for a building height ratio of h 2 ⁄ h 1 = 0.75 and a coupling link height ratio of h 2 ⁄ h c = 0.75 , the active control strategy can reduce the maximum absolute RMS acceleration to 45% of the uncoupled response and reduce the maximum absolute RMS acceleration an additional 40% beyond that of the optimal passive control strategy.
63
Varying Mass and Stiffness As observed above, the placement of the coupling link as well as the relative natural frequencies of the two buildings, as determined by the height ratio, can have an effect on the performance of the coupled building system. The effect of changing the mass density and stiffness ratios should also be studied. To do so, consider the case where the coupling link location is fixed at the roof of building 2, and the mass density and stiffness of building 2 are varied such that m 2 ( EI ) 2 ------ = ------------- ∈ [ 0.02, 3.0 ] m 1 ( EI ) 1
(3.15)
These two ratios, it is reasonable to assume, are scaled roughly in proportion to each other (if the floor mass increases by some ratio, the lateral force demands are also likely to increase by the same ratio). Keeping the ratios equal also has the benefit of leaving the natural frequencies of the system constant to study the effect of varying the mass density and stiffness independently of changes in the natural frequencies. Figure 3.5 illustrates the maximum RMS absolute acceleration and interstory drift ratio percentage of uncoupled responses, achieved by varying the height ratio and the mass density/stiffness ratios. For this analysis, the coupling link is located at the top of building 2 ( h c ⁄ h 2 = 1 ). This location is selected so that the coupling link is not located at the nodes of the dominant modes. The effect of coinciding natural frequencies at height ratios h 2 ⁄ h 1 = 1.0 , 0.6, and 0.4 may again be observed (the latter two are again marked with dashed and dash-dot white lines, respectively). Varying the mass density and stiffness of building 2 has little effect on the maximum absolute accelerations and interstory drift ratios except for at the lower extreme. Again, the performance of the passive and active control strategies, as shown on Figure 3.5, follow the similar trends with respect to regions of degraded performance. Again, the best configuration is one where the building height ratio is such that the natural frequencies of dominant vibratory modes do not coincide with each other. When the build64
Passive Control Strategy absolute acceleration % of uncoupled 8060
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Mass Density and Stiffness Connector Link Height Ratio
Figure 3.5: Effect of mass density and stiffness on coupled building performance.
ing frequencies do coincide, the performance is degraded. As seen in the previous analysis of building height and coupling link height ratios, active control is able to restrict the regions of reduced performance more so than passive control. Although the mass density and stiffness ratios have little effect, where vibratory modes of the two structure may nearly coincide active coupled building control can provide increased performance over passive coupled building control. 65
3.3 Efficacy of Semiactive Coupled Building Control Consider the coupled building configuration of a 50-story high-rise building as 5
defined previously in Section 3.3 ( h 1 = 200 m, ∆h = 4.0 m, m 1 = 4 ×10 kg/m, and 13
( EI ) 1 = 8.18 ×10 density
and
N-m2) connected to a 45-story high-rise building of similar mass
stiffness 13
( EI ) 2 = 8.18 ×10
∆h = 4.0 m,
( h 2 = 180 m,
5
m 2 = 4 ×10 kg/m,
and
N-m2) at the 43rd story of the two buildings ( h c = 172 m). The cou-
pled building system is shown in Figure 3.6. The building configuration chosen here (i.e. h 2 ⁄ h 1 = 0.90 , h c ⁄ h 2 = 0.96 , and m 2 ⁄ m 1 = ( EI ) 2 ⁄ ( EI ) 1 = 1 ) is selected because of it’s similarity to the coupled building configuration of the two taller buildings in the Triton Square complex, where, as described in Section 2.1, h 2 ⁄ h 1 = 0.90 , h c ⁄ h 2 = 0.91 , and m 2 ⁄ m 1 = ( EI ) 2 ⁄ ( EI ) 1 = 1 . The ground excitation, measures of performance, and control strategies are the same as described in Section 2.4. The ground excitation is modeled as a filtered white noise corresponding to the Kanai-Tajimi spectrum, with a prepended filter. The measures of performance are the maximum (over the heights of both buildings) RMS absolute accelerations and maximum RMS interstory drift ratios of the two buildings. In particular, the
Coupled Building Properties h 1 = 200 m
f(t)
1 h1
2
h 2 = 180 m h 3 = 172 m
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(EI)1
(EI)2
ζ1
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m 1 = 4 ×10 kg/m
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5
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.. xg(t)
( EI ) 1 = 8.18 ×10
13
N-m2
( EI ) 2 = 8.18 ×10
13
N-m2)
Figure 3.6: High-rise MDOF coupled building system for semiactive control. 66
maximum absolute RMS acceleration will be reduced while the maximum RMS interstory drift ratio is not allowed to exceed the uncoupled maximum interstory drift ratio. Passive and active coupled building control strategies are presented for comparative purposes. The maximum absolute RMS acceleration and interstory drift ratio over the height of each building are shown in Figure 3.7 as a function of control force. The active and passive control strategies considered here are linear, thus the response of these control strategies varies linearly with the level of excitation. Additionally, the clipped-optimal semiactive control strategy employed here is homogeneous, and the building responses for the semiactive control strategy also vary linearly with the level of excitation. In order to present the results independent of the level of ground excitation, the control force of the coupling link and the maximum absolute RMS acceleration and interstory drift responses are normalized with respect to the ground acceleration. The performance curves in Figure 3.7 illustrate that, for small levels of control force, the improved performance of all three control strategies (passive, active and semiactive) with respect to the uncoupled system is small, and the relative performance difference among the three control strategies is small. As the control force increases, the performance of the control strategies increase, and the relative difference in performance between the passive, active and semiactive control strategies becomes more noticeable. Also note, at low control effort, the absolute RMS acceleration of building 2 is largest, while for more aggressive control strategies, the absolute acceleration of building 1 becomes the larger, more critical response. The uncoupled, optimal passive, active, and semiactive controlled RMS response profiles for the two buildings are illustrated comparatively in Figure 3.8. The uncoupled absolute RMS acceleration responses are influenced by higher mode participation while the uncoupled RMS interstory drift ratios are primarily influenced by the first mode of each building. The maximum absolute RMS story accelerations and maximum RMS interstory drift ratios over the height of the buildings are located at the top stories of both buildings. 67
absolute RMS story acceleration
RMS interstory drift ratio
Control Strategies uncoupled passive active semiactive
building 1
building 2
building 1
building 2
Figure 3.8: RMS response profiles of absolute story acceleration and interstory drift ratio over the height of both buildings for uncoupled and optimal passive, active, and semiactive control strategies.
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Figure 3.7: Semiactive coupled building control RMS responses over range of control forces as compared to passive and active control strategies.
68
8
10
The maximum RMS responses over the height of the coupled buildings are given in Table 3.8 for semiactive control, as well as for uncoupled and passive and active control, along with the response percent of the uncoupled and passive control strategy responses. The efficacy of the semiactive control is measured by its ability to reduce the evaluation responses significantly beyond the responses of the uncoupled and passive control strategy and to nearly the same level of the active control strategy.The semiactive control strategy can reduce maximum absolute RMS acceleration (building 1) to 75% of the performance of the uncoupled system and an additional 8% beyond that of passive coupled building control. An additional benefit of semiactive control is that the interstory drift ratio over the two buildings is reduced to 69% and 53% of the performance of the uncoupled maximum RMS interstory drift for buildings 1 and 2, respectively, corresponding to an additional 22% and 28% reduction beyond the performance of the passive coupled building control strategy. If damage occurs to the structures when the maximum RMS interstory drift ratio TABLE 3.1: PERFORMANCE OF PASSIVE, ACTIVE AND SEMIACTIVE CONTROL STRATEGIES Max. Absolute RMS Acceleration Max. RMS Interstory Drift Ratio RMS Ground Acceleration RMS Ground Acceleration sec2/m Control Strategy (% of Uncoupled) (% of Uncoupled) [% of Passive] [% of Passive] {% of Semiactive} {% of Semiactive} building 1 building 2 building 1 building 2 Uncoupled 5.2204 5.5762 4.3290x10-3 3.7671x10-3 Passive
4.2502 (81%)
2.3139 (42%)
3.3263x10-3 (88%)
3.1825x10-3 (74%)
Semiactive
3.9180 (75%) [92%]
2.2583 (41%) [98%]
2.5834x10-3 (69%) [78%]
2.2879x10-3 (53%) [72%]
Active
3.4079 (65%) [80%] {87%}
2.1980 (39%) [95%] {97%}
1.9462x10-3 (52%) [59%] {75%}
1.6718x10-3 (39%) [53%] {75%}
69
exceeds 1.67x10-3 (corresponding to a maximum peak drift ratio of 5x10-3, Vision 2000 Committee, 1995), then for the coupled building system examined here, the uncontrolled buildings can withstand an earthquake of RMS ground acceleration up to 0.39 m/sec2. The semiactive coupled building system can withstand an earthquake with RMS ground acceleration of 0.65 m/sec2 without sustaining any significant damage . The active control strategy can reduce maximum absolute RMS acceleration to 65% of the uncoupled response an additional 20% beyond that of the passive control strategy. Depending on the building configuration (as shown in Section 3.2), active control can reduce the maximum absolute RMS acceleration to as much as 40% beyond that of the passive control strategy. Additionally, for this configuration, the active control strategy can reduce the maximum RMS interstory drift ratio to 52% of the uncoupled response an additional 41% beyond that of the passive control strategy. The semiactive control strategy can reduce maximum absolute RMS acceleration to 75% of the uncoupled response an additional 8% beyond that of the passive control strategy. The semiactive control strategy can reduce the maximum RMS interstory drift ratio to 69% of the uncoupled response an additional 22% beyond that of the passive control strategy.
3.4 Constraint on Maximum Allowable Control Force In contrast to previously reported studies, an important feature of this study is the ability to place limits on the allowable control forces. The effects of this limit is studied in this section. A maximum controller force is selected that both ensures the control force can be accommodated by the lateral load resisting systems of the structures and can be feasibly produced by a small number of control devices. The coupling link is located near the top of the buildings for the example considered in this section. Unfortunately, the top floors of building structures typically have small lateral load resisting capacities. 70
The maximum control force is set to the same order of magnitude as the minimum design story shear at the levels of the coupling link for the buildings in this study, as determined from Section 1628 - Minimum Design Lateral Forces and Related Effects, of the Uniform Building Code (1994 Uniform Building Code). A maximum peak control force is set to 12,000 kN. A control force of this magnitude can be generated, for example, by placing six 2000 kN actuators in parallel in each coupling link. The maximum controller force is enforced, in an RMS sense, by assuming a maximum RMS control force of 4,000 kN. To enforce the constraint of a maximum control force in the design of the control strategy, a level of excitation must be assumed. This level of excitation is the design earthquake. Although the uncoupled and semiactive coupled buildings will begin yielding at an RMS ground accelerations of 0.39 m/sec2 and 0.65 m/sec2 respectively, based on a maximum allowable peak interstory drift ratio of 0.005, stronger magnitude earthquakes may be wish to be chosen for the design earthquake. In developing the Kanai-Tajimi spectrum to model the coupled building ground acceleration (Section 2.4), four historical earthquakes were considered. The RMS ground accelerations for the significant portions of the El Centro and Hachinohe earthquake records were determined to be 0.65 and 0.41 m/sec2, respectively. The RMS ground accelerations for the significant portions of the Northridge and Kobe earthquakes were determined to be 1.69 and 1.80 m/sec2, respectively. Here, two levels of design earthquakes are defined: one strong level for far-field El Centro and Hachinohe type earthquakes, with RMS ground acceleration of 0.5 m/sec2, and the second level for near-field Northridge and Kobe type earthquakes, with stronger RMS ground acceleration of 1.75 m/sec2. Additionally, a third extreme design earthquake of RMS ground acceleration 3.27 m/sec2, is considered as an upper bound. The three design earthquakes correspond to peak ground accelerations of 1.5 m/sec2, 5.25 m/sec2, and 9.81 m/ sec2, respectively.
71
The maximum normalized control forces (maximum RMS control force / RMS ground acceleration) for the three levels of design earthquake, identified in Figure 3.9, are 6
6 4 ×10 N ------------------------- = 8 ×10 N/(m/sec 2 ) 0.5 m/sec 2 6
6 4 ×10 N ---------------------------- = 2.3 ×10 N/(m/sec 2 ) . 1.75 m/sec 2
(3.16)
6
6 4 ×10 N ---------------------------- = 1.2 ×10 N/(m/sec 2 ) 3.27 m/sec 2
force limit for stronger EQ
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force limit for strong EQ
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RMS Force / Ground Accel [N/(m/sec )]
Figure 3.9: Semiactive performance with identified maximum allowable control force for three levels of excitation as compared to passive and active control strategies.
The maximum RMS responses over the height of the coupled buildings, for these three levels of design earthquake, are given in Table 3.2. For a strong design earthquake of 0.5 m/sec2 with a normalized control force limit 6
of 8.0 ×10 N/(m/sec2), the semiactive control strategy can reduce maximum absolute RMS acceleration to 79% of the performance of the uncoupled system, an additional 4% beyond the performance of passive coupled building control, and to within 3% of the performance of the active control. The maximum RMS interstory drift ratio over the two buildings is reduced quite effectively by semiactive coupled building control for a strong 72
TABLE 3.2: PERFORMANCE OF PASSIVE, ACTIVE AND SEMIACTIVE CONTROL STRATEGIES FOR VARIOUS LEVELS OF GROUND ACCELERATION WITH A CONSTRAINT ON THE MAXIMUM ALLOWABLE CONTROL FORCE Max. Abs. RMS Acceleration Max. RMS Interstory Drift Ratio RMS Ground Acceleration RMS Ground Acceleration sec2/m (% of uncoupled) (% of uncoupled) [% of passive] [% of passive] {% of semiactive} {% of semiactive} building 1 building 2 building 1 building 2 4.3369 2.6219 3.0389x10-3 2.8530x10-3 Passive (83%) (47%) (81%) (66%) -3 4.1425 2.4931 2.2724x10 1.9987x10-3 Semiactive (79%) (45%) (60%) (46%) 0.5 m/sec2 [96%] [95%] [75%] [70%] (strong) -3 4.0238 2.3981 2.1115x10 1.9180x10-3 (77%) (43%) (56%) (44%) Active [93%] [91%] [69%] [67%] {97%} {96%} {93%} {96%} 4.9434 4.2647 2.3023x10-3 2.0764x10-3 Passive (95%) (76%) (61%) (48%) 4.9472 4.0147 2.6430x10-3 2.4413x10-3 Semiactive (95%) (72%) (70%) (56%) 1.75 m/sec2 [100%] [94%] [115%] [118%] (stronger) 4.8627 3.9147 2.9264x10-3 2.8509x10-3 (93%) (70%) (78%) (66%) Active [98%] [92%] [127%] [137%] {98%} {98%} {111%} {117%} -3 5.0780 4.8455 2.4619x10 2.3889x10-3 Passive (97%) (87%) (65%) (55%) -3 5.1363 4.6016 3.0640x10 3.0131x10-3 Semiactive (98%) (83%) (81%) (70%) 3.27 m/sec2 [101%] [95%] [124%] [126%] (extreme) 5.0490 4.5539 3.2740x10-3 3.3632x10-3 (97%) (82%) (87%) (78%) Active [99%] [94%] [133%] [141%] {98%} {99%} {107%} {112%} RMS Design Control Earthquake Strategy Ground Accel.
73
design earthquake, to 60% and 46% of the performance of the uncoupled system and an additional 25% and 30% beyond the performance of passive coupled building control, for buildings 1 and 2, respectively. For the stronger design earthquake of 1.75 m/sec2 with a normalized control force 6
of 2.3 ×10 N/(m/sec2) and the extreme earthquake of 3.27 m/sec2 with a normalized con6
trol force of 1.2 ×10 N/(m/sec2), the semiactive control strategy can reduce maximum absolute RMS accelerations to 95% and 98% of the performance of the uncoupled system, respectively, with no additional increase beyond the performance of passive coupled building control, but within 2% of the performance of active control. For the stronger and extreme design earthquakes, the relative performance difference between the passive, active and semiactive control strategies is negligible. Placing a maximum limit on the coupled building control force provides a more detailed look at the expected performance of the passive, active, and semiactive control strategies. As the control force limit is relaxed, the relative performance of semiactive and active control beyond that of passive control increases to a significant level.
3.5 Low-Rise Coupled Building System Analysis The relative performance of passive, active, and semiactive coupled building control is examined for the low-rise coupled building system. Limits on the allowable control forces are considered, as was done for the high-rise coupled buildings in Section 3.3.
Low-Rise Coupled Building Model To insure that the previous performance of passive, active, and semiactive control strategies is not unique to high-rise buildings, the low-rise coupled building system described in Iemura, et al. (1998) is considered. The system consists of two building frames, 5-stories and 3-stories tall, connected at the 3rd story with a coupling link. The 74
link can provide passive, active, and semiactive type control forces. The properties of the low-rise coupled building system are given in Table 3. The coupled building system does not have dominant natural frequencies that coincide and the coupling link is not placed at the node of a dominant mode, therefore it is a reasonable candidate for coupled building control. TABLE 3.3: SUMMARY OF FULL-SCALE STRUCTURAL FRAME MODELS Building 1 (5-STORY FRAME) (i = 1 )
Building 2 (3-STORY FRAME) (i = 2 )
Total Height ( h i )
17.22 m
10.65 m
Mass of Each Story ( m i )
30 kg
20 kg
2.05 Hz 6.28Hz 11.01 Hz
2.44 Hz 7.63 Hz 12.82 Hz
Natural Freqs.
1st mode 2nd mode 3rd mode
An in-plane dynamic model is developed for the low-rise coupled building system. The low-rise buildings will behave as shear beams, as opposed to the high-rise buildings that behaved as flexural beams. Consequently, the shear buildings are modeled using the finite element method, employing Euler-Bernoulli beam elements and fixing the rotation at the ends of the beam elements and condensing out these rotational degrees-of-freedom. Each beam element corresponds to one building story. The beam element, building model, and building deflection are shown in Figure 3.10. The consistent mass matrix and plane rigid frame stiffness are m = m 1 0 0 1
(3.17)
12EI 1 – 1 k = ----------3 –1 1 L
(3.18)
75
The stiffness is adjusted such that the natural frequencies match the experimental results reported in Iemura, et al. (1998). Viscous damping is assumed for each building model, with 2% of critical damping in the each mode. The lateral displacements of buildings 1 and 2 are combined such that T
x ( t ) = [ x1 ( t )
T
x2 ( t ) ]
T
and the equations of motion for the coupled system are written
Mx˙˙( t ) + Cx˙ ( t ) + Kx ( t ) = – Gx˙˙g ( t ) + Pf ( t )
(3.19)
where M , C , and K are the global mass, stiffness and damping matrices, G = [ ( M1 1 )
T
T T
( M 2 1 ) ] is the loading vector for the ground acceleration ( 1 is a vecT
tor of ones), x˙˙g ( t ) is the ground acceleration, P = [ – P 1
T T
P 2 ] is the loading vector for
the control force where P k is the loading vector of the kth building and consists of a 1 at the degree-of-freedom where the coupling link is attached and zeros elsewhere, and f ( t ) is the control force of the coupling link. Equation (3.19) can be written in state space form, where the states are defined as T
q(t ) = [x (t )
T
T
x˙ ( t ) ] . The outputs are: evaluation responses, y e(t) , which include the
beam element θj=0
xj
j
i
ction defle
defle
ction
coupling link
θi=0 xi
5-story building
3-story building
Figure 3.10: Beam element, 5- and 3-story building models, and building deflection for the low-rise coupled building system. 76
absolute accelerations and interstory drift ratios; measured outputs y m(t) , which include the absolute accelerations of buildings 1 and 2 at the location of the coupling link and the relative displacement of the buildings at the height of the coupling link; and connector response, y c(t) , which is the relative velocity of the buildings at the height of the coupling link. The state space equations are written as q˙ ( t ) = Aq ( t ) + Bx˙˙g ( t ) + Ef ( t )
(3.20)
y e(t ) = C e q ( t ) + F e f ( t )
(3.21)
y m(t) = C m q ( t ) + F m f ( t )
(3.22)
y c(t) = C c q ( t )
A =
where
0
I
–1
–1
(3.23)
,
–M K –M C –1
–1
Ce = –M K Γh
–M C , 0
0
B =
–1
,
E =
–M G –1
0 –1
,
M P –1
–1
C m = – ΛM K – ΛM C , ∆ 0
Fe = M P , 0
–1
F m = ΛM P , and C c = 0 ∆ , 0
where Γ = h
I ⁄ h1
0
0
I ⁄ h2
, ∆= P
T
and Λ =
P1 0
T
.
0 P2
Both passive and active control strategies, as in the previous sections, are considered to couple the 5-story and 3-story frames for response reduction. The maximum RMS evaluation responses are minimized for the control strategies subject to a fixed maximum control force. The relative performance of passive and active control strategies will be examined for different levels of ground acceleration.
77
To assess the effectiveness of each control strategy, stationary RMS responses are determined for the coupled system subjected to a ground acceleration. The ground acceleration is modeled as previously defined in Section 2.4.
Efficacy of Low-Rise Coupled Building System The system is considered for both unlimited control force and for a limited maximum control force. The maximum allowable RMS control force, when enforced, is 20 kN. The three design earthquakes, from the previous section, are considered: a strong level with RMS ground acceleration of 0.5 m/sec2, a stronger RMS ground acceleration of 1.75 m/sec2, and an extreme design earthquake of RMS ground acceleration 3.27 m/sec2. Maximum RMS responses for the passive, active, and semiactive control strategies as well as for the uncoupled system are presented in Table 3.4. The responses have been normalized with respect to the ground acceleration. The maximum absolute RMS accelerations occur at the top floors of the building frames and the maximum RMS interstory drift ratios occur at the first story of the building frames. If damage occurs to the structures when the maximum RMS interstory drift ratio exceeds 1.67x10-3 (corresponding to a maximum peak drift ratio of 5x10-3, Vision 2000 Committee, 1995), then for the low-rise coupled building system, the uncontrolled buildings can withstand an earthquake of RMS acceleration up to 0.48 m/sec2. The semiactive coupled building system can withstand an earthquake of 0.72 m/sec2 before any damage might occur in the buildings. The case of unlimited control force and a strong design earthquake of 0.5 m/sec2 yield the same results. The semiactive control strategy can reduce maximum absolute RMS acceleration to 69% of the performance of the uncoupled response, an additional 9% beyond the performance of passive coupled building control, and to within a little over 9% of the performance of the active control. The maximum RMS interstory drift ratio over the two buildings is not reduced as effectively as for the high-rise buildings. Semiactive coupled building control for a strong design earthquake reduces the maximum RMS interstory 78
TABLE 3.4: COMPARISON OF PASSIVE AND ACTIVE CONTROL STRATEGIES FOR THE LOW-RISE COUPLED BUILDING SYSTEM
RMS Ground Acceleration
uncoupled unlimited control force and 0.5 m/sec2 (strong)
Max. Abs. RMS Acceleration Max. RMS Interstory Drift Ratio RMS Ground Acceleration RMS Ground Acceleration Control sec2/m Strategy (% of uncoupled) (% of uncoupled) [% of passive] [% of passive] building 1 building 2 building 1 building 2 -3 5.6381 5.7825 3.4834x10-3 2.9327x10 Passive
Semiactive
Active
Passive
1.75 m/sec2 Semiactive (stronger) Active
Passive
3.27 m/sec2 Semiactive (extreme) Active
4.2213 (75%) 3.8624 (69%) [91%] 3.5396 (63%) [84%] 4.7738 (85%) 4.7376 (84%) [99%] 4.7119 (84%) [99%] 5.1393 (91%) 5.1381 (91%) [100%] 5.1150 (91%) [100%]
2.6219 (61%) 3.5353 (63%) [103%] 3.7108 (64%) [105%] 4.4213 (76%) 4.5082 (78%) [102%] 4.4953 (78%) [102%] 4.9677 (86%) 5.0322 (87%) [101%] 4.9891 (86%) [100%]
79
2.0377x10-3 (69%)
2.2734x10-3 (65%)
1.9175x10-3 (65%) [94%]
2.3141x10-3 (66%) [102%]
1.7721x10-3 (60%) [87%]
2.3998x10-3 (69%) [106%]
2.4375x10-3 (83%)
2.6845x10-3 (77%)
2.4328x10-3 (83%) [100%]
2.7468x10-3 (79%) [102%]
2.4193x10-3 (82%) [99%]
2.7400x10-3 (79%) [102%]
2.6518x10-3 (90%)
2.9985x10-3 (86%)
2.6680x10-3 (91%) [101%]
3.0445x10-3 (87%) [102%]
2.6446x10-3 (90%) [100%]
3.0190x10-3 (87%) [101%]
drift ratio to 66% of the performance of the uncoupled system and 102% of the performance of passive coupled building control. For the stronger design earthquake of 1.75 m/sec2 and the extreme earthquake of 3.27 m/sec2, the semiactive control strategy can reduce maximum absolute RMS accelerations to 84% and 91% of the performance of the uncoupled system, an additional 0-1% beyond the performance of passive coupled building control. For the stronger and extreme design earthquakes, the relative performance difference between the passive, active and semiactive control strategies is negligible. The relative performance of the passive, active, and semiactive control strategies for the low-rise coupled building frames is similar to what was shown for the high-rise buildings.
3.6 Chapter Summary In this chapter, two coupled building control strategies are proposed. The first proposed control strategy is the active control of coupled buildings using absolute acceleration and relative displacement measurements for feedback control. The second proposed control strategy is semiactive coupled building control, using a clipped-optimal approach with the active strategy as the primary controller. Also within this chapter, the effect of building configuration on the performance of passive and active coupled building control strategies is examined. It is observed, as had been previously alluded to (Klein and Healy, 1987, Seto, et al., 1994a), that the optimal coupled building configuration follow two guidelines. One, where the dominant frequencies of the two coupled buildings do not coincide, and the second, where the coupling link is not placed at the node of a dominant vibratory mode. Failing to follow both of these guidelines leads to a reduction in the performance of the passive and active coupled building control strategies. 80
When building frequencies nearly match, or the coupling link is placed near to a vibratory node, active control can provide improved performance over passive control. For example, for a coupled building system with a building height ratio of h 2 ⁄ h 1 = 0.75 (near the height ratio h 2 ⁄ h 1 = 0.60 where the second mode of building 2 coincides with the third mode of building 1) and a coupling link height ratio of h c ⁄ h 2 = 0.75 (near h c ⁄ h 2 = 0.783 where the coupling link is located at the node of building 2’s second mode), the active control strategy can reduce the maximum absolute RMS acceleration to 45% of the uncoupled response and up to an additional 40% beyond that of the optimal passive control strategy. The performance of the semiactive coupled building control strategy is evaluated for a coupled building system with a building height ratio of h 2 ⁄ h 1 = 0.90 and a coupling link height ratio of h c ⁄ h 2 = 0.96 . For this system, the coupling link is located near a node of building 1’s third mode. The semiactive control is able to reduce the maximum absolute RMS acceleration to between 75% and 98% of the uncoupled maximum absolute RMS acceleration for various levels of design earthquake, and to an additional 8% beyond the optimal passive control strategy (for the case assuming unlimited control force). For larger design level earthquakes, semiactive control provides less additional benefit in reducing the maximum absolute RMS acceleration than passive control. Additionally, for the larger earthquakes, the buildings may be damaged and the performance objective to reduce maximum accelerations may be of less importance. Two low-rise buildings are considered for coupling in order to examine if performance for buildings bending in shear deformation is similar to that for the high-rise buildings. The buildings do not have coinciding natural frequencies, and the coupling link is not placed at the node of a dominant vibratory mode. For the system considered, active control reduces the maximum absolute RMS story acceleration an additional 10% beyond that of the optimal passive control. This is consistent with the additional performance active control provides beyond passive control for high-rise building system observed. 81
Semiactive coupled building control is shown to reduce maximum absolute RMS accelerations to 75% and 41% of the uncoupled buildings and reduce the maximum RMS interstory drift ratios to 69% and 53% of the uncoupled buildings. It is shown that the semiactive control strategy can reduce the maximum absolute RMS acceleration an additional 8% beyond an optimal passive control strategy. Furthermore, placing restrictions on the maximum control force reduces the overall and relative performance of the semiactive control strategy. However, for certain conditions (e.g., building configuration has coinciding natural frequencies, a coupling link placed at a vibratory node, or a strong design earthquake), the relative performance of active control beyond that of passive control is indeed significant. Studies in this chapter have identified scenarios where active control can provide an additional 40% reduction of the maximum absolute RMS acceleration beyond that of the optimal passive control strategy. For this reason, in Chapter 4 the active control strategy proposed in this chapter is experimentally verified.
82
CHAPTER 4: COUPLED BUILDING CONTROL: EXPERIMENTAL VERIFICATION
The primary focus of this research is to examine the semiactive control of civil structures for natural hazard mitigation. To this end, semiactive control of coupled buildings was studied in the previous two chapters. Semiactive coupled building control is shown to reduce maximum absolute RMS accelerations an additional 8% beyond an optimal passive control strategy. The relative performance of active control beyond that of passive control has been shown in this research to provide an additional 40% reduction of the maximum absolute RMS acceleration beyond that of the optimal passive control strategy. Because active control has been shown analytically to provide significant increased performance beyond passive and semiactive control, the active coupled building control strategy proposed in Chapter 3 to reduce the absolute acceleration response of adjacent buildings to seismic excitation is experimentally verified in this chapter. In the area of structural control, it is well-recognized that experimental verification of control strategies is necessary (Housner, et al. 1994a, 1994b). Experimental studies to investigate actively coupled adjacent buildings for response mitigation have traditionally employed displacement feedback. As direct measurement of displacement is difficult, in particular for larger-scale structures, and absolute acceleration measurements are more readably available, acceleration feedback is an appealing control strategy for coupled building control. In this chapter, active coupled building control employing absolute acceleration and connector link displacement feedback, is experimentally verified. The connector link of the experiment here is a DC motor with a ball-screw mechanism, similar to that employed in Triton Square office complex in Tokyo, Japan. In the subsequent sections of 83
this chapter, the experimental setup for the active coupled building control experiment is described, a control oriented design model developed, active control strategy designed, and experimental results presented.
4.1 Coupled Building Experimental Setup A schematic of the experimental setup discussed in this paper is shown in Figure 4.1. Components of the experiment include a coupled building model, shaking table, digital controller, and spectrum analyzer.
Coupled Building Model The coupled building model consists of a pair of 2-story building models, a servomotor control actuator and accelerometers, as pictured in Figure 4.2. The two 2-story building models were manufactured by Quanser Consulting Inc. The buildings are 305 mm by 108 mm in plan and 980 mm tall. The interstory height is 490 mm. The building models are constructed from rigid 12.7 mm thick plexiglas story levels and flexible
shaking table
a x˙˙11
x˙˙g
a x˙˙12
a x˙˙21
a x˙˙22
T
coupled building model
spectrum analyzer
x˙˙12 x˙˙22 ∆x
u
T
digital controller a
x˙˙g - ground acceleration; x˙˙ij - abs. accel. of the jth story of building i; ∆x - relative displacement of the two buildings at height of the coupling link; and u - control signal Figure 4.1: Schematic of coupled building experiment. 84
aluminum strip, 1.59 mm thick, columns. The height and stiffness of the buildings are similar with different story masses. Additional mass is secured to the story levels of building 1 (the building on the left in Figure 4.2) to ensure that the buildings are dynamically dissimilar. The story masses, including the additional mass on building 1 and the mass of the control actuator on the top stories of both buildings, are m 11 = 3.22 kg, m 12 = 3.45 kg, m 21 = 0.47 kg, and m 22 = 0.83 kg ( m ij , where i indicates the building number and j indicates the story level). The buildings are located adjacent to one another and separated by a distance of 75 mm. When dominant natural frequencies of coupled buildings coincide, the ability of a control strategy to reduce responses is significantly degraded as shown in Chapter 3. Thus, the frequencies of building 1 and building 2 are purposely adjusted (by adding mass to building 1 as previously identified) such that the four uncoupled natural frequencies are more evenly spaced. The dynamic properties of the uncoupled buildings are determined
(1,2)
(2,2) control actuator
additional mass
(1,1)
(2,1)
building 1
building 2
Figure 4.2: Two-story coupled building model for experimental verification. 85
with the control actuator disconnected, but left in-place. The natural frequencies of building 1 are 0.90 Hz and 2.70 Hz with corresponding damping ratios of 1% and 0.5% of critical. The natural frequencies of building 2 are 1.85 Hz and 5.73 Hz with corresponding damping ratios of 1% and 0.5% of critical. A control actuator is used to provide the forces to the coupled building system. The control actuator is pictured in Figure 4.3. The two buildings are coupled at the top stories. The actuator, manufactured by Quanser Consulting, is a DC servo-motor and ball-screw mechanism with a stroke of ± 100 mm, as dictated by the length of the threaded rod. The stroke is limited by the distance of separation of the two buildings (75 mm). This allowable stroke is an order of magnitude larger than necessary for sufficient control. A potentiometer is attached to the motor to measure the rotation of the actuator threaded rod. The relative displacement is related to the rotation of the motor through the pitch of the threaded rod attached to the servo-motor and passing through the ball-screw mechanism. The pitch of the threaded rod is 3.18 mm/turn. Because the servo-motor control actuator is inherently open loop unstable, position feedback is employed to stabilize the control actuator. The position control of the coupling link is obtained by a PD controller with displacement feedback provided by the potentiometer. PCB capacitive DC accelerometers, model 3701G3FA3G, are employed to provide evaluation and measurement responses of the building stories. The accelerometers have a
ball-screw mechanism
threaded rod
servo-motor
Figure 4.3: Control actuator, consisting of a servo-motor with ball-screw mechanism. 86
range of ± 3 g and sensitivities of 1000 mV/g. The ground acceleration is measured by a DC accelerometer produced by Quanser Consulting, Inc.
Shaking Table The shaking table used is a small-scale uniaxial earthquake simulator constructed by SMI Technology and located in the Structural Dynamics and Control/Earthquake Engineering Laboratory (SDC/EEL) at the University of Notre Dame. The table has a maximum displacement of ± 120 mm and a maximum acceleration of ± 1 g (with a 11.3 kg test load). The nominal operational frequency range of the simulator is 0-20 Hz. Because the shake table motor is inherently open loop unstable, position feedback, measured from the shake table motor, is employed to stabilize the table. The position control is obtained by a PD controller with displacement feedback.
Digital Controller The digital controller is a PCI MultiQ I/O board1 with the WinCon realtime controller2 installed in a PC. The controller is developed using Simulink and executed in real time using WinCon. The MultiQ I/O board has 13-bit analog/digital (A/D) and 12-bit digital/analog (D/A) converters with eight input and eight output analog channels. Four digital encoders are also available. The Simulink code is converted to C code using the Real Time Workshop in MATLAB and interfaced through the WinCon software to run the control algorithms on the CPU of the PC.
1. 2.
http://www.quanser.com/english/html/solutions/fs_soln_hardware.html http://www.quanser.com/english/html/solutions/fs_soln_software_wincon.html 87
Spectrum Analyzer The spectrum analyzer is a 4-input/2-output PC-based spectrum analyzer manufactured by DSP Technology. The device has a 90 dB signal to noise ratio and includes 8-pole elliptical antialiasing filters, programmable gains on the inputs/outputs, user selectable sample rates and a MATLAB user interface. These features allow for direct acquisition of high quality data and transfer functions for system identification and response analysis.
4.2 Experimental Coupled Building Control-Oriented Design Model A critical precursor to the control design is the development of an accurate dynamic model of the structural system. Here, the approach used for system identification is to construct a mathematical model to replicate the input/output behavior of the system (Dyke, et al. 1996a). As indicated in Figure 4.1, the inputs to the coupled building model are the ground acceleration ( x˙˙g ) and the control input to the actuator (u), and the available a
outputs are the four absolute story accelerations ( x˙˙ij , where i indicates the building number, and j indicates the story height) and the relative displacement of the two buildings at the height of the coupling link ( ∆x ). First, experimental transfer function data is obtained and curve-fit to determine mathematical representations of the frequency responses. The transfer functions are experimentally determined from the ground acceleration and the control input to the absolute accelerations of each story and the relative displacement of the top of the buildings. These ten experimental transfer functions are each curve-fit to determine the poles and zeros of the system. Since the transfer functions represent the input/output relationships for a single physical system, a common denominator, of 8th order, is assumed for the elements of each column of the transfer function matrix. This corresponds to the assumption that each building is modeled with two degrees-of-freedom. Figure 4.4 compares the experiment transfer functions of the coupled building system for the absolute story acceleration out88
40
H
(ω)
a
30
x˙˙11 x˙˙g
magnitude (dB)
magnitude (dB)
30
40
20 10 0
10 0
−30
−30 −40 0
−40 1
2
3
4
5
6
7
−50 0
8
frequency (Hz) 40
H
30
1
2
3
4
5
6
7
8
6
7
8
6
7
8
6
7
8
frequency (Hz) 40
(ω)
a x˙˙12 x˙˙g
H
30
20
magnitude (dB)
magnitude (dB)
(ω)
−20
−20
10 0
a
x˙˙12 u
(ω)
20 10 0
−10
−10
−20
−20
−30
−30 −40 0
−40 1
2
3
4
5
6
7
−50 0
8
frequency (Hz)
H
1
2
3
4
5
frequency (Hz) 40
(ω)
a x˙˙21 x˙˙g
H
30
magnitude (dB)
30
magnitude (dB)
a
x˙˙11 u
−10
−10
20 10
a
x˙˙21 u
(ω)
20 10 0
−10
0
−20
−10
−30 −40
−20 0
1
2
3
4
5
6
7
−50 0
8
frequency (Hz) 40
H
1
2
3
4
5
frequency (Hz) 40
(ω)
a x˙˙22 x˙˙g
H
30
magnitude (dB)
30
magnitude (dB)
20
H
experimental curve-fit
20 10 0
a
x˙˙22 u
(ω)
20 10 0
−10
−10
−20
−20
−30
−30 −40 0
−40
1
2
3
4
5
6
7
−50 0
8
1
2
3
4
5
frequency (Hz)
frequency (Hz)
Figure 4.4: Comparison of the experimental and curve-fit transfer functions. 89
puts to the curve-fit transfer functions used to develop the model. At low frequencies the curve-fit and experimental transfer functions are different. This difference can be attributed to the difficulty in exciting the building system at frequencies, below 1 Hz. At higher frequencies the curve-fit and experimental transfer functions again deviate. This difference results from the high frequency vibration of the buildings’ columns, which are not represented in the curve-fit models. However, the transfer functions do match well within the frequency range of concern, 1-6 Hz. The following transfer function matrix is thus determined: H H H H
a
(ω) H
a
(ω) H
a
(ω) H
a
(ω) H
x˙˙11 x˙˙g x˙˙12 x˙˙g x˙˙21 x˙˙g x˙˙22 x˙˙g
a
(ω)
a
(ω)
a
(ω)
a
(ω)
x˙˙11 u x˙˙12 u x˙˙21 u x˙˙22 u
(4.1)
H ∆xx˙˙g(ω) H ∆xu(ω) Next, the transfer function input/output behavior of the coupled building system is transformed to a multi-input multi-output state space minimal realization. Each column of the transfer function matrix in Equation (4.1) is transformed to a state space realization in controller canonical form and balanced (MATLAB, 1999). The two state space models are combined by simply stacking the two models. The dynamics of the coupled building system are redundantly represented in this combined, stacked, state space model. A minimal realization of the system is found by performing a model reduction on the 16-state system. The resulting 9-state, state space model preserves the salient qualities of the coupled building system and is represented mathematically as
90
x˙(t) = Ax(t) + B
x˙˙g(t) u(t)
y e(t) = C z x(t) + D z
x˙˙g(t)
y m(t) = C y x(t) + D y
(4.2)
u(t) x˙˙g(t)
+ v(t)
u(t)
where A [9x9], B [9x2], C z [4x9], D z [4x2], C y [3x9] and D y [3x2] are the state space matrices determined by the system identification described previously in this section, x(t) is the state space vector, y e(t) = x˙˙a x˙˙a x˙˙a x˙˙a 11 12 21 22 y m ( t ) = x˙˙a12 x˙˙a22 ∆x
T
are the regulated outputs,
T
are the available measurements, and v(t) is the measurement
noise. Control-structure interaction (CSI) has been shown to have a profound effect on the ability for the control actuator to produce control forces at the resonant frequencies of the structures under control. Accounting for CSI is essential to achieving high quality control (Dyke, et al. 1995). By performing system identification in the manner described here, CSI is fully incorporated in the resulting design model.
4.3 Experimental Active Coupled Building Control Strategy The focus of this study is to experimentally verify the coupled building concept using acceleration feedback for the seismic protection of structures. Typically, for tall flexible buildings, the dynamic response of concern is the absolute story accelerations. The objective of the control strategy is to reduce the maximum absolute story accelerations over both buildings. An H2/LQG approach (Spencer, et al. 1994, 1998a; Stengel, 1986) is used to design the active control strategy for the coupled building model. A fourth-order filter is augmented to the model of the structural system to shape the spectral content of the 91
ground excitation in the H2/LQG design and analysis. This filter is the Kanai-Tajimi spectrum with prepended filter as given in Equations (2.21) and (2.22), with parameters ω g = 1.2 ( 2π ) rad/sec and ζ g = 0.3 , ω p = 5.22 ( 2π ) rad/sec and ζ p = 0.2 . The objective function is given by J = lim
τ→∞
1 --- E τ
τ
∫ { ye (t)Qye(t) + u 2(t) } dt T
(4.3)
0
where Q is a weighting matrix for the regulated outputs which is selected such that the responses of interest are minimized and u(t) is the control signal sent to the actuator. The H2/LQG control strategy is designed to minimize absolute root mean square (RMS) story accelerations over both buildings. The selection of the weighting matrix Q, which weights a linear combination of the absolute story accelerations, determines the particular control strategy. The optimal weighting matrix is determined iteratively and for this study takes the form Q = diag 3.6633 3.8125 6.2826 4.7449 . This weighting matrix was selected to insure that the maximum absolute accelerations over both buildings are effectively reduced. The resulting H2/LQG output feedback compensator is given by q˙ ( t ) = A c q ( t ) + B c y m ( t ) u(t) = C c q ( t )
(4.4)
where A c [13x13], B c [13x3] and C c [1x13] are the state space matrices and q(t) is the state space vector for the H2/LQG output feedback compensator. The method of “emulation” is used for the design of the discrete-time controller. Using this technique, the continuous-time controller of Equation (4.4) is emulated with an equivalent digital filter using a bilinear (Tustin) transformation (MATLAB, 1999). The resulting discrete system is given by q ( k + 1 ) = Ad q ( k ) + Bd y ( k ) u ( k ) = Cd q ( k ) 92
(4.5)
where A d [13x13], B d [13x3] and C d [1x13] are the discrete state space matrices of the feedback compensator and q(k) is the discrete state space vector, y m ( k ) is the discrete measurements at the kth time step and u ( k ) is the discrete control signal. The sampling rate of the controller is 0.01 sec, which is greater than 10 times the closed-loop system bandwidth. The equivalent discrete system adequately represents the behavior of the emulated continuous-time system over the frequency range of interest. A consequence of modeling continuous structures with a finite number of modes is that at certain frequencies (for this experiment at frequencies above 6 Hz) the structure is not well represented by the design model. Care must be taken during the design of the controller to insure sufficient roll-off of the control effort at higher frequencies.This was accomplished by analytically determining the loop gain, H ux˙˙g(ω) from Figure 4.1, during the design of the controller and rejecting those controllers where H ux˙˙g(62.8) > – 10 dB.
4.4 Experimental Active Coupled Building Results Two series of tests are conducted to evaluate the performance of the actively controlled coupled building system subjected to ground excitation. First, a frequency domain examination is conducted whereby the transfer functions are observed. Second, the buildings are subjected to simulated earthquakes, and the time histories are considered. Root mean square (RMS) response reduction is observed, which illustrate the increased damping of the active control strategy. To provide a baseline for comparison of the active control strategy, two other configurations are considered: the uncoupled building system and the rigidly connected building system. The uncoupled system is realized by simply disconnecting the screw mechanism from the actuator motor. The actuator components are not removed from the top story of the buildings. The rigidly connected building system is realized with a zero
93
control signal u(t) = 0 to the control actuator, which locks the motor in place, fixing the distance between the buildings’ top stories.
Frequency Domain Analysis The coupled building system is subjected to a 10 Hz bandlimited white noise ground excitation. The frequency response functions from the ground acceleration to the absolute story accelerations are measured for the actively controlled coupled building system as well as for the uncoupled and rigidly connected systems. Transfer functions from the ground acceleration to the absolute story accelerations are shown in Figure 4.5 for the uncoupled, rigidly connected and controlled building systems. The analytical active control transfer function is also shown in Figure 4.5. The analytically expected results compare reasonably well to the experimental active control transfer functions. An H ∞ measure of the performance of the active coupled building is considered. The H ∞ norm of a transfer function is a measure of the upper limit of the ratio of the root mean square (RMS) of the output vector to the RMS of the input (Spencer, et al. 1994). The H ∞ norm is measured as the peak value of the transfer function and it represents the maximum RMS gain of that response. For this reason, an H ∞ measure is associated with a “worst case” control design. Thus, as a measure of performance for the actively controlled building configurations, the peak value of the transfer functions are indicated in Table 4.1. Both peak values for frequency ranges in the neighborhood of resonant peaks (e.g. 0-2 Hz, 2-4 Hz, and 4-8 Hz) and the maximum peak value over all frequencies are provided. When the buildings are uncoupled, the resonant peaks of building 2 ( H x˙˙a
and
a
) are larger in magnitude than the resonant peaks of building 1 ( H
and
a
). Rigidly connecting the two buildings has the effect of reducing the resonant
˙˙g 21 x
H H
x˙˙22 x˙˙g x˙˙12 x˙˙g
a
x˙˙11 x˙˙g
peaks of building 2 by 3% and 14%, while increasing the resonant peaks of building 1 by
94
H
a
x˙˙12 x˙˙g
(iω)
20 10 0
−10
20 10 0
−20 1
2
3
4
5
6
7
8
0
1
2
Frequency (Hz)
H
a
x˙˙11 x˙˙g
3
4
5
6
7
8
6
7
8
Frequency (Hz)
(iω)
H
a
x˙˙21 x˙˙g
(iω)
30
2
1 20 10 0
Magnitude (dB)
30
Magnitude (dB)
(iω)
−10
−20
20 10 0
−10
−10
−20
−20 0
a
x˙˙22 x˙˙g
30
Magnitude (dB)
Magnitude (dB)
30
0
H
uncoupled rigid active active-analytical
1
2
3
4
5
6
7
0
8
1
2
3
4
5
Frequency (Hz)
Frequency (Hz)
Figure 4.5: Experimental transfer functions of ground acceleration to absolute story accelerations.
8% and 7%. Thus, rigidly connecting two adjacent buildings is seen to not benefit the coupled building system as a whole. In contrast, the active control strategy reduces the magnitude of the resonant peaks of all stories over the uncoupled and rigidly connected building systems. The resonant peaks are reduced from 37%-90% over the uncoupled buildings and from 37%-92% over the rigidly connected buildings. The peak values of the active control transfer functions are reduced by 37%, 55%, 80% and 82% over the uncoupled transfer functions. The peak values of the active controlled transfer functions are reduced by 50%, 65%, 78% and 68% 95
TABLE 4.1: PEAK MAGNITUDE OF COUPLED BUILDING SYSTEM TRANSFER FUNCTIONS.
0-2 Hz H x˙˙11 x˙˙g dB 2-4 Hz 4-8 Hz peak value 0-2 Hz H x˙˙12 x˙˙g dB 2-4 Hz 4-8 Hz peak value 0-2 Hz H x˙˙21 x˙˙g dB 2-4 Hz 4-8 Hz peak value 0-2 Hz H x˙˙22 x˙˙g dB 2-4 Hz 4-8 Hz peak value
Coupled Building Configuration
Active % Reduction with respect to:
Uncpld 26 25 -26 28 22 -28 30 -28 30 35 -24 35
Uncpld 37% 90% -37% 55% 84% -55% 80% -87% 80% 82% -89% 82%
Rigid 28 17 -1 28 30 13 15 30 21 6 29 29 30 13 15 30
Active 22 5 -5 22 21 6 -6 21 16 1 10 16 20 3 5 20
Rigid 50% 75% 37% 50% 65% 55% 92% 65% 44% 44% 89% 78% 68% 68% 68% 68%
over the rigidly connected transfer functions. Active coupled building control is seen to significantly reduce the peak value of the transfer functions, as well as all resonant peaks of the coupled building system, providing increased seismic protection.
Simulated Ground Motions The coupled building system is next subjected to simulated earthquakes. The simulated earthquakes are produced by twice integrating the acceleration records, accounting for the integration constant, scaling the signal to an appropriate magnitude for the small scale shake table, and scaling the time by a factor of 1/5 for dynamic similitude. The resulting signal is used as the input signal to the shake table. Unlike a transfer function iteration (Spencer and Yang, 1998b), this method does not exactly reproduce the ground 96
accelerations; however, it does capture the essence of each earthquake sufficiently for the analysis purposes in this study. The coupled building system is subjected to four simulated earthquakes, which are derived from: (i) El Centro. The N-S component recorded at the Imperial Valley Irrigation District substation in El Centro, California, during the Imperial Valley, California earthquake of May 18, 1940. (ii) Hachinohe. The N-S component recorded at Hachinohe City during the Tokachi-oki earthquake of May 16, 1968. (iii) Northridge. The N-S component recorded at Sylmar County Hospital parking lot in Sylmar, California, during the Northridge, California earthquake of January 17, 1994. (iv) Kobe. The N-S component recorded at the Kobe Japanese Meteorological Agency (JMA) station during the Hyogoken Nanbu earthquake of January 17, 1995. The time history responses for the rigidly connected and actively controlled cases are shown in Figures 4.6 through 4.9. The active control strategy provides little help to reduce the peak absolute accelerations. The peak absolute acceleration response is difficult to control and does not provide a good measure of the overall damping added to the system, and thus the effectiveness of the active control strategy. The root mean square (RMS) responses do provide a good measure and are computed for each earthquake, for a 40 second duration beginning at the start of each earthquake. The RMS responses provide an indication of the ability of the active control strategy to add damping to the coupled building system. These absolute RMS accelerations for uncoupled, rigidly connected and actively controlled coupled building configurations are presented in Table 4.2. Similar to the results observed in the frequency analysis, for the uncoupled buildings the larger absolute RMS accelerations for each of the four simulated earthquakes a
a
˙˙ 21 and σ ˙˙ 22 ). When the buildings are rigidly connected the absolute were for building 2 ( σ story accelerations of building 2 are reduced. However, the absolute RMS accelerations at the top floor of rigidly connected building 1 actually increase during the Northridge and Kobe simulate earthquakes. This increase is similar to what was observed in the frequency 97
Displacement (cm)
1
∆x(t)
0.5
active rigid
0
−0.5
a
a
x˙˙12(t)
−1 0
5
10
15
25
5
0
−5
−10 0
30
5
10
15
20
25
30
35
35
40
10
Acceleration (m/sec2)
Acceleration (m/sec2)
20
Time (sec)
10
x˙˙22(t)
5
0
−5
−10 0
40
Time (sec)
5
10
a
25
30
35
40
30
35
40
a
x˙˙21(t)
10
10
2
Acceleration (m/sec2)
1 5
0
−5
5
10
15
20
25
30
35
Time (sec)
5
0
−5
−10 0
40
5
10
15
20
25
Time (sec)
10
Acceleration (m/sec2)
Acceleration (m/sec2)
20
Time (sec)
x˙˙11(t)
−10 0
15
x˙˙g(t) 5
0
−5 El Centro (actual−scaled) El Centro (experimental)
−10 0
5
10
15
20
25
30
35
40
Time (sec)
Figure 4.6: Time history response to El Centro simulated ground acceleration. 98
Displacement (cm)
1
∆x(t)
0.5
active rigid
0
−0.5
a
a
x˙˙12(t)
−1 0
5
10
15
25
30
Time (sec)
5
0
−5
−10 0
5
10
15
20
25
30
35
35
x˙˙22(t)
40
10
Acceleration (m/sec2)
10
Acceleration (m/sec2)
20
5
0
−5
−10 0
40
Time (sec)
5
10
20
25
30
35
40
30
35
40
Time (sec)
a
a
x˙˙11(t)
x˙˙21(t)
10
10
2
2
1
Acceleration (m/sec )
Acceleration (m/sec2)
15
5
0
−5
−10 0
5
10
15
20
25
30
35 10
0
−5
−10 0
40
5
10
15
20
25
Time (sec)
x˙˙g(t)
2
Acceleration (m/sec )
Time (sec)
5
5
0
−5 Hachinohe (actual−scaled) Hachinohe (experimental)
−10 0
5
10
15
20
25
30
35
40
Time (sec)
Figure 4.7: Time history response to Hachinohe simulated ground acceleration. 99
Displacement (cm)
1
∆x(t)
0.5
active rigid
0
−0.5
a
a
x˙˙12(t)
−1 0
5
10
15
25
30
Time (sec)
5
0
−5
−10 0
5
10
15
20
25
30
35
35
x˙˙22(t)
40
10
Acceleration (m/sec2)
10
Acceleration (m/sec2)
20
5
0
−5
−10 0
40
Time (sec)
5
10
20
25
30
35
40
30
35
40
Time (sec)
a
a
x˙˙11(t)
x˙˙21(t)
10
10
2
2
1
Acceleration (m/sec )
Acceleration (m/sec2)
15
5
0
−5
−10 0
5
10
15
20
25
30
35
Time (sec)
5
0
−5
−10 0
40
5
10
20
25
x˙˙g(t)
2
Acceleration (m/sec )
15
Time (sec)
10
5
0
−5 Northridge (actual−scaled) Northridge (experimental)
−10 0
5
10
15
20
25
30
35
40
Time (sec)
Figure 4.8: Time history response to Northridge simulated ground acceleration.
100
Displacement (cm)
1
∆x(t)
0.5
active rigid
0
−0.5
a
a
x˙˙12(t)
−1 0
5
10
15
25
0
−5
5
10
15
20
25
30
35
35
x˙˙22(t)
40
10
Acceleration (m/sec2)
5
−10 0
30
Time (sec)
10
Acceleration (m/sec2)
20
5
0
−5
−10 0
40
Time (sec)
5
10
a
25
30
35
40
30
35
40
a
x˙˙21(t) 10
10
2
2
Acceleration (m/sec )
1
2
Acceleration (m/sec )
20
Time (sec)
x˙˙11(t)
5
0
−5
5
10
15
20
25
30
35
5
0
−5
−10 0
40
5
10
15
20
25
Time (sec)
Time (sec) 10
x˙˙g(t)
2
Acceleration (m/sec )
−10 0
15
5
0
−5 Kobe (actual−scaled) Kobe (experimental)
−10 0
5
10
15
20
25
30
35
40
Time (sec)
Figure 4.9: Time history response to Kobe simulated ground acceleration. 101
TABLE 4.2: RMS PERFORMANCE OF COUPLED BUILDING SYSTEM TO SIMULATED EARTHQUAKES Active % Reduction with respect to:
Coupled Building Configuration Uncpld
Rigid
Active 0.52
Uncpld 64%
Rigid 32%
a
1.42
0.76
a
1.49
1.00
0.51
66%
49%
a
2.29
2.48
1.12
52%
55%
a
2.09
1.03
0.87
59%
16%
a
0.47
0.35
0.15
69%
57%
a
0.46
0.42
0.16
65%
62%
a
1.17
1.11
0.29
75%
73%
a
1.72
0.44
0.31
82%
30%
a
0.68
0.61
0.34
51%
45%
a
0.56
0.67
0.36
35%
46%
a
1.87
1.63
0.73
61%
55%
a
2.30
0.70
0.53
77%
24%
a
1.46
1.28
0.66
55%
48%
a
1.09
1.42
0.80
27%
44%
a
2.18
2.39
1.56
28%
35%
a
1.82
1.46
0.95
48%
35%
El Centro
˙˙ 11 m/sec2 σ ˙˙ 12 m/sec σ ˙˙ 21 m/sec σ ˙˙ 22 m/sec σ Hachinohe
˙˙ 11 m/sec σ ˙˙ 12 m/sec σ ˙˙ 21 m/sec σ ˙˙ 22 m/sec σ Northridge
˙˙ 11 m/sec2 σ ˙˙ 12 m/sec σ ˙˙ 21 m/sec σ ˙˙ 22 m/sec σ
Kobe
˙˙ 11 m/sec2 σ ˙˙ 12 m/sec σ ˙˙ 21 m/sec σ ˙˙ 22 m/sec σ
analysis when the buildings were rigidly connected. Additionally, the absolute RMS accela
˙˙ 21 ) are shown to increase over erations of the first story of rigidly connected building 2 ( σ the uncoupled responses for the El Centro and Kobe simulated earthquakes. Again here, rigidly connecting the buildings results in a trade-off of performance, reducing absolute
102
acceleration responses at some stories while increasing the absolute acceleration responses at other stories, thus not benefiting the coupled building system as a whole. The active control strategy is able to reduce all of the absolute RMS accelerations. The active control strategy reduces the RMS responses of the absolute story accelerations by 52-66% over the uncoupled buildings and by an additional 16-55% over the rigidly connected buildings for the El Centro simulated earthquake. Active control reduces the absolute RMS acceleration responses by 65-82% over the uncoupled buildings and by 3073% over the rigidly connected buildings for the Hachinohe simulated earthquake. Active control reduces the absolute RMS acceleration responses by 35-77% over the uncoupled buildings and by 24-55% over the rigidly connected buildings for the Northridge simulated earthquake and 27-55% over the uncoupled buildings and 35-48% over the rigidly connected buildings for the Kobe simulated earthquake. The active control strategy is seen to significantly reduce all of the coupled building system’s absolute acceleration responses to four different simulated historical earthquakes.
4.5 Chapter Summary This chapter details experimental tests conducted on two 2-story flexible building models, placed adjacent to one another on a shake table and coupled at the top of the building models with a control actuator, to reduce absolute acceleration responses due to seismic excitation. The control actuator is a DC servo-motor with a ball screw mechanism, similar to the 35-ton control actuators coupling the Triton Square buildings. The control design uses absolute acceleration and relative displacement measurements at the top of the building models, where the actuator is located. A control-oriented design model is developed by experimentally measuring the transfer functions of the coupled building system and developing a mathematical model to
103
replicate the input/output behavior of the system. This design model fully accounts for control-structure interaction. Frequency and time domain analyses are performed for system evaluation. The active coupled building system is able to reduce the resonant peaks of the transfer functions of absolute story acceleration to ground acceleration to between 67-18% of the uncoupled system’s resonant peaks and corresponding to between 50-88% over the rigidly connected resonant peaks. The coupled building system is subjected to four simulated earthquakes and absolute RMS story accelerations are computed. The active coupled building system can reduce these absolute RMS story accelerations to as low as 18% of the uncoupled absolute RMS accelerations and to as low as 73% compared to the rigidly connected system. Active control, using readily available absolute acceleration and relative displacement feedback measurements of the coupling link, is shown experimentally to be an effective method of structural control.
104
CHAPTER 5: CABLE DAMPING CONTROL: BACKGROUND
Cables are efficient structural elements that are used in cable-stayed bridges, suspension bridges and other cable structures. These cables are subject to environmental excitations, such as rain-wind induced vibration, and support excitations. Steel cables are flexible and have low inherent damping, resulting in high susceptibility to vibration. Vibration can result in premature cable or connection failure and/or breakdown of the cable corrosion protection systems, reducing the life of the cable structure (Watson, 1988). Additionally, cable vibrations can have a detrimental effect on public confidence in the safety of cable structures. Transmission lines have also demonstrated significant vibration problems, including those caused by vortex shedding, wake-induced oscillation, and iced and ice-free galloping. Fatigue of the transmission lines near clamps or masses (such as aircraft warning spheres) is the principal effect of conductor vibration, though galloping can cause sparkover between lines of different phase (Tunstall, 1997). Cable damping, as studied herein, uses transversely attached passive, active and semiactive dampers to mitigate cable vibration. Suppressing cable vibration has been done, on transmission lines, since the first part of the last century (Stockbridge, 1925). However, cable damping employing transversely attached dampers to the cable structural elements of civil structures is more recent, in fact only over the past two decades. Numerous passive and active cable damping studies have been performed and fullscale applications realized. Semiactive control of a taut cable has recently been proposed (Johnson et al., 1999, 2000a, and 2000b, Baker et al. 1999a, and Baker 1999b). This dissertation extends the semiactive analytical studies to include cables with sag and verifies experimentally cable damping. 105
In this chapter a literature review of cable damping control is presented, a controloriented evaluation model for the in-plane motion of cables with sag is developed, cable excitation and a performance measure are identified, and passive, active and semiactive cable damping control strategies presented.
5.1 Cable Damping Literature Review A number of methods have been proposed to mitigate cable vibrations. For stay cables, tying cables together, aerodynamic cable surface modification, and passive and active axial and transverse cable control have been used to dampen vibration. Tying cables together shortens the effective length of the cables, and is intended to shift the frequencies of the cable out of the range of the excitation. This strategy deteriorates the aesthetics of the cable structure. Changing the surface of the cable to reduce susceptibility to environmental excitations has also been explored, but is impractical for retrofit applications and may increase motion during high winds. For transmission lines, two primary methods are used for reducing vibration. Stockbridge dampers (Stockbridge, 1925), a variety of tuned vibration absorbers, are the most common means today for adding supplemental damping to transmission lines (Tunstall, 1997). An alternate solution for multiple parallel transmission lines is adding dampers to the bundle spacers routinely used for separating conductors (Edwards and Boyd, 1965). Various researchers have proposed passive control of cables using viscous dampers attached transverse to the cables. Kovacs (1982) first identified that an optimal damper size exists and developed optimal damping coefficients for the transverse passive viscous damper control strategy of a taut cable. Sulekh (1990) and Pacheco et al. (1993) numerically developed a “universal” design curve to facilitate the design of passive dampers for stay cables. This nondimensionalized curve can be used to determine the optimal viscous damping properties for a desired mode of any given cable span and fundamental fre106
quency. Krenk (1999) obtained explicit asymptotic results for the optimal damping coefficients, developing an analytical solution for the design curve. These studies indicate that, for a passive linear damper, the maximum supplemental damping ratio is approximately x d /2L, where x d is the distance from the cable anchorage to the damper and L is the length of the cable. Transverse passive viscous dampers have been applied to full-scale applications, including the cables on the Brotonne Bridge in France (Gimsing, 1983), the Sunshine Skyway Bridge in Florida (Watson, 1988) and the Aratsu Bridge in Japan (Yoshimura, 1989). The damper location is typically restricted to be close to the bridge deck for aesthetic and practical reasons. For short cables, a high x d /L ratio is feasible and a passive damper can provide sufficient damping. For increasingly longer bridge cables, passive dampers may not provide enough supplemental damping to eliminate vibration effects, such as rainwind induced motion, without significant changes to the aesthetics of the structure. Several recent papers have shown that semiactive dampers may provide levels of damping far superior to their passive counterparts. Johnson et al. (1999, 2000a, 2000b) and Baker et al. (1999a, 1999b) used a taut string model of in-plane cable vibration and developed a control-oriented model using a static deflection shape in a series expansion for the cable motion. They showed that a “smart” semiactive damper can provide 50 to 80% reduction in cable response compared to the optimal passive linear damper. The level of reduction was most significant when the damper was connected close to the cable anchorage. A passive damper moved close to the end of the cable was shown to rapidly lose any ability to add damping to the system, whereas a semiactive damper retained most of its performance even at damper locations below 1% of the cable length (though with larger forces). The taut string model of cable vibration neglects some cable characteristics that are known to have some effect on passive damper performance. Cable sag effects the dynamics of the cable. In particular, sag modifies the stiffness of the modes that are sym107
metric about the center of the cable. Previous studies have examined the performance of transverse passive viscous dampers on sag cables. For example, using a sine series Galerkin approach, Sulekh (1990) showed that the damping added to the first symmetric mode by passive dampers was notably reduced by sag — by about 14% for a typical stay cable sag level — compared to that predicted by a taut string model. Further, the higher modes were virtually unaffected. An alternate approach by Xu et al. (1998a, 1998b, 1998c), using a spatial discretization, made similar observations, with a 38% decrease in first-symmetric-mode damping for a long (442.6 m) stay cable with slightly larger sag. Semiactive control, employing smart cable dampers, have been proposed for the mitigation of rain-wind induced vibration of cables with sag (Christenson et al. 2001a, 2001b, Johnson et al. 2000c, 2001a, 2001b).
5.2 In-Plane Motion of Cable with Sag Consider the uniform cable suspended between two supports of different heights, as shown in Figure 5.1. This dissertation investigates cables with a flat profile (flat-sag
x θ f(x,t)
gravity
L
z,w
ρ, c, EA z(x
d
xd
)
Fd(t)
Figure 5.1: In-plane static profile z(x) and dynamic loading f(x,t) of inclined cable with sag and transverse damper force. 108
cables); for a horizontal cable, this assumption requires the sag to span ratio be less than 1:8 (for inclined cables the assumption is valid but over a smaller range of sag) (Irvine, 1981). The primary suspension cables for the Golden Gate Bridge have a sag to span ratio of less than 1:8 and can thus be considered as flat-sag cables. Further, the effects of longitudinal flexibility are included and flexural rigidity is ignored. The static profile of the cable can be approximated by a parabolic curve and the in-plane transverse cable motion w(x, t) , relative to the static profile, is given by the nondimensional equation of motion (Irvine, 1981) λ2 1 w ˙˙(x, t) + cw˙ (x, t) – ----2- w″(x, t) + -----2 π
π
1
∫ w(ξ, t) dξ
= f (x, t) + Fd (t)δ (x – x d)
(5.1)
0
in the domain 0 ≤ x ≤ 1 , with boundary conditions w(0, t) = w(1, t) = 0 . c is the viscous damping per unit length, ( )′ and (˙) denote partial derivatives with respect to x and t , respectively, f (x, t) is the distributed load on the cable, Fd (t) is a transverse in-plane damper force at location x = x d , and δ (·) is the Dirac delta function. The nondimensional quantities are related to their dimensional counterparts, shown with overbars, according to the following relations t = ω0 t
x = x⁄L
c = c ⁄ ρω 0
w(x, t) = w(x, t ) ⁄ L
f ( x, t ) = L f ( x, t ) ⁄ π 2 H
δ (x – x d) = Lδ (x – x d)
ω 02 = H π 2 ⁄ ρ L 2
Fd (t) = Fd (t ) ⁄ π 2 H
where L is the length of the cable, ω 0 is the fundamental natural frequency of the taut cable, H is the component of cable tension in the longitudinal x-direction, and ρ is the cable mass per unit length. λ 2 is the nondimensional independent parameter (Irvine, 1981) ρ gL cosθ 2 EAL d 2 EAL λ 2 = ---------------------- ----------- = 64 --- -----------
H
H Le
L H L e
where θ is the inclination angle, L e is the static (stretched) length of the cable
109
(5.2)
1 ρ gL cosθ 2 d 2 ----------------------Le = L 1 + = L 1+8 L H 8
(5.3)
For flat-sag cables, d = dL = ( ρ gL 2 cosθ ) ⁄ 8H
(5.4)
is the peak (dimensional) sag of the parabolic static profile x x z(x) = – 4d --- 1 – --- L L
(5.5)
The effects of cable sag, angle-of-inclination, and axial stiffness on the nondimensional dynamic response of the system enter only though the independent parameter λ2. Stay cables on cable-stayed bridges typically have λ2 values on the order of 1 or smaller (Gimsing, 1983); some stay cables reported in the literature have larger λ2 values such as the 2.2 reported in Pacheco et al. (1993) and the 3.6 reported in Xu et al. (1998a). Typical transmission line characteristics (Tunstall, 1997) give a λ2 in the neighborhood of 90. λ 2 ∈ [ 140, 350 ] is the range typical for the main cable on a suspension bridge (Gimsing, 1983). Specific performance examples will be given below for control of cables with some λ2 values of interest, as well as the general trends as λ2 increases from 0 to 500. Even for large λ2 values such as the λ2 = 1000 shown, the midspan sag-to-length ratio d can be less than the 1/8 required for the flat-sag cable (i.e., parabolic static profile) assumption for horizontal cables (Irvine, 1981). 0.009 d = 0.095
0.044
0
0.020
1 10
100
λ2 = 1000
Figure 5.2: Typical static sag profiles.
Control-Oriented Evaluation Model Determining an accurate and efficient control-oriented design model is the first and fundamental step in the design of a semiactive control strategy. A design model is sought 110
that can capture the salient features of the dynamic system with a relatively small number of degrees-of-freedom (DOFs). Previous transversely-controlled cable models have employed the Galerkin method, using only sine shape functions requiring 350 terms in the series (Sulekh, 1990), as well as hybrid-type finite element methods which also require numerous DOFs to insure accurate results (Xu et al., 1998a). Semiactive control design, as well as the computation of performance criteria through simulation with numerous control strategies, is impractical for systems of such size. Thus, successful semiactive control design is dependant on determining a lower order, control-oriented design model. A lower order model is accomplished here by including a static deflection shape in addition to the sine series in the approximation of the cable motion (Johnson et al., 1999). Using a Galerkin method, the motion of the cable may be computed using a finite series approximation w( x , t ) =
m
∑ φ j (x)q j (t)
(5.6)
j=1
where the q j (t) are generalized displacements and the φ j (x) are a set of shape functions that are continuous with piecewise continuous slope and that satisfy the geometric boundary conditions φ j (0) = φ j (1) = 0
(5.7)
A sine series may be used for the shape functions, though Johnson et al. (2000a,b) showed that the convergence of this series is slow, making it difficult to construct a control-oriented model. However, they also demonstrated that the introduction of a static deflection shape as an additional shape function significantly improved the series convergence and provided an excellent control-oriented model. This approach is used here as well, though it must be extended to account for cable sag. Consider the static deflection of a cable with sag due to a unit load at location x = x d — the same as the equation of motion (5.1) without the dynamic terms and with a unit point load on the right hand side 111
λ2 1 – ----2- wstatic ″ (x) + -----2 π
π
1
∫ wstatic (ξ) dξ
= δ (x – x d)
(5.8)
0
where wstatic (0) = wstatic (1) = 0 . For a given deflection wstatic (x) , the integral term in Equation (5.8) acts like a constant load distributed over the entire length of the cable. Such a load produces a parabolic deflection. The point load, given by the Dirac delta, adds a triangular component. Substituting a linear combination of parabolic and triangular deflections into (5.8) and solving for the unknown coefficients results in the static deflection wstatic (x) =
π2( 1
– x d )x –
π2( x
3λ 2 π 2 x d ( 1 – x d ) - x(1 – x) – x d )H (x – x d) – --------------------------------------12 + λ 2
(5.9)
where H (.) is the Heaviside, or unit step, function. For consistent shape function scaling, Equation (5.9) is normalized to give a maximum deflection of 1,1 resulting in the static deflection shape function 12 + λ 2 x 2 2 x 12 + λ – 3λ x d ( 1 – x d ) d
x H (x – x d) x d 1 – x d
3λ 2 12 + λ 2
φ 1(x) = ---------------------------------------------------------- ----- + 1 – ----- ---------------------- – ------------------ x ( 1 – x )
(5.10)
Note that as the independent parameter λ2 tends to zero, Equation (5.10) reverts to the triangular static deflection
φ 1(x)
λ2 = 0
x ⁄ x , 0 ≤ x ≤ xd d = ( 1 – x ) ⁄ ( 1 – x d ), x d ≤ x ≤ 1
(5.11)
used by Johnson et al. (2000a,b) to model a taut cable (where λ2 = 0). The remaining shape functions are sine functions: φ j + 1(x) = sin π jx , j=1,...,m–1
(5.12)
Substituting the shape functions into the nondimensional equation of motion (5.1) and simplifying results in the matrix equation
1. The peak of wstatic(x) can be shown to always occurs at x = xd.
112
˙˙ + Cq˙ + Kq = f + f Fd (t) Mq
(5.13)
with mass M = [ m ij ] , damping C = cM , and stiffness K = [ k ij ] matrices 1
m ij =
∫ φi(x) φ j (x) dx 0
1 --- δ ij, 2 1 2 - λ [ 4 λ 2 + 60 – 15 ( 12 + λ 2 )x d ( 1 – x d ) ] 48 + ----30 -------------------------------------------------------------------------------------------------------------, [ 12 + λ 2 – 3 λ 2 x d ( 1 – x d ) ] 2 = sin k π x d 12 + λ 2 ----------------------------------- , --------------------------------------------------------2 – 3 λ 2 x ( 1 – x ) x ( 1 – x )k 2 π 2 12 + λ d d d d 2 sin k π x d 12 + λ 12 λ 2 ------------------------------------------------------------------------------------------– -------------------------------- 12 + λ 2 – 3 λ 2 x ( 1 – x ) x ( 1 – x )k 2 π 2 k 3 π 3 ( 12 + λ 2 -) , d d d d
i > 1, j > 1 i = 1, j = 1 otherwise, even k
where k = max { i, j } – 1
otherwise, odd k
where k = max { i, j } – 1
1
c ij = c ∫ φi (x) φ j (x) dx = cm ij 0
k ij =
1 ----π2
k isag
k ijtension
λ2
1 1 1 tension ∫ φi (x) dx ∫ φ j (x) dx + ∫ φi′(x) φ j′ (x) dx = λ 2 k isag k sag j + k ij 0 0 0
6 -, i=1 ----------------------------------------------------------------2 2 [ 12 + λ – 3 λ x d ( 1 – x d ) ] π 2 = --------------------2-, even i (i – 1)π 0, otherwise 1--- ( i – 1 ) 2 δ , ij 2 3 λ 4 ( 3x d2 – 3x d + 1 ) 1 -, -----------------------------2- + ---------------------------------------------------------------------[ 12 + λ 2 – 3 λ 2 x d ( 1 – x d ) ] 2 π 2 xd ( 1 – xd ) π = sin k π x d 12 + λ 2 ------------------------------------------------------------------------------------------ π 2 [ 12 + λ 2 – 3 λ 2 x ( 1 – x ) ] x ( 1 – x ) , d d d d sin k π x d 12 + λ 2 12 λ 2 ------------------------------------------------------------------------------------------– ---------------------------- , π 2 [ 12 + λ 2 – 3 λ 2 x d ( 1 – x d ) ] x d ( 1 – x d ) k π ( 12 + λ 2 ) 113
i > 1, j > 1 i = 1, j = 1 otherwise, even k
where k = max { i, j } – 1
otherwise, odd k
where k = max { i, j } – 1
externally applied load vector f = [ f1 f2 …
fm ] T
1
fi =
∫ f (x, t) φi(x) dx
(5.14)
0
vector q = [ q j ] of generalized displacements, and damper load vector f f = φ(x d) = [ φ 1(x d) φ 2(x d) … φ m(x d) ] T = [ 1 sin ( 1 π x d ) … sin ( { m – 1 } π x d ) ] T
(5.15)
Note that the stiffness in Equation (5.13) is comprised of stiffness due to tension, as in the taut-string model, plus additional stiffness due to the independent parameter λ
2
that only affects the modes not antisymmetric about the center of the cable. Further, note that the mass, damping, and stiffness elements reduce exactly to the corresponding equa2
tions in Johnson et al. (2000b) in the absence of λ . The resulting model captures the salient features of a cable damper system much better than with sine terms alone. With just 11 terms (static deflection plus 10 sine terms), the first several natural frequencies, damping ratios, and modeshapes are more accurate than those computed with 100 sine terms alone. Convergence tests showed this accuracy to be true in the uncontrolled case, in the case with the optimal passive linear viscous damper, and with an active damper. For control design, the system dynamics may be equivalently written in state-space form with input/output relations ˙ = A h + B F (t ) + G f h z z z d y = Cy h + Dy F d(t) + Hy f + v
(5.16)
where h = [ q T q˙ T ] T is the state vector, y = [ w(x d, t)
w ˙˙(x d, t) ] T + v is a vector of
noisy sensor measurements (includes the displacement and absolute acceleration at the damper location), v is a vector of stochastic sensor noise processes, and
114
Az =
Cy =
0
I
–1
–1
–M K
–M C
fT
0
–1
–f T M K
–1
–fT M C
Bz =
Dy =
0 –1
M f 0 –1
fTM f
Gz =
Hy =
0 M
–1
(5.17)
0 fTM
–1
Cable Excitation There are no well established models for rain-wind induced galloping, though it tends to be dominated by one of the first few modes. The cable/damper system is here simulated with a stationary Gaussian white noise excitation shaped by the first mode of the cable with no sag (i.e., a half-sine). Without a supplemental damper, and in the absence of sag, this half-sine excitation would energize just the first mode of the cable.
Measure of Damper Performance Modal damping ratios provide a useful means of determining the effectiveness of linear viscous damping strategies. However, using a semiactive damper introduces a nonlinearity into the combined system. Consequently, performance measures other than modal damping must be used for judging the efficacy of nonlinear damping strategies in comparison with linear (passive or active) dampers. Using the root mean square (RMS) or peak response of the cable at some particular location (or several locations) is one possible measure of damper performance. However, it may be possible for one control strategy to decrease the motion significantly in some regions of a structure but allow other parts to vibrate relatively unimpeded. Thus, the primary measure of damper performance considered herein is the root mean square (RMS) cable deflection integrated along the length of the cable, defined by
115
1
σ displacement(t) =
E
∫ v 2(x, t) dx
=
E [ q T(t)Mq(t) ]
0
=
(5.18)
trace { M 1 / 2 E [ q(t)q T(t) ]M 1 / 2 }
where M 1 / 2 is a square symmetric matrix such that M 1 / 2 M 1 / 2 = M . The corresponding RMS cable velocity may be computed from the generalized velocities σ velocity(t) =
E [ q˙ T(t)Mq˙ (t) ] =
trace { M 1 / 2 E [ q˙ (t)q˙ T(t) ]M 1 / 2 }
(5.19)
For stationary response to a stationary stochastic excitation, these performance measures are not functions of time, but become constants.
5.3 Cable Damping Control Strategies Three types of dampers are considered in this study. The damper of primary interest is a general semiactive device, one that may exert any required dissipative force. However, comparison with passive linear viscous dampers, similar to the oil dampers that have been installed in numerous cable-stayed bridges (Gimsing, 1983; Watson, 1988; Yoshimura, 1989), is considered to demonstrate the improvements that may be gained with semiactive damping technology. Additionally, comparison with active control devices is useful as they bound the achievable performance.
Passive Viscous Damper If the damping device is a passive linear viscous damper, then the damper force is Fd (t) = – c dw˙ (x d, t)
(5.20)
where c d is a nondimensional damping constant, and w˙ (x d, t) is the nondimensional velocity at the damper location
116
w˙ (x d, t) =
m
∑ q˙ i(t) φi ( xd )
= f T q˙ = [ 0 T f T ]h
(5.21)
i=1
The modal damping may be determined via a straightforward eigenvalue analysis. Note that the optimal passive damper supplies pure damping; stiffness tends to degrade the damper performance (Sulekh, 1990; Xu et al., 1998b).
Active Damper The optimal passive viscous damper provides one benchmark against which to judge semiactive dampers. The other end of the spectrum of control possibilities is an ideal active damper, which may exert any desired force. The performance of the actively controlled systems give a performance target for semiactive control. One family of H2/LQG control designs is considered in this study. This family of 2
controllers performed well for cables with λ = 0 (Johnson et al., 2000a,b). These controllers use force proportional to an estimate of the state of the system, Fdactive(t) = – Lhˆ , where L = R –1 B T P is the feedback gain that minimizes the cost function 2 2 2 + σ velocity ) + R σ force J = 1--2- ( σ displacement T
= lim
T →∞
1 E --- ∫ ( 1--2- q T Mq + 1--2- q˙ T Mq˙ + R Fd2) dt T
(5.22)
0
where P satisfies the algebraic Riccati equation A T P + PA – PBR –1 B T P + Q = 0
(5.23)
By varying the control weight R, a family of controllers that use varying force levels can be designed. A standard Kalman filter observer is used to estimate the states of the system ˙ˆ = ( A – L KF C y )hˆ + L KF y + ( B – L KF D y ) Fd (t) h
117
(5.24)
where L KF = ( P˜ C yT + GQ KF H T ) ( R KF + HQ KF H T ) –1 is the estimator gain and P˜ is computed from the Riccati equation AP˜ + P˜ A T – ( P˜ C yT + GQ KF H yT ) ( R KF + HQ KF H T ) –1 ( C y P˜ + H y Q KF G T ) = – GQ KF G T
(5.25)
where Q KF is the magnitude of the excitation spectral density S ff (ω ) , R KF the magnitude of noise spectral density S vv(ω ) , E [ f ] = 0 , E [ v ] = 0 , where E [ · ] is the expectation operator, and excitation f and sensor noise v are uncorrelated.
Semiactive Damper Unlike an active device, a semiactive damper, such as a variable-orifice viscous damper, a controllable friction damper, or a controllable fluid damper (Spencer and Sain, 1997; Housner et al., 1997), can only exert dissipative forces. Herein, a generic semiactive device model is assumed that is purely dissipative. Essentially, this requirement dictates that the force exerted by the damper and the velocity across the damper must be of opposite sign; i.e., Fd (t)w˙ (x d, t) must be less than zero. Figure 5.3 shows this constraint graphically. A clipped optimal strategy is used, with a primary controller based on the same family of H2/LQG designs used for the active damper, and a secondary controller to account for the nonlinear nature of the semiactive device
Fd(t) semiactive device nondissipative
per dam s u visco dissipative
dissipative .. w(xd,t) nondissipative
Figure 5.3: Ideal semiactive damper dissipative forces. 118
F active(t) Fdactive(t)v˙(x d, t) < 0 Fd (t) = d 0 otherwise
(5.26)
Here, the secondary controller simply clips non-dissipative commands. For implementation, a bang-bang controller with force feedback has been shown to be effective (Dyke et al., 1996a). The semiactive device introduced here and used for the analysis in Chapter 6 is an ideal semiactive device. Of course actual semiactive devices, such as the smart damper examined in Chapters 7 and 8, may be limited in its performance and may not be capable of achieving all forces in the first and third quadrants of the force versus velocity plot shown in Figure 5.3. This limit is identified and discussed in Chapter 8. Having noted this, the ideal semiactive device does, however, provide an “upper bound” on the performance one could expect from a semiactive device.
5.4 Chapter Summary The effects of cable sag, inclination, and axial stiffness are introduced into the dynamic model of transverse in-plane cable vibration. These parameters are described 2
completely by the independent parameter λ . The Galerkin approach using 20 sine functions and a static deflection shape are used to accurately provide a low-order model of the cable system. The static deflection shape used is the static profile of a flat-sag cable with a point load applied to the cable at the location of the damper. Passive, active and semiactive control strategies are presented. The effects of the sag on the performance of the cable system are examined in Chapter 6 using the cable model defined here.
119
CHAPTER 6: CABLE DAMPING CONTROL: EFFECTS OF CABLE SAG
In this chapter, the effect of cable sag on cable damping control, in particular damping ratio, cable response, and damper location, is examined. The cable is assumed to have virtually no inherent damping without the supplemental damper, about 0.005% in the first mode. Root mean square (RMS) responses to the excitation are computed via a Lyapunov solution for linear (passive and active) strategies and from simulation for semiactive dampers. A 1% RMS sensor noise corrupts each sensor measurement (modelled as Gaussian pulse processes).
6.1 Effects of Sag on Damping Ratio Before examining RMS responses with passive and semiactive dampers, it is worth studying the modal properties of the controlled system. Since the semiactive system is, by definition nonlinear, the active system will be used to compute modal properties and will be compared with passive modal damping. The RMS responses of the optimal active and optimal semiactive damping strategies will be seen below to be quite similar. Thus, the modal properties of the active system are a good indication of “equivalent” modal properties for the semiactive system. The modal damping that can be provided to the fundamental mode of the cable/ damper system by passive and active dampers is shown in Figure 6.1 for a damper location xd = 0.02. (The reader may note that the five markers, whether filled or not, denote different levels of λ2, whereas dashed lines with open markers denote the passive, and solid lines with filled markers denote the active results.) In the absence of sag, the maximum damp120
2.2
1.8 1.6 1.4 1.2 1
Active Passive
-1
Damping Ratio
First Symmetric Mode
Frequency
2
10
-2
10
-3
10
-4
10
2.3
2
λ = 0 2 λ = 1 2 λ = 30 2 λ = 42.5 2 λ = 50
Frequency
2.2 2.15 2.1 2.05 2
Damping Ratio
First Antisymmetric Mode
2.25
-1
10
-2
10
-3
10
-4
10
-1
10
0
10
1
10
RMS Damper Force Figure 6.1: Natural frequency and damping ratio in the first two modes for the linear designs for xd = 0.02. 121
ing in the first symmetric mode provided by a passive damper is 1.03%, whereas the active damper provides over 36% of critical damping. With small sag, λ2 = 1, the passive damping is degraded slightly to 0.91% (a factor of 0.88); the active system drops to 33.6% (a factor of 0.93). For a larger sag λ2 = 30, the passive damper is less effective, providing only 0.04% damping (a factor of 0.039 compared to no sag). The active device, however, still provides almost a 1.6% damping ratio (a factor of 0.044 compared to no sag). For λ2 = 42.5, the passive damper is ineffective for the first symmetric mode, providing only
0.002% damping. The active device for this particular level of sag is also severely reduced, providing only 0.04% damping. For yet larger sag at λ2 = 50, a passive damper can provide 0.04% damping and the active strategy can provide 1% damping in the first symmetric mode. The natural frequency of the first symmetric mode for larger sag increases to over twice the value at small sag, which gives the cable somewhat smaller displacements with the same excitation, but does not degrade the improvements seen with active and (below) semiactive dampers. The natural frequencies remain relatively constant over the range of xd. Sag has virtually no effect, except for sag levels in the immediate vicinity of λ2 = 39.5 and 41.93 (note that λ2 = 42.5 even appears unaffected by sag), on passive
damping for the first antisymmetric mode; this result is consistent with previous studies (Sulekh, 1990), with the damping remaining about 1% of critical. Similarly, the active control of the first antisymmetric mode is unaffected by the inclusion of sag, again except for sag levels in the immediate vicinity of λ2 = 39.5 and 41.93, achieving 30% damping. The significance of λ2 = 39.5 and 41.93 are identified in the subsequent discussion on the effects of sag and inclination on modal characteristics of the controlled system, and will be explained in a discussion of the effects of sag on RMS cable response in Section 6.2. Figure 6.1 does indicate that an optimal level of control does exists for both the passive control and for the active control strategy.
122
TABLE 6.1: COMPARISON OF PEAK MODAL DAMPING RATIOS WITH A LINEAR PASSIVE VISCOUS DAMPER AT XD = 0.02 λ λ2
2
=0 λ2 = 0.245 λ2 = 1 λ2 = 1.20 λ2 = 3.63
mode
Sulekh (1990)
Xu et al. (1998a)
Krenk (2001)
this research
first (symmetric) first (symmetric) first (symmetric) first (symmetric) first (symmetric)
1.10% -0.95% ---
-0.98% -0.85% 0.64%
1.00% 0.97% 0.89% 0.87% 0.66%
1.03% 1.00% 0.91% 0.89% 0.68%
The passive results computed here are comparable to those in previous studies, thus further verifying the control-oriented model used herein. Table 6.1 shows a comparison of the peak modal damping ratio that can be achieved with a passive damper for several sag levels, comparing to the results of Sulekh (1990) and Xu et al. (1998a) for xd = 0.02. The former used a Galerkin approach, requiring 350 sine shape functions, whereas the latter used a numerical method in which the cable was discretized into 400 segments for solution purposes. Comparing the results of these two previous studies to those found in this study, it is clear that the design oriented model used here is both efficient, requiring only 21 degrees-of-freedom, and accurate, resulting in damping values bounded by the Sulekh and Xu studies. Additionally, a recent paper by Krenk (2001) finds an explicit analytical approximate solution to the maximum modal damping for cables with sag using asymptotic relations. The results of optimal passive damping from Krenk’s approximate solution, also presented in Table 6.1, are consistent with the control oriented model developed in this research. The effects of sag and inclination on modal characteristics of the controlled system, in particular on the first symmetric and first antisymmetric modes of vibration, may be better seen in Figure 6.2. As λ2 approaches 40, the passive control of both symmetric and antisymmetric modes is significantly reduced — indeed, it is ineffective at λ2 = 39.5 and 41.93. (Reasons for these regions of decreased performance are explained in
123
Section 6.2.) The symmetric mode is more greatly affected in regions nearby λ2 = 40 than is the first antisymmetric mode. The active control damping is similarly affected, although the active strategy is capable of providing significantly increased performance in general. Crossover of the controlled symmetric and antisymmetric natural frequencies does occur at certain levels of λ2. For levels of sag below λ2 = 39.5, it is observed that the increase in sag results in a significant decrease in damping in the first two modes for both passive and active control strategies. Increasing the sag beyond λ2 = 42 increases the damping in these modes, eventually to values near that of the taut cable. Both control strategies result in increased natural frequencies as the sag is increased. Figures 6.3 and 6.4 show the frequency and damping ratio of the first symmetric and antisymmetric modes, respectively, over a range of damper locations and for several levels of the independent parameter λ2. The symmetric mode is affected by sag, particu-
Frequency
3
Active Passive
2.5 2 1.5 1
Damping Ratio
-1
10
-2
10
-3
10
-4
st
10
1st Symmetric 1 Antisymmetric 0
5
10
100
500
2
independent parameter λ
Figure 6.2: Modal frequency and damping ratios over a range of sag with a damper at xd = 0.02. 124
larly for certain combinations of λ2 and damper location. For example, λ2 = 42.5 drops to minimal damping near xd = 0.025 for both passive and active strategies. The antisymmetric mode is somewhat different; active control is quite effective in adding damping to this mode over a wide range of sag and damper location. The passive has some areas, near xd = 0.025 and 0.075, where it does not perform well.
6.2 Effects of Sag on RMS Cable Response The RMS cable displacement, defined in Equation (5.18), as well as the RMS cable velocity and RMS damper force, were computed using a Lyapunov solution for passive and active control strategies, but through simulation for the semiactive system. Due to minimal damping in less aggressive control strategies, which require longer impractical simulation times for the computation of performance criteria, only several semiactive controllers in the family of possible controllers are simulated here. The responses with the semiactive are shown using large bold markers (the same markers as the active and passive for a given value of the independent parameter λ2). Figure 6.5 shows the RMS cable displacement as a function of the RMS damper force for a damper at xd = 0.02 at several levels of sag. For strategies using small forces, the passive and active performance are nearly the same — this trend was also seen in Johnson et al. (2000b) where it was also observed that the semiactive strategy had similar performance to passive and active. However, at some point, the passive damper begins to have diminished gains in spite of larger damper forces. This trend is due to the damper only “knowing” local information, that is, the cable velocity at the damper location. Effectively, the passive damper starts to lock the cable down at that point — certainly limiting the cable motion at the damper location — but allowing the rest of the cable to vibrate nearly unimpeded. The active and semiactive strategies, however, are able to take advantage of larger force levels in such a way that they do not lock the cable down, but rather continue to dissipate energy. The effect is that the con125
2
λ = 0 2 λ = 1 2 λ = 30 2 λ = 42.5 2 λ = 50
Frequency
2
Active Passive
1.5
Damping Ratio
1 -1
10
-2
10
-3
10
-4
10
0
0.02
0.04
0.06
0.08
0.1
damper location xd Figure 6.3: Frequency and damping ratios of first symmetric mode as a function of damper location xd for several sag levels.
Frequency
2.5 2.4
2
λ = 0 2 λ = 1 2 λ = 30 2 λ = 42.5 2 λ = 50
Active Passive
2.3 2.2 2.1
Damping Ratio
2 -1
10
-2
10
-3
10
-4
10
0
0.02
0.04
0.06
0.08
0.1
damper location xd Figure 6.4: Frequency and damping ratios of first antisymmetric mode as a function of damper location xd for several sag levels.
126
trollable semiactive damper is able to achieve a 50% to 80% displacement reduction, depending on the sag, compared to the optimal passive linear viscous damper. Figure 6.6 shows the RMS cable displacement for the passive linear viscous damper, and the optimal active and semiactive dampers versus the independent parameter λ2. Without sag (λ2 = 0), the semiactive damper can provide about a 71% response
decrease compared to the best passive device. With small sag (λ2 = 1), the RMS displacements decrease little for all three damping strategies. For λ2 = 30, the control performance for passive, active, and semiactive strategies begin to degrade and around λ2 = 40, the same region where the damping in the first two modes was significantly decreased, the RMS performance is poor. Increasing λ2, the performance improves, but there are additional regions where all methods are ineffective. This phenomenon will be discussed in detail in the next section. Nevertheless, the semiactive damper always decreases response compared to the best passive damper, by as much as 60% to 80%. To observe what happens near the peaks of reduced performance, Figure 6.7 provides a closer look. Indeed, the region of decreased performance around λ2 = 40 consists of two peaks of poor performance with a valley of better performance. The pairing of these two peaks is found for each of the three regions of decreased performance in the [0,500] range of λ2 values studied here. The peaks of poor performance occur at λ2 values of 4π2, 41.93, 16π2, 167.79, 36π2 and 377.59. Similar results are seen in RMS cable velocity in Figure 6.8 (though there is a small increase from no sag to small sag). Thus, it may be concluded that a “smart” damper may provide superior damping to cables for a large range of cable sag. Note, however, that the benefit comes with larger damper forces, though these force levels (Johnson et al., 2000b) are still well within the capabilities of current damper technology.
127
RMS Displacement
1
10
Semiactive Active Passive 2
λ = 0 2 λ = 1 2 λ = 30 2 λ = 42.5 2 λ = 50
0
10
-1
0
10
1
10
10
RMS Force
Figure 6.5: RMS displacement for a semiactive, passive viscous, or active dampers at xd = 0.02 as a function of the RMS force.
Semiactive Active Passive Uncontrolled
RMS Displacement
1
10
0
10
-1
10
0
5
10
100
500
2
independent parameter λ
Figure 6.6: Minimum RMS displacement for a semiactive, passive viscous, or active dampers at xd = 0.02. 128
RMS Displacement
1
10
Semiactive Active Passive Uncontrolled 36
38
40
42
44
46
2
independent parameter λ
1
RMS Displacement
10
Semiactive Active Passive Uncontrolled
0
10
145
150
155
160
165
170
175
180
390
400
independent parameter λ2
1
RMS Displacement
10
0
10
Semiactive Active Passive Uncontrolled 330
340
350
360
370
380 2
independent parameter λ
Figure 6.7: Minimum RMS displacement expanded views near three pairs of peaks (xd = 0.02). 129
6.3 Effects of Sag on Damper Location Previous studies with zero sag indicated that, as the damper location approached the support end, semiactive control strategies provided increased performance over the optimal passive strategies. Figure 6.9 shows the RMS displacement of semiactive, active and passive control strategies for various damper locations and for various level of sag. What is again observed here is that, even for damper locations very near the cable support, semiactive control can provide increased performance for various levels of sag. There are some damper location and sag levels, i.e., some combinations of (xd, λ2), that give poor performance for all three vibration mitigation strategies, such as for λ2 = 42.5 and 50 near xd = 0.025 and 0.075, respectively (these combinations are discussed in the next section). Even so, the optimal semiactive damper always outperforms the passive, usually by a wide margin. Similar trends may also be observed for RMS velocity (not shown here). To better highlight the relative improvements of a semiactive damper compared to the optimal passive linear viscous damper, Figure 6.10 shows the RMS displacement, relative to that of the optimal passive linear viscous damper, of the active and semiactive strategies for several sag levels and over a range of damper locations. For damper locations around xd = 0.05, the response with a semiactive damper is 55% to 70% less than with the passive damper. For most levels of sag, the superior relative performance only gets better for a damper closer to the end of the cable (except when it is very close to the end of the cable).
6.4 Effects of Sag on Cable Modes The (xd, λ2) regions of poor performance by all three damping strategies (passive, active, and semiactive) are based on specific changes in the underlying dynamics of the cable alone. These changes are explored here to explain the specific performance results given above, both in terms of modal properties and RMS response. 130
RMS Velocity
Semiactive Active Passive Uncontrolled
1
10
0
10
0
5
10
100
500
independent parameter λ2
Figure 6.8: RMS velocity for minimum displacement with a semiactive, passive viscous, or active damper xd = 0.02. 2
λ = 0 2 λ = 1 2 λ = 30 2 λ = 42.5 2 λ = 50
RMS Displacement
Semiactive Active Passive
1
10
0
10
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
damper location xd
Figure 6.9: RMS displacement with a semiactive, passive viscous, or active damper at various damper locations. 131
In the absence of a supplemental damper, the mode shapes of the cable without sag (λ2 = 0) are sine functions, with integer natural frequencies. However, as the independent parameter λ2 increases, the mode shapes that are symmetric about the center of the cable change significantly, while the antisymmetric mode shapes remain the same. These effects are discussed in depth elsewhere (e.g., Irvine, 1981), but as these changes ultimately affect the performance of a damper, some details are given here to explain the variations in damper performance that was seen above. Figure 6.11 shows the first six natural frequencies of a sag cable as a function of the independent parameter λ2. Note particularly that due to the increased stiffness on the symmetric modes, there are a number of frequency crossover points, where two modes have identical natural frequencies. These crossovers occur at λ2 = 4π2, 16π2, 36π2, etc. — i.e., at λ2 = (2iπ)2, i = 1, 2, 3, ... (Irvine, 1981). At these points, passive, active, and semi-
RMS Displacement (relative to optimal passive)
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 2
λ = 0 2 λ = 1 2 λ = 30 2 λ = 42.5 2 λ = 50
0.2 0.1 0
Semiactive Active 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
damper location x
d
Figure 6.10: RMS displacement, relative to the optimal passive linear damper, with an active or semiactive damper at various damper locations. 132
7
λ2 = 36π2 nondimensional frequency
6
5
4
λ2 = 16π2 3
2
λ2 = 4π2 1
Symmetric Antisymmetric 0 0
5
10
100
independent parameter λ2
1000
Figure 6.11: Natural frequencies as a function of the independent parameter λ2 for sag cables. active damper difficulties may be expected, since the manifold defined by the two modes with identical frequency can have controllable and uncontrollable subspaces with respect to a single point-located damping device. Indeed, the addition of damper force will cause the two “crossing” mode shapes (symmetric and antisymmetric) to combine such that a node will occur at the damper location, with no possibility of controlling that mode. Further conditions that may give rise to poor damper performance are when a mode has a node at the damper location. Without sag, this phenomenon will only occur for rational xd and only for mode m if an integer i exists such that i = xdm; for small xd, this phenomenon will only occur for higher frequency modes. However, with sag, it is possible for the first several symmetric modes to have a node at a typical damper location. Figure 6.12 shows the mode shapes of the first six modes for ten values of λ2. Consider the thick line representing the first symmetric mode of the cable. At small λ2, it is sinusoidal in 133
λ2 = 0
λ2 = 50
ωs1 = 1 ωs2 = 3 ωs3 = 5
ωs1 = 2.1626 ωs2 = 3.1354 ωs3 = 5.0203
λ2 = 1
λ2 = 157.91
= 16π2
ωs1 = 1.0402 ωs2 = 3.0015 ωs3 = 5.0003
ωs1 = 2.7375 ωs2 = 4 ωs3 = 5.1222
λ2 = 30
λ2 = 167.8
ωs1 = 1.8264 ωs2 = 3.0632 ωs3 = 5.0111
ωs1 = 2.7486 ωs2 = 4.0816 ωs3 = 5.1400
λ2 = 39.478
=
λ2 = 355.3
4 π2
= 36π2
ωs1 = 2 ωs2 = 3.0934 ωs3 = 5.0153
ωs1 = 2.8205 ωs2 = 4.7777 ωs3 = 6
λ2 = 41.93
λ2 = 377.6
ωs1 = 2.0406 ωs2 = 3.1023 ωs3 = 5.0164
ωs1 = 2.8234 ωs2 = 4.7955 ωs3 = 6.1224
π2 16
.93
π
=
16 7.8
2
λ
1 =4
1st symmetric mode 2nd symmetric mode 3rd symmetric mode
λ2
=3
77 .6
2 4 λ =
2
λ2
=
λ2
λ2
=3
6π 2
Figure 6.12: Cable mode shapes at various sag levels. The antisymmetric modes are shown in gray. The natural frequencies (in nondimensional rads/sec) are given for the symmetric modes.
0.02 L
λ2 = 50
0.075 L
Figure 6.13: Expanded view of some cable modeshapes.
134
shape, but the slope at the ends flattens out with increasing sag. At λ2 = 42, the end slope is zero, as may be seen in the expanded view in Figure 6.13. As λ2 increases beyond 42, the first symmetric mode has a node near each end of the cable. When λ2 reaches 41.93, the node is at x = 0.02; a damper placed at xd = 0.02 would be unable to control the first mode in this case. Similarly, a damper at xd 0.025 and 0.075 could not control the first symmetric mode of a cable with λ2 42.5 and 50, respectively. Nodes will occur near the end of the cable in the second and third symmetric modes for λ2 > 162 and λ2 > 362, respectively, causing the second of each pair of response peaks in Figures 6.6, 6.7, and 6.8.
6.5 Chapter Summary The effect of cable sag, through the independent parameter λ2, is examined for the cable damping system. In general, as sag increases, two phenomena occur in the cable modes that effect the system performance. First, as sag increases, the natural frequencies of the symmetric modes increase. At sag levels of λ2 = (2iπ)2, the ith symmetric and antisymmetric natural frequencies cross. When this occurs, the two modes are uncontrollable, as a linear combination of the two mode shapes can form a node anywhere along the length of the cable where a control force would be applied. The second phenomenon occurs slightly after the crossover point and is a result of the newly formed node, which is traveling from the support to the center of the cable as the sag continues to increase, passes through the location of the cable damper. Again, if the damper is located at a vibratory node, that mode is uncontrollable. The effect of sag on the achievable damping ratios, RMS response and damper location are all a result of the frequency crossover and nodes of the symmetric modes. In regions of cable sag near where these phenomena occur, control performance is reduced. In general, active and semiactive cable damping can provide significantly improved performance over passive cable damping, in particular as bridge designs require 135
longer cables and the damper location, relative to the length of the cable, is reduced. However, the research indicating this increased performance in this chapter assumes ideal active and semiactive control devices. For the semiactive control to achieve similar performance to active control, the semiactive device must exhibit excellent performance. For less than perfect semiactive damper performance, the performance of the semiactive control strategy will be effected. The next chapter, Chapter 7, considers a realistic semiactive smart fluid damper for experimental verification of semiactive cable damping.
136
CHAPTER 7: CABLE DAMPING CONTROL: EXPERIMENTAL VERIFICATION
To experimentally verify the performance of semiactive dampers in mitigating cable responses, a medium-scale cable experiment is built. The experiment represents an inclined flat-sag cable with a semiactive damper attached transverse to the cable near the bottom support to reduce cable vibration. The environmental excitation is produced in the laboratory with a shaker attached to a point on the cable near the top support. Two measurements, the cable displacement at the location of the damper and the damper force, are available to the control strategy. Two additional measurements, the cable displacement at a location near the midpoint of the cable and the shaker force, are used for evaluation. In this chapter, the experimental setup for the smart cable damping control experiment is described, a control-oriented design model developed, semiactive control strategy identified, and experimental results presented.
7.1 Cable Damping Experimental Setup A schematic of the experimental setup used in this study is shown in Figure 7.1. Components of the experiment include the flat-sag cable, semiactive “smart” damper, digital controller, shaker, and spectrum analyzer.
Flat-Sag Cable The flat-sag cable is a stainless steel wire rope, comprised of 19 strands, 4 mm in diameter with brass weights attached every 10 cm. Figure 7.2 shows the flat-sag cable.
137
shaker
[we fd fs]T
Fs flat-sag cable
spectrum analyzer
wd
Fd
smart damper
u
digital controller
(Fs - shaker force; Fd - damper force; we - evaluation displacement; wd - damper displ.; u - control signal) Figure 7.1: Schematic of smart cable damping experiment.
Figure 7.2: Flat-sag cable experimental setup. 138
The cable is attached at one end to a base plate secured to the floor and
brass disk
assembled brass weight
attached at the other end to a wall plate attached to a sufficiently thick masonry wall. The cable is inclined at 20.53° from the horizontal. At the base and wall plates, the wire rope passes over frets to insure the bound“C”-clip
ary conditions are simply supported and to allow for proper calculation of cable length. The wire rope is 12.65 m
Figure 7.3: Brass weights to insure dynamic similitude.
in length. The brass weights, as pictured in Figure 7.3, are two piece disks held together with “C”-clips. The mass of each weight is 0.034 kg and they are attached to the cable every 10 cm to so that the cable achieves dynamic similitude with full-scale stay cables.
Smart Shear Mode Magnetorheological Fluid Damper Cable vibration is mitigated by a “smart” damper attached transverse to the cable. A magnetorheological (MR) shear mode damper (Carlson, 1994, Yi et al., 2001) is constructed to provide controllable damping forces. The damper consists of two pairs of parallel plates between which a steel paddle passes. The paddle is connected to the cable and has MR fluid-soaked sponge rubber of either face. A magnetic field is produced by an electromagnetic consisting of a coil of copper wire at one end of each of the steel plates. The damper force is varied by varying the magnetic field. This damper, shown in Figure 7.4, has a maximum force level of approximately ± 10 N. The damper is positioned 0.253 m (2% of the cable length) from the bottom support and provides in-plane forces transverse (nearly vertical) to the cable. The cable displacement is measured at the location of the smart damper with a Keyence LB-70(W) series laser displacement sensor with a ± 10 139
cable steel paddle
coils
load cell
Direction of Motion
MR fluid saturated foam
Figure 7.4: Smart shear mode magnetorheological fluid damper. mm measuring range. A PCB Series 208 force sensor, with dynamic range of ± 450 N, is used to measure the control forces of the damper.
Digital Controller The controller is implemented digitally on a MultiQ I/O board with the WinCon realtime controller. The controller is developed using Simulink (1998) and executed in real time using WinCon. The MultiQ I/O board has a 13-bit analog/digital (A/D) and 12-bit digital/analog (D/A) converters with eight input and eight output analog channels. The Simulink control model is automatically converted to C code and interfaced through the WinCon software to run the control algorithms on the CPU of the PC.
Exciter The cable is excited with a Ling Dynamics permanent magnet shaker, capable of producing 90 N of force. The shaker is attached at a location 0.362 m from the top support and provides transverse in-plane cable excitation. A PCB Series 208 force sensor, with a dynamic range of ± 450 N, is positioned in series with the shaker to measure the dynamic
140
Figure 7.5: Permanent magnet shaker. forces used to excite the cable. The shaker is capable of exciting the cable across the dynamic range of interest, namely the first three vibratory modes of the cable (2-10 Hz).
Performance Evaluation/Spectrum Analyzer The evaluation measurement is the displacement of the cable at a location 4.12 m from the bottom support. The first several modes of vibration are of concern for wind-rain induced stay-cable vibration. The evaluation measurement point is not located at a vibratory node of the first three modes and, thus, should be a good indicator of the vibratory motion of the entire cable for these lower modes. The displacements are measured with a Keyence LB-72(W) series laser displacement sensor with a ± 40 mm measuring range. A 4-input/2-output PC-based spectrum analyzer, manufactured by DSP Technology, is used to acquire the measurement signals. The spectrum analyzer has a 90 dB signal to noise ratio and includes 8-pole elliptical antialiasing filters, programable gains on the inputs/outputs, user selectable sample rates and a MATLAB (1999) user interface. These features 141
allow for direct acquisition of high quality data and transfer functions for system identification and response analysis.
7.2 System Identification of Cable Damping Model A critical precursor to control design is development of a low order, high-quality model of the system. In this section, the system input/output characteristics are analytically derived by physically modeling the cable and damper. Flat-sag cable and shear mode magnetorheological damper models are developed and combined to model the semiactive cable damping system.
Flat-Sag Cable Model with Point Load Excitation Using the flat-sag cable model identified previously, a model for the 12.65 meter cable, shown in Figure 7.6, is developed. The parameters for the cable tested here are: ρ = 0.407 kg/m, L = 12.65 m, θ = 22.53° , H = 2172 N, E = 1.9 ×1011 N/m2,
A = 1.26 ×10–5 m2,
ζ 1 = 0.0015 ,
ζ 2 = 0.003 ,
ζ 3 = 0.005
and
ζ i ≠ 1, 2, 3 = 0.0005 ,and ω 0 = 2.89 Hz. The effects of cable sag, angle-of-inclination, and axial stiffness on the nondimensional dynamic response of the system enter only though the independent parameter λ 2 . For the experiment in this study, λ 2 was determined to be 0.59 . This value is within the [0,1] range typical for cable stayed-bridges. The sag ratio for this cable is 0.28%. Using a Galerkin method, the transverse motion of the cable relative to the static profile may be approximated with a finite series w( x , t ) =
m
∑ q j (t ) φ j ( x )
(7.1)
j=1
where the shape functions, φ j (x) , include a sine series as well as static deflection shape functions for the cable with sag introduced in Johnson et al. (2001b) to account for the 142
L
z,w
EA
Fs(t)
,ρ
) z(x xs
x θ
xd
, ,c
Fd(t)
gravity
Figure 7.6: In-plane static profile z(x) and dynamic loading f(x,t) of inclined cable with sag and transverse damper force. point loads on the cable acting at the location of the damper and the location of the shaker. The shape functions are given as 12 + λ 2 x 12 + λ – 3λ x d ( 1 – x d ) x d
x H (x – x d) x d 1 – x d
3λ 2 12 + λ
x H ( x – x s) x s 1 – x s
3λ 2 12 + λ 2
- ----- + 1 – ----- ---------------------- – -----------------2- x ( 1 – x ) φ 1(x) = --------------------------------------------------------2 2 12 + λ 2 x 2 2 12 + λ – 3λ x s ( 1 – x s ) x s
φ 2(x) = --------------------------------------------------------- ---- + 1 – ---- ---------------------- – ------------------ x ( 1 – x )
(7.2)
j = 1, 2, ..., m – 1
φ j + 2(x) = sin π jx
Substituting the shape functions into the nondimensional equation of motion and simplifying, results in the matrix equation ˙˙ + Cq˙ + Kq = f s F s(t) + f d Fd (t) Mq with mass M = [ m ij ] , damping C = cM , and stiffness K = [ k ij ] matrices 1
m ij =
∫ φi(x) φ j (x) dx 0
1
c ij = c i ∫ φi (x) φ j (x) dx 0
143
(7.3)
k ij =
1 ----π2
λ2
1 1 1 ∫ φi (x) dx ∫ φ j (x) dx + ∫ φi′(x) φ j′ (x) dx 0 0 0
vector q = [ q j ] of generalized displacements, and shaker and damper load vectors f s and f d f s = f(x s) = [ φ 1(x s) φ 2(x s) … φ m(x s) ] T f d = f(x d) = [ φ 1(x d) φ 2(x d) … φ m(x d) ] T
(7.4)
The resulting model captures the salient features of a cable damper system with 22 terms (2 static deflection shapes plus 20 sine terms). The system dynamics may be equivalently written in state-space form with input/ output relations ˙˙ = A h + B F (t) + G F (t) h z z s z d y = Cy h + Dy F d(t) + Hy F s(t) + v
(7.5)
where h = [ q T q˙ T ] T is the state vector, y = w(x d, t) + v is a vector of noisy sensor measurements (includes the displacement at the damper location), v is the stochastic sensor noise process, and
Az =
0 2
–1
–ω0 M K
Cy = f dT
0
I
0
–1
–ω0 M C
Bz =
–1
1 ------- M f d ρL Dy = 0
0 Gz =
1 –1 ------- M f s ρL Hy = 0
(7.6)
The model is verified by comparing analytical transfer functions from the two force inputs to two displacement outputs of the model to experimentally collected data. The experimental transfer functions are obtained by applying a white noise excitation to the shaker and a white noise current to the damper, measuring the shaker and damper forces and damper and evaluation displacements and applying
144
H wd fs H we fs
=
H wd fd H we fd
–1
G fs fs G fd fs G fs fd G fd fd
G wd fs G we fs
(7.7)
G wd fd G we fd
where H xy is the transfer function and G xx and G xy are the auto- and cross-spectral density functions (Bendat and Piersol, 1986). The analytical and experimental transfer functions are shown in Figure 7.7. 10
10
0
−20
−30
−40
−50
−20
−30
−40
−50
−60
−60
−70
−70
−80
0
1
2
3
4
5
H we fs(ω)
−10
mag (dB) − mm/N
−10
mag (dB) − mm/N
0
H wd fs(ω)
6
7
8
9
−80
10
0
1
2
3
frequency (Hz) 10
H wd fd(ω)
6
7
8
9
10
−10
−20
−30
−40
−50
8
9
10
−20
−30
−40
−50
−60
−60
−70
−70
0
1
2
3
4
5
H we fd(ω)
0
mag (dB) − mm/N
−10
mag (dB) − mm/N
5
10
0
−80
4
frequency (Hz)
6
7
8
9
10
−80
frequency (Hz)
0
1
2
3
4
5
6
7
frequency (Hz)
Figure 7.7: Transfer functions comparing flat-sag cable model (black) to experimental data (grey).
Shear-Mode Magnetorheological Damper Model The cable is controlled with a smart magnetorheological (MR) shear mode damper. A schematic of the shear mode MR damper is shown in Figure 7.4. The damper is controlled by varying the current sent to the damper coils, which in turn varies the magnetic 145
field enveloping the MR saturated paddle. As the yield strength in the MR fluid increases, as a result in an increase in the magnetic field, it becomes more difficult for the paddle to pass between the two steel plates, resulting in increased damper force. A phenomenological model of the shear mode MR damper is developed (Fu, 1999). viscous damper
Bouc-Wen model
The model uses a Bouc-Wen model in parallel with a viscous damping element (Spencer, et al., 1997) as shown in Figure 7.8. The force of the damper is
Figure 7.8: Phenomenological model of shear mode magnetorheological damper.
F d = c 0 w˙ (x d) + αz
(7.8)
where c0 is the damping coefficient of the damper and the evolutionary variable z is governed by z˙ = – γ w˙ (x d) z z
n–1
n
– βw˙ (x d) z + Aw˙ (x d) .
(7.9)
The parameters γ , n, β and A set the hysteretic behavior of the damper. A least squares fit of the analytical model to experimental force versus displacement plots is con5
5
ducted. The parameters are determined to be: γ = 1.3 ×10 , n = 1 , β = 1.3 ×10 , and A = 200 . The parameters of Equation (7.8) are proposed to be linear functions of the command signal (in volts) sent to the damper c 0 = c 0(u) = c 0a + c 0b u
and
α = α(u) = α a + α b u
(7.10)
where u is the command signal. The coefficients c 0(u i) and α(u i) are determined from a least squares fit of experimental data for i constant levels of u over the range of 0-4 Amps. Linear regression is performed and the coefficients are determined to be c 0a = 50 , c 0b = 125 , α a = 70 , and α b = 700 for this particular damper.
146
The resistance and inductance present in the electromagnetic circuit introduces dynamics into the command signal. The dynamics are observed to be a first order time lag to changes in the command input and are replicated by a first order filter u˙ = – κ(u – v c)
(7.11)
where vc is the control signal of the bang-bang controller (in volts) and κ affects the time lag. A least squares fit of the experimental data of the damper under random excitation and bang-bang control is used to determine an appropriate value for κ . For this circuit, κ = 70 is determined. The results of the shear mode damper model are compared to actual data taken during a smart damping control test of the cable. The damper force for this 3 second time period is compared for the analytical and experimental dampers in Figure 7.9. The results show quite good agreement between the analytical model and the shear mode MR damper.
10
8
6
force (N)
4
2
0
−2
−4
−6
−8
−10
0
0.5
1
1.5
2
2.5
3
time (sec) Figure 7.9: Comparison of shear mode MR damper analytical model (black) and experimental data (grey).
147
Cable Excitation The cable is excited with a point load excitation produced by the permanent magnet shaker. The control of the exciter is open loop. The excitation considered in this chapter is intended to excite the first symmetric mode of the cable, near 2.89 Hz. The open loop control to excite this first mode is accomplished by sending a white noise process through a series of filters, as shown schematically in Figure 7.10. The permanent magnet shaker is driven by an audio amplifier. The audio amplifier has a roll-off below 8-10 Hz. This low frequency roll-off is counteracted by passing the command signal to be sent to the amplifier through a low-pass filter of the form 1 H lowpass(ω) = ---------------------------------------------------2 2 – ω + 2ξ L ω L jω + ω L
(7.12)
where ω L = 4 ( 2π ) and ξ L = 1 . A second order filter is employed to shape the spectral content of the control signal such that the permanent magnet shaker excites primarily first symmetric mode of the cable. The filter for the exciter takes the form 2
ωs H excitation(ω) = -------------------------------------------------2 2 – ω + 2ξ s ω s jω + ω s
(7.13)
where ω s = 2.89 ( 2π ) and ξ s = 0.2 . The final filter used to condition the exciter command signal is a low-pass filter. As the modes of interest in these experimental studies are below 10 Hz, an 8-pole elliptical low-pass filter with cutoff frequency of 15 Hz is used to attenuate the signal at higher frequencies.
white noise
low-pass filter Eq. (7.12)
excitation filter Eq. (7.13)
elliptical low-pass filter
command signal to shaker
Figure 7.10: Schematic of control signal to permanent magnet shaker.
148
The resulting frequency content of the excitation force produced by the permanent magnet shaker is shown in Figure 7.11 as compared to the target frequency content. Over the frequency range of interest, from 2-10 Hz, the actual frequency content matches closely with the target frequency content to excite primarily the first symmetric mode of the cable. 2.89 Hz
20
10
10 Hz
0
magnitudedB (dB)
−10
excitation filter
−20
−30
−40
actual frequency content
−50
2 Hz
−60
−70
−80
0
2
4
6
8
10
12
14
16
18
20
frequency (Hz)
Figure 7.11: Comparison of frequency content of actual (experimental) shaker force to target (analytical). The cable can also be excited in primarily the first antisymmetric mode, ω2 = 5.77 Hz, and in the second symmetric mode, ω3 = 8.66 Hz. To command the permanent magnet shaker to excite first antisymmetric and second symmetric modes, the parameters for the excitation filter of Equation (7.13) are changed. The frequency content of the two additional excitations are shown in Figure 7.18, comparing the frequency content of the actual shaker force, measured experimentally, to the excitation filter.
Experimental Estimation of RMS Cable Deflection Integrated along the Length of Cable The evaluation measurement identified in Section 7.1 is the displacement of the cable at a location 4.12 m from the bottom support. The evaluation measurement point is 149
2 Hz
2 Hz
5.77 Hz
30
20
8.66 Hz
30
20
10 Hz
10
10 Hz
10
magnitude (dB)
magnitude (dB)
0
−10
−20
−30
0
−10
−20
excitation filter −30
−40
−40
−50
actual frequency content −50
−60
−70
0
2
4
6
8
10
12
14
16
18
20
−60
0
2
4
frequency (Hz)
6
8
10
12
14
16
18
20
frequency (Hz)
Figure 7.12: Comparison of frequency content of analytical (solid) and experimental (grey) shaker force. located away from the significant vibratory nodes, however, it is not certain that this measurement will be a good indicator of the vibratory motion of the entire cable. A control strategy might decrease the motion significantly in the region of the evaluation measurement, but allow other parts of the cable to vibrate relatively unimpeded. For this reason, in Section 5.2, the damper performance was defined as a measure of the length of the cable, the mean square cable deflection integrated along the length of the cable. Clearly, the mean square cable deflection integrated along the length of the cable is not directly available by measurement in the experiment. The performance measure is estimated with a Kalman Filter. The Kalman Filter uses all available measurements and estimates q , the generalized displacements at each time step. A standard Kalman filter observer is used to estimate the states of the cable model ˙ hˆ = ( A – L KF C KF ) hˆ + L KF y KF + ( B KF – L KF D KF )u KF
(7.14)
T + G Q H T )(R T – 1 is the estimator gain, where L KF = ( P˜ C KF KF KF KF KF + H KF Q KF H KF )
u KF = F s(t) F d(t)
T
are the inputs, y KF = w d w e
T
are the measurements, the matri-
ces B KF = B G and G KF = B G are defined from Equation (7.5), the matrices 150
C KF =
fd 0
, where f e = f(x e) = [ φ 1(x e) φ 2(x e) … φ m(x e) ] T , D KF = 0 , and 0 fe 0
T T H KF = 0 , QKF = E [ ww ] is the process noise covariance matrix, R KF = E [ vv ] is 0
the measurement noise covariance matrix, and P˜ is computed from the Riccati equation ˜ T + G Q H T )(R T –1 T AP˜ + P˜ A T – ( P˜ CKF KF KF KF KF + H KF Q KF H KF ) ( C KF P + H KF Q KF G KF ) T = – G KF Q KF G KF
(7.15)
The estimate for the generalized displacements, qˆ , is qˆ = I 0 hˆ
(7.16)
Since the generalized displacements, q , and thus the product q T(t)Mq(t) , are ergodic, the ensemble average (expected value operator) is equal to the time average. The mean square cable deflection can be estimated, in the experiment, using the estimates of the general displacements as σˆ w(t) =
1 Tf E [ q T(t)Mq(t) ] ≅ ----- ∫ qˆ T(t)Mqˆ (t) dt Tf 0
(7.17)
The process for the estimation of experimental values of the deflection integrated along the length of the cable is illustrated in Figure 7.13.
Fs Fd wd we
Kalman FIlter Eq. (7.14)
qˆ
1 Tf ----- ∫ qˆ T(t)Mqˆ (t) dt Tf 0 Eq. (7.17)
σˆ w
Figure 7.13: Schematic of process to calculate experimental performance measure.
151
7.3 Passively-Operated Smart Damping Control Strategy One method of operating smart cable dampers is in a purely passive capacity, supplying the dampers with constant optimal voltage. The advantages to this strategy are the relative simplicity of implementing the control strategy as compared to a smart or active control strategy and that the dampers are more easily optimally tuned in-place, eliminating the need to have passive dampers with unique optimal damping coefficients for each cable of a cable-stayed bridge. The limitations to this method are the limited increase in performance over optimally tuned passive dampers, and the dependence of the optimal voltage on excitation magnitude and frequency content. The limited performance as well as amplitude and frequency dependence of the optimal passively-operated smart damping voltage is shown in this section.
Amplitude Dependence The dependence on the amplitude of the shaker force is shown in simulation and experimentally. Here the cable is excited in the first mode and the RMS shaker force is varied. The performance of the passively-operated smart damper control strategy versus damper voltage is plotted in Figure 7.14. Both the analytical curves, using the smart damper model of Section 7.2, and experimental points are presented. From Figure 7.14 it is more clear that different damper voltages will be optimal for the different levels of shaker excitations. The optimal voltage versus excitation level is shown in Figure 7.15, for the analytical and experimental systems. During experimental tests, voltage to the damper was varied in 0.2 volt increments. The experimental data verifies the analytical results, showing a definite dependence of damper voltage on the level of excitation. If the optimal passively-operated smart damping voltage is determined using the 2 N RMS excitation level, the optimal voltage is 0. The performance of this system, opti152
analytical 2 N shaker force analytical 4 N shaker force analytical 6 N shaker force experimental 2 N shaker force experimental 4 N shaker force experimental 6 N shaker force
0.75
Norm. RMS Cable Response − we
rms
rms
/ wunct
0.8
0.7
0.65
0.6
0.55
0.5 −1 10
0
10
Damper Voltage (V)
Figure 7.14: Passively-operated smart damper cable response versus damper voltage for various levels of excitation.
0.5
optimal voltage ( 0.4 ± 0.1 volt)
0.45
Damper Voltage (V)
0.4
0.35
0.3
optimal voltage ( 0.2 ± 0.1 volt)
0.25
0.2
analytical experimental
0.15
0.1
optimal voltage ( 0 + 0.1 volt)
0.05
0
0
1
2
3
4
5
6
7
8
9
10
RMS Shaker Force (N)
Figure 7.15: Optimal passively-operated smart damper voltage versus excitation level.
153
mally designed for a 2 N RMS excitation, during larger amplitude excitations (e.g. 4 N and 6 N RMS excitations) will degrade, achieving around 95% of the potential performance.
Frequency Dependence To observe the passively-operated smart damping control strategy’s dependence on the frequency of the excitation, consider three excitations of different frequency content. Using the shaker excitations described in Section 7.2, the cable is excited near: (i) the first symmetric mode; (ii) the first antisymmetric mode; and (iii) the second symmetric mode. The excitation level is held to a constant 4 N RMS for each of the three excitations, such that what is observed is the effect of the frequency content of the excitation and not the excitation amplitude, as observed previously. The cable response versus damper voltage is shown in Figure 7.16. The optimal voltage changes with the frequency content of the excitation. The analytical results indicate that the optimal voltage for the first symmetric, first antisymmetric, and second symmetric mode excitations are 0.4, 0.3, and 0.1 volts. Experimentally, it was observed that the optimal voltage levels for the first three modes of excitation are 0.2, 0.4, and 0 volts. If the optimal passively-operated smart damping voltage is determined using an excitation of the first symmetric mode, the optimal voltage is 0.2 volts. The performance of this system, optimally designed for a first symmetric mode excitation, during other mode excitations (e.g. first antisymmetric and second symmetric mode excitations) will degrade, achieving around 98% of the optimal potential performance. The optimal voltage of the passively-operated smart damping control strategy is dependent on the excitation amplitude and frequency content. For the cable and damper system examined in this research, the excitation amplitude and frequency content had a small effect on the overall performance of the system. The optimal system in one case is able to achieve within 95% of the potential performance for the other types and levels of 154
Norm. RMS Cable Response − wrms / wrms e unct
0.9
analytical first mode excited analytical second mode excited analytical third mode excited exp. first symmetric excited exp. first antisymmetric excited exp. second symmetric excited
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5 −1 10
0
10
Damper Voltage (V)
Figure 7.16: Passively-operated smart damper cable response versus damper voltage for various modes excited. excitation. However, different cables and different dampers, as well as a combination of different amplitude and frequency changes in the excitation, may result in more significant loss of performance. These concerns should be considered when applying the passivelyoperated smart damper control strategy.
7.4 Experimental Semiactive Cable Damping Control Strategy An ideal controllable-fluid damper (Spencer and Sain, 1997; Housner et al., 1997) can only exert dissipative forces. For control design, a generic semiactive device model is assumed that is purely dissipative. Essentially, this requirement dictates that the force exerted by the damper and the velocity across the damper must be of opposite sign; i.e., Fd (t)w˙ (x d, t) must be less than zero. A clipped optimal strategy is used, with a pri155
mary controller based on an LQG design and a secondary controller to account for the nonlinear nature of the semiactive device. The controller is designed to reduce the cable displacement over the length of the cable.
Control Objective Modal damping ratios provide a useful means of determining the effectiveness of linear viscous damping strategies. However, using a semiactive damper introduces a nonlinearity into the combined system. Consequently, performance measures other than modal damping must be used for judging the efficacy of nonlinear damping strategies in comparison with linear (passive or active) dampers. The measure of damper performance considered herein is the square root of the mean square cable deflection at the evaluation point (4.12 m from the bottom support) and the square root of the mean square cable deflection integrated along the length of the cable, as estimated with a Kalman filter.
Primary Controller An H2/LQG control design, as presented in Section 5.3, is considered for the primary controller of the cable experiment. The cost function employed here is given as T
2 2 J 1 = σ displacement + R σ force = lim
T →∞
1 E --- ∫ ( q T Mq + R Fd2) dt T
(7.18)
0
A second order filter is augmented to the model of the structural system, Equation (7.5), to weight the spectral content of the shaker excitation in the H2/LQG control design. The second order filter is 2
ωf H design(ω) = ------------------------------------------------2 2 – ω + 2ξ f ω f jω + ω f
156
(7.19)
The parameters ω f = 15 ( 2π ) and ξ f = 0.66 are chosen to weight the first three modes of the cable, from 0 to 10 Hz. The frequency content of the second order weighting filter of Equation (7.19) is shown in Figure 7.17. 1
10 Hz
0
magnitude (dB)
−1
ωf = 15 Hz ξf = 0.6
−2
−3
−4
−5
−6
0
2
4
6
8
10
12
14
16
18
20
frequency (Hz)
Figure 7.17: Control design filter to weight the spectral content of the shaker excitation in the H2/LQG control design.
Secondary Controller A secondary controller is used to implement the desired control force by the smart damper in a clipped optimal fashion. The secondary controller used is a bang-bang controller (Dyke et al., 1996). The bang-bang controller determines the current sent to the damper as follows max
v c(t ) = v c max
where v c
H (F dmeas(t) [ F dactive(t) – F dmeas(t) ])
(7.20)
= 3 volts is the voltage sent to the current driver to insure the magnetic field
is saturated and H (.) is the Heaviside step function.
Damper Lock-up Resolution The damper is prone to lock up off-center at larger control forces and vibrate about a nonzero mean. The resulting desired control forces that use this damper displacement measurement have a nonzero mean and the performance of the control strategy suffers. To 157
mitigate this problem, a highpass filter is used to remove the static offset. The highpass filter takes the form 3
– jω H highpass(ω) = --------------------------------------------------------------------------3 2 – jω – 3 ω hp ω + 3ω hp jω + ω hp
(7.21)
where ω hp = 2π . Figure 7.18 illustrates how this filter is able to remove the static displacement component of the damper measurement before this measurement is used by the control strategy.
0.3
displacement (mm)
0.2
0.1
0
−0.1
−0.2
−0.3 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time (sec) Figure 7.18: Actual (grey) damper displacement and zero-mean (black) damper displacement used by control strategy.
7.5 Experimental Semiactive Cable Damping Results The 12.65 meter cable is tested and the experimental results are provided. The cable is excited with an RMS shaker force of approximately 4 N. The uncontrolled RMS cable vibration at the evaluation point is 11.4 mm. The experimental results for the semi158
active, smart damping control strategy are compared to experimental results for a passively-operated smart damper (discussed previously in Section 7.3) and to analytical results using similar shaker force and uncontrolled displacement levels. The damper force is normalized by the shaker force and the cable displacement normalized by the uncontrolled cable displacement corresponding to the measured shaker force. Figure 7.19 shows the RMS displacement at the evaluation point, 4.12 m from the base of the cable, as well as the “averaged” displacement measurement over the entire length of the cable. The control strategy results are given in Table 7.1. The measures of RMS
performance provided are w e
, the RMS displacement at the evaluation location nor-
malized by the uncontrolled response, and σ w , the RMS displacement over the length of the cable normalized by the uncontrolled response. The RMS displacement at the evaluaRMS
tion point, w e
, is reduced by the smart damping strategy to 46% of the uncontrolled
RMS displacement, and for the passively-operated smart strategy to 56% of the uncontrolled RMS displacement. The RMS displacement over the length of the cable, σ w , is reduced by the smart damping strategy to 44% of the uncontrolled RMS displacement over the length of the cable, and for the passively-operated smart strategy to 55% of the uncontrolled RMS displacement over the length of the cable. The smart strategy is able to reduce the RMS displacement an additional 18% and 20% beyond the passively-operated RMS
smart strategy for the two measures of RMS displacement, w e
and σ w , respectively.
TABLE 7.1: CONTROL PERFORMANCE FOR CABLE DAMPER EXPERIMENT Analytical Passive Viscous Damper
Experimental
Passively Operated Smart Ideal Active Smart Damping Semiactive Control Damper
Passively Operated Smart Smart Damping Damper
RMS
0.53
0.57
0.41
0.09
0.09
0.56
0.46
σw
0.53
0.57
0.42
0.11
0.11
0.55
0.44
we
159
rms
0.9
0.8
Norm. RMS Cable Response − we
Displacement at Evaluation Location
rms
/ wunct
1
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 −2 10
−1
0
10
10
Normalized RMS Damper Force − Frms / Frms d
1
10
s
analytical passive analytical passively-operated smart analytical active analytical smart damping experimental smart damping experimental passively-operated rms Norm. RMS Cable Response − σwwrms ⁄ σ/ww
unct unct
1
0.8
e
Displacement Over Length of Cable
0.9
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 −2 10
−1
0
10
10
rms
Normalized RMS Damper Force − Fd
rms
/ Fs
1
10
Figure 7.19: Controller performance at evaluation point and over length of cable.
160
From the results presented in Figure 7.19 and Table 7.1, the evaluation measurement is observed to be a good descriptor of the overall cable performance. Further perforRMS
mance will be presented in terms of the evaluation measurement, w e
.
Additional Primary Controllers Additional cost functions, Equation (7.18), and shaping filters, Equation (7.19), are considered to examine the potential for increased performance by the smart damping control strategy. The additional cost functions considered include: T
J2 =
2 σ velocity
+
2 R σ force
= lim
T →∞
1 E --- ∫ ( q˙ T Mq˙ + R Fd2) dt T 0
(7.22) T
1 E --- ∫ ( 1--2- q T Mq + 1--2- q˙ T Mq˙ + R Fd2) dt T
2 2 2 + σ velocity ) + R σ force = lim J 4 = 1--2- ( σ displacement
T →∞
0
In addition to different cost functions, the filter used to weight the spectral content of the excitation in the control design is also varied. The original filter considered, as shown in Figure 7.17, weights the first three modes of the cable. Three additional weighting filters are considered here as shown in Figure 7.20. The first of these filters emphasizes
ωf = 8.66 Hz ξf = 0.03
30
20
ωf = 5.77 Hz ξf = 0.2
magnitude (dB)
10
0
−10
ωf = 2.89 Hz ξf = 0.07
−20
−30
−40
0
2
4
6
8
10
12
14
16
18
20
frequency (Hz)
Figure 7.20: Additional control design filters to weight the spectral content of the shaker excitation in the H2/LQG control design. 161
predominately the first mode of the cable, the second additional filter emphasizes predominately the second mode of the cable, and the final filter emphasizes predominately the third mode of the cable. Five additional control strategies (#2-#6), consisting of various combinations of weighting functions and shaping filters, are considered. The control strategies are identified in Table 7.2. TABLE 7.2: CONTROL STRATEGY COST FUNCTION AND SHAPING FILTER COMBINATIONS Control Strategy #1
Cost Function
Weighting Filter
displacement (J1)
first three modes (ωf = 15 Hz, ξf = 0.66)
#2
velocity (J2)
first three modes (ωf = 15 Hz, ξf = 0.66)
#3
displacement and velocity (J4)
first three modes (ωf = 15 Hz, ξf = 0.66)
#4
displacement (J1)
first mode (ωf = 2.89 Hz, ξf = 0.07)
#5
displacement (J1)
second mode (ωf = 5.77 Hz, ξf = 0.2)
#6
displacement (J1)
third mode (ωf = 8.66 Hz, ξf = 0.03)
Figure 7.21 shows the RMS displacement at the evaluation point, 4.12 m from the base of the cable, as well as the displacement measurement over the entire length of the cable for the additional controllers. The results are summarized in Table 7.3. In the table the RMS displacement at the evaluation point are presented for control strategies #2-#6, as defined in Table 7.2. At best, considering control strategy #3, the RMS displacement at the evaluation point is reduced to 45% of the uncontrolled RMS displacement. For these additional controllers considered, the smart strategy still able to reduce the RMS evaluation displacement to 80% of the passively-operated smart strategy. Since the performance of these additional control strat162
Strategy #2
Strategy #5 Norm. RMS Cable Response − wrms / wrms
unct
0.9
0.9
0.8
e
0.8
e
Norm. RMS Cable Response − wrms / wrms
1
unct
1
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 −2 10
−1
10
0
10
rms
Normalized RMS Damper Force − Fd
rms
/ Fs
1
10
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 −2 10
−1
10
Strategy #3
10
rms
10
/ Fs
rms
/ Fs
1
unct
Norm. RMS Cable Response − wrms / wrms
rms
/ wunct
0.9
0.8
Norm. RMS Cable Response − we
e
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 −2 10
−1
10
0
10
rms
Normalized RMS Damper Force − Fd
rms
/ Fs
1
10
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 −2 10
−1
10
0
10
Normalized RMS Damper Force − Fd
Strategy #4 rms
/ wunct
1
0.9
0.8
Norm. RMS Cable Response − we
rms
1
rms
rms
Strategy #6
1
0.9
rms
0
10
Normalized RMS Damper Force − Fd
analytical passive
0.7
analytical passively-operated smart
0.6
analytical active analytical smart damping
0.5
experimental smart damping
0.4
experimental passively-operated
0.3
0.2
0.1
0 −2 10
−1
10
0
10
rms
Normalized RMS Damper Force − Fd
rms
/ Fs
1
10
Figure 7.21: Controller performance at evaluation point for additional controllers.
163
1
egies is similar, selecting one control strategy, namely the first control strategy, is done for further studies. RMS
TABLE 7.3: CONTROL PERFORMANCE, w e , FOR ADDITIONAL CONTROL STRATEGIES Control Strategy #2 #3 #4 #5 #6
ANALYTICAL EXPERIMENTAL Smart Smart Damping Damping 0.41 0.46 0.41 0.45 0.46 0.46 0.42 0.46 0.48 0.46
Additional Cable Excitation The performance of the smart damper, using the first control strategy (#1) is compared to analytical results as well as a passive strategy whereby the shear mode damper is supplied varying levels of constant current such that the damper behaves as a nonlinear passive damper. The controller performance is illustrated in Figure 7.22. The results are summarized in Table 7.4. In the table both the RMS displacement at the evaluation point and the estimated RMS displacement over the length of the cable are presented. RMS
TABLE 7.4: CONTROL PERFORMANCE, ( w e ), FOR 1ST ANTISYMMETRIC AND 2ND SYMMETRIC MODE EXCITATIONS Excitation
Analytical
Experimental
Passively Passively Passive Operated Smart Ideal Active Operated Smart Viscous Smart Damping Semiactive Control Smart Damping Damper Damper Damper 1st antisymmetric 2nd symmetric
0.61 0.69
0.64 0.69
0.51 0.58
164
0.12 0.49
0.64 0.70
0.51 0.58
Norm. RMS Cable Response − wrms / wrms e unct
1
0.9
Second Mode Excitation
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−1
0
10
10
Normalized RMS Damper Force − Frms / Frms d s
1
10
analytical passive analytical passively-operated smart analytical active analytical smart damping experimental smart damping experimental passively-operated Norm. RMS Cable Response − wrms / wrms e unct
1
0.9
Third Mode Excitation
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 −2 10
−1
0
10
10
rms
Normalized RMS Damper Force − Fd
rms
/ Fs
1
10
Figure 7.22: Controller performance at evaluation point for additional cable excitation. 165
The experimental results show that the RMS displacement at the evaluation point for the smart damping strategy is reduced to 51% and 58% of the uncontrolled RMS displacement for the first antisymmetric and second symmetric mode excitations, respectively. The passively-operated smart strategy is able to reduce the displacement to 64% and 70% of uncontrolled RMS displacement at the evaluation location for the first antisymmetric and second symmetric mode excitations, respectively. For the first antisymmetric and second symmetric mode excitations, the smart strategy is able to reduce the RMS RMS
evaluation displacement, w e
, an additional 20% and 17% beyond that of the passively-
operated smart strategy.
7.6 Chapter Summary In this chapter, a 12.65 m flat-sag cable is implemented with a semiactive control strategy employing a shear mode magnetorheological fluid damper. System identification is performed and models for the cable and smart damper are developed. A semiactive control strategy is proposed and implemented that uses damper displacement and force measurements for feedback control. A practical issue observed in implementing the control is when the damper locks up off-center and begins vibrating about off-center point, resulting in a loss of control performance. The off-center vibration issue is resolved by passing the measurement signals through a highpass filter, eliminating the DC component of the signal. Experimental results are confirmed by analytical studies including the damper model, that smart cable damping achieves reduces cable displacement an additional 20% beyond the performance of passive control. The passive control examined experimentally involves sending a constant control signal to the smart damper. The frequency and amplitude dependency of this control strategy are identified; although not significant for this particular cable system, this dependency should be considered on a case by case basis.
166
In the next chapter, Chapter 8, control-structure interaction, identified in this section, is examined in further detail.
167
CHAPTER 8: INVESTIGATING EXPERIMENTAL AND SIMULATION CABLE DAMPING CONTROL PERFORMANCE
The experimental and smart damping simulation performance of Chapter 7 was shown to be less than the ideal semiactive damper studied in Chapter 6. This chapter investigates factors that may explain the difference in performance. Two factors are considered to have a possible effect. First, the bending stiffness of the cable, neglected in the simulation studies, is examined. Next, the properties of the semiactive damper are examined. This investigation offers an explanation to the difference in cable damping performance and suggests a solution to experimentally regain this performance. The analytical cable model developed in Chapter 5 and the simulation results of Chapters 6 and 7 consider a flat-sag cable model where the bending stiffness of the cable is neglected. The point load of the damper acting on the cable model will result in a kink at the point of application. Cable bending stiffness reduces this kinking action. Since the actual cable used in the experiment has an associated bending stiffness, it is important to consider the effect of the bending stiffness on the performance of cable damping control. The second factor investigated is the difference between the ideal cable model used in Chapter 6 and the shear mode magnetorheological (MR) damper used in Chapter 7. The issue of controller discretization is identified and the effect on the control studied. An alternative arctangent damper model is considered to be more representative of the physical system than the ideal model. A stiffness element is added in series to the arctangent damper model to observe the effect of device compliance on the performance of the cable damper. The investigation provides insight for future damper design for experimental and full-scale cable damping control strategies. 168
8.1 Investigating Cable Bending Stiffness When damper forces are applied to the cable, the cable deforms locally. For the analytical cable model developed in Chapter 5 where bending stiffness is neglected, as well as in previous studies of cable damping (Fujino et al. 1993, 1994, 1995, Johnson et al. 2000c, 2001a, 2001b), the cable is allowed to form a kink at the location of the point load. Physically, with some bending stiffness inherently present in the cable, the slope of the cable deflection cannot be discontinuous, as required to form a kink. In this section, bending stiffness is included in the cable model to examine, analytically, the effect of cable bending stiffness on cable damping control. Figure 8.1 shows the cable model of Chapter 7 without bending stiffness under control with a damper located at 2% the total length of the cable (the cable is excited by another point load at the shaker at x = 0.97 ). The first profile shown is at time 3.35 sec, just prior to the control force being applied to the cable. This profile represents the first mode vibration of the cable, supported at both ends. The second profile, at 3.36 sec, shows the cable just as the control force is applied. The next two profiles at 3.4 and 3.45 sec show the cable with an applied control force and the resulting kink that is formed in the cable at the location of the cable damper. When the control force is turned off, just prior to 3.50 sec, the kink disappears. Including bending stiffness in the cable model will eliminate the kinking that occurs at the damper. In what follows, the effect of cable bending stiffness on passive and active cable damping control are examined as they bound the performance of semiactive “smart” cable damping control.
Flat-Sag Cable Model Incorporating Bending Stiffness Bending stiffness typical for stay cables is included in the cable model to observe the effect on cable damping control. The nondimensional parameter 1 ⁄ γ is used as a measure of the bending stiffness and is defined as:
169
t = 3.35 sec
2
Displacement − w(x,t) w (mm)mm
Displacement − w (mm)
t = 3.36 sec
6
0
−2
−4
4
prior to control
t = t3.40 sec = 3.36 sec
2
t = 3.40 sec 0
during control −2
t = 3.50 sec
−4
after control
t = 3.50 sec
−6
t = 3.45 sec −8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Length of Cable − x [0,1]
−6
Figure 8.1: Profile of cable at different instances in time for smart cable damping control strategy.
t = 3.45 sec
−8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Length of Cable − x [0,1] EI 1 ----2- = ----------2 γ HL
0.8
(8.1)
Irvine (1981) identifies typical values for γ in “the vast majority of cable problems” of order 1000. For the cable experiment of Chapter 7, γ = 373 . This assumes 4
I = ( π ⁄ 64 )r , which is a conservative assumption where the individual wire stands behave as one strand with radius r equal to the radius of the cable. Large values of γ indicate low cable bending stiffness. Small values of γ indicate significant bending stiffness is present. The nondimensional equation of motion for the cable with bending stiffness and sag is 1 λ2 1 --------2 w″″ ( x , t ) – w″ ( x , t ) + w ˙˙(x, t) + cw˙ (x, t) + ---------2 2 π2 π γ π = Fs (t) δ(x – x s) + Fd (t) δ(x – x d) 170
1
∫ w(ξ, t) dξ 0
(8.2)
The inclusion of bending stiffness will affect the stiffness matrix in Equation (5.13). Applying integration by parts, the stiffness, including bending stiffness, is k ij =
1 ----π2
λ2
1 1 1 1 1 ′′ ′ ′ --- ∫ φi (x) dx ∫ φ j (x) dx + ∫ φi (x) φ j (x) dx + 2 ∫ φi (x) φ j′′(x) dx γ 0 0 0 0
(8.3)
tension + k bending = λ 2 k isag k sag j + k ij ij
To apply the Galerkin method, the second derivative of the shape functions must be continuous. The second derivative of the static deflection shape function in Equation (7.2) are discontinuous and cannot be employed here (only the first derivative is continuous). As an alternate static deflection shape, consider the deflection of a cable with bending stiffness due to a unit load at location x = x d . The point loads at the damper and shaker locations, given by the Dirac delta, add a triangular-like deflection component. Lx A 1 sinh ( kxL ) + -----------2 ( 1 – x d ) EI k φ 1(x) = Lx d B 1 sinh ( kxL ) – B 1 cosh ( kxL ) + -----------2 ( 1 – x ) EI k Lx A 2 sinh ( kxL ) + -----------2 ( 1 – x s ) EI k φ 2(x) = Lx s B 2 sinh ( kxL ) – B 2 cosh ( kxL ) + -----------2 ( 1 – x ) EI k
x ≤ xd (8.4) x > xd
x ≤ xs (8.5) x > xs
1 sinh ( k x d L ) cosh ( kL ) 2 H - – cosh ( k x d L ) , where k = ------ , A 1 = -----------3 ------------------------------------------------- EI sinh ( kL ) EI k 1 sinh ( k x s L ) cosh ( kL ) 1 sinh ( k x d L ) cosh ( kL ) - – cosh ( k x s L ) , B 1 = -----------3 --------------------------------------------------- , A 2 = -----------3 ------------------------------------------------- sinh ( kL ) sinh ( kL ) EI k EI k 1 sinh ( k x s L ) cosh ( kL ) and B 2 = -----------3 --------------------------------------------------- . For consistent shape function scaling, sinh ( kL ) EI k
171
Equations (8.4) and (8.5) are normalized to result in a maximum deflection of 1. The remaining shape functions are sine functions: φ j + 2(x) = sin π jx , j=1,...,m–2
(8.6)
The equation of motion and resulting state space model are developed as in Section 7.2, using the shape functions identified in Equations (8.4), (8.5), and (8.6) in place of those identified in Equation (7.2).
Effect of Bending Stiffness on Cable Damping Control The effect of cable bending stiffness on the performance of passive and active cable damping control strategies is illustrated in this section. Figure 8.2 shows the optimal nondimensional viscous damping per unit length for a passive control strategy versus the nondimensional bending stiffness γ for the cable system identified in Chapter 7. As noted previously, the nondimensional bending stiffness of the cable used in the experiment is γ = 373 . The value of γ = 40 , which is the smallest value of bending stiffness considered corresponds to a cable with similar material and configuration properties but with a radius 3 times larger than the cable employed in the experiment (a 12 mm diameter cable). The optimal damping coefficient reduces the cable displacement measured over the length of the cable to a point load excitation. When γ decreases this indicates that the cable bending stiffness increases. As the cable bending stiffness increases the optimal damping coefficient increases. For a cable with larger bending stiffness a stronger damper is required to reduce cable displacement. Figure 8.3 shows the maximum first mode modal damping for an optimal passive and an optimal active LQG control strategy. The modal damping for the passive cable control increases nearly notably for the damper locations studied. For example, when xd=0.02 the achievable modal damping increases from 1.2% for no bending stiffness to 2.3% for γ = 40 . The achievable modal damping for active control remains 35% (24% for
172
6
xd = 0.02 xd = 0.01 xd = 0.005 xd = 0.001
5
10
4
10
no bending stiffness
damping coefficient - copt
10
3
10
2
10 1 10
2
10
γ
3
10
Figure 8.2: Effect of bending stiffness ( γ ) on optimal damping coefficient of passive cable damper for various damper locations. xd=0.001), for γ over the range considered. Again, for xd=0.02, the achievable modal damping for active control increases from 35.2% for no bending stiffness to 35.6% for γ = 40 . Passive cable control, with smaller damping levels, is more greatly affected by bending stiffness than is active cable control. Figure 8.4 considers the normalized (with respect to the uncontrolled cable) root mean square (RMS) cable displacement at the evaluation point for the passive and active control as bending stiffness is varied. The performance of the passive control is more greatly affected than the active control strategy, as was expected from the results of the damping study. The closer the damper is located to the deck, the larger the effect of bending stiffness on the control performance. The relative performance between the active and passive control strategies decreases as γ decreases (i.e., the bending stiffness increases). The bending stiffness, over a wide range of stiffness and damper locations, is shown to have a significant effect on cable damping performance. For the cable model identified in Chapter 7 with a damper location of 2% of the total length of the cable, 173
40
25
{
20
xd = 0.02 xd = 0.01 xd = 0.005 xd = 0.001
15
10
5
passive 0 1 10
2
10
γ
2.5
2
{ no bending stiffness
modal damping (%)
passive 1.5
3
10
no bending stiffness
modal damping (%)
30
no bending stiffness
active
35
1
0.5
0 1 10
2
10
γ
3
10
Figure 8.3: Effect of bending stiffness ( γ ) on achievable modal damping for passive and active optimal control strategies, and various damper locations. including the cable bending stiffness will increase the performance of the passive control an additional 5%, but will have a minimal effect on the active cable damping control.
8.2 Investigating Semiactive Cable Damper The performance of the cable damping strategy can be linked to the ability of the damper to produce the desired control forces. Figure 8.5 shows a time history of control forces for a period of vibration of the cable. The ideal semiactive damper is able to instantaneously produce dissipative active control forces; in doing so it is able to retain most or 174
0.6
0.5
0.4
no bending stiffness
0.7
no bending stiffness
0.8
xd = 0.02 xd = 0.01 xd = 0.005 xd = 0.001 passive
0.9
0.3
0.2
0.1
0 1 10
active
Norm. RMS Cable Response - wrms / wrms unctld
1
2
10
3
10
γ Figure 8.4: Effect of bending stiffness ( γ ) on the reduction of RMS response for various damper locations. all of the performance of the active control strategy. The Bouc-Wen model representing the shear mode MR damper of the experiment does not achieve the full force, resulting in reduced performance.
Ideal Semiactive Damper The simulation studies in Chapter 6 indicate that semiactive control can achieve performance similar to active control, an additional 80% reduction in cable RMS displacements beyond that of optimal passive control for a damper located at 2% of the total length of the cable. The experimental and corresponding simulation results employing the BoucWen model in Chapter 7 indicate that, for a damper location of 2%, the semiactive “smart” damping can reduce the cable RMS displacement by approximately 20% beyond the passive control. Figure 8.6 compares the performance of the ideal semiactive damper used in Chapter 6 to the smart damper model developed in Chapter 7 over a range of damper locations. 175
6
active ideal semiactive Bouc-Wen
Damper Force (N)
4
2
0
−2
−4
−6
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Time (sec)
Figure 8.5: Force for active, ideal semiactive, and smart dampers.
Norm. RMS Cable Response - wrms / wrms unctld
1
0.9
passive Bouc-Wen active
0.8
0.7
0.6
ideal semiactive
0.5
0.4
0.3
0.2
0.1
0
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Damper Location - xd / L Figure 8.6: Cable damping performance versus damper location including damper dynamics.
176
At a damper location of 10% the total length of the cable, the Bouc-Wen damper model is able to reduce RMS cable response by 23% relative to the response of the optimal passive control strategy, whereas the ideal semiactive damper reduces it by 60% compared to passive control. As the damper location moves closer to the cable support, at 2-10% of the total length of the cable, the relative performance of the Bouc-Wen damper model increases slightly as it is able to reduce cable response by 25-30% compared to the optimal passive control strategy. In contrast, the ideal semiactive control performance provides a reduction of up to 84%. At damper locations closer to the cable support, less than 2% of the total length of the cable, the ideal semiactive control improved performance becomes even more significant, but the Bouc-Wen damper model performance deteriorates. Reconciling this difference in performance between smart damping and ideal semiactive cable damping is examined in this section. Another difference between the ideal damper system of Chapter 6 and the experimental and Bouc-Wen damper model strategy of Chapter 7 is the implementation of the controller. In Chapter 6 the controller is implemented in the simulation as a continuous time controller. In the experiment and simulation of Chapter 7 the digital controller is implemented as a discrete time controller. For the digital controller used in the experiment a discrete time step of 0.005 sec was the fastest the controller was able to run. All simulations in Chapter 7 use this time step for discretization. The discretization does not affect the performance of the Bouc-Wen model simulations. However, the ideal semiactive damper performance decreases when implemented at a discrete time step of 0.005 sec. For this study, the ideal damper models are implemented with continuous time controllers. Additionally, the simulations in Chapter 7 include a time lag for the control signal to produce the desired force which is neglected in the ideal damper of Chapter 6. For comparative purposes, the performance of the damper models with the time lag is computed for the ideal semiactive damper as well as the proposed damper models in the following studies.
177
The performance of the ideal semiactive damper is reduced from a normalized RMS damper displacement of 0.08 to 0.125 for the ideal damper with time lag.
Ideal Semiactive Arcrtangent Damper The ideal semiactive system, as observed in Figure 5.3, allows for discontinuous force at zero damper velocity. This discontinuity cannot be physically realized. An ideal semiactive arctangent model is introduced to be more representative of the physical system than the ideal model. The semiactive tangent damper model is given as 2 active --- atan [ µw˙ (x d) ] F (t) Fd (t) = π d 0
Fdactive(t)w˙ (x d) < 0
(8.7)
otherwise
where Fdactive is the damper force of the fully active system identified in Section 5.3, w˙ d is the damper velocity, and µ is a constant that determines the slope of the inverse tangent. The performance of the ideal semiactive arctangent system is studied for a range of µ values. Figure 8.7 examines the performance of the ideal semiactive arctangent damper model. This model is compared to both the experimental results and to simulation results using Bouc-Wen damper model, ideal semiactive damper model, and ideal semiactive damper model with time lag. For µ greater than 100 sec/m, the performance of the ideal semiactive arctangent system is nearly identical to the ideal semiactive damper (both with time lag). As the slope parameter µ is reduced, the performance degrades. To understand the significance of this range of values of µ , the parameter µ is determined for the smart cable damper in Chapter 7. Figure 8.8 shows the experimental force versus velocity plot for the smart damper in the 12.65 m cable experiment with maximum control signal and a sinusoidal excitation. Also on this figure is a plot of the ideal semiactive arctangent damper for µ = 500 sec/m which is representative of the shear mode MR damper. Note, from Figure 8.7, that µ = 500 sec/m is sufficient for the ideal arctangent damper to achieve the full performance of ideal semiactive damper. However, 178
Norm. RMS Cable Response - wrms / wrms unctld
1
experimental Bouc-Wen ideal semiactive
0.9
0.8
0.7
ideal semi. (time lag) ideal semiactive arctan.
0.6
0.5
0.4
0.3
0.2
0.1
0 0 10
1
2
10
3
10
10
µ sec/m Figure 8.7: Cable damping performance versus arctangent slope parameter µ .
5
4
3
µ = 500
force (N)
2
1
0
−1
experimental ideal semiactive arctan.
−2
−3
−4
−5
−6
−4
−2
0
2
4
6
velocity (mm/sec) Figure 8.8: Comparison of ideal semiactive arctangent damper model to experimental data.
179
there is a clear difference between the experimental shear mode MR damper data and the ideal semiactive arctangent damper model force versus velocity curves. The ideal semiactive arctangent damper does not capture the hysteretic behavior of the experimental shear mode MR damper. This next study proposes a modified semiactive arctangent damper model that is able to capture the hysteretic behavior of this damper.
Semiactive Arctangent Damper with Compliance This study proposes a modified semiactive arctangent damper model, compares the model to experimental data, and examines the performance of the damper model. Physically, the contact surface of the MR damper is supported by saturated foam rubber. This foam rubber is a source of compliance and can be represented as a stiffness element in series with the damping element, as shown in Figure 8.9. The first order differential equation for the ideal semiactive arctangent damper model with compliance is given as 1 π k z˙ = – --- tan --- ---------- z + w˙ (x d) µ 2 β(u)
(8.8)
The parameter β(u) is assumed to be a linear functions of the command signal. Note that while k may also be a function of the command signal, for illustrative purposes it is assumed constant here. Experimental data from the shear mode MR damper operated at
w(xd)
cable ideal arctan model 2 β(u) --- atan ( µ [ w˙ (x d) – z˙] ) π
z
stiffness k
Figure 8.9: Schematic of semiactive arctangent damper model with compliance and corresponding force of each element. 180
u = 0 and u = 4 amps and excited by a sinusoidal shaker force are used to determine 4
k = 2.5 ×10 N/m and β(u) = β a + β b u
(8.9)
where β a = 1.75 N, and β b = 0.6 N/Amp. A comparison of the hysteretic force versus velocity loops of the semiactive arctangent damper model with compliance to the experimental data is shown in Figure 8.10. Including compliance allows the ideal arctangent model to capture hysteretic behavior similar to that experimentally observed for the shear mode MR damper. 5
5
4
4
3
3
2
2
1
1
force (N)
force (N)
u=0
0
−1
0
−1
−2
−2
−3
−3
−4
−4
−5 −8
−6
−4
−2
0
2
4
6
8
u=4
−5 −8
−6
velocity (mm/sec)
experimental
−4
−2
0
2
4
6
8
velocity (mm/sec)
ideal semiactive arctan.
semiactive arctan. with compliance
Figure 8.10: Comparison of semiactive arctangent damper model with compliance to experimental data.
The semiactive cable damping control strategy is evaluated using the arctangent damper model with compliance developed here. The results are shown in Figure 8.11 as compared to the other damper models and the experimental smart damping results. Including compliance in the damper model results in performance comparable to what was seen in the experiment. With little compliance, the performance the damper will approach the full performance of the ideal semiactive damper. Thus it appears that compliance is a key design feature in the cable damper. Future studies are required to validate this conjecture. 181
Norm. RMS Cable Response - wrms / wrms unctld
experimental Bouc-Wen ideal semiactive
1
0.9
ideal semi. (time lag) ideal semiactive arctan.
0.8
0.7
0.6
semiactive arctan. model with compliance
0.5
0.4
0.3
0.2
0.1
0 0 10
1
2
10
10
3
10
µ sec/m Figure 8.11: Performance of semiactive arctangent damper model with compliance compared to previous damper models and experimental results.
8.3 Chapter Summary When damper forces are applied to the cable, the cable forms a kink at the damper location. The flat-sag cable model used in the previous studies neglected the bending stiffness of the cable. The bending stiffness is modeled and passive and active control strategies examined. The bending stiffness, over a wide range of stiffness and damper locations, is shown to have a significant effect on cable damping performance, in particular for passive control. For the experimental cable model including the cable bending stiffness (with γ = 373 ) increases the performance of the passive control an additional 5% and has a minimal effect on the active cable damping control. Bending stiffness leads to an increase
182
in cable damping performance and thus does not account for the experimental performance of the smart damper. For values of γ on the order of 1000, as Irvine identified typical for cable structures, the bending stiffness has little effect for both passive and active control. Therefore, the previous models neglecting bending stiffness are shown here to be valid. A series of damper models are presented to identify the areas for improvement in the semiactive damper design. An ideal semiactive damper model with a time lag is presented that is able to reduce the cable response 76% beyond that of optimal passive control for a damper location of 2% of the total length of the cable. An ideal semiactive arctangent damper model is used to better represent the force behavior at zero velocity. The performance of this damper model is shown to approach that of the ideal semiactive damper model. Compliance is modeled for the arctangent damper to better represent the hysteretic behavior observed experimentally in the smart shear mode MR damper. The compliance is shown to reduce performance of the ideal semiactive arctangent damper. It appears that damper compliance is a key design feature in the cable damper. Future studies should examine and validate the level of compliance required to achieve sufficient semiactive cable damping performance in experimental and full-scale applications.
183
CHAPTER 9: CONCLUSIONS
This research investigated semiactive control of civil structures for natural hazard mitigation. The research has two components, the seismic protection of buildings and the mitigation of wind-induced vibration in cable structures. Analytical, numerical, and experimental methods are employed. The dynamic characteristics of the structures, the high-rise buildings and flexible cables, are modeled and examined. The effect of structural configurations on control performance is examined. Viable semiactive control strategies are proposed and the relative merits are compared with optimal active and passive control strategies, which provide an upper and lower bound, respectively, on the achievable performance of semiactive control strategies. Control concepts are demonstrated experimentally for both the seismically excited coupled building system and the environmentally excited cable damping system. Experimental results were presented to verify the proposed models and control strategies. In what follows, conclusions for the two specific components of this research, namely coupled building control and cable damping control are provided. Following the conclusions, future studies are identified.
9.1 Coupled Building Control Conclusions Active and semiactive coupled building control strategies were proposed to reduce the building responses due to seismic excitation. Previous research has identified passive, active and, recently, semiactive coupled building control strategies. The control strategies presented here employ readily available absolute acceleration and relative displacement 184
measurements at the location of the coupling link for feedback control. The active control strategy employs H2/LQG control, using measurements of absolute acceleration and actuator displacement feedback. The semiactive control strategy uses clipped-optimal H2/ LQG control requiring these same feedback measurements. A two-degree-of-freedom coupled building model is used to demonstrate the effect of coupling on the system dynamics. For passive control, it is shown that an optimal level of connector damping exists. A multi-degree-of-freedom coupled building model is used to demonstrate the effect of building configuration on coupled building performance. Identified are two concerns regarding the coupled building configurations. First, when natural frequencies of the coupled buildings become similar, the performance of the system is reduced. Second, when the coupling link is located near the node of a vibratory mode, the performance of the system is similarly reduced. The efficacy of the proposed active and semiactive control strategies are examined. For buildings similar in configuration to those coupled in the Triton Square office complex, active control is shown to reduce the maximum absolute root mean square (RMS) acceleration to 65% of the uncoupled building absolute acceleration and to 20% beyond that of passive control. Semiactive control is shown to reduce the maximum absolute RMS acceleration to 75% of the uncoupled building absolute acceleration and to 8% beyond that of passive control. Limiting the maximum allowable control force is shown to reduce the performance of all control strategies. As the constraint on the maximum allowable control force becomes more restrictive, the relative difference in performance between active, semiactive, and passive control is shown to be reduced. Also, for larger earthquakes, the buildings may be damaged and the performance objective to reduce maximum accelerations may be of less importance than reducing the interstory drift. When the dominant coupled building natural frequencies become similar or the coupling link is located near the node of a dominant vibratory mode, the relative performance difference between active and passive control is shown to be more significant. In 185
these cases, active control is shown to reduce the maximum absolute RMS acceleration up to 40% beyond that of passive control. The active coupled building control proposed in this research was experimentally demonstrated using a control actuator with a ball-screw mechanism, similar to the 35-ton control actuators coupling the Triton Square buildings. A model of the system is developed for the control design that fully accounts for control-structure interaction. The active coupled building system is shown to reduce the resonant peaks of the transfer functions of absolute story acceleration to ground acceleration to 18-67% of the uncoupled system’s resonant peaks and to 22-50% of the rigidly connected resonant peaks. The coupled building system is shown to reduce the maximum absolute RMS story accelerations due to simulated ground motions to as little as 18% of the uncoupled absolute RMS accelerations and 45% of the rigidly connected system.
9.2 Cable Damping Control Conclusions A low-order model is developed to include the effect of cable sag into the dynamic model of transverse in-plane cable vibration. The cable model is used to demonstrate the effect of sag on cable damping performance. Two concerns for sag in cable damping are identified. First, when λ2 = (2iπ)2, symmetric and antisymmetric natural frequencies crossover and the performance of cable damping is reduced due to an uncontrollable manifold. Second, when the level of sag is slightly larger than the frequency crossover levels, the newly formed node in the symmetric mode coincides with the damper location, again resulting in reduced performance for the system. Active and semiactive cable damping control is shown analytically to provide significantly increased performance corresponding to passive control. Semiactive control is shown to reduce cable displacement an additional 80% beyond that of passive control for a damper location at 2% the total length of the cable. These results assume an ideal semi186
active damper and serve to provide an “upper bound” on the performance one could expect from semiactive cable damping. Semiactive cable damping, using a shear mode magnetorheological (MR) fluid damper, is experimentally verified on a 12.65 m inclined cable that is dynamically similar to a typical stay cable on a cable-stayed bridge. Combined models for the cable and smart damper are established for the control design. Experimental results showed that semiactive control is able to reduce the cable displacement an additional 20% beyond the performance of passive control. The effect of including bending stiffness in the cable model is studied over a range of stiffness. The experimental cable model including the cable bending stiffness (with γ = 373 ) increases the performance of the passive control an additional 5% and has a minimal effect on the active cable damping control. Bending stiffness leads to an increase in cable damping performance and thus does not account for the experimental performance of the smart damper. For values of γ on the order of 1000, as Irvine (1982) identified typical for cable structures, the bending stiffness has little effect for both passive and active control. Therefore, the previous models neglecting bending stiffness are shown here to be valid. The difference between the ideal semiactive and experimental shear mode MR damper performance is studied. An ideal semiactive damper model with a time lag is presented that is able to reduce the cable response 76% beyond that of optimal passive control. An ideal semiactive arctangent damper model is used to better represent the force behavior at zero velocity. The performance of this damper model is shown to approach that of the ideal semiactive damper model. Compliance is modeled for the arctangent damper to better represent the hysteretic behavior observed experimentally in the smart shear mode MR damper. The compliance is shown to reduce performance of the ideal semiactive arctangent damper and is considered a key design feature in the cable damper.
187
9.3 Future Studies •
The performance of coupled building control is limited by the maximum allowable control force of the coupling link, particularly for larger magnitude design earthquakes. Further studies on the detailing required to increase the maximum control force that may be applied to a high-rise building near the top of the structure can be examined.
•
The coupled building models considered in this research are in-plane models of two adjacent buildings. Future studies should consider in-plane, out-of-plane, and torsional motion of the coupled buildings. Additionally, the performance of active, passive, and semiactive control for asymmetric building clusters as well as asymmetric buildings can be examined.
•
Active coupled building control using absolute acceleration and relative displacement measurements at the location of the coupling link is shown analytically and in a small-scale test to be an effective method of control for high-rise buildings. Experimental tests using building models to consider in-plane, out-of-plane, and torsional motion for seismic excitation can be further verified.
•
The Triton Square office complex in Tokyo, Japan, was coupled using active control actuators earlier this year (2001). Demonstrating the active and even semiactive control strategies proposed in this dissertation on full-scale applications can serve to verify the concepts proposed here as well as provide a comparison to the performance of the existing system in place on the Triton Square office complex.
•
A clipped-optimal H2/LQG control is proposed for smart cable damping control. This type of controller is shown to have good performance for the ideal semiactive cable damper. The particular shear-mode damper employed in the experimental studies was not able to achieve this ideal performance. The control design for the experimental studies did not incorporate the dynamics of the combined actuator and cable in the control design process. Future studies should be pursued to 188
develop new control algorithms, possibly nonlinear control strategies, to capture the dynamics of the MR damper. •
Identifying the damper characteristics that limit overall performance can help to develop smart dampers to provide for increased performance in cable damping applications. In particular, compliance appears in this research to have a detrimental effect on cable damping performance. The level of compliance required to achieve sufficient semiactive cable damping performance needs to be verified for experimental and full-scale applications.
189
APPENDIX A: ROOT MEAN SQUARE RESPONSES OF A FIRST ORDER LINEAR SYSTEM USING THE SOLUTION TO THE LYAPUNOV EQUATION
The ground excitation for the coupled building system in Chapters 2-4 and the cable excitation for the smart cable damping system in Chapters 5-8 are random excitations. These excitations are modeled in this work as filtered zero-mean stationary Gaussian white noise processes. Since the excitation is a stochastic process, the output will also be a stochastic process. In fact, since the system is linear, it will also be a Gaussian stochastic process, fully defined by the mean vector and covariance. The mean of the output is zero. Thus the root mean square (RMS) response is a good measure of the system performance. This appendix outlines the process of determining RMS responses of a state space system of equations (Soong and Grigoriu, 1993). The covariance of the state vector of the first order linear state space system z˙(t) = Az(t) + Bw(t)
(A.1)
can be written as T
T
G zz = E [ zz – m z m z ]
(A.2)
where E [ . ] is the expected value and m z is the mean of the state vector. The displacements and velocities of the buildings have a zero mean. Therefore, the states of the system are all zero mean processes ( m z = 0 ), and the covariance reduces to the mean square value of the state T
G zz(t) = E [ z(t)z (t) ]
(A.3)
Taking the time derivative of the covariance, as defined in Equation (A.3), and applying the chain rule to the right hand side results in 190
T T G˙ zz(t) = E [ z˙z + zz˙ ]
(A.4)
Equation (A.4) can be expanded using the state equation to T T G˙ zz = AG zz + G zz A + BG wz + G zw B
(A.5)
The cross-correlation terms G wz and G zw are now determined realizing that the response can be formulated as the following integral of z˙(t) (and assuming z(t 0) = 0 ) z( t ) =
∫t z˙(u) du t
0
=
∫t ( Az(u) + Bw(u) ) du t
=
0
∫t Az(u) du + ∫t Bw(u) du t
t
0
(A.6)
0
The cross-correlation function G zw can be written using the relationship in Equation (A.6) as G zw(t) = E [ z(t)w(t) ] = E [ ∫t Az(u)w(t) du + ∫t Bw(u)w (t) du ] t
T
t
0
(A.7)
0
The expected value of the first integral on the right hand side of Equation (A.7) is equal to zero by the argument that the response z(u) is independent of the excitation w(t) for u ≤ t (the system is causal). This equality can be understood as the excitation at some time in the future, or even at the exact instant, will not affect the state at the present time. Also recall that E [ w(t) ] = 0 and E [ z(u) ] = 0 . Over the interval of the integral, u ≤ t , the expected value of the state and the excitation can be written as E [ ∫t Az(u)w(t) du ] = A ∫t E [ z(u)w(t) ] du = A ∫t E [ z(u) ]E [ w(t) ] du = 0 t
t
0
t
0
(A.8)
0
The expected value of the second integral on the right hand side of Equation (A.7) can be rewritten, using the covariance matrix of the white noise defined as T
E [ w(u)w (t) ] ≡ 2πS o δ ( u – t ) , as E [ ∫t Bw(u)w (t) du ] = B ∫t E [ w(u)w (t) ] du = 2πBS o ∫t δ ( u – t ) du T
t
T
t
0
0
t
(A.9)
0
Taking into account the value of the integral in Equation (A.9) (recall that u ≤ t )
∫t δ ( u – t ) du t
0
= 0.5
191
(A.10)
The cross-correlation function of Equation (A.7) can be written using Equations (A.7) through (A.9) as G zw(t) = 0 + 2πBS o ( 0.5 ) = πBS o
(A.11)
Additionally, it can be shown that G wz(t) = πS o B
T
(A.12)
Substituting Equations (A.10), (A.11) and (A.12) into Equation (A.5) and combining terms results in the following T T G˙ zz = AG zz + G zz A + 2πBS o B
(A.13)
Assuming the vector z is a stationary process (the transient effects have died out), then G˙ zz = 0 and Equation (A.13) takes the form T
T
AG zz + G zz A + 2πBS o B = 0
(A.14)
which is in the form of a Lyapunov equation that can be solved in MATLAB using the function lyap to determine the value of G zz , the covariance matrix of the state vector z . The covariance of the output can be determined from the covariance of the state, where y e = Cz , as T
T
T
T
T
T
G ye ye = E [ y e y e ] = E [ Cz ( Cz ) ] = E [ Czz C ] = CE [ zz ]C = CG zz C
T
(A.15)
The RMS responses are the square root of the covariance of the responses, the diagonal terms of G ye ye determined as rms
ye
=
diag ( G ye ye )
192
(A.16)
APPENDIX B: MODELING TALL ADJACENT BUILDINGS USING THE GALERKIN METHOD
The Galerkin method is employed to model the coupled building system. The equations of motion for the coupled building system are determined using the Galerkin method. The in-plane motion for the kth building, subjected to ground acceleration x˙˙g ( t ) and the coupling force f ( t ) , is given by the equation of motion for a uniform flexural cantilevered beam with mass proportional viscous damping (Clough and Penzien, 1993). 2
4
∂ x k ( y, t ) ∂x k ( y, t ) ∂ x k ( y, t ) = – m x˙˙ – δ(x – h ) f (t) ----------------------------------------k g ∑ c + c + EI m k ---------------------k k 2 4 ∂t ∂t ∂x
(B.1)
Applying the Galerkin method, the response of the system is assumed to be represented by the finite series n
x k ( y, t ) =
∑ fk ( y )qk ( t ) i
i
T
= f k ( y )q k ( t )
(B.2)
i=1 i
where x k is the horizontal displacement of the kth building, fk is the ith trial function of i
the kth building q k is the ith generalized coordinate of the kth building and n is sufficiently large. i
The trial functions, fk ( y ) , are taken as the closed-form eigenfunctions of an Euler-Bernoulli fixed-free beam (cantilever beam). The closed-form eigenfunctions are determined as follows.
193
Using simple beam theory, the 4th-order partial differential equation governing the flexural vibration of an undamped uniform Euler-Bernoulli beam with no external forcing function (Meirovitch, 1986) is – EI
∂
4
∂y
4
x ( y, t ) = m
∂ ∂t
2
2
x ( y, t )
(B.3)
where E is the modulus of elasticity, I is the moment of inertia, m is the mass per unit length of the beam, and x ( y, t ) is the lateral response of the building as a function of the height, y, and time, t. The solution of this problem requires two boundary conditions at each end of the beam. Each building, modeled as a cantilever beam, is fixed at the ground ( y = 0 ) and free at the roof ( y = L ). This results in two geometric boundary conditions (resulting from the system geometry) from the fixed end and two natural boundary conditions (resulting from the force and moment equilibrium) from the free end. The boundary conditions are x ( 0, t ) = 0 ,
x yy ( L, t ) = 0
x y ( 0, t ) = 0 ,
x yyy ( L, t ) = 0
(B.4)
∂ x ( y, t ) is employed. ∂y Using separation of variables, a solution of the form x ( y, t ) = f( y )T ( t ) is
where the notation x y ( y, t ) =
sought. The general solution of T(t) is T ( t ) = C 1 sin ωt + C 2 cos ωt
(B.5)
and is periodic with a frequency of ω . To determine the two constants of Equation (B.5), initial conditions for the displacement and velocity are required. For this study, it is not necessary to determine these particular constants, as the focus will be on determining the eigenfunction of the system to be used as a trial function in the Galerkin method. The solution of f( y ) takes the form f( y ) = C 3 sin by + C 4 sinh by + C 5 cos by + C 6 cosh by 194
(B.6)
2
4 ω m where b = ----------- . Using the four boundary conditions of Equation (B.4), three of the four EI constants C3-C6 can be determined. The values for b i L are solved for numerically from
the equation cos b i L cosh b i L = – 1
(B.7)
An approximation of the above equation, providing accuracy to four significant digits for i > 3 is given as π b i L ≅ --- ( 2i – 1 ) 2
(B.8)
The first five solutions are bL =[1.8751, 4.6941, 7.8548, 10.996, 14.137]T, where the first three terms are found using Equation (B.7) and terms four and five are estimated from Equation (B.8). Once we have solved for bL the natural frequencies, ω i , of the system are found as ωi = ( bi L )
2
EI --------4 L m
(B.9)
The eigenfunctions of the system described in Equation (B.3) are determined from the boundary conditions as i
f ( y ) = a i [ ( sin b i L – sinh b i L ) ( sin b i y – sinh b i y )
(B.10)
+ ( cos b i L + cosh b i L ) ( cos b i y – cosh b i y ) ] where the constant a i , is a product of the unsolved constant of C3-C6 and can be set to any L
value to produce a desired norm (e.g., ∫ mf k ( y ) dy = 1 ). At this point, closed form solu2
0
tions for the mode shapes of cantilevered beams are known, Equation (B.10). T
Substituting Equation (B.2) into Equation (B.1), premultiplying by f k ( y ) , and integrating over the height of the building, the combined equations of motion for the coupled building system are Mq˙˙( t ) + Cq˙ ( t ) + Kq ( t ) = – Gx˙˙g ( t ) + P f (t)
(B.11)
where the mass, damping and stiffness matrices are diagonal as a result of the orthogonality of the trial functions, and all of the matrices are defined as 195
M=
M1 0
,
C =
0 M2
q(t ) =
q1 ( t ) q2 ( t )
C1 0
,
K1 0
K=
0 C2
,
G =
G2
0 K2 hk
,
Mk = mk ∫
where
G1
,
P1
P =
,
and
P2 hk
T f k ( y )f k ( y ) dy ,
0
G k = m k ∫ f k ( y ) dy , T
0
hk
∂2 T ∂2 P k = f k ( h c ) , and K k = ( EI ) k ∫ 2 f k ( y ) 2 f k ( y ) dy , and where the modal ∂x 0 ∂x damping C k for each building is determined as follows. The undamped natural frequencies are found either from Equation (B.9), or by solving the eigenvalue problem 2
( M k Λ k + K k )F k = 0
(B.12)
where Λ k = diag ω k ω k ... ω k . 1 2 n The modal damping matrix for the kth building is then 2ζ 1, k ω 1, k Ck =
0 0 0
0
0
0
2ζ 2, k ω 2, k 0
0
0 0
(B.13)
... 0 0 2ζ n, k ω n, k
where ζ i, k and ω i, k are the model damping ratio and the undamped natural frequency, respectively, for the ith mode of the kth building, and the damping matrix for the kth building is –1
Ck = Mk Fk Ck Fk .
196
(B.14)
APPENDIX C: MODELING TALL ADJACENT BUILDINGS USING THE FINITE ELEMENT METHOD
An in-plane finite element model is θj
now developed for the coupled building sys-
xj
j
tem. Each building is modeled as a series of beam elements stacked end to end. The number of nodes is varied. Each node contains two degrees-of-freedom (DOFs): lateral and rotational, corresponding to the flexibility of an Euler-Bernoulli beam. The length (L), moment i
of inertia (I), modulus of elasticity (E) and mass per unit length ( m ) are defined for each
θi xi
Figure C.1: Degrees-of-freedom for beam element.
element. These values are constant. The elemental, or local, mass and stiffness matrices are determined as functions of these properties. Each element, modeled as a beam element, contains two nodes, i and j, and four degrees-of-freedom. The consistent mass matrix and plane rigid frame stiffness for the beam element identified in Figure C.1 are 156 22L
54 – 13L
2
mL 22L 4L 13L – 3L 2 -------m = 420 54 13L 156 – 22L 2
– 13L – 3L – 22L 4L
197
2
(C.1)
12 6L – 12 6L EI 6L 4L 2 – 6L 2L 2 ----k = 3L – 12 – 6L 12 – 6L 2
6L 2L – 6L 4L
(C.2)
2
with respect to x i θ i x j θ j (Cook, et al. 1989). Global mass and stiffness matrices for each building are assembled from the local mass and stiffness matrices by summing the mass and stiffness associated with each degree-of-freedom for each element of the each building. In this fashion, a global mass matrix, M k , and global stiffness matrix, Kk , for each building separately ( k = 1, 2 ) are determined. The combined equations of motion for the coupled building system are Mx˙˙( t ) + Cx˙ ( t ) + Kx ( t ) = – Gx˙˙g ( t ) + P f (t)
(C.3)
where the mass, damping and stiffness matrices are diagonal as a result of the orthogonality of the trial functions, and all of the matrices are defined as
M=
M1 0
,
0 M2
x(t ) =
x1 ( t ) x2 ( t )
C =
C1 0
K=
,
0 C2
K1 0 0 K2
,
G =
G1
,
G2
where x k ( t ) = x 1, k θ 1, k x 2, k θ 2, k ... x n, k θ n, k
T
P =
P1
,
and
P2
,
and where G k is the load vector for the ground acceleration applied to the horizontal DOFs, P k is the load vector for the coupling force applied to the horizontal DOFs corresponding to the location of the coupling link, and where the modal damping C k for each building is determined in the following manner. The undamped natural frequencies and eigenvectors, F k , are found by solving the eigenvalue problem 2
( M k Λ k + K k )F k = 0 where Λ k = diag ω 1, k ω 2, k ... ω n, k . 198
(C.4)
The modal damping matrix for the kth building is then 2ζ 1, k ω 1, k Ck =
0 0 0
0
0
0
2ζ 2, k ω 2, k 0
0
0 0
(C.5)
... 0 0 2ζ n, k ω n, k
where ζ i, k and ω i, k are the model damping ratio and the undamped natural frequency, respectively, for the ith mode of the kth building, and the damping matrix for the kth building is –1
Ck = Mk Fk Ck Fk .
199
(C.6)
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