Accuracy and stability of the substructure algorithm with sub step control

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

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

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

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 1 to Chapter 4 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 5 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 6 to Chapter 9 The methodologies of accuracy and stability analyses, methods for development of error compensations and the verification approaches are stated in Chapter 6 In addition, the lists of cases for accuracy and stability analyses and the lists of development as well as verifications are presented in Chapter 7 Moreover, the results of investigation on accuracy and stability are presented in Chapter 8 Meanwhile, the developments of two error compensations are described in Chapter 9 Finally, the last chapter includes conclusions and future research and applications

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

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 Echtzeit-Substrukturtest 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 Echtzeit-Substrukturtest 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 5 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

Phasenverzögerungskompensation Die Abweichungskraftskompensation minimiert

Phasenverzögerungskompensation die Phasenverzögerung im hydraulischen System reduziert

Sowohl die durchgeführten Untersuchungs- und Entwicklungsverfahren als auch die daraus resultierenden Ergebnisse werden von Kapitel 6 bis Kapitel 9 detailliert dargestellt Die Methodologien der Genauigkeits- und Stabilitätsanalysen, das Entwicklungsverfahren der Kompensationen und Verifikationsannäherungen werden dann im Kapitel 6 besprochen Weiterhin werden im Kapitel 7 die Fälle der

Genauigkeits- und Stabilitätsanalyse und dazu die Verifikationen präsentiert; veranschaulicht werden die Ergebnisse zu diesen Analysen im Kapitel 8 Außerdem werden die Entwicklungen zweier Kompensationen im Kapitel 9 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

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

1 INTRODUCTION TO SUBSTRUCTURE TESTING 1

1.1 Principle of substructure method and substructure testing 1

1.2 History of substructure testing 3

1.3 State of the art of real-time substructure testing 5

2 CONTROL OF SUBSTRUCTURE TEST 9

2.1 Introduction 9

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

3 ERRORS AND THEIR EFFECTS IN SUBSTRUCTURE TESTING 25

3.1 Errors of modeling and loading assumption 25

3.2 Errors in control hardware 27

3.3 Errors of measurement and conversion 30

4 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

5 SCOPE OF THESIS AND CONTRIBUTIONS 43

6 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

7 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

8 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 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 92

8.1.7 Effect of the mistuned-frequency phenomenon on the errors in the case of typical TMD system 93

8.1.8 Effect of the mass ratio on the errors in the case of coupled system with large mass ratio 94

8.1.9 Effect of the frequency ratio on the errors in the case of coupled system with large mass ratio 95

8.2 Effect of phase lag on the accuracy of the substructure solution 96

8.2.1 Effect of phase lag on the transfer function in the case of typical TMD system96 8.2.2 Effect of phase lag on the errors in the case of typical TMD system 97

8.3 Verification of the substructure solution 99

8.4 Results of the stability analysis 103

8.4.1 Introduction on the results 103

8.4.2 Stability when varying the damping ratio of the experimental substructure and the dimensionless frequency 103

8.4.3 Stability when varying the damping ratio of the numerical substructure and the dimensionless frequency 106

8.4.4 Stability when varying the frequency ratio and the dimensionless frequency.109 8.4.5 Stability when varying the mass ratio and the dimensionless frequency 111

8.5 Verification of the stability analysis 114

8.5.1 The amplification matrix in a special case without coupler 114

8.5.2 Verification of the stability regions 116

8.5.3 Verification of the effectiveness of force compensation in stabilizing the substructure solution 120

8.6 Summary of accuracy and stability analyses 121

8.6.1 Summary of accuracy analysis 121

8.6.2 Summary of stability analysis 123

9 DEVELOPMENT OF COMPENSATION METHODS 125

9.1 New error force compensation 125

9.1.1 Description and formulation 125

9.1.2 Discussion on accuracy and convergence 127

9.1.3 Discussion on features and applications 129

9.2 New phase lag compensation 130

9.2.1 Description and formulation 130

9.2.2 Discussion on accuracy and convergence 135

9.2.3 Discussion on features and applications 136

9.3 Implementation of the compensations and the substructure algorithm in a control system 136

9.4 Verification of the compensations by using virtual substructure test 138

9.4.1 Control and measurement system for virtual substructure tests 138

9.4.2 Description of the substructure system and loading condition for virtual substructure tests 141

9.4.3 Results of the error force compensation 145

9.4.4 Results of phase lag compensation 146

9.4.5 Comparison and discussion on the test results 149

9.5 Summary on development of the error force and phase lag compensations 153

10 CONCLUSIONS AND OUTLOOK 155

10.1 Conclusions 155

10.2 Outlook 159

APPENDICES 161

REFERENCES 174

CONFERENCE PAPERS 180

TERMS AND SYMBOLS

Phrases

Latin Uppercase letters

substructure

1

H(f) transfer function of the hydraulic system at frequency f

I

X(f) input signal at frequency f

Y(f) output signal at frequency f

Latin lowercase letters

g

f l loading vector of the numerical substructure

h1 , h2e two peaks of the exact transfer function

h(ω), h(r) transfer function at an angular frequency ωor at the frequency ratio r

o

LIST OF FIGURES

Figure 1.1: The simplest case of substructures 1

Figure 1.2: Substructures with nonlinear simulation and experimental substructure 2

Figure 1.3: Substructure test using actuator for displacement control and load cell for force measurement 2

Figure 2.1: The shape functions over time (Zienkiewicz 1977) 10

Figure 2.2: Classification of integration methods in different weighting functions and integration parameters (Zienkiewicz 1977) 11

Figure 2.3: Numerical damping for three-step and four-step integration methods (Zienkiewicz 1977, extended by Dorka 2002) 13

Figure 2.4: Period distortion in free vibration of a SDOF system for different integration methods (Zienkiewicz 1977, Nakashima 1984, extended by Dorka 2002) 13

Figure 2.5: Substructure algorithm with digital feed back and error force compensation (Roik and Dorka 1989, Dorka et al. 1998, Dorka 2002) 21

Figure 3.1: Realistic load and a linear assumed load of the numerical substructure 26

Figure 3.2: A typical transfer function of single hydraulic actuator in low frequency range 28

Figure 3.3: Typical phase lag of a hydraulic actuator 30

Figure 3.4: The basic components of a channel in the measurement system 30

Figure 3.5: Two different sinusoids that fit the same set of samples 33

Figure 4.1: Prediction of displacement one time lag ahead, illustration with a time lag as three time steps and extrapolation with the third order 37

Figure 4.2: Diagram of displacement control with feedback and feed forward controls 40

Figure 6.1: Substructure model for accuracy and stability analyses 49

Figure 6.2: A special case of the substructure model when m sapproached infinity, the experimental substructure was a simple specimen including a spring and a damper 49

Figure 6.3: Substructure simulation or virtual substructure test 50

Figure 6.4: Amplitude and frequency errors of the substructure solution 53

Figure 6.5: The procedure to calculate the substructure solution for stability analysis in the case of k sub= 1 56

Figure 6.6: The procedure to calculate the substructure solution for stability analysis in the cases of k sub > 1 59

Figure 6.7: The substructure test system using hydraulic system without phase lag compensation 63

Figure 6.8: Substructure testing with phase lag compensation 64

Figure 6.9: Black-box data model 66

Figure 6.10: The Box-Jenkin model 67

Figure 6.11: Output Error (OE) model 67

Figure 6.12: ARMAX model 67

Figure 6.13: ARX model 68

Figure 6.14: FIR model 68

Figure 6.15: Validation of the substructure solution 73

Figure 8.1: Transfer function of the substructure solution in series of ηT with 2 = sub k , typical TMD system (ηm =5 % , ζ = 5 % , ζs =10 % ), case 2 in Table 7-3 85

Figure 8.2: Transfer function of the substructure solution in series of ηT with 5 = sub k , typical TMD system (ηm =5 % , ζ = 5 % , ζs =10 % ), case 2 in Table 7-3 85

Figure 8.3: Transfer function of the substructure solution in series of ηT with 10 = sub k , typical TMD system (ηm =5 % , ζ = 5 % , ζs =10 % ), case 2 in Table 7-3 86

Figure 8.4: Transfer function of the substructure solution in series of k sub, with 1 0 = T η , typical TMD system (ηm =5%, ζ = 5 %, ζs =10%), case 2 in Table 7-3 87

Figure 8.5: Transfer function of the substructure solution in series of k sub, with 2 0 = T η , typical TMD system (ηm =5%, ζ = 5 %, ζs =10%), case 2 in Table 7-3 87

Figure 8.6: Period errors of the substructure solution at two frequencies, in series of k sub, typical TMD system (ηm=5%, ζ = 5 %, ζs =10%, fo f η η = ), case 2 in Table 7-3 89

Figure 8.7: Amplitude errors of the substructure solution at two frequencies, in series of k sub, typical TMD system (ηm =5%, ζ = 5 %, ζs =10%), case 2 in Table 7-3 89

Figure 8.8: Period errors of the substructure solution in series of mass ratio ηm, typical TMD systems (ζ = 5 %, ζs =10%), cases 1, 2 and 3 in Table 7-3 90

Figure 8.9: Amplitude errors of the substructure solution, in series of mass ratio m η , typical TMD systems (ζ = 5 %, ζs =10%), cases 1, 2 and 3 in Table 7-3 90

Figure 8.10: Period errors of the substructure solution in series of damping ratio ζ, typical TMD system (ηm=5%, ζs =10%), cases 4, 5 and 2 in Table 7-3 91

Figure 8.11: Amplitude errors of the substructure solution in series of damping ratio

ζ, typical TMD system (ηm =5%, ζs =10%), cases 4, 5 and 2 in

Table 7-3 91 Figure 8.12: Period errors of the substructure solution, in series of damping ratio ζs

in typical TMD systems (ηm =5%, ζ = 5 %), cases 6, 7 and 2 in Table 7-3 92 Figure 8.13: Amplitude errors of the substructure solution, in series of the damping

ratio ζs in typical TMD systems (ηm =5%, ζ = 5 %), cases 6, 7 and 2

in Table 7-3 92 Figure 8.14: Period errors of the substructure solution in series of frequency ratio

f

η , typical TMD system (ηm =5%, ζ = 5 %, ζs =10%), cases 8, 2

and 9 in Table 7-3 93 Figure 8.15: Amplitude errors of the substructure solution in series of frequency

ratio η , typical TMD system (f ηm =5%, ζ = 5 %, ζs =10%), cases

8, 2 and 9 in Table 7-3 93 Figure 8.16: Period errors of the substructure solution in series of mass ratio ηm,

coupled system (ζ = 5 %,ζs =5%, ηf = 1) with large mass ratio,

cases 10, 11 and 12 in Table 7-3 94 Figure 8.17: Amplitude errors of the substructure solution in series of mass ratios

m

η , coupled system (ζ = 5 %,ζs =5%,ηf = 1) with large mass ratio,

cases 10,11,12 in Table 7-3 94 Figure 8.18: Period errors of the substructure solution, in series of frequency ratio

f

η , coupled system ηm =50%, ζ = 5 %, ζs =5%), cases 13, 14, 10,

15, 16 in Table 7-3 95 Figure 8.19: Amplitude errors of the substructure solution, in series of frequency

ratio η , coupled system (f ηm =50%, ζ = 5 %, ζs =10%), cases 13,

14, 10, 15, 16 in Table 7-3 96 Figure 8.20: Transfer function in series of the phase lag ratio η with p k sub=2,

1.0

=

T

η , typical TMD system (ηm =5%, ζ = 5 %, ζs =10%) 97 Figure 8.21: Period errors in series of phase lag ratio η with p k sub=2, for typical

TMD system (ηm =5%, ζ = 5 %, ζs =10%) 97 Figure 8.22: Amplitude errors in series of the phase lag ratio η with p k sub=2,

typical TMD system (ηm =5%, ζ = 5 %, ζs =10%) 98 Figure 8.23: Period errors in series of the phase lag ratio η with p k sub=10, typical

TMD system (ηm =5%, ζ = 5 %, ζs =10%) 99 Figure 8.24: Amplitude errors in series of the phase lag ratio η with p k sub=10,

typical TMD system (ηm =5%, ζ = 5 %, ζs =10%) 99

Figure 8.25: Base acceleration as sine sweep signal for validation of substructure

solution 100 Figure 8.26: Comparison of the substructure solutions and the exact solution 101 Figure 8.27: Zoom view at the peaks of the transfer functions: (a) at the first

frequency, (b) at the second frequency 101

Ω, typical TMD substructure (ηm =5%, ζ = 5 %, ηf = 1), number of

sub steps k sub=1 104

Ω, typical TMD substructure (ηm =5%, ζ = 5 %, ηf = 1), number of

sub steps k sub=2 104

Ω, typical TMD substructure (ηm =5%, ζ = 5 %, ηf = 1), number of

sub steps k sub=3 105

Ω, typical TMD substructure (ηm =5%, ζ = 5 %, ηf = 1), number of

sub steps k sub=4 105

Ω, typical TMD substructure (ηm =5%, ζ = 5 %, ηf = 1), number of

sub steps k sub=5 106 Figure 8.33: Maximum spectral radius when varying ζ and Ω, typical TMD

substructure (ηm =5%, ζs =10%, ηf = 1), number of sub steps

1

=

sub

k 107 Figure 8.34: Maximum spectral radius when varying ζ and Ω, typical TMD

substructure (ηm =5%, ζs =10%, ηf = 1), number of sub steps

2

=

sub

k 107 Figure 8.35: Maximum spectral radius and stable regions when varying ζ and Ω,

typical TMD substructure (ηm =5%, ζs =10%, ηf = 1), number of

sub steps k sub=3 108 Figure 8.36: Maximum spectral radius and stable regions when varying ζ and Ω,

typical TMD substructure (ηm =5%, ζs =10%, ηf = 1), number of

sub steps k sub=4 108 Figure 8.37: Maximum spectral radius and stable regions when varying ζ and Ω,

typical TMD substructure (ηm =5%, ζs =10%, ηf = 1), number of

sub steps k sub=5 109

Figure 8.38: Maximum spectral radius and stable regions when varying η and Ω, f

typical SDOF substructure with (ηm =5%, ζ = 5 %, ζs =10%),

number of sub steps k sub=1 110

substructure with (ηm =5%, ζ = 5 %, ζs =10%), number of sub

steps k sub=2 110

substructure with (ηm =5%, ζ = 5 %, ζs =10%), number of sub

steps k sub=5 111

substructure with (ζ = 5 %, ζs =10%, ηf = 1), number of sub steps

1

=

sub

k 112

substructure with (ζ = 5 %, ζs =10%, ηf = 1), number of sub steps

2

=

sub

k 112

substructure with (ζ = 5 %, ζs =10%, ηf = 1), number of sub steps

3

=

sub

k 113

substructure with (ζ = 5 %, ζs =10%, ηf = 1), number of sub steps

4

=

sub

k 113

substructure with (ζ = 5 %, ζs =10%, ηf = 1), number of sub steps

5

=

sub

k 114 Figure 8.46: Maximum spectral radius ρ of the amplification matrix in the case of

the case of k sub=1, typical TMD system with ηm =5%, ζ = 5 %,

1

=

f

η 117 Figure 8.48: Position of the checked points for validation of the stability analysis in

the case of k sub=5, typical TMD system with ηm=5%, ζ = 5 %,

1

=

f

η 117 Figure 8.49: Substructure solution at the point A1 in the case of k sub=1: stable

solution 118

Figure 8.50: Substructure solution at the point A2 in the case of k sub=1: limited

stability 119 Figure 8.51: Substructure solution at the point A3 in the case of k sub=1: unstable

solution 119 Figure 8.52: Substructure solution at the point B1 in the case of k sub=5: stable

solution 119 Figure 8.53: 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 is well stable 120 Figure 9.1: Algorithm for substructure control with error force compensation 126 Figure 9.2: Contribution of a datum in the past to the cost function of the

estimation 129 Figure 9.3: Substructure testing with phase lag compensation 130 Figure 9.4: Scheme of identification and estimation of the phase lag

compensation, illustration with a number of estimation steps k lag = 5

and order n u =3 132 Figure 9.5: Variations of θ and ∆uwithin an identification time interval: (a) step

change of θ, (b) sudden change of ∆udue to the step change of θ, (c)

linear variation of θ 133 Figure 9.6: Phase lag compensation in substructure testing 134 Figure 9.7: Implementation of the error force compensation and the phase lag

compensation in substructure testing 137 Figure 9.8: The test system for virtual substructure tests without hydraulic system

(S1 is set at the position 2) or with hydraulic system (S1 is set at

position 1) and without compensation, with or without noise of force

measurement (S2 is set at position 1 or 2) 138 Figure 9.9: Diagram of a VST using a hydraulic system with the error

compensation and noise of force measurement 140 Figure 9.10: Diagram of a VST using a hydraulic system with phase lag

compensation and noise of force measurement 140 Figure 9.11: Diagram of the control and measurement systems for VST 141 Figure 9.12: The substructure system for VSTs 142 Figure 9.13: Base displacement ug in the time (left) and frequency domain (right) 143 Figure 9.14: Base velocity u.g in the time (left) frequency domain (right) 143

Figure 9.15: Comparison of the residual errors in different orders n u of the error

force compensation (n u=2,3,4,5,8,12; λ=0.98), with noise of force

measurement 145 Figure 9.16: The compensating force is compared with its ideal value, the noise of

force measurement and the error force in a VST with the error fore

compensation (n u =4, λ= 0 98), upper: in the time domain, lower: in the frequency domain 146 Figure 9.17: Response of the hydraulic cylinder in a virtual substructure test

without phase lag compensation 146 Figure 9.18: Comparison of different orders for the phase lag compensation on a

hydraulic cylinder, order n u= 3, 4, 5, 6, 8 and 10; λ = 0.98 147 Figure 9.19: Response of the hydraulic cylinder in virtual substructure test with

phase lag compensation with n u =6 and λ=0.98 148 Figure 9.20: Comparison of the displacement errors between uncompensated and

λ=0.98 148 Figure 9.21: Comparison of the hysteresis loops between uncompensated and

λ=0.98 149 Figure 9.22: Comparison between the displacements of the VST without

compensation and the reference solution 149 Figure 9.23: Comparison between the coupling forces of the VST without

compensation and the reference solution 149 Figure 9.24: Comparison between the displacements of the VST with the force

compensation and the reference solution 150 Figure 9.25: Comparison between the coupling forces of the VST with the force

compensation and the reference solution 150 Figure 9.26: Comparison between the displacements of the VST with the phase lag

compensation and the reference solution 151 Figure 9.27: Comparison between the coupling forces of the VST with the phase

lag compensation and the reference solution 151 Figure 9.28: Comparison between the displacements of the VST with both force

and phase lag compensations and the reference solution 152 Figure 9.29: Comparison between the coupling forces of the VST with both force

and phase lag compensations and the reference solution 152 Figure 9.30: Force comparison in VSTs with different configurations of the error

force and phase lag compensations and the reference solution 153

LIST OF TABLES

Table 6-1: Features of FIR, ARX, ARMAX, OE and B-J models 69 Table 7-1: List of parameters for stability and accuracy analyses 75 Table 7-2: Parameters for investigation on amplitude increments and period

elongations of the substructure solution 77 Table 7-3: Combination of parameters for accuracy analysis 79 Table 7-4: Combination of parameters for investigation on the effect of phase lag 80 Table 7-5: Combination of the parameters for stability analysis 80 Table 7-6: List of parameters in the error force compensation 81 Table 7-7: Parameters for validation of the substructure solution and accuracy

analysis 82 Table 7-8: Parameters for validation of stability analysis 83 Table 7-9: Virtual substructure tests for validation of the error compensations 84 Table 8-1: Parameters of the two-DOF system for verification of the substructure

solution 100 Table 8-2: Exact dynamic data of the considered two-DOF system for 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 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 of the

compensations 144 Table 9-3: Parameters of the exact direct integration for the experimental substructure 145

1 INTRODUCTION TO SUBSTRUCTURE TESTING

1.1 Principle of substructure method and substructure testing

The concept of substructure method is to divide the entire structural system into several substructures during a test (or simulation) in order to investigate the response

of the whole structure The basic concept behind substructuring can be seen in Figure 1.1 In which, the first substructure represents a part of the system that can be simulated properly, while the second substructure is the part that cannot be simulated

by using the same simulation as the first substructure

Figure 1.1: The simplest case of substructures

Depending on how the response of the second substructure is obtained, there are two different approaches of substructures The first approach is substructure simulation in which the second substructure can be simulated by using an appropriate simulating technique This approach is usually used for analysis of structures where one or more parts of the structure are nonlinear and therefore, they need to be treated separately from the first substructure The substructure simulation approach is also carried out in the context of distributed computing for simulation of very large structures The second approach is the so-called substructure test in which the response of the second substructure is tested by using an actuator The substructure test is performed when there is no possibility to model and simulate adequately the considered substructure Substructure tests may be performed in order to investigate dynamic behaviors of unknown nonlinear structures or to validate the performance of devices and/or control algorithms in integrated engineering systems

In more general cases, the substructure system may include both nonlinear simulation and experimental substructure as depicted in Figure 1.2 Because the displacements

(u c1 and u c2), which are applied on the nonlinear simulation and on the experiment, are generally different, the geometric conversion 1 is used to distribute displacement values to the two substructures On the other hand, the geometric conversion 2 is used

f l

f c

u c

First substructure

Second substruture

to combine local coupling forces (f c1 and f c2) to form the feedback coupling force in the global coordinate

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 for calculation of the next step By carrying out the process of step-by-step integration, the response of the whole structure is obtained

In order to perform a substructure test, an actuator is used to impose the computed movement u c to the experimental substructure and a load cell is used in order to measure the coupling force f c between the numerical and experimental substructures (Figure 1.3)

Figure 1.3: Substructure test using actuator for displacement control and load cell for force

measurement

Depending on how the dynamic effects in the specimens are tested, the substructure tests can be classified into two major kinds: Pseudo Dynamic (PsD) testing and dynamic testing

In PsD testing, acceleration-depending and velocity-depending forces in the specimen are simulated adequately while the stiffness-depending forces are measured The PsD testing method can be used to test nonlinear restoring forces of complete structures or substructures However, when the inertial and damping effects cannot be simulated

Geometric conversion 1

Nonlinear simulation

Experimental substructure

Numerical

substructure

Geometric conversion 2

properly, PsD test cannot be used For instance, the PsD method is not applicable for tests of concrete structures with high strain rate where velocity-depending force is in the nonlinear range When these nonlinearities, acceleration-depending, velocity-depending and displacement-depending effects in specimens are measured, the test is called dynamic testing

In terms of control, substructure tests can be classified as discontinuous and continuous tests Traditional PsD testing is controlled in discontinuous modes (stop-and-go) in which the actuators are controlled to move to a position and stop there in certain duration for positioning and measuring, then they move to the next step In continuous testing, the actuators are controlled to move continuously and smoothly by

using a control loop with relatively short time step such 2 ms or less (Magonette et al

1998, Magonette 2001) In general, discontinuous PsD tests require much more time to complete than continuous PsD tests

When considering the time scale of a test, there are two different types of substructure tests: time-scale substructure test and real-time substructure test The time-scale substructure test is a substructure test with an appropriate time scale (different from 1/1) at which the dynamic effects of coupling force can be reproduced, measured and appropriately converted to the equivalent in the real time case For example, a substructure test of linear damper maybe carried out in an extended time scale since the linear time-dependent effect can be reproduced appropriately in a time-scale test When coupling forces depending on velocity and acceleration are considered, the substructure test must be performed in a real-time fashion This type of substructure test is called real-time substructure test (RTST) In a RTST, all testing tasks including calculating the response of the numerical substructure, control of the actuators,

measuring coupling force, etc., must be done in a certain time step When a step is

finished, the next step is carried out and the physical phenomenon in specimen is reproduced in the same time scale with the considered phenomenon in the real world

1.2 History of substructure testing

PsD tests were performed as early as 1969 by Hakuno et al in Japan, according to Horiuchi et al (1996) Further research and development took place in Japan, United

States and Europe Shing and Mahin (1987), Thewalt and Mahin (1987), Takanashi

and Nakashima (1987), Roik and Dorka (1989), Dorka et al (1991, 1998), Nakashima

et al (1990), Combescure and Pegon (1997), Williams et al 1998 had contributed

important developments on general overviews of the method, formulation and practical implementations

Since the PsD method has several advantages, it is widely used in civil engineering for evaluating large structures under earthquakes PsD tests may be performed with expanded time scales to allow the use of relatively low power actuators Tests of large-scale specimens for investigation on real structures can be carried out within the limited capabilities of most large structural testing laboratories (Nakashima 1995) In addition, PsD tests with very small time scale allow detailed observation of the behavior and failures of the structure

However, the PsD method has some disadvantages One shortcoming is that it does not test the dynamic effect of a distributed mass in the specimen Another disadvantage is that PsD method cannot be used to test any time-dependent nonlinear effect because the PsD method uses quasi-static control

In dynamic situations, the dynamic effects of the distributed masses and/or the nonlinear damping in structures under dynamic loads (earthquake, wind and/or traffic loads) are important and need to be measured Therefore, another method is required for such applications RTST is a more advanced testing method that can be used to measure time-dependent nonlinear effects as well as dynamic interactions in the structural systems by testing only critical substructures In these cases, RTSTs provide more precise and reliable results than PsD tests

RTST usually requires advanced control systems because all testing tasks, including computation, control and measurement, have to be completed in a critical time step and the real-time displacement control should be highly accurate In addition, RTSTs for civil engineering applications usually require relatively high performance hydraulic cylinders and large hydraulic power These are some major reasons why applications

of RTST have just been developed for the last twenty years, about two decades after the development of the PsD method

Early RTSTs have been reported by Nakashima et al (1992) and Horiuchi et al

(1996) RTSTs of nonlinear stiffness-depending structures were performed by using simple methods such the Central Difference Method (CMD) and some other tests are conducted by using more advanced algorithms such as the Operator-Splitting (OS) or

Alpha-Operator-Splitting (α-OS) method (Nakashima et al 1990, Combescure and Pegon 1997) or the predictor-corrector methods (Bonelli et al 2002a, 2002b, Ghaboussi et al 2006, Zhang et al 2005) Many other RTSTs of vibrating

substructures have been carried out by using the substructure algorithm with sub-step

control (Roik and Dorka 1989, Dorka et al 1998, 2006, 2007, Dorka 2002, Bayer et

al 2000a, 2000b, 2002, 2005)

1.3 State of the art of real-time substructure testing

After being developed and investigated in the last decades, research on RTST has gained important achievements and the research is currently facing challenges on both numerical and experimental problems Achievements on RTST can be seen in the development of algorithms and experimentations for testing and the collaboration between structural laboratories in the world

Various integration schemes are currently used for RTST such as the CDM, OS and

α-OS methods (Nakashima et al 1990, Combescure and Pegon 1997, Tada et al 2007), predictor-corrector method (Ghaboussi et al 2006), Newmark-β implicit with sub-step control (Roik and Dorka 1989, Dorka et al 1998) The simplest and most popular

method is CDM; however, it is not suitable for testing structural systems with high range of frequencies because of its conditional stability Implicit methods such α-OS, predictor-corrector method, etc., are more accurate because these methods use highly accurate integration schemes to obtain the implicit solution of the numerical part at the next step Different techniques are used in implicit 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 1998) can be used to deal with such interaction between

the numerical and the experimental substructure within a time step

Although several algorithms have been developed, substructure research is still facing with some problems on accuracy and stability Implicit integrations as Newmark-β implicit schemes with low numerical damping effect (or no numerical damping) are usually preferred because of their high accuracies However, numerical errors in the integration processes and errors due to time lag in control system may cause instability

in tests of low damping structures In contrast, other algorithms with certain numerical damping can assure stability, however they usually induce high numerical damping in high frequency range This may cause poor accuracy in the substructure simulation at high frequencies

Experimentally, there are some problems on real-time control of hydraulic systems for RTSTs The first problem is that the response of hydraulic system is relatively good in low frequencies but poor in the high frequency range The considered range of frequencies in dynamic response of structures in civil engineering is about from 0.1 Hz

to 20 Hz On the other hand, both, the hydraulic system and the experimental structures, are usually run as nonlinear systems Thus, a traditional Proportional - Integral - Differential (PID) control cannot provide highly accurate response in the required range of frequencies Some advanced control methods such as adaptive

inverse control (Wildrow et al 1995), Minimal Synthesis Control (MCS) (Stoten et al

1990, 1994, 1998, 2001, Hodgson et al 1999, Bonnet et al 2007) and high gain adaptive control (Bobrow et al 1995) can be used to improve response of hydraulic

system in higher range of frequencies

The second issue is time lag phenomenon in hydraulic system As presented by

Horiuchi et al (2001), time lag introduces negative damping into the substructure test

system The effect of negative damping results in an increasing response of the tested structure and may cause instability in RTST if the negative damping is larger than the structural damping There are some phase lag compensations such as time delay

prediction (Horiuchi et al 2001), Darby's time delay compensation (Darby et al 2001) and model based method (Spencer at al 2007) These methods still have some

disadvantages (see section 4.2) Therefore, phase lag compensation is still a critical issue in RTST

Currently, topics relating to substructure simulation are discussed and investigated in earthquake engineering laboratories as well as in collaborations among Japan, United States, European countries, China and Taiwan An interesting topic is geographically distributed testing among laboratories in the world The major advantage of distributed testing is to enhance testing capabilities and to exchange and share experiences among laboratories In order to do this, a software framework is needed for distributing the several numerical and experimental activities of a test into different simulation systems and experimental sites and connecting them together via Intranet or Internet

(Pearlman et al 2004, Pan et al 2005, 2006, Mosqueda et al 2005, Kwon et al 2005, Yang et al 2007, Wang et al 2007) The research programs with intensive

investigation on RTST and network collaboration are hybrid simulation at NEES (the Network for Earthquake Engineering Simulation in USA (http://www.nees.org/), E-

Defense in Japan (Ohtani et al 2003), E-FAST and SRIES projects in Europe Some

distributed tests have just been performed in Japan (Pan et al 2006), in Taiwan (Yang

et al 2007) and in France (Dorka et al 2007)

For a short summary on the state of the art, RTST has been actively developed in the last decades and attained a number of achievements on the testing method as well as test facilities However, the research on RTST is still in need of further development

2 CONTROL OF SUBSTRUCTURE TEST

The main purpose of this chapter is to review the background of numerical methods and control algorithms for substructure testing From discussion on the methods, an advanced substructure control algorithm will be selected for further investigation and development in this thesis

2.1 Introduction

Equilibrium equation of the numerical substructure at time t is presented as Eq (2.1)

) ( ) ( ) ( )

( )

t f t f t u K t u C t u

where M, C and K are the mass, damping and stiffness matrices of the numerical

substructure; u (t), 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 transformation of the equation into different forms for MDOF system With a step-by-step integration procedure, the numerical solution is

obtained at discrete times 0, ∆t , 2∆t , …, i∆t , …, n∆t , where i = 1 , 2 , … , n; n is an

integer number

Numerous integration methods can be used as an integration process in substructure control There are two common approaches: explicit and implicit integrations An explicit scheme does not require an external force at the next step in order to calculate the response of the numerical substructure while an implicit one needs information of the coupling force at the next step With an implicit scheme, the response of the numerical substructure at the next step is obtained by solving the equilibrium equation

at time t+∆t

The explicit scheme is sometimes preferred because the implementation is simple and

its computational speed is fast (Blakeborough et al 2001) In contrast to this,

implementation of an implicit scheme is more complicated because the measured coupling force at the next step is not available before calculating the implicit displacement In order to drive the specimen from the current step to the next step, it is needed a special control mechanism that can deal with the interaction between the experimental and numerical substructures within a time step

There are two feedback techniques for dealing with this interaction The first one uses the analog feedback technique (Thewalt and Mahin 1987) while the second one uses

digital feedback technique (Roik and Dorka 1989, Dorka et al 1998) The analog

feedback mechanism is an ideal feedback mechanism since the equation of motion is satisfied continuously within a time step and there is no theoretical error in the feedback mechanism However, there are some difficulties in the implementation of the analog feedback for different applications In the digital feedback control, a linear control equation is used to control the displacement via a number of sub steps within a time step The digital feedback method uses an error force compensation to minimize the error force of the equilibrium equation at the end of each step

2.2 General time discretisation for integration

A general solution of the equilibrium equation of the numerical substructure can be derived using a weighted residual formulation and a finite element approach in the time domain (Zienkiewicz 1977, Dorka 2002):

1 1

*

1 1

1 1

1 1

+

+ +

+

+ +

+ +

−

−

+ +

−

−

+ +

−

−

+ +

N u N u N u K

N u N u N u C

N u

N u N u

M W

1 i i

* i i

* 1 i i

1 i i i i 1 i i

1 i dt d i i dt d i 1 i dt d i

1 i dt d i i dt d i 1 i dt d i

2 2 2

2 2

2

(2.2)

where u is the displacement vector at discrete time steps i ; W is a weighting function

(in Figure 2.2) ; ξ = /t ∆t; f*= f l + f c ; N i−1, N i and N i+1 are shape functions as Eq (2.3) (see Figure 2.1)

Figure 2.1: The shape functions over time (Zienkiewicz 1977)

(1 )/2

11

2/1

1

1

ξξ

ξξ

ξξ

N N N

(2.3)

It is worth noticing that, the same sharp functions are applied on both internal and external forces This consistency would yield to the nature of the physical phenomenon of structures

Performing the integration with various weighting functions yields all major time stepping algorithms that use three time steps:

∆+

−+

∆+

∆

−++

∆+

=

− +

−

− +

1

*

2 2

1

*

2 2

1 1

* 2

1 2 2

1

2 2

1

1 2 1

2

.1

.2

212

i

i i

i i

i

f t

f t f

t

u K t tC

M

u K t tC

M K

t tC M

u

γβ

γββ

γβγ

γβγ

β

with constants defining a particular algorithm as Eq (2.5)

Figure 2.2: Classification of integration methods in different weighting functions and integration

β

ξξ

ξγ

Wd d

W

Wd d

Linear acceleration

Fox Goodwin Average acceleration Galerkin

The formulation can be extended to integration with four time steps A four-step procedure with the appropriate selection of a weighting function was suggested by

Hilber, Hughes and Taylor (Hilber et al 1977) In the same form with Eq (2.4), the solution of the four-point scheme is given as Eq (2.6) (Dorka et al 1998)

−

−+

⋅

∆

⋅+

−+

⋅

∆

⋅+

−+

−+

⋅

∆

⋅

−+

−+

⋅+

−+

⋅

∆

⋅+

−+

⋅

∆

⋅

−++

−+

⋅

∆

⋅+

−+

−

−

− +

2

* 2 1

* 2

* 2 1

* 2

2 2

1 2

2

1 2 1

)16

116

1()

32

52

1

(

)2

322

1()

3

12

16

1

(

)16

116

1()

6

1122

1()2(

)32

52

1()

352

3()53

(

)2

322

1()

2

342

3()3

4

(

)3

12

16

1()

3

12

1()

1

(

i i

i i

i i

i i

f t f

t

f t f

t

u K t C

t M

u K t C

t M

u K t C

t M

K t C

t M

u

γβαγ

βα

γβαγ

βα

γβαγ

βγ

γβαγ

βγ

γβαγ

βγ

γβαγ

βγ

ξ

ξξα

d W

d W

ξ

ξξβ

d W

d W

d W

d W

(2.7)

With different parameter combinations, numerous integration schemes can be compared in terms of numerical accuracy in a linear Single-Degree-Of-Freedom (SDOF) system It is noted that a linear Multi-Degree-Of-Freedom (MDOF) system can be decoupled in different SDOF systems on which the numerical accuracy of integration schemes can be evaluated

To investigate numerical damping and accuracy, Zienkiewicz (1977) has studied the amplification factor which is defined as ρu as Eq (2.8) for free vibration of SDOF system The oscillation is bounded and stable if |ρu|≤1 When |ρu|=1 there is no damping and the solution is exact; when |ρu|<1 the integration induces certain numerical damping In difference with this, when |ρu|>1 occurs, the numerical solution is not stable and there is negative damping in the integration scheme

i u i

u

Applying three-point schemes on SDOF system, the amplification factor is the root of the Eq (2.9)

22

12

2 2

1

2 2

1 2

2

=

∆

−++

∆

−

−+

∆+

−+

∆

−+

−+

∆+

∆+

k t tc

m

k t tc

m k

t tc

u

γβγ

γβγ

ρβ

γρ

(2.9)

With different integration parameters, a comparison of numerical damping between different integration schemes can be seen in Figure 2.3

Figure 2.3: Numerical damping for three-step and four-step integration

methods (Zienkiewicz 1977, extended by Dorka 2002)

Figure 2.4: Period distortion in free vibration of a SDOF system for different integration

methods (Zienkiewicz 1977, Nakashima 1984, extended by Dorka 2002)

|

|ρu

With different cases of weighting factors (see Figure 2.2), the accuracy and stability features of the schemes are different By using Figure 2.3 and Figure 2.4, constants β and γ can be chosen to optimize accuracy or to satisfy stability condition

Various integration schemes have been used in substructure testing with different control mechanisms In a substructure test using an explicit integration, the displacement solution at the next step is known before being imposed to the experiment Thus, explicit integrations need no interaction mechanism to deal with the interaction between the numerical and the experimental substructures In contrast with this, the displacement solutions of implicit schemes are unknown before being applied

to the experiment and they depend on the coupling forces at the end of the next step Therefore, implicit schemes require appropriate control algorithm for dealing with the numerical-experimental interaction

Several integration schemes with different numerical features and different control aspects in substructure testing are discussed in the following

2.3 Integration schemes

2.3.1 The Central Difference Method (CDM)

The CDM is the simplest case (γ = 1 / 2, β = 0) of the general time discretisation scheme above This method is so-called Central Difference Method due to the nature

of its weighting function (see Figure 2.2) and its approximation for velocity and acceleration is given as Eq (2.10)

− +

1 1

2

1 1

2 1

2 1

i i i

i

i i i

u u u

t u

u u t u