Control of Structures

This book is an outgrowth of several years of teaching and research of the two authors in the field of structural dynamics and control. The content of the book is based on structural dynamics, classical and modern control theory and involves also recent developments that took place with respect to the control of systems with distri­ buted masses. It is hoped that the book will serve the researcher and the practicing engineer in the areas of ci vi 1, mechanical and aeronautical engineering. It may also be of interest to applied mathematicians and to physicists. There is no question that the book can be used as a reference book for advanced courses in the above mentioned areas. The numerous examples will provide students with the necessary material for exercising themselves and for self­ studying. Thanks are due to Mrs. Cynthia Jones for preparing patiently and competently the typescript of the book. The services of Mrs. Linda Strouth, Solid Mechanics Division, University of Waterloo, rendered in producing the camera-ready copy of the book with great skill and devotion, are gratefully acknowledged. Special thanks go to Mr. Ir. Ad. C. Plaizier, at Martinus Nijhoff/Dr. W. Junk, who with much understanding and enthusiasm supervised the production of the book as Publisher. May the readers of it enjoy it, and may they have the fee ling of having gained something in turning to it and using it.

1 — Introduction to Structural Control.- 1.1 Introduction.- 1.2 Structural Control.- 1.3 Classification of Structural Control.- 1.4 Elaboration of Vibration Control.- 1.4.1 Response to a Constant Force with Finite Rise Time.- 1.4.2 Effect of Damping.- 1.5 Active Control of ODOF Systems.- 1.5.1 Control Using an Auxiliary Mass.- 1.5.2 Control Using Auxiliary Tendons (Springs).- 1.5.3 Control Using an Auxiliary Damper.- 1.6 The Control Devices.- 1.6.1 Electro-Hydraulic Servomechanism.- 1.6.2 Proportional Gain Controller.- 1.6.3 Automatic Gain Controller.- 1.7 References.- Appendix A — Classical Control Theory.- A. 1 Introduction.- A. 2 Feedback Control Systems.- A. 3 Design of a Control System.- A. 3.1 Stability.- A.3.1.1 Routh’s Criterion.- A. 3.1.2 Root Locus.- A.3.1.3 Bode Plots.- A.3.1.4 Polar Plots and Nyquist’s Stability Criterion.- A.3.1.5 Nichols’ Charts.- A.3.2 Steady-State Accuracy.- A.3.3 Satisfactory Transient Response.- A.3.4 Satisfactory Frequency Response.- A. 4 References.- 2 — Morphology Of Structural Control.- 2.1 Preliminary Remarks.- 2.2 On the Basics and the Actual Nature of Structural Control Problems.- 2.3 Examples.- 2.4 The Modal Approach.- 2.5 References.- 3 — Automatic Active Control of Simple Span Bridges.- 3.1 Mathematical Models of Simple Span Bridges.- 3.2 Active Control Mechanisms in Bridges.- 3.2.1 Control through Auxiliary Masses.- 3.2.2 Control through Auxiliary Tendons.- 3.2.3 Control through Auxiliary Dampers.- 3.2.4 Control using Aerodynamic Appendages.- 3.3 Active Control by Classical Control Methods.- 3.3.1 Equation of Motion of the System.- 3.3.2 Control of the Fundamental Mode of Vibration.- 3.3.2.1 Stability Requirement.- 3.3.2.2 Damping Requirement.- 3.3.2.3 Steady State Error Requirement.- 3.3.2.4 Sensitivity Requirement.- 3.3.2.5 Controlled System’s Response.- 3.3.2.6 Summary and Conclusion.- 3.3.3 Control of Three Modes of Vibration.- 3.3.3.1 Controlled Systems’ Response.- 3.3.3.2 Design of the Control System using a Servomechanism.- 3.3.3.3 Design of the Control System using a Proportional Controller.- 3.3.3.4 Summary and Conclusions.- 3.4 Active Structural Control by Pole Assignment Method.- 3.4.1 The Pole Assignment Method.- 3.4.1.1 Controllability.- 3.4.1.2 Observability.- 3.4.1.3 Design Procedure.- 3.4.1.4 Design of State Feedback.- 3.4.1.5 Design of Output Feedback.- 3.4.1.6 Design of the Observer.- 3.4.2 Application of the Pole Assignement Method.- 3.4.2.1 Example 1.- 3.4.2.2 Example 2.- 3.4.2.3 Example 3.- 3.4.3 Summary and Conclusion.- 3.5 Active Structural Control by Optimal Control Methods.- 3.5.1 The Optimal Control Problem.- 3.5.2 Solution of the Optimal Control Problem.- 3.5.3 Numerical Applications.- 3.5.3.1 Application of the Regulator Problem.- 3.5.3.2 Application of the Tracking Problem.- 3.5.4 Summary and Conclusions.- 3.6 Study of Secondary Effects.- 3.6.1 Inertia Effect of the Moving Load.- 3.6.1.1 Uncontrolled Response.- 3.6.1.2 Controlled Response using a Regulator Control System.- 3.6.1.3 Controlled Response using a Tracking Control System.- 3.6.2 Normal Force Effect.- 3.6.2.1 Passive Control Response.- 3.6.2.2 Active Control Response using a Regulator Control System.- 3.6.2.3 Active Control Response using a Tracking Control System.- 3.6.3 Effect of Uneveness of the Bridge Deck.- 3.6.3.1 Uncontrolled Response.- 3.6.3.2 Active Control Response using a Regulator Control System.- 3.6.3.3 Active Control Response using a Tracking Control System.- 3.6.4 Stochastic Control Against Uneveness.- 3.6.4.1 Stochastic Control Problem.- 3.6.4.2 Numerical Application.- 3.6.4.3 Summary and Conclusions.- 3.6.5 Effect of Various Moving Loads.- 3.6.5.1 Pulsating Force Moving with Constant Speed.- 3.6.5.2 Uniform Load Moving with Constant Speed.- 3.6.5.3 Concentrated Load Moving with Decelerated Speed.- 3.6.5.4 Concentrated Load Moving with Accelerated Speed.- 3.6.6 Summary and Conclusions.- 3.7. Optimal Control and Sensors Locations.- 3.7.1 The Optimal Observer.- 3.7.2 The Design Method.- 3.7.3 Numerical Examples.- 3.7.3.1 Example.- 3.7.3.2 Example.- 3.7.4 Conclusions.- 3.8 References.- 4 — Automatic Active Control of Tall Buildings.- 4.1 Introduction.- 4.2 Building Response due to Wind Forces.- 4.2.1 Wind Forces.- 4.2.2 Stochastic Response.- 4.2.3 Deterministic Analysis in the Time Domain.- 4.2.4 Stochastic Analysis.- 4.3 Active Control of Tall Buildings.- 4.3.1 Movable Appendages.- 4.3.2 Tuned Mass Dampers.- 4.3.3 Active Tendons.- 4.3.4 Practical Control Mechanisms.- 4.4 Control Using Active Tendons.- 4.4.1 Design of the Control Force.- 4.4.2 Response Analysis.- 4.5. Control Using Tuned Mass Dampers.- 4.5.1 Design of the Active Control Law.- 4.5.2 Controlled Response.- 4.5.3 Effectiveness of Active TMD.- 4.5.4 Concluding Remarks.- 4.5.5 Design of an Active TMD.- 4.5.5.1 Stochastic Analysis.- 4.5.5.2 The Design Process for an Active TMD.- 4.5.5.3 Numerical Investigation.- 4.5.5.4 Conclusions.- 4.6 Active Control By Means of Appendages.- 4.6.1 Design of Optimal Appendage Movement.- 4.6.2 Contributions.- 4.6.3 Example.- 4.6.3.1 Klein’s Method.- 4.6.3.2 Soong’s Method.- 4.6.3.3 The Authors’ Method.- 4.6.4 Extension.- 4.6.5 Example.- 4.7 Stochastic Control of Tall Buildings.- 4.7.1 Introduction.- 4.7.2 Equations of Motion.- 4.7.3 Spectrum of Wind Forces.- 4.7.4 Design of Filters.- 4.7.5 Design of a Stochastic Control Law.- 4.7.6 Numerical Investigations for Tendon Control.- 4.7.7 Numerical Investigation of Tuned Mass Damper Control Application.- 4.7.8 Control by Appendages.- 4.7.9 Conclusions.- 4.8 Control of Tall Buildings Taking Time Delay into Account.- 4.8.1 Equations of Motion.- 4.8.2 Design of Control Laws.- 4.8.3 Numerical Applications to Tendon Control.- 4.8.4 Numerical Application to TMD Control.- 4.9 Feasibility of Active Control of Tall Buildings.- 4.9.1 Introduction.- 4.9.2 Feasibility of Active Tendon Control.- 4.9.3 Feasibility of Active Tuned Mass Damper Control.- 4.9.4 Feasibility of Active Control by Appendages.- 4.10 Finding a Practical Control Mechanism for Tall Buildings.- 4.10.1 Introduction.- 4.10.2 The Uncontrolled Building Response.- 4.10.3 Combined Passive Tendon and PTMD.- 4.10.4 Combined Active Tendon and Passiv Passive TMD.- 4.10.5 Combined Passive Tendon and Active TMD.- 4.10.6 Combined Active Tendon and Active TMD.- 4.10.7 Combined Appendages, Passive Tendon and PTMD.- 4.10.8 Combined Appendage, Active Tendon and PTMD.- 4.10.9 Combined Appendage, Active Tendon and ATMD.- 4.10.10 Conclusions.- 4.11 References.- 5 — A Review of Methods in the Control of Continuous Systems.- 5.1 Introduction.- 5.2 Considerations of Existence and Performance of a Control.- 5.3 A Closed Form Solution to the Control Problem.- 5.4 A Liapunov-Like Assessment of a Spillover-Safe Design.- 5.5 References.