Accuracy and stability of the substructure algorithm with sub step control

Van Thuan Nguyen Accuracy and Stability of the Substructure Algorithm with Sub-step Control This work has been accepted by the Faculty of Civil and Environmental Engineering of the University of Kassel as a thesis for acquiring the academic degree of Doktor der Ingenieurwissenschaften (Dr.-Ing) Supervisor: Prof Dr -Ing Uwe Dorka First reviewer: Prof Dr -Ing Uwe Dorka Second reviewer: Prof Dr.-Ing Ekkehard Fehling Defense day: 17 December 2008 ii ACKNOWLEDGEMENTS I would like to express my heartfelt gratitude to my advisor, Prof Dr -Ing Uwe Dorka for the huge amounts of time, help and guidance that he has generously bestowed upon my doctoral research Especially, Professor Dr -Ing Uwe Dorka provided an advanced substructure algorithm from which I have investigated and developed further in this thesis I would also like to thank the second reviewer of my thesis - Prof Dr -Ing Ekkehard Fehling My gratitude is also extended to the Laboratory of the Institute of Construction Engineering in University of Kassel where I could conduct experimental investigation on hydraulic systems in substructure tests I am particularly indebted to the Ministry of Education and Training - Vietnam which have given financial support to my doctoral study via Project No 322 - the Vietnamese Overseas Scholarship Program Last, but not least important, I would like to express my gratefulness to my family and relatives They have given great support, love and inspiration for my doctoral study in Germany i ii ABSTRACT Although substructure testing has been used in civil, mechanical and aerospace engineering for the last decades, the testing method is currently needed to be investigated and developed further In real-time substructure testing, the control algorithm plays the most important role Currently, there are two main problems in control of substructure testing Firstly, although advanced substructure algorithms have been developed so far, substructure algorithms still have certain disadvantages and limitations The accuracy and stability of substructure algorithms are critically concerned in substructure testing Secondly, high accuracy in control of hydraulic actuators is required in real-time substructure testing The first four chapters from Chapter to Chapter provide a review on the principle, algorithms, error effects and error compensations of the substructure testing method This review presents the fundamentals of the testing method and discusses numerical and experimental problems in real-time substructure testing Chapter states the scope of the investigation and development in this thesis The investigation is focused on analyzing the accuracy and stability of a substructure algorithm and the development is aimed at proposing new error compensations for improving accuracy and stability of real-time substructure tests The considered algorithm for the investigation on accuracy and stability is a substructure algorithm with sub-step control The proposed compensation methods are the error force compensation and the phase lag compensation The error force compensation minimizes the force error in the substructure algorithm while the phase lag compensation reduces the displacement error due to phase lag in the hydraulic system The processes of investigation and development as well as the consequent results are presented in detail from Chapter to Chapter The methodologies of accuracy and stability analyses, methods for development of error compensations and the verification approaches are stated in Chapter In addition, the lists of cases for accuracy and stability analyses and the lists of development as well as verifications are presented in Chapter Moreover, the results of investigation on accuracy and stability are presented in Chapter Meanwhile, the developments of two error compensations are described in Chapter Finally, the last chapter includes conclusions and future research and applications iii There are three major achievements in this thesis Firstly, the results of accuracy and stability analysis are helpful in understanding accuracy and stability of the substructure algorithm and useful in selecting appropriate parameters for real-time substructure testing Secondly, new error force compensation is developed for compensating error force in the substructure algorithm Thirdly, new phase lag compensation is developed for the control of hydraulic actuators in real-time substructure testing Both compensation methods are based on on-line system identification Indeed, the proposed compensation methods have some advantages such as having adaptive capability and no requirement of a pretest for system identification iv KURZFASSUNG Obwohl Substrukturtests im Bauingenieurwesen, in der Mechanik und auch in der Luft- und Raumfahrttechnik in den letzten Jahrzehnten verwendet wurden, bedürfen diese Testmethoden weiteren Erforschungen und Entwicklungen Beim EchtzeitSubstrukturtest spielt der Steuerungsalgorithmus die wichtigste Rolle Heutzutage bestehen zwei Hauptprobleme bei der Steuerung der Substrukturtests Das erste Problem ist, dass alle Steuerungsalgorithmen trotz bisher fortgeschrittener Entwicklung noch gewisse Nachteile und Anwendungsgrenzen aufweisen Die Genauigkeit und Stabilität der Substrukturalgorithmus sind kritisch im Substrukturtests betroffen Das zweite Problem liegt darin, dass eine hohe Genauigkeit in der Steuerung von hydraulischen Zylindern beim EchtzeitSubstrukturtest erforderlich ist Die ersten vier Kapiteln geben einen Überblick über die Prinzipien, die Algorithmen, die Abweichungen und Abweichungskompensationen von der Substrukturtest-Methode Dieser Überblick präsentiert die Grundlagen von Substrukturtests und darüber hinaus die Diskussionen über die numerischen und experimentellen Probleme in dem Echtzeit-Substrukturtest Im Kapitel wird insbesondere über den Umfang bzw den Inhalt der Untersuchung sowie der Entwicklung der genannten Test-Methode besprochen Die Untersuchung fokussiert auf die Analyse der Genauigkeit und die Stabilität von den Substrukturalgorithmen währenddessen die Entwicklung auf die Empfehlung neuer Abweichungskompensationen zur Erhöhung der Genauigkeit und der Stabilität von den Echtzeit-Substrukturtests zielt Der betrachtete Algorithmus für die Untersuchung der Genauigkeit und der Stabilität der Tests ist ein Substrukturalgorithmus Kompensationsmethoden mit sind Zwischenschritten Die die Abweichungskrafts- empfohlenen und die Phasenverzögerungskompensation Die Abweichungskraftskompensation minimiert die Abweichungskraft in den Substrukturalgorithmen, wobei die Phasenverzögerungskompensation die Phasenverzögerung im hydraulischen System reduziert Sowohl die durchgeführten Untersuchungs- und Entwicklungsverfahren als auch die daraus resultierenden Ergebnisse werden von Kapitel bis Kapitel detailliert dargestellt Die Methodologien der Genauigkeits- und Stabilitätsanalysen, das Entwicklungsverfahren der Kompensationen und Verifikationsannäherungen werden dann im Kapitel besprochen Weiterhin werden im Kapitel die Fälle der v Genauigkeits- und Stabilitätsanalyse und dazu die Verifikationen präsentiert; veranschaulicht werden die Ergebnisse zu diesen Analysen im Kapitel Außerdem werden die Entwicklungen zweier Kompensationen im Kapitel beschrieben Abschließend wird ein Fazit im letzten Kapitel gezogen und es werden künftige Forschungs- und Anwendungsmöglichkeiten dargestellt Diese Dissertation soll drei Leistungen erbringen: Erstens sollen die Ergebnisse der Genauigkeits- und Stabilitätsanalyse helfen beim Verstehen der Genauigkeit und Stabilität des Substrukturalgorithmus, darüber hinaus sind sie auch sinnvoll für die Auswahl entsprechender Testparameter Zweitens wird eine neue Methode empfohlen zur Kompensation der Abweichungskraft im Substrukturalgorithmus Nicht zuletzt wird eine neue Kompensation der Phasenverzögerung zur Steuerung der hydraulischen Zylinder beim Echtzeit-Substrukturtest entwickelt Beide Kompensationsmethoden basieren auf der Online-Systemidentifikation und haben ohne Zweifel gewisse Vorteile, beispielsweise Adaptionsfähigkeit und ein extra Test der Systemidentifikation vorab ist nicht mehr erforderlich vi TABLE OF CONTENTS ACKNOWLEDGEMENTS I ABSTRACT III KURZFASSUNG V TABLE OF CONTENTS VII TERMS AND SYMBOLS XI LIST OF FIGURES XVII LIST OF TABLES XXIV INTRODUCTION TO SUBSTRUCTURE TESTING 1.1 Principle of substructure method and substructure testing 1.2 History of substructure testing 1.3 State of the art of real-time substructure testing CONTROL OF SUBSTRUCTURE TEST 2.1 Introduction 2.2 General time discretisation for integration .10 2.3 Integration schemes 14 2.3.1 The Central Difference Method (CDM) 14 2.3.2 The Newmark-β implicit Method 15 2.3.3 The Hilber, Hughes and Taylor (HHT) method .16 2.3.4 Other implicit methods .16 2.3.5 Summary of integration schemes .17 2.4 Control methods for implicit schemes .18 2.4.1 The operator splitting methods 18 2.4.2 The analog feedback method 19 2.4.3 The sub-step control method 20 ERRORS AND THEIR EFFECTS IN SUBSTRUCTURE TESTING 25 3.1 Errors of modeling and loading assumption 25 vii 3.2 Errors in control hardware 27 3.3 Errors of measurement and conversion 30 ERROR COMPENSATION METHODS IN SUBSTRUCTURE TESTING 37 4.1 Phase lag compensations based on prediction 37 4.2 The phase lag compensation using model-based control 39 4.3 Error force compensation in sub-step control 40 SCOPE OF THESIS AND CONTRIBUTIONS 43 METHODOLOGIES 47 6.1 Theory of accuracy and stability analyses 47 6.1.1 Introduction on accuracy and stability analyses 47 6.1.2 Method of accuracy and stability analyses 49 6.1.3 Limitations of the accuracy and stability analyses 61 6.2 Methodology for development of the error compensations 62 6.2.1 Error force compensation based on estimation 62 6.2.2 The method of new phase lag compensation 63 6.2.3 Requirements of the estimations 65 6.2.4 Theory of data model 65 6.2.5 Theory of on-line system identification 70 6.3 Verification approaches 72 LIST OF ANALYSES, DEVELOPMENTS AND VERIFICATIONS 75 7.1 List of investigations on accuracy and stability of the substructure solution 75 7.1.1 Ranges of the parameters 75 7.1.2 List of investigations on the amplitude and period errors 77 7.1.3 List of investigations on the effect of phase lag on the amplitude and period errors 79 7.1.4 List of investigations on stability of the substructure solution 80 7.2 List of developments of compensation methods 80 7.2.1 The work on development of error force compensation 81 7.2.2 The work on development of phase lag compensation 81 7.3 List of verifications 82 viii q55 = ∂ xs i +1 ∂ xs = i C m + C a c + C a8 k − C8 m s c s d (A3.25) where the denominator d is given as Eq (A3.26) (A3.26) d = m + a c + a8 k where a1 , …, a10 are the integration constants specified in Appendix A5 while C1, …, C8 are given in Appendix A1 166 Appendix A4: Coefficients of the matrix Q in the case of ksub>1 The coefficients qsw (s=1; w=1,2, ,8) are specified as follow: q1w = when w ≠ q1w = when w = (A4.1) The coefficients qsw (s=2; w=1,2, ,8) are specified as follow: q2w = when w ≠ q2w = when w = (A4.2) The coefficients qsp (s=3; w=1,2, ,8) are specified as follow: q3w = when w ≠ q3w = when w= (A4.3) By using the procedure in Figure 6.6, the substructure solution ( u i+1, u i+1, u i+1, x si+1, i+1 x s ) at the next step can be obtained and then the coefficients qsw (s =1, ,8; w=1, ,8) of the matrix Q can be calculated in through the following equations: q 41 = q 44 = ∂u i +1 ∂u i −1 ∂u i +1 i ∂u q 47 = ∂u q 51 = i +1 ∂u i −1 q 54 = i ∂u q 48 = i ∂u i −1 ∂u i +1 i q 52 = ∂u i +1 ∂u i −1 ∂u i +1 ∂u i (A4.4) i i −1 q 53 = i ∂ xs i −1 q 56 = ∂u ∂u i +1 ∂u i +1 ∂u ∂u i +1 ∂u ∂u q 58 = q 46 = i +1 ∂u ∂u q 55 = q 43 = i +1 ∂u ∂ xs i +1 ∂ xs i +1 ∂u ∂u i +1 ∂u q 57 = i ∂u ∂u q 45 = i +1 ∂ xs q 42 = i 167 i +1 i (A4.5) q 61 = ∂u i +1 ∂u i −1 q 64 = ∂u q 67 = ∂u i q 74 = q 77 = i +1 ∂u i −1 ∂ xs i +1 ∂u i ∂ xs i +1 ∂ xs q 81 = i +1 ∂u i −1 ∂ xs i +1 ∂u i q 87 = i ∂ xs q 84 = i ∂ xs ∂ xs q 63 = i −1 ∂u q 65 = i +1 i i +1 i −1 i +1 ∂u ∂u q 66 = i q 68 = ∂ xs ∂u q 75 = q 78 = q 73 = i −1 ∂ xs i +1 ∂u i ∂ xs i +1 ∂ xs (A4.6) ∂ xs ∂ xs i +1 ∂u i −1 ∂ xs i +1 (A4.7) i i q 83 = ∂ xs ∂u i +1 ∂ xs ∂u i +1 q 86 = i ∂u q 88 = q 76 = i −1 ∂u q 85 = i i i +1 ∂ xs q 82 = i +1 i +1 ∂u ∂ xs q72 = ∂u ∂u i +1 ∂ xs ∂u ∂u i +1 ∂ xs q 71 = q 62 = i +1 ∂u ∂u ∂ xs ∂u i +1 i −1 i +1 (A4.8) i i +1 ∂ xs i It is worth noticing that when the number of sub steps ksub increases from to 5, the coefficients qij (i=1,…,8; j=1,…,8) are more and more complicated In order to process these equations, mathematical tools such Maple or any other Mathcad Software should be used to process the Eqs (A4.4) to (A4.8) 168 Appendix A5: Integration constants of the Newmark-β β integration The equilibrium equation can be solved in three different schemes in order to firstly obtain displacement, acceleration or velocity (Dorka et al 1998) The integration constants of these schemes are listed as follows: a0 = α∆t a1 = δ α∆t a2 = α∆t a3 = −1 2α a4 = δ −1 α a5 = ( δ ∆t − 2) α a6 = (1 − δ )∆t a7 = δ∆t a8 = α∆t a9 = ∆t a10 = ( − α )∆t 2 a11 = a12 = α∆t δ a13 = δ −1 (A5.1) δ∆t a14 = (1 − α )∆t δ α a15 = ( − )∆t 2 δ For unconditional stability, the constants are chosen as:  δ ≥  α ≥ ( + δ )  (A5.2) For non-numerical damping and least period elongation, the Newmark-β with average constant acceleration is used, which implies the following values:  α = β =  δ =  (A5.3) 169 Appendix A6: The amplification matrix of the Newmark-β β solution for a SDOF system The equilibrium equation for a SDOF system with load can be read as Eq (A6.1) i +1 i +1 u im +2ζ imωim u im +ωim uim i +1 = f im i +1 (A6.1) With the Newmark-β integration, the solution at step (i+1) can be shown as Eq (A6.2) (Bathe et al., 1976) uim i +1    u im i +1  =   u im i +1    uim i    i i +1 Au im  + A f f im   u im i    (A6.2) where A and Af are specified as Eqs (A6.3) - (A6.6)  1  1  ∆t  − α −  − α αβ − 2(1 − δ )ακ  2    2    1  A = ∆t 1 − δ −  − α δβ − 2(1 − δ )δκ  2      −  − α  β − 2(1 − δ )κ    β= κ= 1 2ζδ + +α 2 ω ∆t ω∆t ∆t (1 − αβ − 2ακ ) − βδ − 2δκ (− β − 2κ ) ∆t  (1 − αβ )  − βδ   ∆t  − β  ∆t  (A6.3) (A6.4) ζβ ω∆t (A6.5)  β   ω ∆t    βδ   Af =  ω ∆t   βα     ω  (A6.6) For free vibration (r=0), using the relationship between the angular frequency ω and the dimensionless frequency Ω as Eq (A6.7) 170 ω= Ω ∆t (A6.7) and the integration constants as α=0.25, δ=0.5, one can obtain after some manipulations that the Eq (A6.2) can be written as Eq (A6.8) − U i+1 − = AU i (A6.8) where the numerical states are given as Eqs (A6.9) and (A6.10) and the matrix A is given as Eq (A6.11) − i − i+1 U = [ u im U i = [ u im ∆t u im i ∆t2 u im i]T i+1 (A6.9) ∆t u im i+1 ∆t2 u im i+1]T  4(1 + ξΩ)  Ω + 4ξΩ +   − 2Ω ) A=   Ω + 4ξΩ +  − 4Ω   Ω + 4ξΩ + 2(2 + ξΩ) Ω + 4ξΩ + 4 − Ω2 Ω + 4ξΩ + − 4Ω(Ω + 2ξ ) Ω + 4ξΩ + (A6.10)  Ω + 4ξΩ +    Ω + 4ξΩ +  − Ω(Ω + 4ξ )   Ω + 4ξΩ +  A is called the amplification matrix of the numerical solution 171 (A6.11) Appendix A7: Data of the testing system in virtual substructure tests The real-time control hardware: Specifications of the real-time control system ADwin Pro II are listed below: Processing module: Pro-CPU-T11-ENET Processor: ADSP TS101S Clock time interval: 10/3 ns (300 MHz) Internal memory: 768 kB External memory: 256 MB RAM Interface: Ethernet 100 Mps Analog-digital converter: Pro-Ain-F-8/14 Rev B Digital resolution: 14 bits Analog input range: +/-10 Vols Maximum sampling rate: 2200 ksamples/s Digital-analog converter: Pro-AOut-8/16 Rev C Digital resolution: 16 bits Analog output range: +/-10 Vols Settling time: µs Made by: Jäger Computergesteuerte Messtechnik, Germany The hydraulic cylinder: Type: Dynamic Servo Cylinder SLZ-250-400 Maximum displacement: +/- 200 mm Maximum force: 250 kN Working pressure: 280 bar Working area of piston: 92 mm2 Built-in transducer: Inductive displacement transducer, sensitivity 80 mV/V Made by: SANDNER-Messtechnik GmbH, Germany 172 The servo valve: Proportional hydraulic servo valve model 550 Maximum flow: 60 l/min (at ∆p=70 bar) Maximum pressure: 315 bar Current control: +/-30 mA Hysteresis: < 3% Made by: Hydraulic Star The load cell: Type S9-10K Maximum force +/- 10 kN Sensitivity: 2.00 mV/V Accuracy class: 0.05 Made by: Hottinger Baldwin Messtechnik GmbH, Germany The amplifier: Amplifier system MGCplus with control pannel AB22A/AB32 Input module: AP01i Input type: inductive transducers, bridge transducers Output range: +/-10 Vols Made by: Hottinger Baldwin Messtechnik GmbH, Germany 173 REFERENCES Bathe, K J and Wilson, E L (1976), Numerical method in Finite Element Analysis, Prentice Hall Bayer, V., Dorka, U E and Füllekrug, U (2000a), A New Algorithm for Real-Time SubStructure Pseudo-Dynamic Tests The 12th World Conf on Earthquake Eng., Auckland, New Zealand Bayer, V and Dorka, U E (2000b), Qualification of TMDs by real-time SubPSD-testing, The 2nd European Conference on Structural Control, Champs sur Marne, France Bayer, V., Dorka, U E., Füllekrug, U and Gschwilm J (2002), Realisation real-time pseudodynamic sub-structure testing, Int conf on noise and vibration eng (ISMA2002), Leuven, Sept 2002 Bayer, A., Dorka, U E., Füllekrug, U and Gschwilm, J (2005), On real-time pseudodynamic substructure testing: algorithm, numerical and experimental results, Aero Sc and Tech., 9, 223-232 Blakeborough, A., Williams, M.S., Darby, A.P and Williams, D M (2001), The development of real-timesubstructure testing, Philosophical Transaction of the Royal Society: Theme Issue on Dynamic Testing of Structures, A 359, 1869-1891 Bobrow, J E and Lum, K (1995), Adaptive, High Bandwidth Control of a Hydraulic Actuator Proceedings of the American Control Conference, 1995, Washington, USA Bonelli, A., Bursi, O S., Erlicher, S., Vulcan, L., (2002a), Analyses of the generalized-alpha method for linear and non-linear forced excited systems, Structural DynamicsEURODYN'02, 379-386 Bonelli, A., Bursi, O S., Mancuso, M (2002b), Explicit predictor-multicorrector time discontinuous Galerkin methods for non-linear dynamics, Journal of Sound and Vibration, 256, No.4, 695-724 Bonnet, P A., Lim, C N., Williams, M S., Blakeborough, A., Neild, S A., Stoten, D P and Taylor, C A (2007), Real-time hybrid experiments with Newmark integration, MCSmd outer-loop control and multi-tasking strategies, Earthquake Engng Struct Dyn., 36, 119–141 Bursi, O.S and Shing, P.B (1996), Evaluation of some implicit time-steing algorithms for pseudodynamic tests, Earthquake Engng Struct Dyn., 25: 333-355 Carrion, J.E and Spencer, B.F (2008), Real-time Hybrid Testing Using Model-based Delay compensation, Smart Structures and Systems, 4(6), 809–828 Combescure, D., Pegon, P., Magonette, G (1995), Numerical investigation of the impact of experimental errors on various pseudo-dynamic integration algorithms Proceedings of the 10th European Conference on Earthquake Engineering, Duma G (ed.) Balkema, Rotterdam, The Netherlands, 2479-2484 174 Combescure, D and Pegon, P (1997), α-Operator splitting time integration technique for pseudodynamic testing error propagation analysis Soil Dynamics and Earthquake Engineering, 16, Issue: 7-8, Publisher: Elsevier, 427–443 Darby, A P , Blakeborough, A and Williams, M S (2001), Improved control algorithm for real-time substructure testing, Earthquake Engng Struct Dyn., 30 (3), 431–448 Dorka, U E and Heiland, D (1991), Fast online earthquake simulation using a novel PC suorted measurement and control concept Proc 4th Int Conf Structural Dynamics, Southhampton, 636-645 Dorka, U.E and Füllekrug, U (1998), Report of DFG-project No Do 360/7: Sub-PSD Tests, "Algorithmen für real-time pseudo-dynamische Substrukturtests", University of Kaiserlautern, Germany Dorka U.E (2002), Hybrid experimental - numerical simulation of vibrating structures Proceedings of The International Workshop WAVE 2002, Okayama, Japan, 183-191 Dorka, U E., Queval, J C., Nguyen, V T and Maoult, A L (2006), Real-time sub-structure testing on distributed shaking tables in CEA Saclay, Proceedings of 4th World Conference on Structural Control and Monitoring, San Diego, USA Dorka, U E., Queval, J C., Nguyen, V T and Maoult, A L (2007), Substructure testing on distributed shaking tables, Proceedings of the 2nd International Conference on Advances in Experimental Structural Engineering, Shanghai, China Forssell, U and Ljung, L (1999), Closed-loop identification revisited, Automatica, 35, No.7, 1215-1241 Ghaboussi, J., Yun G J and Hashash, Y M A (2006), A novel predictor–corrector algorithm for sub-structure pseudo-dynamic testing, Earthquake Engng Struct Dyn., 35, 453–476 Hilber, H M., Hughes, T J R and Taylor, R L (1977), Improved numerical dissipation for time integration algorithms in structural mechanics, Earthquake Eng Struct Dyn., 5(3), 283–292 Hodgson, S P., Stoten, D P (1999), Robustness of the minimal control synthesis algorithm to non-linear plant with regard to the position control of manipulators, International Journal of Control, 72, No.14, 1288-98 Horiuchi, T., Nakagawa, M., Sugano, M and Konno, T (1996), Development of a real-time hybrid experimental system with actuator delay compensation, Proceedings of the 11th World Conf Earthquake Engineering, paper No 660 Horiuchi, T., Inoue, M., Konno, T and Namita, Y (1999), Real-time hybrid experimental system with actuator delay compensation and its alication to a piping system with energy absorber, Earthquake Engng Struct Dyn., 28(10), 1121-1141 175 Horiuchi, T and Konno, T (2001), A new method for compensating delay in time hybrid experiments, Phil Trans R Soc Lond , A359, 1893-1909 Jäger Computegeteuerte Messtechnik (2006), ADwin Pro – System and hardware description, Germany Kreyszig, E (1999), Advanced engineering mathematics, 8th edn New York: Wiley Kwon, O S and Nakata, N (2005), Technical note a framework for multi-site distributed simulation and alication to complex structural systems, Journal of Earthquake Engineering, (no.5), 741–753 Ljung, L (1999), System Identification - Theory for the users, Prentice Hall Magonette, G., Pegon, P., Molina, F J and Buchet, P H (1998), Development of fast continuous substructuring tests, Proceedings of the 2nd World Conference on Structural Control Magonette, G (2001), Development and application of large scale continuous pseudodynamic testing techniques, Phil Trans R Roc Lond A 359, 1771-1799 Morari, M and Lee, J H (1999), Model predictive control: past, present and future, Computers and Chemical Engineering, 23, 667–682 Mosqueda, G., Stojadinovic, B., Hanley, J., Sivaselvan, M and Reinhorn, A (2005), Fast hybrid simulation with geographically distributed substructures, Proceedings of The first International Conference on Advances Experimental Structural Engineering, Japan Nakashima, M (1984), Stability and accuracy of integration techniques in pseudo dynamic testing, Buiding Research Center, Ministry of Construction, Japan Nakashima, M., Kato, H (1987), Experimental error growth behavior and error growth control in online computer test control method, Building Research Institute, BRIReport No 123, Ministry of Construction, Tsukuba, Japan Nakashima, M., Kaminosomo, T., Ishida, M., Ando, K (1990), Integration techniques for substructuring pseudodynamic test, the Fourth US National Conference on Earthquake Engineering, Palm Springs, USA Nakashima, M., Kato H., Takaoka E (1992), Development of real-time pseudo dynamic testing, Earthquake Eng & Struct Dyn., 21, 79-92 Nakashima, M (1995), Strain-hardening behavior of shear panels made of low-yield steel I: Test, J Struct Eng., ASCE:121(12), 1742–1749 Nakashima, M and Masaoka, N (1999), Real time on-line test for MDOF systems, Earthquake Engng Struct Dyn., 28 (no.4), 393-420 176 Negro, P., Verzeletti, G., Magonette, G E and Pinto, A V (1994), Test on a four-storey fullscale R/C frame designed according to Eurocodes and 2: Preliminary Report, EUR 15879 EN, JRC Newmark N.M (1959), A method of computation for structural dynamics Journal of Engineering Mechanics (ASCE) 1959; 85:67–94 Nguyen, V T and Dorka, U E (2007), Unbalanced force compensation in substructure testing based on online system identification, Proceedings of the second International Conference on Advances in Experimental Structural Engineering, Shanghai, China Nowak, R D (2002), Nonlinear system identification, Circuits Systems Signal Processing, 21 (No I), 109-122 Ohtani, K., Ogawa, N., Katayama, T and Shibata, H (2003), Project “E-Defense”Introduction of E-Defense, Pacific Conference on Erthquake Engineering Proceedings Pacific Conference in Earthquake Engineering, 2003, New Zealand Paz, M (1997), Structural dynamics: Theory and Computation, Chapman & Hall, USA Pan, P., Tada, M and Nakashima, M (2005), Online hybrid test by internet linkage of distributed test-analysis domains, Earthquake Engng Struct Dyn., 34, 1407–1425 Pan, P., Tomofuji, H., Wang, T., Nakashima, M., Ohsaki, M and Mosalam, K M (2006), Development of peer-to-peer (P2P) internet online hybrid test system, Earthquake Engng Struct Dyn., 35, 867–890 Pearlman, L., Kesselman, C., Gullapalli, S., Spencer, B.F., Futrelle, J., Ricker, K., Foster, I., Hubbard, P., and Severance, C (2004), Distributed Hybrid Earthquake Engineering Experiments: Experiences with a Ground-Shaking Grid Alication Technical Report NEESgrid-2004-42, NEESgrid, USA Pegon, P and Pinto, A V (2000), Pseudo-dynamic testing with substructuring at the ELSA Laboratory, Earthquake Engng Struct Dyn., 29, 905-925 Pegon, P., Molina, F J., Magonette, G (2008), Continuous pseudo-dynamic testing at ELSA in Hybrid Simulation; Theory, Implementation and Alications, Eds Saouma VE, Sivaselvan MV, Taylor & Francis/Balkema, 79-88 Pinto, A V., Pegon, P., Magonette, G and Tsionis, G (2004), Pseudo-dynamic testing of bridges using non-linear substructuring, Earthquake Engng Struct Dyn., 33, 11251146 Plett, G L (2002), Adaptive inverse control of unmodeled stable SISO and MIMO linear systems, Int J Adapt Con Sig Pro., 16, 243-272 Roik, K and Dorka, U E (1989), Fast online earthquake simulation of friction damped systems SFB151 Report No 15, Ruhr-University Bochum, Germany Rossiter, J A (2004), Model Predictive Control: A practical Aproach, CRC Press 177 Söderström, T and Stoica, P (1989), System Identification, Prentice Hall Stoten, D P and Benchoubane, H (1990) Robustness of a minimal controller synthesis algorithm, Int Journal of Control, 51(No.4), 851-861 Stoten, D P and Bulut, S (1994), Alication of the MCS algorithm to the control of an electrohydraulic system, IEEE-07802-1328-3/94, 1742-1747 Stoten, D P and Hodgson, S P (1998), Passivity-based analysis of the decentralized and extended minimal control synthesis algorithms, Int J Control, 69(No 5), 633-646 Stoten, D P and Gomez, E G (2001), Adaptive control of shaking tables using the minimal control synthesis algorithm, Phil Trans R Soc Lond , A359, 1697-1723 Shing, P.B and Mahin, S (1987), Cumulative experimental errors in pseudodynamic tests, Earthquake Engng Struct Dyn., 15, 409-424 Shing, P.B., Nakashima, M., Bursi, O (1996), Aplication of pseudodynamic test method to structural research, Earthquake Spectra, 12, 29-56 Spencer, B F and Carrion, J E (2007) Real-time hybrid testing of semi-actively controlled structure with MR damper, Proceedings of 2nd Int Conf on Advances in Experimental Structure Engineering, China Tada, M and Pan, P (2007), A modified operator splitting (OS) method for collaborative structural analysis (CSA), Int J Numer Meth Engng., 72, 379–396 Takanashi, K and Nakashima, M (1987), Japanese activies on on-line testing Journal of Engineering Mechanics, 113 (No 7), 1014–1032 Thewalt, C R and Mahin, S A (1987), Hybrid solution techniques for generalized pseudodynamic testing, Report No UCB/EERC-87/09, Univ of California Berkeley, USA Wallace, M I., Wagg, D J and Neild, S A (2005), An adaptive polynomial based forward prediction algorithm for multi-actuator real-time dynamic substructuring, Proc R Soc A 461, 3807–3826 Wang, K J., Tsai, K C., Wang, S J., Cheng, W C and Yang, Y S (2007), ISEE: Internetbased Simulation for Earthquake Engineering - Part II: The alication protocol aroach, Earthquake Engng Struct Dyn., 36 (No 15), 2307-2323 Wei, H L., Billings, S A and Liu, J (2004), Term and variable selection for non-linear system identification, Int J Control, 77 (No 1), 86-110 Widrow, B and Walach, E (1995), Adaptive Inverse Control, Prentice-Hall, Englewood Clis, NJ Williams, A S., Blakeborough, A., Houlsby, J T and Williams, D M (1998), Development of a real-time hybrid dynamic testing system Seismic Design Practice into the Next Century, Booth ed., ISBN 9054109343, 373-379 178 Yang, Y E., Hsieh, S H., Tsai, K C., Wang, S J., Wang, K J., Cheng, W C and Hsu, C W (2007), ISEE: Internet-based Simulation for Earthquake Engineering - Part I: Database aroach, Earthquake Engng Struct Dyn., 36, Issue 15 , 2291-2306 Zhang, Y., Sause, R., Ricles, J M and Naito, C J (2005), Modified predictor–corrector numerical scheme for real-time pseudo dynamic tests using state-space formulation, Earthquake Engng Struct Dyn., 34, 271–288 Zhou, Q G and Cluett, W (1996), Recursive identification of time-varying systems via incremental estimation, Automatica, 32 (No 10), 1427-1431 Zhou, Y., Han, A., Yan, S and Chen, X (2006), A fast method for online closed-loop system identification, Int J Adv Manuf Technol , 31, 78-84 Zienkiewicz, O C (1977), The Finite Element Method, McGraw-Hill 179 CONFERENCE PAPERS Nguyen, V T and Dorka, U E (2006), Aplication of digital technique in a control system for real-time sub-structure testing, Proceedings of 4th World Conference on Structural Control and Monitoring, San Diego, USA Dorka, U E., Queval, J C., Nguyen, V T and Maoult, A L (2006), Real-time sub-structure testing on distributed shaking tables in CEA Saclay, Proceedings of 4th World Conference on Structural Control and Monitoring, San Diego, USA Nguyen, V T and Dorka, U E (2007), Unbalanced force compensation in substructure testing based on online system identification, Proceedings of the second International Conference on Advances in Experimental Structural Engineering, Shanghai, China Dorka, U E., Queval, J C., Nguyen, V T and Maoult, A L (2007), Substructure testing on distributed shaking tables, Proceedings of the second International Conference on Advances in Experimental Structural Engineering, Shanghai, China 180 [...]... RESULTS OF ACCURACY AND STABILITY ANALYSES 85 8.1 Accuracy of the substructure solution without phase lag 85 8.1.1 Variation of the transfer function with different dimensionless time steps in the case of typical TMD system 85 8.1.2 Variation of the transfer function with different numbers of sub steps in the case of typical TMD system 86 8.1.3 Effect of the number of sub steps... two peaks of the exact transfer function h(ω), h(r) transfer function at an angular frequency ω or at the frequency ratio r k stiffness of the numerical substructure klag number of sub steps for identification ks stiffness of the experimental substructure ksub number of sub steps m mass of the numerical substructure ms mass of the experimental substructure n an integer number nb order of the polynomials... verification of the substructure solution 101 Table 8-3: Two peaks of the transfer functions of the substructure solution and their errors 102 Table 8-4: Comparison of the errors in the validation with the errors of the accuracy analysis 102 Table 8-5: Substructure system for validation of the stability analysis 116 Table 8-6: Data of the checked points for validation of. .. number of sub steps on the errors in the case of typical TMD system 88 8.1.4 Effect of the mass ratio on the errors in the case of typical TMD system 90 8.1.5 Effect of the damping ratio of the numerical substructure on the errors in the case of typical TMD system 91 8.1.6 Effect of the damping ratio of the experimental substructure on the errors in the case of typical TMD system... Experimental substructure fc fc2 Figure 1.2: Substructures with nonlinear simulation and experimental substructure In a substructure test, an integration method is used to calculate the response of the numerical substructure at each step (computed displacement), which is then applied to the experimental substructure; the coupling force at the coupler is measured and fed back to the numerical substructure. .. , u (t ) and u (t ) are respectively the acceleration, velocity and displacement vectors of the numerical substructure at time t; f l (t ) is the loading vector of the numerical substructure; f c (t ) is the vector of coupling forces between the experimental and numerical substructures To obtain the response of the numerical substructure, the equation of motion (Eq 2.1) is solved directly without prior... Substructure solution at the point B2 in the case of k sub= 5 : limited stability 119 Figure 8.54: Substructure solution at the point B3 in the case of k sub= 5 : unstable solution 120 Figure 8.55: Substructure solution at the point B3 in the case of k sub= 5 , the substructure algorithm uses a simple PID error force compensation with P = 1, D = 0, I = 0: the substructure solution... integration schemes to deal with the interaction between substructures The OS and α-OS methods apply their prediction displacements as explicit terms on the experiment, correct the restoring force by using stiffness and solve for the implicit response of the numerical substructure at the end of the next step An analog feedback (Thewalt and Mahin 1987) or digital feedback with sub- step control (Dorka et al... Pseudo-Linear Regression xii T period of vibration T0 the period of free vibration of at the initial time Tn period of the numerical substructure T1 , T2 periods of the numerical solution at the first and second frequencies T1e , T2 e periods of the exact solution at the first and second frequencies U numerical state of the substructure solution − U numerical state of the substructure solution in which all... validation of the stability analysis in the case of k sub= 1 117 Table 8-7: Data of the checked points for validation of the stability analysis in the case of k sub= 5 118 Table 8-8: Summary of the stability validation at the checked points 118 Table 9-1: Mass and stiffness of the considered system in VSTs 144 Table 9-2: Eigenmodes and damping of the full system in the verification