advances in shake table control and …...fully dynamic environment with base excitation of the test...

ADVANCES IN SHAKE TABLE CONTROL AND

SUBSTRUCTURE SHAKE TABLE TESTING

by

Matthew Joseph James Stehman

A dissertation submitted to The Johns Hopkins University in conformity with the

requirements for the degree of Doctor of Philosophy.

Baltimore, Maryland

June, 2014

c© Matthew Joseph James Stehman 2014

All rights reserved

Abstract

Shake table tests provide a true representation of seismic phenomena including a

fully dynamic environment with base excitation of the test structure. For this rea-

son shake tables have become a staple in many earthquake engineering laboratories.

While shake tables are widely used for research and commercial applications, further

developments in shake table techniques will allow researchers to use shake tables in a

broader range of studies. This dissertation presents recent advances in the use of shake

tables for the seismic performance evaluation of civil engineering structures. Develop-

ments include theoretical and experimental investigations of substructure shake table

testing, a technique where the entire test structure is separated into computational

and experimental substructures. Challenges in meeting the boundary conditions be-

tween the substructures have limited the number of implementations of substructure

shake table testing to date; thus in this dissertation, appropriate methods of address-

ing the boundary conditions between experimental and computational substructures

are presented and evaluated. Also, a novel strategy for acceleration control of shake

tables is presented to enhance the acceleration tracking performance of shake tables

ii

ABSTRACT

to be used in substructure shake table testing. Results are presented that show the

promise in using the developed techniques over traditional shake table testing meth-

ods.

Primary Reader: Narutoshi Nakata

Secondary Readers: Benjamin Schafer and Judith Mitrani-Reiser

iii

Acknowledgments

I would like to sincerely thank my advisor, Professor Narutoshi Nakata, for all of

his helpful advice and mentoring during my time at Johns Hopkins. I would also like

to thank my colleagues John Hinchcliffe and Richard Erb, for helping me design and

build the test setups necessary for the experimental part of my research. I would like

to extend a special thanks to Nick Logvinovsky for always helping me find the right

tool for the job.

Finally, I would like to acknowledge the National Science Foundation for their

financial support. The research presented in this dissertation was fully supported

by the grant entitled “CAREER: Advanced Acceleration Control Methods and Sub-

structure Techniques for Shaking Table Tests (grant no. CMMI- 0954958)”.

iv

Dedication

To my family, for all the support and encouragement they have given me.

v

Contents

Abstract ii

Acknowledgments iv

List of Tables xi

List of Figures xii

1 Introduction 1

1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Shake Table Testing . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 Real-Time Hybrid Simulation . . . . . . . . . . . . . . . . . . 7

1.1.3 Substructure Shake Table Testing . . . . . . . . . . . . . . . . 10

1.2 Overview of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Shake Table Dynamics 15

2.1 Dynamics of Shake Tables with Hydraulic Actuators . . . . . . . . . . 16

vi

CONTENTS

2.2 Acceleration Relationships . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Substructure Shake Table Testing of Upper Stories in Tall Buildings 24

3.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.2 Compatibility Requirements . . . . . . . . . . . . . . . . . . . 27

3.1.3 Concept of Substructure Shake Table Testing . . . . . . . . . 29

3.2 Experimental Setup and Modeling . . . . . . . . . . . . . . . . . . . . 30

3.2.1 Uni-Axial Shake Table . . . . . . . . . . . . . . . . . . . . . . 30

3.2.2 Control and Data Acquisition System . . . . . . . . . . . . . . 31

3.2.3 Experimental Substructure . . . . . . . . . . . . . . . . . . . . 32

3.2.4 Computational Substructure . . . . . . . . . . . . . . . . . . . 33

3.2.5 Measurement of Base Shear . . . . . . . . . . . . . . . . . . . 34

3.3 Acceleration Control Performance . . . . . . . . . . . . . . . . . . . . 35

3.3.1 Issues of Acceleration Control and Control-Structure-Interaction 35

3.3.2 Propagation of Input Acceleration Errors . . . . . . . . . . . . 38

3.4 Substructure Shake Table Test System with Error Compensation . . . 39

3.4.1 Numerical Integration for the Computational Substructure . . 40

3.4.2 State Observer and Kalman Filter . . . . . . . . . . . . . . . . 42

3.4.3 Model-Based Actuator Delay Compensation . . . . . . . . . . 44

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CONTENTS

3.4.4 Corrector for Errors in Base Shear Induced by Input Accelera-

tion Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.5.1 Harmonic Ground Excitation Inputs . . . . . . . . . . . . . . 48

3.5.2 Earthquake Ground Excitation Input . . . . . . . . . . . . . . 54

3.6 Advanced Model-Based Shake Table Compensation Techniques . . . . 58

3.6.1 Feedforward Compensation using Derivatives of Reference Signal 58

3.6.2 IIR Compensation Technique for Significant Control-Structure-

Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.6.3 Experimental Investigation of Model-Based Delay Compensa-

tion Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 62


4 Substructure Shake Table Testing of Lower Stories in Tall Buildings 71

4.1 Interface Compatibility using a Controlled Mass . . . . . . . . . . . . 72

4.1.1 Simulation Models . . . . . . . . . . . . . . . . . . . . . . . . 78

4.1.2 Numerical Investigation . . . . . . . . . . . . . . . . . . . . . 81

4.2 Interface Compatibility using a Force Controlled Actuator . . . . . . 91

4.2.1 Actuator Control Scheme . . . . . . . . . . . . . . . . . . . . 91

4.2.2 Numerical Case Study . . . . . . . . . . . . . . . . . . . . . . 95


viii

CONTENTS

5 Acceleration Feedback Control of Shake Tables 102

5.1 Acceleration Feedback Control with Force Stabilization . . . . . . . . 103

5.1.1 Control Architecture . . . . . . . . . . . . . . . . . . . . . . . 103

5.1.2 Hardware Requirements . . . . . . . . . . . . . . . . . . . . . 105

5.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.3 Experimental Investigation of the Proposed Acceleration Control Strat-

egy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.3.1 Experimental Modeling of the Shake Table System . . . . . . 110

5.3.2 Design of the Feedback Controllers and Pre-Gains . . . . . . . 112

5.3.3 Experimental Validation of the Proposed Acceleration Control

Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.4 Impact of Input Acceleration Errors in Shake Table Tests on Structural

Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120


6 Conclusions and Future Work 125

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.2.1 Near Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.2.2 Long Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Appendix A 131

ix

CONTENTS

References 133

Vita 142

x

List of Tables

3.1 Dynamic properties of the entire 10-story RTHS structure . . . . . . 343.2 Summary of shake table performance from substructure shake table

testing using 3 different compensation techniques. . . . . . . . . . . . 66

4.1 Properties and dynamic characteristics of the 5-story structure. . . . 794.2 RMS differences for simulation responses under different earthquake

simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.1 Dynamic characteristics of three story test structure. . . . . . . . . . 1085.2 Analytical representations of the open loop shake table dynamics. . . 1125.3 Errors between measured and reference shake table accelerations. . . 1205.4 Errors between measured and reference top floor structural accelerations.123

xi

List of Figures

1.1 Schematic of a early stage hand powered shake table, CREDIT: Severn(2011). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Schematic of the E-Defense shake table system, CREDIT: Ogawa et al.(2001). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Early implementation of pseudo-dynamic testing, CREDIT: Nakashimaet al. (1992). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Implementation of RTHS where the experimental substructure is asingle damper, CREDIT: Carrion et al. (2009). . . . . . . . . . . . . . 9

1.5 Concept of substructure shake table testing including a tuned massdamper, CREDIT: Igarashi et al. (2000). . . . . . . . . . . . . . . . . 11

2.1 Schematic of a uni-axial shake table with linear structure. . . . . . . 162.2 Block diagram of shake table system including hydraulic actuator, test

structure and feedback controller. . . . . . . . . . . . . . . . . . . . . 19

3.1 Schematics of substructure shake table testing in comparison with theentire simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 A block diagram for the concept of substructure shake table testing. 303.3 A three-story steel frame structure on the uni-axial shake table at

Johns Hopkins University. . . . . . . . . . . . . . . . . . . . . . . . . 313.4 Frequency response curves of the three-story steel experimental sub-

structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.5 Frequency response curves of closed-loop (reference to measured) dis-

placement and acceleration: (a) displacement magnitude; (b) acceler-ation magnitude; (c) displacement phase; and (d) acceleration phase. 37

3.6 Frequency response curve and coherence from the table acceleration tomeasured base shear. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.7 A block diagram of the substructure shake table test system with com-pensation techniques for experimental errors. . . . . . . . . . . . . . 40

xii

LIST OF FIGURES

3.8 Acceleration and base shear time histories under 2.0 Hz harmonicground excitation: (a) the entire acceleration time histories; (b) azoomed section of the acceleration time histories; (c) the entire baseshear time histories; and (d) a zoomed section of the base shear timehistories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.9 Structural responses under 2.0 Hz harmonic ground excitation: (a),(d), and (g), relative floor displacement at the 10th, 6th and 2nd floor,respectively; (b), (e), and (h), absolute floor displacement at the 10th,6th and 2nd floor, respectively; and (c), (f), and (i), absolute flooracceleration at the 10th, 6th and 2nd floor, respectively. . . . . . . . . 51

3.10 Acceleration and base shear time histories under 6.0 Hz harmonicground excitation: (a) the entire acceleration time histories; (b) azoomed section of the acceleration time histories; (c) the entire baseshear time histories; and (d) a zoomed section of the base shear timehistories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.11 Structural responses under 6.0 Hz harmonic ground excitation: (a),(d), and (g), relative floor displacement at the 10th, 6th and 2nd floor,respectively; (b), (e), and (h), absolute floor displacement at the 10th,6th and 2nd floor, respectively; and (c), (f), and (i), absolute flooracceleration at the 10th, 6th and 2nd floor, respectively. . . . . . . . . 54

3.12 Acceleration and base shear time histories under the 1995 Kobe earth-quake excitation: (a) the entire acceleration time histories; (b) a zoomedsection of the acceleration time histories; (c) the entire base shear timehistories; and (d) a zoomed section of the base shear time histories. . 55

3.13 Structural responses under the 1995 Kobe earthquake excitation: (a),(d), (g), (j), and (m), relative displacement at the even floors from topto bottom (10th to 2nd); (b), (e), (h), (k), and (n), absolute displace-ment at the even floors from top to bottom (10th to 2nd); and (c), (f),(i), (l), and (o), absolute displacement at the even floors from top tobottom (10th to 2nd). . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.14 Experimental reference to measured frequency response functions forshake table with 3DOF experimental substructure: (a) and (c) dis-placement tracking magnitude and phase; (b) and (d) accelerationtracking magnitude and phase. . . . . . . . . . . . . . . . . . . . . . . 63

3.15 Absolute shake table response from substructure shake table tests with3DOF experimental substructure subjected to Kobe earthquake record:(a) and (c) full displacement and acceleration time histories; (b) and(d) zoomed-in views of displacement and acceleration time histories. . 65

xiii

LIST OF FIGURES

3.16 Shake table tracking errors from substructure shake table tests with3DOF experimental substructure subjected to Kobe earthquake record:(a) and (d) displacement and acceleration errors using feedforwardcompensator; (b) and (e) displacement and acceleration errors usingIIR compensator; (c) and (f) displacement and acceleration errors us-ing iterative compensator. . . . . . . . . . . . . . . . . . . . . . . . . 68

4.1 Schematics of the Entire and substructure systems. . . . . . . . . . . 734.2 A block diagram of the substructure shake table test using controlled

masses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.3 Closed-loop frequency response function of the 911-D actuator with a

mass of 45kg: (a) magnitude; (b) phase. . . . . . . . . . . . . . . . . 804.4 Displacement comparison for controlled mass system in a simulation

using Kobe ground motion. . . . . . . . . . . . . . . . . . . . . . . . 834.5 Comparison of force achieved by the controlled mass system and target

computational base shear in a simulation using the Kobe ground motion. 844.6 Comparison of top floor structural responses: (a) accelerations; (b)

velocities; (c) displacements in a simulation using the Kobe groundmotion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.7 Comparison of top floor acceleration frequency response functions forsubstructured systems with no filtering and ideal actuator model, fil-tering and ideal actuator model, filtering and realistic actuator modelto entire structure: (a) magnitude and (b) phase. . . . . . . . . . . . 89

4.8 Comparison of top floor acceleration frequency response functions forsubstructured systems with separation floor 1 and separation floor 2to entire structure: (a) magnitude and (b) phase. . . . . . . . . . . . 90

4.9 Experimental substructure using a force controlled actuator to applythe computational base shear. . . . . . . . . . . . . . . . . . . . . . . 92

4.10 Block diagram of substructure shake table test method including ac-tuator control system. . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.11 Performance of experimental setup during step input tests: a.) shaketable displacement; b.) force from second actuator. . . . . . . . . . . 96

4.12 Shake table acceleration during Kobe simulation: a.) entire record; b.)zoomed-in view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.13 Experimental force during Kobe simulation: a.) entire record; b.)zoomed-in view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.14 4th floor absolute acceleration during Kobe simulations: a.) time his-tories; b.) Fourier Transform of time histories. . . . . . . . . . . . . . 100

5.1 Block diagram of proposed acceleration control strategy. . . . . . . . 1045.2 Schematic of a uni-axial shake table setup for the proposed acceleration

control strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

xiv

LIST OF FIGURES

5.3 The shake table in the SSHT lab at Johns Hopkins: (a) shake tablewith three-story structure; (b) view of restoring springs. . . . . . . . . 109

5.4 Open loop dynamics for shake table system: (a) valve command tomeasured acceleration; (b) valve command to measured force. . . . . 111

5.5 Controller design for the proposed acceleration control strategy: (a)acceleration feedback controller; (b) acceleration closed-loop frequencyresponse function; (c) force feedback controller; (d) force closed-loopfrequency response function. . . . . . . . . . . . . . . . . . . . . . . . 113

5.6 Closed-loop frequency response functions of shake table system usingthe proposed acceleration control strategy: (a) reference accelerationto measured force magnitude; (b) reference acceleration to measuredacceleration magnitude. . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.7 Results with the 1995 Kobe ground motion as the reference acceler-ation: (a) shake table acceleration tracking comparison; (b) close upview of table accelerations; (c) frequency domain comparison of tableaccelerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.8 Measured table force from the acceleration control strategy using theKobe reference acceleration. . . . . . . . . . . . . . . . . . . . . . . . 118

5.9 Comparison of measured table displacements with the Kobe referenceacceleration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.10 Comparison of structural responses with the 1995 Kobe ground motion:(a) top floor structural acceleration comparison; (b) close up view ofstructural accelerations; (c) frequency domain comparison of top floorstructural accelerations. . . . . . . . . . . . . . . . . . . . . . . . . . 122

xv

Chapter 1

Introduction

Each year earthquakes cause a significant amount of structural damage, economic

turmoil and ultimately human causalities. The United States Geological Survey es-

timates that every year several million earthquakes occur world wide, with only a

small percentage that are actually measured, USGS (2011). Even though it is nearly

impossible to predict when and where an earthquake will strike, society depends on

civil engineering structures to keep us safe during such extreme events.

While a large percentage of recorded earthquakes have small magnitudes and cause

mostly superficial damage, the small number of quakes that are more powerful tend

to be detrimental. It is under these extreme circumstances that our structures are

pushed to the limits of their design where their response may not be well known.

Although many advanced computational techniques have been developed to reduce

the uncertainty in analyzing structures under extreme loading conditions, they tend

1

CHAPTER 1. INTRODUCTION

to be computationally demanding and capable of capturing only general responses.

To this extent a large research effort has be placed on the experimental evaluation

of civil engineering structures subjected to earthquake environments. With the recent

onset of technological advances, experimental tests have become more sophisticated

and thus allow for a more realistic representation of earthquake phenomena. The

most significant breakthrough was the development of the shake table, Severn (2011),

which tests the structure through base excitation. While shake table testing is a

preferred experimental technique recent advances continue to make it more appealing

to earthquake engineering researchers. This dissertation presents recent efforts to

advance the current state of shake tables and furthermore improve the data that is

generated from such tests.

1.1 Literature Review

This section presents a brief literature review for academic and commercial re-

search related to shake table testing. Included herein are previous efforts related to

shake table control, use of shake tables, Real-Time Hybrid Simulation (RTHS) and

substructure shake table testing.

2


1.1.1 Shake Table Testing

Shake table testing has gained a lot of attention from the experimental earthquake

engineering community since the development of the first hand-powered shake table in

the late 19th century Severn (2011), see Figure 1.1. A modern shake table consists of

hydraulic actuators that drive a large platform to which a structure can be attached,

Figure 1.2. The inherent true-rate dynamic nature of shake table tests, which include

all inertial effects of the test structure, is ideal for seismic performance evaluation of

structures. Shake tables have become a staple in many of the earthquake engineer-

ing laboratories (Conte et al., 2004; Ogawa et al., 2001; Reinhorn et al., 2004) and

results from shake table tests continue to serve for the improvement of the design

specifications on today’s new and existing buildings (Deierlein et al., 2011; Elwood

and Moehle, 2003; Filiatrault et al., 2001; Jacobsen, 1930).

While shake table testing is now accepted as a structural testing technique, re-

cent developments are allowing researchers to explore new applications of shake table

testing. However, shake table testing presents some unique challenges which need

to be overcome to allow shake table testing to reach its full potential. Some of the

challenges are due to practical matters and economical constraints. For example,

although full-scale shake table tests with multi-directional loading might be desir-

able, such tests are rarely possible because of limited access to experimental facilities

and lack of funds. Such practical and economical challenges are often beyond the

engineer’s control. However, some of the other challenges in shake table testing are

3


Figure 1.1: Schematic of a early stage hand powered shake table, CREDIT: Severn(2011).

Figure 1.2: Schematic of the E-Defense shake table system, CREDIT: Ogawa et al.(2001).

4


technical issues that should be addressed by engineers.

Technical challenges in shake table testing range from actuator control to efficient

post-processing of test results: reproduction of desired table accelerations; shake ta-

ble nonlinearities; control of multiple actuators for multi-dimensional loading and

asynchronous ground motions; instrumentations to capture structural responses of

interest; boundary conditions such as soil-structure interaction and interaction with

surrounding members that are omitted in shake table testing; difficulties to incor-

porate substructure techniques; extrapolation of scale effects; interpretation, impli-

cation, and generalization of test results; and so on. While all of these challenges

require attention and consideration, accurate acceleration control of shake tables is

the focus of the work presented in this dissertation.

The challenge in acceleration control of shake tables is mainly due to the inherent

dynamics of the control system (i.e., servo hydraulic actuators) and its interaction

with the test structures, often referred to as control-structure interaction (Dyke et al.,

1995). In general, the hydraulic actuators used in shake tables are displacement-

controlled with proportional-integral-differential (PID) controllers; where reference

displacements are determined a priori by integrating the acceleration time history and

removing drifting components. In some cases, velocity and acceleration feedback are

added to the displacement controller (e.g., three-variable controller by MTS (Nowak

et al., 2000; Tagawa and Kajiwara, 2007)). Iterative approaches are often incorpo-

rated in commercial shake table controllers to compensate for the dynamics of the

5


control system by modifying the reference displacement outside the servo control loops

(Spencer and Yang, 1998). Displacement control in shake tables provides reasonable

performance in the low frequency range if evaluated in the frequency domain (e.g.,

performance spectrum often used in manufacturer specifications). However, displace-

ment control generally produces poor acceleration tracking in the time domain and

does not provide adequate repeatability in generated accelerations (Nakata, 2010).

Efforts to improve acceleration control accuracy of shake tables over the conven-

tional displacement control strategies can be found in literature. Stoten and Gomez

(2001) proposed adaptive control using the minimal control synthesis (MCS) algo-

rithm for shake tables. The MCS algorithm improves the control of shake tables when

using the first order strategy of displacement control in the low-medium frequency

range. Kuehn et al. (1999) developed a feedback/feed-forward control strategy based

on receding horizon control (RHC). Experimental results showed that the RHC based

controller had better phase characteristics in the acceleration transfer function than

a feedback control using the linear quadratic regulator control. Other approaches

adopt feed-forward/feedback control using additional reference signals (Nowak et al.,

2000; Phillips et al., 2013; Tagawa and Kajiwara, 2007). Some studies utilize ad-

vanced feedback controllers for improvement of shake table acceleration control with

heavy test structures, Stehman and Nakata (2013). Trombetti and Conte (2002)

employed a method that combines displacement feedback, velocity feed-forward and

differential pressure feedback control in a test-analysis comparison study. Their sen-

6


sitivity analysis showed the effectiveness of the velocity feed-forward term in the

magnitude characteristics of the displacement transfer function. Nakata (2010) de-

veloped an acceleration trajectory tracking control strategy that combines accelera-

tion feed-forward, displacement feedback, command shaping, and a Kalman filter for

the displacement measurement. Experimental results showed superior performance

and repeatability of the acceleration trajectory tracking control over the conventional

displacement control. While those methods improve control accuracy of shake tables

to some extent, acceleration tracking in a wide range of frequencies, particularly in

high frequency range, is still a challenging problem.

1.1.2 Real-Time Hybrid Simulation

While shake table testing is the preferred experimental testing technique, size con-

straints typically limit researchers to the use of scaled experiments. Real-time hybrid

simulation (RTHS) is a promising new experimental technique that enables systems-

level performance assessment of structures at the true dynamic loading rate. In RTHS,

responses of entire structural systems are simulated combining computational mod-

els (computational substructures) and physical tests (experimental substructures); in

general, only structural members of which responses are difficult to model are exper-

imentally tested. It was expanded upon from pseudo-dynamic testing incorporating

real-time computational processes and an experimental substructure using hydraulic

actuators to apply boundary forces, (Mahin and Shing, 1985; Nakashima et al., 1992),

7


see Figure 1.3. Pseudo-dynamic testing allows for testing of restoring members and

does not include inertial components in the experiment, thus tests are run at slow

speed and the system dynamics are simulated. RTHS offers a cost-effective means for

performance assessment of entire structural systems with fully incorporated physical

tests of structural members. In particular, RTHS is advantageous for simulations of

systems with rate-dependent structural members that are not accurately evaluated by

conventional slow-speed pseudo-dynamic testing. Those structural members include

dampers (Carrion et al., 2009; Christenson et al., 2008; Zapateiro et al., 2010), Fig-

ure 1.4 depicts a traditional RTHS partitioning scheme, and bearings (Igarashi et al.,

2009; Pan et al., 2005). Although the advantages of RTHS have been recognized by

many earthquake engineers, research efforts on RTHS are still limited to date. Chal-

lenges in RTHS that have yet be addressed by the community include but are not

limited to: tests where the experimental substructure has a significant influence on

the computational substructure and tests where experimental substructures include

all inertial components. Further advances in methodologies and more applications

are needed to promote this emerging experimental technique.

One of the main challenges in RTHS is the ability of the experimental loading

system (typically hydraulic actuators) to impose the boundary conditions accurately.

Since both experimental and computational substructures form a closed loop, any

errors introduced by the experimental loading system are propagated through the

system. Depending on the system configuration and parameters, the added energy

8


Figure 1.3: Early implementation of pseudo-dynamic testing, CREDIT: Nakashimaet al. (1992).

Figure 1.4: Implementation of RTHS where the experimental substructure is a singledamper, CREDIT: Carrion et al. (2009).

9


from such errors may not be dissipated fast enough resulting in an unstable simula-

tion. Such challenges typically limit RTHS to applications where the experimental

substructure is simple and includes significant damping. Recent research has made

steps toward RTHS implementations with more challenging experimental conditions.

Some of the most notable advancements in this area include the use of shake tables

in RTHS implementations.

1.1.3 Substructure Shake Table Testing

One of the more recent and promising advancements in the use of shake tables

is the incorporation of shake tables into real-time hybrid simulation, known as sub-

structure shake table testing (some refer to it as real-time substructured systems,

Neild et al. (2005), or hybrid shake table testing, Schellenberg et al. (2013)). Sub-

structure shake table testing combines the fully dynamic nature of shake table testing

with the effective computational modeling in hybrid simulation. Substructure shake

table testing in an active research area because most shake tables can be upgraded

to enable RTHS, and a number of researchers have investigated substructure shake

table testing to date. Igarashi et al. (2000) introduced a substructure shake table

test method that allows for tuned mass dampers to be tested on the top floor of

a computational substructure represented by a shake table, Figure 1.5. Lee et al.

(2007) introduced a method where the shake table included the upper stories of the

test structure while the lower stories were computational. Shao et al. (2011) and

10


Figure 1.5: Concept of substructure shake table testing including a tuned massdamper, CREDIT: Igarashi et al. (2000).

Nakata and Stehman (2012) investigated substructure shake table testing where the

experimental substructure and shake table represented the lower floors of the test

structure. Recently Mosalam and Gunay (2013) have used substructure shake table

testing to examine the performance of large-scale electrical equipment. Nakata and

Stehman (2014a) introduced an array of compensation techniques that can be added

to existing substructure shake table systems to increase accuracy and ensure stability

of the tests.

Like any RTHS, substructure shake table testing is very sensitive to time delays

11


introduced by hydraulic actuators. Actuator delay compensation has been a major

research topic for the stabilization and improvement of RTHS methods. Many new

delay compensation techniques have been developed and implemented since the topic

was first addressed (Horiuchi et al., 1999; Horiuchi and Konno, 2001; Nakashima

and Masaoka, 1999). Many of the delay compensation techniques use models of the

test structure or actuator dynamics. Carrion and Spencer (2007) introduced two

model-based delay compensation techniques: the first uses computational models of

the RTHS substructures to predict future actuator commands from iterative simula-

tion. The second technique uses an inverse model of the actuator to compensate the

closed loop actuator dynamics (including magnitude roll off and phase/time delay).

Chen and Ricles (2009) have completed comparative studies to analyze the perfor-

mance of popular delay compensation techniques. Gao et al. (2013) proposed an

H∞ approach for actuator control and delay compensation that handles uncertainties

in the experimental system. Phillips and Spencer (2011) expanded on Carrion and

Spencer’s second method by realizing the actuator inverse model as a weighted series

of derivatives of the reference signal. While delay compensation techniques have been

widely studied for RTHS applications, most implementations have been limited to

experimental substructures with little to no inertial effects (i.e.: single dampers and

structures with negligible mass). Substructure shake table testing often includes test

structures with substantial floor mass and delay compensation techniques need to be

studied for such cases with significant inertial components.

12


The presence of inertial components (e.g., mass of experimental substructures) as

well as restoring members affects the dynamics of the actuator (known as Control-

Structure-Interaction (CSI), Dyke et al. (1995)). Several studies in shake table control

have addressed the adverse effects of CSI and actuator dynamics. While approaches

that can compensate for CSI are available for shake table control, their primary

objective is the improvement of the magnitude characteristics. Compensation tech-

niques for shake tables that focus on phase characteristics (equivalently time delay),

are necessary to fully enable substructure shake table testing due to the challenging

experimental conditions.

1.2 Overview of Dissertation

This dissertation contains a collection of works that focus on advanced techniques

to enhance the capabilities of shake tables. The works presented in this dissertation

are either previously published or under peer review (Nakata and Stehman, 2012,

2014a,c; Stehman and Nakata, 2013, 2014).

Chapter 2 provides an introduction to shake table dynamics and traditional con-

trol using displacement feedback. The existence of control-structure-interaction (CSI)

is derived using theoretical relations for the dynamics of the hydraulic actuator, shake

table platform and test structure. Chapter 3 discusses substructure shake table test-

ing of upper stories in tall buildings using a shake table. Compensation techniques

13


are developed to enable the use of substructure shake table testing in the presence of

errors in the the experimental setup. Chapter 4 introduces substructure shake table

testing of the lower floors in a building using a shake table. The boundary forces in

the experimental substructure are applied in two fashions: inertial masses and force

controlled actuators. Chapter 5 introduces a novel control algorithm to improve the

acceleration tracking performance of shake tables over traditional methods. Unlike

traditional techniques this method does not require displacement information which

typically degrades the acceleration performance. Chapter 6 will conclude the disser-

tation with general remarks, technical conclusions and openings for future research.

14

Chapter 2

Shake Table Dynamics

The work presented throughout this dissertation relies on the use of shake tables.

This chapter provides a basic introduction to the dynamic relations for shake tables

including electro-hydraulic actuators, table platform, test structure and standard

feedback control system. The purpose of this chapter is to familiarize the reader with

the dynamics of a shake table system as well as the challenges that arise due to said

dynamics. The contents of this Chapter were previously published in Stehman and

Nakata (2014).

15

CHAPTER 2. SHAKE TABLE DYNAMICS

mn

cn kn

m1

c1 k1

x1

xn

xtft mt

Figure 2.1: Schematic of a uni-axial shake table with linear structure.

2.1 Dynamics of Shake Tables with Hy-

draulic Actuators

Many research topics, from traditional shake table testing to substructure shake

table testing, rely on accurate acceleration tracking of shake tables to generate ac-

ceptable test results. In order to understand the challenges and limitations of shake

table reference tracking, shake table dynamics are discussed. This section derives the

equations of motion and dynamic relationships for a shake table system including a

hydraulic actuator and a multi-degrees of freedom, MDOF, test structure. This study

uses the linearized dynamics for both the test structure and hydraulic actuator. A

schematic of a uni-axial shake table supporting a linear n-story shear type building

is shown in Figure 2.1.

16


In Figure 2.2, mi, ci and ki are the i-th floor mass, story damping coefficient and

story stiffness of the test structure; xi is the i-th floor absolute displacement; mt is

the mass of the table platform; xt is the absolute displacement of the platform and ft

is the force applied by the hydraulic actuator. Using Newton’s second law of motion,

the equations of motion for the shake table platform and test structure are written

as:

mt 01×n

0n×1 M

xt

xs

+

c1 cT

c C

xt

xs

+

k1 kT

k K

xt

xs

=

ft

0n×1

(2.1)

where M, C and K are the mass, damping and stiffness matrices of the test structure;

the over dots represent time differentiation; xs is the vector of structural displacements

(i.e.: xs = [x1 . . . xn]T ); c and k are the damping and stiffness coupling terms, defined

as:

c =

−c1

0n×1

and k =

−k1

0n×1

(2.2)

The Laplace Transform of Equation 2.1 yields the system transfer functions from

the shake table and structural displacements to the external forces (i.e. actuator

force), and these relationships are written as:

mt 01×n

0n×1 M

s2 +

c1 cT

c C

s+

k1 kT

k K

Xt(s)

Xs(s)

=

Ft(s)

0n×1

(2.3)

17


Where s is the Laplace variable. Equation 2.3 can be partitioned into a single transfer

function matrix acting on the system displacements:

H ttfx(s) Hts

fx(s)

Hstfx(s) Hss

fx(s)

Xt(s)

Xs(s)

=

Ft(s)

0n×1

(2.4)

Where H ttfx(s) is the transfer function of the table platform; Hts

fx(s) and Hstfx(s) are

the coupling transfer function matrices (such that: Htsfx(s) = (Hst

fx(s))T ) and Hssfx(s)

is the structural transfer function matrix.

Using Equation 2.4, the transfer function that directly relates the shake table

displacement to the actuator force is obtained through static condensation as:

Ft(s) =(H tt

fx(s) − Htsfx(s)

(Hss

fx(s))−1

Hstfx(s)

)Xt(s) = Hftxt(s)Xt(s) (2.5)

Equation 2.5 represents the system dynamics, incorporating the dynamics of the table

platform as well as the test structure. The relation that gives the resulting shake table

displacement from an input actuator force is obtained from the inverse of Equation

2.5 as:

Xt(s) =1

Hftxt(s)Ft(s) = Hxtft(s)Ft(s) (2.6)

The previous equations addressed the dynamic relationships for the shake table plat-

form with test structure. The relationships show that the force-displacement transfer

function for the table, Hxtft(s), is influenced not only by the properties of the table

18


Hxt ft(s)

Shake Table Actuator

Xt (s) Xm(s)Ft (s)Q(s)U (s)E(s)Xr (s) C(s) Hqv (s)Shake Table Controller

Table Platform & Structure

LVDT

1/ ke

skl

sA

Sx (s)

Sa (s)Am(s)At (s)

Accelerometer W (s)

N2(s)

N1(s)

s2

V (s)

Servo Valve

Figure 2.2: Block diagram of shake table system including hydraulic actuator, teststructure and feedback controller.

platform but also the dynamics of the test structure.

The complete dynamics of the shake table system can be obtained by incorporating

a model for the hydraulic actuator that applies the force to the shake table platform.

Figure 2.2 shows the block diagram of the shake table system including actuator

dynamics, servo valve, test structure, and feedback controller. In this study, the shake

table is assumed to be operating in displacement feedback control with a conventional

proportional-integral-derivative (PID) feedback controller, whose transfer function is

denoted here as C(s).

In Figure 2.2, Xr(s) is the reference displacement for the shake table, Xm(s)

is the measured shake table displacement; E(s) = Xr(s) − Xm(s) is the tracking

error for the shake table; C(s) is the shake table PID controller; U(s) is the control

command sent to the actuator servo valve; V (s) is the actual command the servo

valve receives after being corrupted by the process noise, W (s); Hqv(s) is the transfer

function of the servo valve from actual command to oil flow through the actuator

19


chambers, Q(s); ke is the force-flow coefficient; kl is a system constant defined by

the properties of the hydraulic fluid as well as the volume and cross-sectional area of

the actuator chambers, A; Xt(s) is the true displacement of the shake table; N1(s)

represents the displacement measurement noise; Sx(s) is the transfer function of the

displacement transducer (LVDT); At(s) is the true acceleration of the shake table;

N2(s) represents the acceleration measurement noise; Sa(s) is the transfer function

for the shake table accelerometer and Am(s) is the measured shake table acceleration.

Further descriptions of actuator dynamics can be found in (Dyke et al., 1995; Merritt,

1967; Nakata, 2012).

Analysis of the block diagram in Figure 2.2 yields the overall dynamics of the shake

table system including actuator and structural dynamics. The transfer function from

the actual servo valve command, V (s), to shake table displacement, Xt(s), is obtained

through a block diagram reduction as:

Xt(s) =Hxtft(s)

ke + (kl + AHxtft(s)) sHqv(s)V (s) = Hxtv(s)V (s) (2.7)

The appearance of Hxtft(s) in the actuator transfer function was identified by

Dyke et al. (1995) as control-structure-interaction, CSI. Because Hxtft(s) appears in

both the numerator and denominator of Equation 2.7, the transfer function is likely

to have near pole-zero cancelations depending on the specific values of ke, kl and A.

This type of CSI is often seen in shake table testing (Conte and Trombetti, 2000;

20


Phillips et al., 2013; Stehman and Nakata, 2013).

Next the overall relationship between the reference and measured shake table

displacements are determined. Figure 2.2 indicates the measured displacement is not

only influenced by the reference displacement but also the process noise, W (s), and

the displacement measurement noise, N1(s), which are always present in shake table

testing. Including these effects, the measured shake table displacement is represented

as:

Xm(s) =Sx(s)Hxtv(s)C(s)

1+Sx(s)Hxtv(s)C(s)Xr(s) +

Sx(s)Hxtv(s)

1+Sx(s)Hxtv(s)C(s)W (s) + Sx(s)

1+Sx(s)Hxtv(s)C(s)N1(s)

(2.8)

Equations 2.7 and 2.8 indicate that the closed loop displacement tracking performance

of the shake table is influenced by the dynamics of the structure, through CSI, as

well as actuator dynamics including adverse effects by the existence of process and

measurement noises.

2.2 Acceleration Relationships

In addition to displacement tracking, a very important performance requirement

in shake table testing is acceleration tracking performance of the shake table. Tradi-

tionally the reference displacement, Xr(s), is obtained from a conversion of the true

21


reference acceleration, Ar(s), where the relationship between the two is:

Xr(s) = T (s)Ar(s) (2.9)

T (s) is the transfer function from reference acceleration to reference displacement,

T (s) includes double integration often accompanied by a high pass filter to eliminate

displacement drift. With this relationship, the measured acceleration of the shake

table can be expressed as:

Am(s) =s2Sa(s)Hxtv(s)C(s)T (s)

1 + Sx(s)Hxtv(s)C(s)Ar(s) +

s2Sa(s)Hxtv(s)

1 + Sx(s)Hxtv(s)C(s)W (s)

− s2Sa(s)Sx(s)Hxtv(s)C(s)

1 + Sx(s)Hxtv(s)C(s)N1(s) + Sa(s)N2(s) (2.10)

Equation 2.10 indicates that the measured shake table acceleration is not only influ-

enced by the reference table acceleration but also by the process, displacement and

acceleration measurement noises. The middle two terms in Equation 2.10 demon-

strate the complex interactions between the various noises and the measured shake

table acceleration. The acceleration relationships indicate that displacement feedback

control of shake tables can lead to undesirable measured acceleration responses of the

table.

The results ultimately indicate that displacement feedback control does not guar-

antee accurate acceleration tracking of the table in the presence of noise. However

22


most laboratories use displacement feedback control of shake tables because many

other factors outweigh the downside of performance limitations. Therefore in this

dissertation, displacement control of shake tables is used for substructure shake ta-

ble testing Chapters 3 and 4. A novel acceleration control strategy for shake tables

is developed in Chapter 5 that enhances shake table performance over traditional

displacement control strategies.

2.3 Summary and Discussion

This chapter derived theoretical relations for a uni-axial shake table with a multi-

degrees of freedom test structure. The equations included the dynamics of the hy-

draulic actuator, feedback control system, table platform and test structure. It was

shown that the dynamics of the test structure can influence the performance of the hy-

draulic actuator (control-structure-interaction). The relationships also revealed that

accurate acceleration tracking performance of shake tables is not guaranteed when

using displacement feedback control, due to the influence of various types of noise

in the system. The complex acceleration dynamics can limit displacement-controlled

shake tables and thus make substructure shake table testing difficult with existing

technologies.

23

Chapter 3

Substructure Shake Table Testing

of Upper Stories in Tall Buildings

Some researchers are interested in experimentally testing the response of non-

structural components in tall buildings during earthquakes. Such studies may wish

to test responses of furnishings, (Ji et al., 2009; Shi et al., 2014), human response

to vibration or the effectiveness of energy dissipation devices, Igarashi et al. (2000).

Researchers in this area are typically interested in the response of the upper stories in

buildings subjected to earthquakes since large accelerations are typically experienced

at the top of tall structures.. This chapter presents techniques that enable shake

table testing of the upper stories of a building during an earthquake. The contents

of the Chapter were previously published in Nakata and Stehman (2014a); Stehman

and Nakata (2014).

24

CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS

3.1 Formulation

In this study, we consider RTHS where the lower part of the structure is compu-

tationally simulated while the upper part of the structure is experimentally tested on

a shake table. Just for convenience of naming, we simply refer to RTHS using shake

tables as substructure shake table testing in the rest of the study. To derive com-

patibility requirements, this section presents the underlying dynamics of substructure

shake table testing.

3.1.1 Equations of Motion

Figure 3.1 shows two schematics of a multistory building subjected to earthquake

ground motions: (a) entire system and (b) substructure system. The entire system is

an n-story shear-type building where the dynamic response is viewed as a reference

for the substructure system. The equations of motion for each floor of the entire

system can be expressed as:

mixi +Ri(di, di) −Ri+1(di+1, di+1) = −mixg(i = 1, . . . , n and Rn+1 = 0) (3.1)

where mi is the mass of the i-th floor; xi is the i-th floor relative displacement

with respect to the ground; di is the i-th floor story drift that can be expressed as

xi − xi−1; Ri is the i-th floor restoring force including damping; and xg is the ground

acceleration. Note that Ri is the function of the relative story velocity, xi − xi−1 ,

25


m1

mi

mn

R1

Ri

Rn

xg

x1

xi

mi+1 Ri+1xi+1

xn

mc_1

mc_ nc

Rc_1

Rc_ nc

xg _ c

xc_1

xc_ nc

fc_ nc

me_ ne Re_ ne

xg _ e

me_1 Re_1xe_1

xe_ ne

(a) Entire Simulation (b) Substructure Shake Table Hybrid Simulation

Computational Substructure

Experimental Substructure

Shake Table

Figure 3.1: Schematics of substructure shake table testing in comparison with theentire simulation.

and the story drift, xi − xi−1. In the entire simulation, the ground acceleration is the

only input to the dynamic system.

The entire structure is divided into two structures in the substructure system: nc-

story computational substructure and ne-story experimental substructure that repre-

sent the lower and upper parts of the entire system, respectively (n = nc + ne). The

equations of motion of the substructure system can be expressed as:

mc ixc i +Rc i(dc i, dc i)−Rc i+1(dc i+1, dc i+1) = −mc ixg c(i = 1, . . . , nc − 1) (3.2)

mc ixc i +Rc i(dc i, dc i)− fc nc = −mc ixg c(i = nc) (3.3)

26


me ixe i +Re i(de i, de i) −Re i+1(de i+1, de i+1) = −me ixg e(i = 1, . . . , ne and Re ne+1 = 0)

(3.4)

where mc i and me i are the i-th floor mass of the computational and experimental

substructures, respectively; xc i and xe i are the i-th floor relative displacement of the

computational and experimental substructures, respectively; dc i and de i are the i-th

floor story drift of the computational and experimental substructures, respectively;

Rc i and Re i are the i-th floor nonlinear restoring forces of the computational and

experimental substructures, respectively; and xg c and xg e are the ground acceler-

ation of the computational and experimental substructures, respectively; and fc nc

is the interaction force from the experimental substructure at the nc-th floor of the

computational substructure. In the substructure system, Equation 3.2 is solved solely

computationally while Equation 3.3 contains both computational and experimental

components. Equation 3.4 should be experimentally evaluated using a shake table.

3.1.2 Compatibility Requirements

For the substructure system to have the equivalent dynamics as the entire sys-

tem, model assumptions have to be clarified and compatibility conditions have to

be identified. The first requirement is that the model properties in the entire and

substructure systems are identical; that is, mc i = mi and Rc i = Ri for i = 1, . . . , nc,

and me i = mi and Re i = Ri for i = 1, . . . , ne. With the above model assumptions,

the remaining conditions that have to be satisfied are input compatibility conditions.

27


First, the input ground acceleration to the computational substructure has to be the

same as the one to the entire system (computational acceleration compatibility), that

is,

xg c = xg (3.5)

The computational acceleration compatibility is straightforward since the input ground

acceleration in the computational substructure is known in advance and can be di-

rectly incorporated in the computational simulation.

Second, the input acceleration to the experimental substructure has to be the

absolute acceleration at the top floor of the computational substructure (experimental

acceleration compatibility), that is,

xg e = xc nc + xg c (3.6)

The experimental acceleration compatibility implies that the reference ground accel-

eration to the shake table is not known in advance and has to be accurately imposed

in the experimental process.

Finally, the interaction force at the top floor of the computational substructure

has to be equal to the base shear in the experimental substructure (interface force

compatibility), that is,

fc nc = Re 1(de 1, de 1) (3.7)

28


The interface force compatibility requires accurate measurement or estimation of the

base shear in the experimental substructure during shake table tests. All of these

compatibility conditions (Equations 3.5-3.7) have to be satisfied at any given instance

during the simulation.

3.1.3 Concept of Substructure Shake Table Test-

ing

A block diagram of the concept for substructure shake table testing is shown in

Figure 3.2. The entire process of substructure shake table testing can be described by

two blocks with input-output relations. The first block represents a computational

process that simulates response of the computational substructure from two inputs,

the ground acceleration and the interaction force from the experimental substructure.

The output from the computational process is the top floor absolute acceleration

of the computational substructure that is sent to the experimental process as the

input. Then, the experimental process imposes this acceleration to the experimental

substructure using a shake table. The base shear in the experimental substructure

is treated as the output in the experimental process and should be sent back to the

computational process as the interaction force.

As shown here, the concept of substructure shake table testing is rather simple.

However, actual implementation with computational and experimental processes is

29


Comp. structure

fc_ nc

Exp. structure

Re_1

xg _ c xg _ exc_ nc + xg _ cxg

Figure 3.2: A block diagram for the concept of substructure shake table testing.

challenging. Required techniques to enable substructure shake table testing, RTHS

using shake tables, are those that ensure accurate data processing in the block diagram

without errors and time delays.

3.2 Experimental Setup and Modeling

To develop the required techniques to enable RTHS using shake tables, this study

utilizes an experimental setup at the Johns Hopkins University. Figure 3.3 shows a

photo of the experimental set up. The setup consists of a uniaxial shake table; a three-

story steel frame structure as the experimental substructure; and control and data

acquisition systems. In addition to the description of the experimental setup, this

section presents experimentally identified dynamic properties of the shake table and

the experimental substructures as well as parameters for the computational model.

3.2.1 Uni-Axial Shake Table

The shake table has a 1.2m x 1.2m aluminum platform driven by a Shore Western

hydraulic actuator (Model: 911D). The actuator has a dynamic load capacity of 27kN

30


Figure 3.3: A three-story steel frame structure on the uni-axial shake table at JohnsHopkins University.

and a maximum stroke limit of 7.6cm. An MTS 252 series dynamic servo valve is

used to control the fluid flow through the actuator chambers. The specifications of

the shake table are: maximum velocity of +-5.1 cm/s, maximum acceleration of 3.8

g; and maximum payload of 1.0 ton. The actuator is equipped with an embedded

displacement transducer and an inline load cell to measure the force on the actua-

tor. A general-purpose accelerometer is installed on the table to measure absolute

acceleration of the shake table.

3.2.2 Control and Data Acquisition System

The control hardware for the shake table includes a National Instruments 2.3 GHz

high-bandwidth dual-core PXI express controller (PXIe-8130), a windows-based host

PC and other accessories. The data acquisition system consists of a 16-bit high-speed

31


multifunction data acquisition board (PXI-6251), a signal conditioner (SCXI-1000),

and various analog input modules. Programs for the control and data acquisition

are written in NI LabVIEW, and are deployed on a real-time operating system on

the PXIe-8130. The PXIe-8130 is a real-time controller that is capable of running

multiple independent digital processes up to 10 kHz. The integrated control and

data acquisition system enables simultaneous sampling of all of the input and output

signals, and user-defined control and signal processes. More details of the control and

data acquisition system as well as the shake table can be found in Nakata (2011).

3.2.3 Experimental Substructure

The experimental substructure is a 700mm tall three-story steel frame with a floor

size of 304 mm x 610 mm. Each floor has four identical steel columns (5.08cm wide

W8x13 I-beams) that are bolted to the floors. At each floor, five steel plates are

placed as an additional masses of 90.7 kg. The total mass of the structure including

columns and support connections is approximately 300kg, that is more than double

the mass of the shake table platform.

Dynamic properties of the experimental substructure are examined using a band-

limited white noise excitation from the shake table. Figure 3.4 shows the frequency

response curves from the shake table acceleration to the absolute floor accelerations.

Distinct peaks appear at 6.9 Hz, 21.9 Hz, and 34.5 Hz in all of the transfer func-

tions, indicating the first, second, and third natural frequencies of the experimental

32


Figure 3.4: Frequency response curves of the three-story steel experimentalsubstructure.

substructure, respectively. Damping ratios for the first, second, and third vibration

modes are 1.1%, 0.8%, and 2.8%, respectively. In this study, it is assumed that

the structure remains linear elastic during the experiments; however the concept of

substructure shake testing is still valid for nonlinear test structures.

3.2.4 Computational Substructure

The computational substructure is a linear elastic seven-story shear building with

the story mass of 226 kg, floor stiffness of 1.76 × 103 kN/m, and floor damping

coefficient of 17.6 kN s/m. The first three natural frequencies of the computational

structure are 2.92 Hz, 8.64 Hz, and 14.0 Hz, and the corresponding damping ratios are

9.2%, 27.2%, and 43.9%, respectively. Combined with the experimental substructure,

the entire structural model has the dynamic properties listed in Table 3.1.

33


Table 3.1: Dynamic properties of the entire 10-story RTHS structure

Mode Natural Frequencies Damping Ratios1st 2.52 Hz 7.76 %2nd 6.80 Hz 11.11 %3rd 9.60 Hz 19.28 %

3.2.5 Measurement of Base Shear

Measurement of the base shear from the experimental substructure is required for

the interface force compatibility in substructure shake table testing. However, the

base shear is not directly measured in the current test setup; in order to directly

measure, load cells need to be installed either between the base of the structure and

the shake table or all of the columns. In this study, the base shear is obtained as the

sum of the inertial forces of the upper floors (i.e., sum of the mass times absolute

floor acceleration) as:

Re 1 = −3∑

i=1

me i(xe i + xg e) (3.8)

The above form can be derived from the sum of the equations of motion in Equation

3.4. It should be mentioned that this approach is valid only for lumped mass systems

of which dynamic responses can be expressed in the equations of motion in Equation

3.4. Because the experimental setup herein has significant mass at each floor, the

lumped mass assumption is considered appropriate.

It is worth mentioning that another approach for the measurement of the base

shear is examined in this study using the force measurement from the load cell on the

actuator and the table acceleration. However, the load cell is subjected to inevitable

34


friction between the bearings and the linear guides of the shake table. Therefore, this

study adopts the base shear that is obtained using the absolute floor accelerations.

3.3 Acceleration Control Performance

To meet the experimental acceleration compatibility in substructure shake table

testing, shake tables have to provide perfect tracking of the absolute top floor ac-

celeration of the computational substructure. However, acceleration control of shake

tables is extremely difficult mainly due to limitations in displacement control as ex-

plained in Chapter 2. Prior to implementation of substructure shake table testing, a

preliminary investigation of acceleration control performance and influence of input

acceleration errors on structural responses is performed.

3.3.1 Issues of Acceleration Control and Control-

Structure-Interaction

While shake tables are designed to produce reference accelerations, primary con-

trollers, inner-loop servo controllers, for actuators are displacement control. In al-

most all the cases, the inner-loop servo controllers are proportional-integral-derivative

(PID) controllers or PID with additional feedbacks (e.g., differential pressure feed-

back). In practice, a command shaping controller/filter such as the inverse dynamics

35


compensation techniques is added to cancel out the dynamics of the inner-loop con-

trol system, Spencer and Yang (1998). Basically, command shaping is an off-line

process that alters the reference displacements to produce the closest possible refer-

ence accelerations. To assess a possible use of such command shaping techniques for

substructure shake table testing, the control performance of the shake table with the

experimental substructure is discussed.

Figure 3.5 shows the frequency response curves (FRC) of the closed-loop (reference

to measured) displacement (a and c) and the closed-loop (reference to measured)

acceleration (b and d) at the proportional gain of 8.5. For a reference, the FRCs of

the bare table at the proportional gain of 20 are shown in the plots. As shown in the

magnitude plots, the bare table FRCs provide relatively smooth, wide and flat regions

in both displacement and acceleration. Because of their smoothness, the magnitude

responses in both displacement and acceleration are possibly improved up to around

25 Hz with a low-order command shaping compensation technique. On the other

hand, the FRCs with the experimental substructure show peculiar responses in both

displacement and acceleration with pairs of peaks and valleys around 7 Hz and 22 Hz.

These frequencies correspond to the first and the second natural frequencies of the

experimental substructure, indicating significant control-structure-interaction. As a

result, the reliable band-width of the acceleration FRC is limited to 6 Hz. The reason

that the gain has to be lower than the gain for the bare table is because of stability;

due to the spike and the phase drop at 7 Hz, the phase margin becomes much smaller

36


Figure 3.5: Frequency response curves of closed-loop (reference to measured) displace-ment and acceleration: (a) displacement magnitude; (b) acceleration magnitude; (c)displacement phase; and (d) acceleration phase.

than that of the bare table. This stability assessment is also confirmed experimentally

with uncontrollable 7 Hz vibration that occurs at the proportional gain of higher than

8.5. Therefore, the inner-loop control performance in displacement and acceleration

cannot be further improved with a tuning of PID gains.

An application of the inverse compensation techniques is examined. However,

it turns out that because of uncertainties in the high frequency range and inability

to compensate for the complex dynamic characteristics of the closed-loop responses

without introducing further delay, alteration of the reference displacement will amplify

vibration at the first and second natural frequencies of the structure. Because of these

reasons, the proportional controller with the gain of 8.5 is the only controller used in

37


this study. The characteristics of the acceleration FRC in Figure 3.5 indicate that

input acceleration errors are present in this control system.

3.3.2 Propagation of Input Acceleration Errors

To assess the possible response errors induced by the erroneous input acceleration

of the shake table, the dynamic relationship between the input acceleration and the

base shear are discussed. Figure 3.6 shows the frequency response curves and coher-

ence of the measured base shear from the table acceleration. The magnitude plot in

Figure 3.6a shows distinct peaks at the natural frequencies of the experimental sub-

structure. The phase plot in Figure 3.6b exhibits that the phase characteristics of the

base shear has a complex relation with that of the table acceleration. These dynamic

characteristics of the base shear are similar to those of the floor accelerations in Figure

3.4, indicating the relationship between the input acceleration and the base shear is

a multi-degrees-of-freedom dynamic system. Most importantly, the acceleration-base

shear relationship is highly correlated up to 40 Hz as shown in Figure 3.6c. This highly

correlated dynamic relationship reveals that input acceleration errors will propagate

and appear in the base shear measurement with amplified magnitude and varying

phase characteristics depending on its frequency contents.

38


Figure 3.6: Frequency response curve and coherence from the table acceleration tomeasured base shear.

3.4 Substructure Shake Table Test Sys-

tem with Error Compensation

All of the compatibility requirements have to be satisfied during real-time compu-

tational and experimental processes in the substructure shake table test. However, as

discussed in the previous section, errors in the input acceleration and the base shear

are inevitable in the experimental process. To enable accurate dynamic response

analysis of the entire structure through the substructure shake table test, those er-

rors have to be properly compensated for. This section presents a complete set of

techniques developed for substructure shake table testing that compensate for errors

in the experimental acceleration and interface force compatibilities.

Figure 3.7 shows a schematic of the substructure shake table test system with

the compensation techniques for experimental errors. The overall system consists of

a computational simulation of the computational substructure; measurement force

39


Comp. structure

Re_1xe

xp

xr

Delay Compensator

fc_ ncState

Observer

Force Corrector

xg _ cExp.

structure

xm

xc

Re_1

+

_

+

+

ΔRe_1

Figure 3.7: A block diagram of the substructure shake table test system with com-pensation techniques for experimental errors.

corrector; state estimator for the experimental substructure; and actuator delay com-

pensation for the shake table. Details of each process are discussed herein.

3.4.1 Numerical Integration for the Computational

Substructure

A numerical solution algorithm is an essential component in RTHS to solve the

governing equations of motion. In conventional hybrid simulation, only restoring

forces are experimentally evaluated while the rest of the entire structure including

the mass and damping of the experimental substructure are simulated computation-

ally. Therefore, equations of motion for the entire structure, Equations 3.2-3.4, can

be solved with an estimated stiffness of the experimental structure. The Newmark

family and predictor-corrector type numerical integration algorithms (e.g., alpha-OS)

40


are often used to solve for the future response of the entire structure and specify

the reference displacement. In the case of substructure shake table testing, all of the

dynamic effects of the experimental substructure including inertia, damping and stiff-

ness terms are experimentally incorporated. Therefore, it makes more logical sense to

solve for the response of only computational substructure in the numerical algorithm

incorporating the interface force from the experimental substructure as an additional

input. Solutions for such dynamic processes can be obtained using a discrete-time

state space approach.

The procedure for the computational substructure is as follows. Given that the

state vector xc and the input vector uc are available at the j-th step, the output

vector yc at the j-th step and the state vector at the (j + 1)-th step are calculated

as:

xc[j + 1] = Acxc[j] + Bcuc[j] (3.9)

yc[j] = Ccxc[j] + Dcuc[j] (3.10)

where Ac, Bc, Cc, and Dc are the discrete-time system, input, output and

feedthrough matrices of the computational substructure, respectively; The input vec-

tor consists of the ground acceleration and the interface force from the experimental

substructure, and the output vector consists of the top floor absolute acceleration and

41


displacement of the computational substructure. That is,

uc[j] =

[xg[j] fc nc [j]

]T(3.11)

yc[j] =

[xc nc [j] + xg c[j] xc nc [j] + xg c[j]

]T(3.12)

Note that the entries in the output vectors in Equation 3.12 are requisite minimums for

the proposed procedure in this study. Using the above state space representation, the

response of the computational substructure is simulated incorporating the interaction

force from the experimental substructure. The output at the j-th step is used in the

following actuator delay compensation technique.

3.4.2 State Observer and Kalman Filter

If the top floor absolute displacement, xc nc [j] + xg c[j], is sent to the shake table

controller as the reference at the j-th step, the measured table acceleration, xm[j],

at this step will not match the reference acceleration xr[j] that is the top floor ab-

solute acceleration, xc nc [j] + xg c[j], due to the inherent actuator delay. To reduce

the input acceleration errors caused by the actuator delay, a delay compensation

technique needs to be implemented. This study adopts a model-based delay com-

pensation technique that is similar to Carrion and Spencer (2007). The difference is

that the complete state of the experimental substructure that is required for the ini-

tial conditions in the delay compensation process is not available in the substructure

42


shake table testing; while all of the nodal displacements and velocities are known at

the end of each step in the conventional real-time hybrid simulation, not all of the

structural responses are available in the substructure shake table testing because they

are neither computed in the computational process nor measured in the experimental

process. Therefore, a state observer using a Kalman filter is adopted to estimate the

state variables for the experimental substructure.

With the measured input ue and output ye in the experiment and the estimated

state vector xe at the j-th step, the state vector at the (j+1)-th step can be estimated

as:

xe[j + 1] = (Ae − LCe)xc[j] + (Bc − LDe)ue[j] + Lye[j] (3.13)

where Ae, Be, Ce, and De are the discrete-time system, input, output and

feedthrough matrices of the analytical experimental substructure, respectively; and

L is the Kalman gain. The Kalman gain is determined based on estimates for the

covariance of the experimental measurement and process noises along with the accu-

racy of the model for experimental substructure. The measured input and output in

the experiment are the shake table acceleration and the base shear, respectively.

ue[j] = xg e[j] (3.14)

ye[j] = Re 1[j] (3.15)

43


Furthermore, the measured output in the experiment can be filtered to reduce the

influence of the process and measurement noises as:

Re 1[j] = ye[j] = Cexe[j] + Deue[j] (3.16)

Where Re 1[j] is the filtered base shear in the experiment that is used in the force

correction technique.

3.4.3 Model-Based Actuator Delay Compensation

The idea of the model-based delay compensation technique is to predict the future

response of the entire structure and send the reference displacement to the shake table

ahead of time. If the actuator delay constant is δt and the sampling in the iteration

process is dt, the number of required iterations is h = δt/dt . The iteration sampling,

dt, has to be selected such that the number of iterations, h, can be completed within

the simulation sampling, (i.e. a single time step in the RTHS). The model-based

delay compensation begins with initialization of the input and state vectors:

ue[0, j] = P1yc[j] = xc nc [j] + xg c[j] where P1 =

[1 0

](3.17)

xe[0, j] = xe[j] (3.18)

xc[0, j] = xc[j] (3.19)

44


Then, the processes at the k-th iteration (k = 0, . . . , h) in the delay compensation

technique for the j-th step reference displacement to the shake table are expressed

as:

xe[k + 1, j] = Aexe[k, j] + Beue[k, j] (3.20)

ye[k, j] = Cexe[k, j] + Deue[k, j] (3.21)

uc[k, j] =

[xg c[j + k] ye[k, j]

]T(3.22)

xc[k + 1, j] = Acxc[k, j] + Bcuc[k, j] (3.23)

yc[k, j] = Ccxc[k, j] + Dcuc[k, j] (3.24)

ue[k + 1, j] = P1yc[k, j] (3.25)

The delay compensation technique repeats the above processes (Equations 3.20-3.25)

h times for every simulation time step j. At the end of the h-th iteration, the predicted

displacement to the shake table at the j-th step, xp, is specified as:

xp[j] = P2yc[h, j] = xc nc [h, j] + xg c[j + h] where P2 =

[0 1

](3.26)

Where xc nc [h, j] is the top floor relative displacement of the computational sub-

structure at the h-th future step predicted from the current j-th simulation step;

and xg c[j + h] is the ground displacement at the computational substructure at the

(j + h)-th step.

45


3.4.4 Corrector for Errors in Base Shear Induced

by Input Acceleration Errors

To reduce the effect of erroneous response in the base shear induced by the prop-

agation of the input acceleration error, the force correction technique is implemented

as a part of substructure shake table testing. Dynamics of the propagation of the

input error can be expressed as:

xc[j + 1] = Aexe[j] + Beue[j] (3.27)

ye[j] = Cexe[j] + Deue[j] (3.28)

where ue[j], ye[j] and xe are the input, output, and state vectors for the error correc-

tion process. The input and output in this process can be expressed as:

ue[j] = xr[j] − xm[j] (3.29)

ye[j] = ∆Re 1[j] (3.30)

where xr[j] − xm[j] is the input acceleration error and ∆Re 1[j] is the erroneous base

shear induced by the input acceleration error. The corrected force is the sum of the

46


filtered and error-induced base shear as:

fc nc [j] = Re 1[j] + ∆Re 1[j] (3.31)

This corrected force is used in the computational process in Equation 3.11, and the

substructure shake table test system is now completely closed.

3.5 Experimental Results

All of the developed techniques for substructure shake table testing are imple-

mented in the control system at the Johns Hopkins University. A series of substruc-

ture shake table tests are conducted using the techniques developed in this study.

It should be mentioned that the same series of substructure shake table tests were

attempted without the developed techniques. However, tests could not be completed

because of stability issues, and comparable and representable results were not ob-

tained. Therefore, the test results presented in this section are only those with the

developed techniques. Data from the tests can be accessed in Nakata and Stehman

(2014b). Basic parameters for the simulation are as follows: sampling of the entire

simulation, 0.004s; actuator delay, δt = 0.068s; sampling of the iteration process in

the delay compensation technique, dt = 0.004s; and number of iterations in the delay

compensation, h = 17.

47


Figure 3.8: Acceleration and base shear time histories under 2.0 Hz harmonic groundexcitation: (a) the entire acceleration time histories; (b) a zoomed section of theacceleration time histories; (c) the entire base shear time histories; and (d) a zoomedsection of the base shear time histories.

3.5.1 Harmonic Ground Excitation Inputs

The first series of tests presented here are harmonic ground excitation tests. The

main objectives of the harmonic excitation tests are to assess stability, propagation

of errors, and validity of substructure shake table testing. Figure 3.8 shows the entire

and zoomed sections of the acceleration and base shear time histories when the entire

structure is subjected to 10 cycles of 2.0 Hz harmonic ground motion with the peak

ground acceleration of 0.041 g and the peak ground displacement of 2.54 mm. For a

comparative purpose, results from a pure numerical simulation are also shown in the

plots.

Firstly, it can be observed from Figure 3.8a that the measured acceleration tracks

the reference acceleration well in a large simulation time scale. The zoomed section

48


of the acceleration in Figure 3.8b also reveals that reasonable acceleration tracking at

2.0 Hz is achieved. However, the measured acceleration contains high frequency vi-

brations that are not present in the reference and numerical accelerations. Note that

the reference acceleration is the one that is computed in the substructure shake table

test whereas the numerical acceleration is from pure numerical simulation. Because

the substructure shake table test incorporates the experimental base shear, the ref-

erence acceleration has some discrepancy from the numerical acceleration. It should

be pointed out that the peak time of the measured acceleration matches that of the

reference, demonstrating effectiveness of the model-based delay compensation tech-

nique. The difference between the reference and measured accelerations is the effect

of the input acceleration error on the base shear is accounted for in this substructure

shake table test.

Figures 3.8c and 3.8d show that the measured base shear is mostly 2.0 Hz har-

monic. However, as can be seen in Figure 3.8d, the measured base shear also contains

vibration at approximately 20 Hz. The measured base shear is filtered and then

corrected based on the input acceleration error during the substructure shake table

testing. The corrected base shear shows very good agreement with the numerical base

shear. Although the numerical base shear is not a reference that has to be followed,

this agreement and smoothness in the corrected base shear indicates that the errors

due to process and measured noises as well as those that are induced by the input

acceleration errors are effectively reduced by the error compensation techniques in

49


substructure shake table testing.

A comparison between the substructure shake table test (labeled as ‘Hybrid’) and

the numerical simulation in terms of structural responses under the 2.0 Hz harmonic

excitation is shown in Figure 3.9. Relative displacement, absolute displacement, and

absolute acceleration at the 2nd, 6th, and 10th floors are shown in the plots. Note that

the displacement responses at the 10th floor (the top floor of the experimental sub-

structure) are recovered from the acceleration response. These plots illustrate several

important features of the substructure shake table test that can be summarized as

follows: while some discrepancies between the substructure shake table test and the

numerical simulation are seen at floors in the experimental substructure, almost all

of the structural responses in the substructure shake table test show very good agree-

ment with those in the numerical simulation. Furthermore, it can be seen that all

types of responses show gradual increase with the increase of the floor number. Be-

cause the input frequency of 2.0 Hz is relatively close to the first natural frequency of

the entire structure, overall structural responses in the substructure shake table test

seem reasonable, meaning that the structural responses at the first vibration mode or

equivalent can be accurately simulated. Thus, the substructure shake table test here

provides promising results as a potential means to simulate the structural responses.

Next, simulation results under a harmonic ground excitation at 6.0 Hz that is close to

the second natural frequency of the entire structure is presented. Figure 3.10 shows

the entire and zoomed sections of the acceleration and base shear time histories when

50


Absolute Acceleration (g)Relative Displacement (mm) Absolute Displacement (mm)

Figure 3.9: Structural responses under 2.0 Hz harmonic ground excitation: (a), (d),and (g), relative floor displacement at the 10th, 6th and 2nd floor, respectively; (b), (e),and (h), absolute floor displacement at the 10th, 6th and 2nd floor, respectively; and(c), (f), and (i), absolute floor acceleration at the 10th, 6th and 2nd floor, respectively.

51


Figure 3.10: Acceleration and base shear time histories under 6.0 Hz harmonic groundexcitation: (a) the entire acceleration time histories; (b) a zoomed section of theacceleration time histories; (c) the entire base shear time histories; and (d) a zoomedsection of the base shear time histories.

the entire structure is subjected to 10 cycles of 6.0 Hz harmonic ground motion with

the peak ground acceleration of 0.11 g and the peak ground displacement of 0.76 mm.

Unlike in the previous simulation, the acceleration time histories in Figures 3.10a and

3.10b show large discrepancy with the reference acceleration, containing high fre-

quency vibration of approximately 30 Hz. Because of the poor acceleration tracking,

the input acceleration errors are present at this frequency as expected from the ob-

servation in the previous section. The measured base shear shown in Figures 3.10c

and 3.10d has large discrepancy with the numerical base shear. However, despite the

large differences, the corrected base shear shows good agreement with the numerical

base shear. This agreement is because the erroneous responses in the measured base

shear are mostly induced by the input acceleration errors, and the propagation of

52


the input acceleration errors is accurately traced by the techniques developed in this

study. Thus, the proposed compensation techniques are shown to be effective even

with a significant level of input acceleration errors.

Structural responses under the 6.0 Hz harmonic excitation are shown in Figure

3.11. As is the case with the previous simulation with the 2.0 Hz excitation, discrep-

ancies between the substructure shake table test and the numerical simulation are

seen at floors in the experimental substructure. But, good agreement between the

substructure shake table test and the numerical simulation in terms structural re-

sponses is also obtained in this simulation with the 6.0 Hz excitation. It is interesting

to see that the 6th floor accelerations are smaller than those at the 2nd floor and do

not contain much of the 6.0 Hz vibration despite of the 6.0 Hz excitation frequency.

This observation seems to make sense because the input frequency of 6.0 Hz is close

to the second natural frequency of the entire structure; the 6th floor is close to a

node in the second vibration mode. Thus, the test results here demonstrate that the

overall responses of the entire structure under a relative high frequency around the

second natural frequency are also simulated reasonably well using the substructure

shake table test.

The experimental simulations using harmonic excitations in this section proved

that though experimental errors including input acceleration errors are present, the

substructure shake table tests are successfully completed using the developed compen-

sation techniques. The substructure shake table tests are stable and valid, indicating

53



Figure 3.11: Structural responses under 6.0 Hz harmonic ground excitation: (a), (d),and (g), relative floor displacement at the 10th, 6th and 2nd floor, respectively; (b), (e),and (h), absolute floor displacement at the 10th, 6th and 2nd floor, respectively; and(c), (f), and (i), absolute floor acceleration at the 10th, 6th and 2nd floor, respectively.

that experimental errors are not propagated through the real-time hybrid simulation

processes.

3.5.2 Earthquake Ground Excitation Input

Substructure shake table tests are performed using earthquake ground excitations.

In this paper, results from the 1995 Kobe earthquake are presented and discussed.

Figure 3.12 shows the entire and zoomed sections of the acceleration and base shear

time histories when the entire structure is subjected to the 1995 Kobe earthquake with

the peak ground acceleration of 0.23 g and the peak ground displacement of 17.8

54


Figure 3.12: Acceleration and base shear time histories under the 1995 Kobe earth-quake excitation: (a) the entire acceleration time histories; (b) a zoomed section ofthe acceleration time histories; (c) the entire base shear time histories; and (d) azoomed section of the base shear time histories.

mm. The measured acceleration shows good tracking to the primary low frequency

vibrations in the reference acceleration including the phase characteristics. However,

notable high frequency vibrations due to the imperfect acceleration tracking can also

be observed. It should be mentioned that while tracking performance is improved with

the increase of the excitation level, input acceleration errors are still unavoidable due

to the imperfection of the acceleration tracking using displacement control. As in

the case with the previous harmonic excitation simulations, the measured base shear

shows discrepancy with the numerical base shear. However, once the influence of

the input acceleration errors is addressed, the corrected base shear agrees well with

the numerical base shear. This good agreement demonstrates that the developed

compensation techniques effectively reduce the influence of the experimental errors

during the earthquake excitation simulation.

55


Figure 3.13 shows the structural responses under the Kobe earthquake during the

substructure shake table test. Relative displacement, absolute displacement and ab-

solute acceleration at the even floors are shown in the plots. It can be seen that while

some discrepancies between the substructure shake table testing and the numerical

simulation are seen at the upper floor responses, the overall structural responses in

the substructure shake table test show good agreement with the numerical simulation.

The responses of each type are approximately proportional with the increase of floor

number, indicating that the entire structural responses are mostly the first vibra-

tion mode. This observation seems reasonable because the primary frequency of this

earthquake is close to the first natural frequency. Thus, the simulation results here

demonstrate that substructure shake table testing with the developed compensation

techniques for experimental errors successfully simulate the response of the 10th-story

structure under the earthquake ground excitation input. It should be mentioned that

although results are not presented in the paper, more substructure shake table tests

were conducted using different earthquakes and the same level of agreement with

numerical simulation are obtained.

56


Co

mp

uta

tio

nal

Su

bst

ruct

ure


Exp

erim

enta

l Su

bst

ruct

ure

Figure 3.13: Structural responses under the 1995 Kobe earthquake excitation: (a),(d), (g), (j), and (m), relative displacement at the even floors from top to bottom(10th to 2nd); (b), (e), (h), (k), and (n), absolute displacement at the even floors fromtop to bottom (10th to 2nd); and (c), (f), (i), (l), and (o), absolute displacement atthe even floors from top to bottom (10th to 2nd).

57


3.6 Advanced Model-Based Shake Table

Compensation Techniques

The shake table delay compensation technique described in the previous sections

does not address the magnitude performance of the control system. Thus any mag-

nitude limitations introduces by poor shake table control can not be overcome. To

address such limitations, shake table delay compensation techniques that address the

dynamics of the control system are developed herein.

This section describes two delay compensation techniques, which operate by mod-

ifying the original reference displacement before it is sent to the closed loop shake

table system. Both techniques are model-based and involve an inverse model of the

reference to measured displacement relationship of the shake table, Equation 2.8.

The first of the presented techniques was developed by Phillips et al. (2013) and the

second was developed for this specific implementation as an expansion of the effort

presented in Carrion and Spencer (2007).

3.6.1 Feedforward Compensation using Derivatives

of Reference Signal

The first technique considered in this study is a feedforward compensator devel-

oped by Phillips et al. (2013), as an extension from the work by Carrion and Spencer

58


(2007), to compensate for actuator dynamics and time delay in RTHS for a magne-

torheological damper used as the experimental substructure. This technique uses an

inverse model of the closed loop shake table system to account for time delay and

high frequency actuator dynamics (magnitude roll off at high frequencies). Here the

closed loop model is determined through experimental system identification of the re-

lationship between reference and measured shake table displacements. The resulting

model is then curve-fitted to obtain an analytical transfer function of the physical

relationship, denoted here as Hxmxr(s).

The curve fitting is completed using a transfer function with three poles, such

that:

Hxmxr(s) =1

a3s3 + a2s2 + a1s+ a0(3.32)

While this technique can capture the general actuator characteristics, since the model

does not contain a numerator polynomial, pole-zero cancelations and thus CSI from

the inertial components cannot be modeled. The compensator is formed as the inverse

of the Equation 3.32. However since the curve fit contains only poles, the inverse

would be improper and hence cannot be implemented in RTHS. To overcome this

challenge, the inverse is realized as a time domain representation where the modified

reference signal, xp, is obtained as a weighted series of derivatives of the true reference

displacement, xr = xc nc + xg c:

xp[j] = a3...x r[j] + a2xr[j] + a1xr[j] + a0xr[j] (3.33)

59


In this form, the compensator is stable since derivatives can be easily and accurately

calculated in real time. This technique can also accommodate higher order systems

by using higher order derivatives, however since derivatives amplify noise, additional

derivatives can lead to high frequency amplification. The feedforward compensator

falls into the category of Finite-Impulse-Response (FIR) compensators because the

output of the compensator (modified reference signal) depends only on the input to

the compensator (original reference signal).

3.6.2 IIR Compensation Technique for Significant

Control-Structure-Interaction

While the compensation method introduced by Phillips et al. (2013) can com-

pensate for time delays and high frequency actuator dynamics, it cannot address the

inertial effects from CSI which often appear in substructure shake table testing, Equa-

tion 2.7. In order to account for low frequency near pole-zero cancelations due to CSI,

the curve-fitting procedure described in Carrion and Spencer (2007) is enhanced to

include a numerator polynomial.

Similar to the previous feedforward technique, the proposed compensator contains

an inverse model of the displacement tracking transfer function, Hxmxr(s), however the

proposed technique has no limitations on the order for the numerator or denominator

in the transfer function used in the curve fitting procedure. To ensure a stable closed

60


loop compensator, additional poles may be added to the inverse approximation model

to make it proper. The number of poles added is the minimal number to make the

closed loop compensator proper. Thus if:

Hxmxr(s) =bls

l + bl−1sl−1 + . . . b1s+ b0

amsm + am−1sm−1 + . . . a1s+ a0with m > l (3.34)

The number of poles needed to stabilize the compensator is r = m− l+ 1. To ensure

effectiveness of the compensator, the additional poles must not significantly alter

(Hxmxr(s))−1. The additional poles are thus determined from an rth order high-pass

filter, Hf (s). That is, the modified reference signal is calculated as:

Xp(s) =(Hxmxr(s)

)−1Hf (s)Xr(s) = F (s)Xr(s) (3.35)

where F (s) is the transfer function for the proposed compensator. This approach can

accommodate any choice of high pass filter provided it does not have a numerator

polynomial. Since this compensator has both a numerator and denominator, the

modified reference signal is dependent on characteristics of both the original reference

and the modified reference signals. This type of compensator is thus termed an

Infinite-Impulse-Response (IIR) compensator. Since the IIR compensator in this case

is a proper transfer function, direct implementation into RTHS is feasible.

Incorporating the IIR compensator, the displacement performance, (Figure 2.2

61


and Equation 2.8) , of the shake table can now be expressed as:

Xm(s) =Sx(s)Hxtv(s)C(s)F (s)

1+Sx(s)Hxtv(s)C(s)Xr(s) +

Sx(s)Hxtv(s)

1+Sx(s)Hxtv(s)C(s)W (s) + Sx(s)

1+Sx(s)Hxtv(s)C(s)N1(s)

(3.36)

From Equation 3.36, it is clear that the IIR compensator only influences the rela-

tionship between reference and measured displacements. Therefore this compensator

can only modify reference tracking and does not influence robustness or stability of

the closed loop system. The parameters of the IIR compensator should be chosen

such that in the low frequency range F (s) is the inverse of the closed loop shake ta-

ble relationship and that the compensated closed loop system does not amplify high

frequency vibration.

3.6.3 Experimental Investigation of Model-Based

Delay Compensation Techniques

To test the efficacy of these advanced compensation techniques, the same earth-

quake simulation that was presented in Section 3.5.2 is completed using both com-

pensation techniques. Results from all simulations are then compared and discussed.

The tracking performance of the shake table with 3DOF test structure is shown

in Figure 3.14. The shake table has significant effects from CSI with near pole/zero

cancelations occurring at the natural frequencies of the 3DOF test structure (6.9 Hz,

62


0 10 20 30 40

10−2

100

Frequency (Hz)

Ma

gn

itu

de

(m

/m)

Displacement Tracking

Uncompensated

Feedforward

IIR

0 10 20 30 40−200

−100

0

100

200

Frequency (Hz)

Ph

ase

(D

eg

.)

0 10 20 30 40

10−2

100

Frequency (Hz)

Ma

gn

itu

de

(g

/g)

Acceleration Tracking

Uncompensated

Feedforward

IIR

0 10 20 30 40−200

−100

0

100

200

Frequency (Hz)

Ph

ase

(D

eg

.)

(a) (b)

(c) (d)

Figure 3.14: Experimental reference to measured frequency response functions forshake table with 3DOF experimental substructure: (a) and (c) displacement trackingmagnitude and phase; (b) and (d) acceleration tracking magnitude and phase.

21.9 Hz and 34.5 Hz). As shown in Figure 3.14 the uncompensated shake table has

a bandwidth of 2Hz with an estimated time delay of 68 ms. The shake table PID

gain was kept low to ensure stability, resulting in limited tracking performance. The

tracking performances incorporating both the feedforward and IIR compensators are

also shown in Figure 3.14. The displacement tracking of the shake table is signifi-

cantly improved when compensation is added. The time delay is brought down to 2ms

using feedforward compensation and 7ms using IIR compensation. The displacement

bandwidth is also increased to 20Hz with IIR compensation. To avoid significant high

frequency amplification with the feedforward compensator, the low frequency mag-

nitudes are undercompensated; even-still slight amplification is observed around 25

Hz. This amplification is because the feedforward compensator cannot compensate

such pole-zero cancelations. It should be mentioned that several different feedforward

63


compensators were implemented and this specific compensator was the only choice

that yielded stable RTHS. The acceleration tracking response is shown in Figures

3.14b and 3.14d. The responses are similar to their displacement counterparts with

over amplification at high frequencies; this phenomenon is a results of the noise in

Equation 2.10. It is worth noting that the response using iterative compensation

(Section 3.4.3) is also available from Figure 3.14. The closed loop performance using

iterative compensation has the same magnitude response as the uncompensated sys-

tem (since actuator dynamics are ignored) and the modified reference displacement

is calculated from predicting the future reference displacement 68ms in advance.

To test the effectiveness of the three compensation techniques, the 10-story RTHS

structure was subjected to the 1995 Kobe earthquake JMA record (Section 3.5.2).

The time histories of the shake table displacement and acceleration from the 3 sim-

ulations are shown in Figure 3.15. For comparison, results from pure numerical

simulation of the RTHS structure (without shake table) are used as the reference

response for the RTHS structure. In all three simulations, the shake table was able

to track the numerical response with reasonable accuracy, Figures 3.15a and 3.15b.

The simulations with feedforward and iterative compensation showed the least time

delay for the shake table and the IIR compensation still showed slight delay behind

the numerical reference. However, the IIR compensator resulted in the most accurate

magnitude response with both iterative and feedforward techniques unable to achieve

the full displacements. As expected, due to the existence of noise and the acceleration

64


0 1 2 3 4 5 6 7

−0.02

−0.01

0

0.01

0.02

Dis

pla

cem

ent (m

)

Time (s)

Numerical

RTHS, Feedforward

RTHS, IIR

RTHS, Iterative

2.5 2.6 2.7 2.8 2.9 3

−5

0

5

10

15

x 10−3

Time (s)

0 1 2 3 4 5 6 7

−0.5

0

0.5

Accele

ration (

g)

Time (s)2.5 2.6 2.7 2.8 2.9 3

−0.4

−0.2

0

0.2

0.4

Time (s)

(a) (b)

(c) (d)

Figure 3.15: Absolute shake table response from substructure shake table tests with3DOF experimental substructure subjected to Kobe earthquake record: (a) and (c)full displacement and acceleration time histories; (b) and (d) zoomed-in views ofdisplacement and acceleration time histories.

65


Table 3.2: Summary of shake table performance from substructure shake table testingusing 3 different compensation techniques.

Feedforward IIR IterativeCompensation Compensation Compensation

Shake Table Displacement RSME 21 % 8.5 % 27 %Shake Table Acceleration RSME 50 % 28 % 47 %

amplification at higher frequencies, as shown in Figure 3.14b, the acceleration track-

ing results are much more distorted. Results show that all compensation techniques

yielded reasonable tracking of the major frequency contents in the reference accel-

eration, however the simulation using feedforward compensation produced the most

high frequency vibration. The performance of the shake table from each simulation is

summarized in Table 3.2 using root mean squared error (RSME) as the performance

indicator.

The results from further error analysis studies are presented in Figure 3.16 for each

simulation with a different compensation technique. Figure 3.16 plots the reference

versus measured displacements and accelerations for each test. These plots give the

overall performance of the individual compensation techniques in two regards: if

the slope of the plot is 45 degrees than perfect magnitude tracking was achieved

and if the plot generates a thin trajectory than there is little error due to time

shifting of the data (i.e.: perfect delay compensation with no measurement noise

would result in a single line). Analysis of the displacement error plots reveals that

the feedforward compensator (Figure 3.16a) achieved a small delay however the plot

has a slope of less than 45 degrees indicating the magnitude was undercompensated,

66


which agrees with the data presented earlier. The IIR compensator (Figure 3.16b)

generated the most accurate magnitude tracking performance however the width of

the plot signifies some time delay is still present (this was seen in the time histories,

Figure 3.15b). Figure 3.16c shows that the iterative compensator had larger time

delays than the feedforward compensator. Such discrepancies are present because the

iterative technique predicts future delays at a fixed time increment and thus cannot

address the fact that the actuator exhibits a different time delay at different operating

frequencies (due to the non-constant slope of the phase plot in Figure 3.14c). Under-

compensation of the displacement magnitude is also verified in Figure 3.16c. The

acceleration error plots are slightly more distorted due to the influence of process

and measurement noise on the measured accelerations, however general trends can

still be observed. The IIR compensator generated the thinnest acceleration error plot

(Figure 3.16e), this indicates the IIR compensation produced the least amount of high

frequency acceleration. This observation agrees with the data presented in Table 2.

Similar to the displacement error plots, the feedforward and iterative compensators

generally undercompensate the accelerations (Figures 3.16d and 3.16f), while very

good acceleration magnitude tracking is observed from IIR Compensation.

Results from the experimental studies indicated that the IIR compensation tech-

nique was able to produce more accurate shake table tracking performance during

substructure shake table testing when compared to two existing compensation tech-

niques. The increased performance is due to the ability of the IIR compensator to

67


−0.02 0 0.02

−0.02

−0.01

0

0.01

0.02

Refe

rence D

ispla

cem

ent (m

)

Measured Displacement (m)

Feedforward Error

−0.02 0 0.02

−0.02

−0.01

0

0.01

0.02

Refe

rence D

ispla

cem

ent (m

)


IIR Error

−0.02 0 0.02

−0.02

−0.01

0

0.01

0.02

Refe

rence D

ispla

cem

ent (m

)


Iterative Error

−0.5 0 0.5−0.5

0

0.5

Refe

rence A

ccele

ration (

g)

Measured Acceleration (g)−0.5 0 0.5

−0.5

0

0.5

Refe

rence A

ccele

ration (

g)

Measured Acceleration (g)−0.5 0 0.5

−0.5

0

0.5R

efe

rence A

ccele

ration (

g)

Measured Acceleration (g)

(a)

(d)

(b)

(e)

(c)

(f)

Figure 3.16: Shake table tracking errors from substructure shake table tests with3DOF experimental substructure subjected to Kobe earthquake record: (a) and (d)displacement and acceleration errors using feedforward compensator; (b) and (e) dis-placement and acceleration errors using IIR compensator; (c) and (f) displacementand acceleration errors using iterative compensator.

68


account for complex shake table CSI. While results from this study are not exhaus-

tive, the results have shown that a complex case of CSI can significantly limit the

performance of existing delay compensation techniques that do not address CSI.


This study presented a real-time hybrid simulation technique using shake tables

including compensation techniques for experimental errors. The developed techniques

included compensation techniques for response errors induced by erroneous input ac-

celeration, model-based actuator delay compensation with state observer, and force

correction using Kalman filter. The effectiveness of those techniques was experimen-

tally verified through a series of RTHS using a uni-axial shake table and three-story

steel frame structure at the Johns Hopkins University.

While the chapter presented mostly successful parts of the study, unbiased fair

discussions need to be provided. To pursue further research along this direction,

remaining challenges that have to be addressed in the future study are listed below.

As demonstrated, substructure shake table testing with the developed techniques

made it possible to perform reliable simulations that were not possible without them.

However, it is owing to a relatively large damping of the computational structure to

some extent. When the RTHS using shake table were performed using computational

structures with smaller damping, simulations were unstable with and without the

69


compensation techniques. In the future, the compensation techniques have to be

further refined to take effects under more strict conditions.

The compensation techniques for response errors induced by the input acceleration

errors were effective even if the input errors were significant. This is because the test

structure had high correlation between the ground acceleration and the base shear.

If test structures are nonlinear or have less correlation between the ground shaking

and the response, the same level of improvement cannot be expected. Future research

needs to address such limitations in the current approach. A possible approach is the

model updating technique that can capture nonlinearities of the experimental model.

While challenges are still remaining, this study addressed the issues of the response

errors induced by erroneous and inevitable input acceleration errors in RTHS using

shake tables and developed compensation techniques for such experimental errors.

The author believes that the developed compensation techniques can serve as the

initial effort to address such inevitable experimental errors.

70

Chapter 4


of Lower Stories in Tall Buildings

Another particular interest in the earthquake engineering community is the dy-

namic evaluation of structural members located in the lower stories of tall buildings.

These members have potential to see extreme forces during earthquakes to resist base

shear and over turning moments due to the movement of the mass of the entire struc-

ture. However the experimental setups to evaluate these members are difficult to

implement due to size constraints. The concept of substructure shake table testing is

a viable candidate to enable experimental evaluation of such conditions.

This chapter discusses techniques that allow for the testing of lower stories in tall

buildings where the stories of interest are placed on a shake table and the remainder

of the building is modeled computationally. Two techniques are discussed within this

71

CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS

chapter to enable substructure shake table testing of the lower stories of building when

subjected to earthquakes. The contents of this Chapter were previously published in

Nakata and Stehman (2012, 2014c).

4.1 Interface Compatibility using a Con-

trolled Mass

Figure 4.1 shows schematics of an entire system and a substructured system with a

controlled mass. The entire system is an n-story shear building. In the substructured

system, the lower i-stories of the building are experimentally evaluated while the

upper (n − i)-stories are substituted by a controlled mass. Equations of motion for

the j-th floor can be written as:

Entire System:

mj

j∑k=1

xk+Rj(xj, xj)−Rj+1(xj+1, xj+1) = −mjxg(j = 1, . . . , n and Rn+1 = 0) (4.1)

Substructured System:

mj

j∑k=1

xk +Rj(xj, xj) −Rj+1(xj+1, xj+1) = −mjxg e(j = 1, . . . , i− 1) (4.2)

mi

i∑k=1

xk +Ri(xi, xi) − fm = −mixg e(j = i) (4.3)

72


Entire System

1m

mf

ɺɺxg

ix

1ix+

nx

1x

R1x1, ɺx1( )

Rnxn, ɺxn( )

Ri+1xi+1, ɺxi+1( )

Rixi, ɺxi( )

im

1im

+

nm

1m

ɺɺxg _ e

ix

1ix+

nx

1x

R1x1, ɺx1( )

Rnxn, ɺxn( )

Ri+1xi+1, ɺxi+1( )

Rixi, ɺxi( )

im

1im

+

nm

mx

mm

ɺɺxg _ c

Shake Table

!"

Substructured System

Controlled Mass

Experimental Substructure Computational Substructure

Figure 4.1: Schematics of the Entire and substructure systems.

mj

j∑k=1

xk +Rj(xj, xj) −Rj+1(xj+1, xj+1) = −mjxg c(j = i+ 1, . . . , n and Rn+1 = 0)

(4.4)

where mj is the mass of the j-th floor; xj is the relative displacement of the j-th floor

with respect to the (j−1)-th floor; Rj(xj, xj) is the nonlinear restoring force from the

j-th floor; xg is the ground acceleration; xg c and xg e are the input acceleration to

the computational and experimental substructures, respectively; and fm is the force

due to the controlled mass. Note that Equations 4.1 and 4.3 are with respect to

the experimental substructure and Equation 4.4 is with respect to the computational

substructure.

For the substructured system to have the equivalent dynamics as the entire system,

the following three conditions have to be satisfied. First, the input ground acceleration

to the experimental substructure has to be the same as the one to the entire system

73


(experimental acceleration compatibility), i.e.,

xg e = xg (4.5)

Second, the input acceleration to the computational substructure has to be the abso-

lute acceleration at the i-th floor (computational acceleration compatibility), i.e.,

xg c = xg e +i∑

k=1

xk (4.6)

And last, the force due to the controlled inertial mass at the i-th floor in the exper-

imental substructure has to be equal to the base shear at the (i + 1)-th floor in the

computational substructure, referred to as the computational base shear (interface

force compatibility), i.e.,

fm = Ri(xi+1, xi+1) (4.7)

All of the above compatibility conditions have to be met simultaneously at every

instance during the simulation.

Technical difficulties and available resources vary for each compatibility condi-

tion. Here, we look into feasible approaches for the compatibility conditions with a

particular focus on implementation.

The experimental acceleration compatibility requires acceleration control of shake

tables. Needs of the high-performance control of shake tables have been addressed by

74


a number of researchers, and several methods have been developed to date (Nakata,

2010; Stehman and Nakata, 2013; Trombetti and Conte, 2002)) Since some of the

methods have proven to be promising, this paper does not discuss details of the

acceleration control of shake tables that can be found elsewhere; therefore, interested

readers are recommended to refer to relevant literature. This study presumes that

one or more of the acceleration control methods are available in the experimental

system used for the substructure shake table tests.

The computational acceleration compatibility requires application of the measured

i-th floor absolute acceleration to the computational substructure. Such processes can

be performed with a commercially available real-time operating system (e.g., SCRAM-

Net) that provides real-time data acquisition, signal processing, and computation of

structural response. Use of the real-time processes can be found in applications of

structural control, real-time hybrid testing, etc. The interface force compatibility

implies that the force of the inertial mass has to be controlled with reference to the

computationally simulated structural response. However, direct dynamic force con-

trol in structural testing is extremely challenging; therefore, this study adopts a series

of conversions to achieve the requirement.

The first step is to convert the reference base shear of the computational substruc-

ture to an equivalent inertial force of the mass. From the kinetic relationship, the

75


equation of motion of the mass can be expressed:

mm

(xgc +

∑xk + xm

)= fm (4.8)

where mm is the mass of the controlled mass and xm is the relative acceleration of

the controlled mass with respect to the i-th floor. Note that the mass is assumed to

be a rigid body.

If the relative acceleration of the mass is controlled to meet the above equation, the

interface force compatibility is fulfilled. In this study, further steps from the relative

acceleration are taken because of the following reason. Unlike ground accelerations

in the experimental acceleration compatibility, the relative acceleration in Equation

4.8 is not known a priori; therefore, feedforward techniques often used in acceleration

control of shake tables cannot be applied, and accuracy is not ensured.

The next step is to convert the relative acceleration of the mass to the relative

displacement. This conversion process can be digitally performed using the numer-

ical integration of the relative acceleration. Using the trapezoidal rule, the relative

displacement can be obtained as follows:

xm[n] = xm[n− 1] +∆t

2(xm[n] + xm[n− 1]) (4.9)

xm[n] = xm[n− 1] +∆t

2(xm[n] + xm[n− 1]) (4.10)

76


where xm[n], xm[n] and xm[n] are the n-th step relative acceleration, velocity, and

displacement of the mass, respectively; and ∆t is the time increment in the numerical

integration.

The relative displacement obtained from the numerical integration is not always

stable due to the existence of DC and low frequency components in the measured

acceleration data. To stabilize the relative displacement of the mass, a high-pass

filter is applied. The filtered relative displacement can be expressed in the following

form.

xm fil[z] = Hfil[z]xm[z] (4.11)

where Hfil is the discrete transfer function of the high-pass filter; xm fil is the filtered

relative displacement of the mass; and z is the z-transform variable. Finally, the

filtered relative displacement is fed to the control system of the inertial mass (a.k.a.,

actuator).

The overall proposed implementation diagram for the substructure shake table

test using controlled masses is shown in Figure 4.2. As shown in the diagram, the

proposed substructure shake table test method consists of a feedback of a series of

experimental and computational processes. Accuracy and stability of the method rely

on the precision and time delay of each process.

77


Exp. Acc. Compatibility

Computational

Substructure

Experimental

Substructure

ɺɺxg _ c

= ɺɺxg _ e

+ ɺɺxk

k =1

i

∑

i+1R

( )i+1m Rf =

ɺɺxg _ e

= ɺɺxg( )

ɺɺxm

Numerical

Integration

High-pass

Filter xm

Actuator/

Mass

Acceleration

Conversion _m fil

x

Comp. Acc. Compatibility Interface Force Compatibility

Figure 4.2: A block diagram of the substructure shake table test using controlledmasses.

4.1.1 Simulation Models

The fundamental concept of the proposed substructure shake table test method

is relatively straightforward. However, accuracy and stability are not guaranteed due

to technical difficulties to meet all of the compatibility conditions. In particular, the

boundary force compatibility condition is challenging because it is governed by the

dynamics of the control system for the inertial mass. Feasibility of the method is

considered to be also dependent on the height of the structure, substructure configu-

ration and input ground motion. In this study, numerical simulations are performed

to investigate the influence of simulation parameters on the accuracy and stability as

well as identify limitations of the method. This section presents details of the models

used in the numerical simulations.

STRUCTURAL MODEL

78


Table 4.1: Properties and dynamic characteristics of the 5-story structure.

Structural PropertiesFloor mass 50kgStory stiffnes 97.5KN/mDamping ratio 6.50%

Natural frequencies1st mode 2.00Hz2nd mode 5.84Hz3rd mode 9.20Hz4th mode 11.83Hz5th mode 13.49Hz

For this initial study a 5-story structure is chosen as the structure of interest. The

structure is separated at the second floor such that the lower 2-stories comprise the

experimental substructure and the upper 3-stories form the computational substruc-

ture: refer to Figure 4.1 with i=2 and n=5. Specifications of the structure along with

structural characteristics are presented in Table 4.1. The scale of the structural model

is selected so that physical construction is feasible in an experimental investigation

of the proposed method.

CONTROLLED MASS SYSTEM

As described in the previous section, the relative displacement of the mass is

controlled with respect to the i-th floor of the experimental substructure for the

interface force compatibility. In this study, it is assumed that the mass is controlled

by a hydraulic actuator; hydraulic actuators meet high-speed and high-force demand

in structural tests and are often used in dynamic applications.

A fatigue-rated, high-speed, dynamic actuator manufactured by Shore Western

Inc. (model number: 911D) is adopted. The 911D actuator has dynamic force

79


0 5 10 15 20 25 3010

−1

100

Frequency (Hz)

Ma

gn

itu

de

(m

/m)

0 5 10 15 20 25 30

−180

−160

−140

−120

−100

−80

−60

−40

−20

0

Frequency (Hz)

Ph

ase

(D

eg

)

(a)

(b)

Figure 4.3: Closed-loop frequency response function of the 911-D actuator with amass of 45kg: (a) magnitude; (b) phase.

capacity of 27 kN and stroke of ±78 mm. A closed loop displacement frequency

response function of the 911D actuator with 45 kg mass is shown in Figure 4.3. The

transfer function of the actuator is given by Equation 4.12:

H(s) =3.653 · 108

s4 + 172.2s3 + 8.588 · 104s2 + 6.787 · 106s+ 3.786 · 108(4.12)

This transfer function is experimentally obtained through system identification of

the controlled mass system including proportional controller. For more details on

actuator mechanics, refer to (Conte and Trombetti, 2000; Dyke et al., 1995; Nakata,

2010, 2012). This transfer function is converted into a state space model for use in the

80


numerical simulations to represent the interface force from the computational sub-

structure back to the experimental substructure. This study investigates the effects

of the dynamics of the controlled mass system on the accuracy and stability of the

substructure shake table test method.

NUMERICAL INTEGRATION ALGORITHM

In this study, Newmark’s method is adopted as the numerical time step integration

algorithm for both experimental and computational substructures. The parameters

in the numerical integration algorithm are γ = 12

and β = 16. The sampling rate of

the simulation is selected at 0.001 sec.

HIGH-PASS FILTER

A 4th order Butterworth filter is adopted as the high-pass filter. A cutoff frequency

of 0.1 Hz is selected not to distort the vibration characteristics of the structure while

eliminating any problematic low frequency contents in the actuator displacements.

4.1.2 Numerical Investigation

Numerical simulations are conducted to investigate the stability and accuracy of

the proposed substructure shake table test method under various ground motions.

Numerical simulations here are performed to emulate the data flow in an actual

experimental implementation as shown in Figure 4.2. This time domain process

explicitly includes a step delay of input force to the experimental substructure by the

actuator. Stability of the method will be discussed based on the response of both

81


the structure and the control system. Accuracy of the method will be evaluated with

reference to entire simulations in which the structure is modeled and analyzed as a

whole. Matlab (2011) was employed to perform the numerical simulations.

PERFORMANCE UNDER EARTHQUAKE EXCITATION

The performance of the proposed substructure method under earthquake ground

excitations is evaluated here. As an example simulation, numerical results of the sub-

structured simulation and the entire simulation using the 1995 Kobe ground motion

are presented first.

To demonstrate the simulation process, the results are discussed in order. Once the

computational substructure is excited by the top floor acceleration of the experimental

substructure, the computational base shear is converted to the relative acceleration

of the controlled mass. This acceleration is numerically integrated to obtain the

relative displacement required to produce the equivalent inertial force (Equations

4.8-4.10). As mentioned earlier, a high-pass filter is applied to the directly-integrated

displacement to ensure stability. The filtered displacement is used as the target

displacement for the actuator. Then, the actuator produces the achieved displacement

with reference to the target. These three displacements are shown in Figure 4.4. This

figure shows the effect of the high-pass filter and the difference between the target

and achieved displacements due to the actuator dynamics. As shown in the figure,

the target and achieved displacements are within the actuator stroke limit.

A comparison of the computational base shear and the inertial force of the mass

82


6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8−15

−10

−5

0

5

10

15

Time (s)

Dis

pla

ce

me

nt

(mm

)

Direct−Int. Disp.

Target Disp.

Achieved Disp.

Figure 4.4: Displacement comparison for controlled mass system in a simulation usingKobe ground motion.

is shown in Figure 4.5. The inertial force of the mass is the result from the motion

of the actuator discussed previously. While some variances exist, the achieved force

produced by the actuator provides excellent agreement with the computational base

shear. These time histories indicate the control system in the proposed method is

capable of producing the required interface forces. Structural responses including floor

displacements, velocities and accelerations from entire and substructured simulations

are shown in Figure 4.6. These comparisons are made at the top floor of the 5-story

structure. While some discrepancies can be seen in the acceleration time histories, the

velocity and displacement responses in the substructured simulation show comparable

results with the entire simulation. Based on these simulation results, the proposed

substructure method has a potential to serve as an alternative to the conventional

83


6 6.5 7 7.5 8−50

−40

−30

−20

−10

0

10

20

30

40

50

Time (s)

Forc

e (

N)

Comp. Base Shear

Achieved Force

Figure 4.5: Comparison of force achieved by the controlled mass system and targetcomputational base shear in a simulation using the Kobe ground motion.

testing of entire structures. To further investigate the versatility of the proposed

method, simulations are performed with the same 5-story structure subjected to 9

more earthquake ground motions. The peak ground accelerations of the selected

ground motions are scaled to 0.1g. The suite of earthquakes was chosen to subject the

5-story structure to earthquakes with a variety of characteristics including different

frequency contents. The results from these simulations are assessed based on the

Root Mean Squared (RMS) difference between responses of the substructured and

entire simulations given by Equation 4.13.

∆ =

√√√√ 1

m

m∑i=1

(φE[l] − φS[l])2 (4.13)

Where ∆ is the RMS difference; φ is the simulation response; subscripts E and S

84


6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8

−0.2

0

0.2

Time (s)

Accele

ration (

m/s

2)

Entire

Substructured

6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8

−0.02

−0.01

0

0.01

0.02

Time (s)

Velo

city (

m/s

)

6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8

−2

−1

0

1

2

Time (s)

Dis

pla

cem

ent (m

m)

(a)

(b)

(c)

Figure 4.6: Comparison of top floor structural responses: (a) accelerations; (b) ve-locities; (c) displacements in a simulation using the Kobe ground motion.

85


Table 4.2: RMS differences for simulation responses under different earthquakesimulations.

Controlled mass response Top floor structural response

Force Displacement Acceleration

∆(N) PCBS(N) ∆/PCBS ∆(mm) PTFD(mm) ∆/PTFD ∆(g) PTFA(g) ∆/PTFA

Chi Chi, 1999 2.627 40.510 0.065 0.095 1.642 0.058 0.121 0.305 0.396

Coalinga, 1983 3.610 64.290 0.056 0.138 2.595 0.053 0.166 0.398 0.416

Duzce, 1999 1.264 20.250 0.062 0.042 0.808 0.052 0.051 0.208 0.248

El Centro, 1940 7.360 48.320 0.152 0.255 2.127 0.120 0.343 0.405 0.847

Imperial Valley, 1940 2.638 38.130 0.069 0.103 1.585 0.065 0.117 0.292 0.401

Kobe, 1995 1.042 43.330 0.024 0.055 1.895 0.029 0.034 0.288 0.117

Kocaeli, 1999 3.750 46.120 0.081 0.133 2.023 0.066 0.173 0.332 0.519

Landers, 1992 2.658 40.970 0.065 0.104 1.873 0.055 0.119 0.379 0.314

Loma Prieta, 1989 1.486 23.130 0.064 0.061 0.954 0.064 0.063 0.160 0.394

Morgan, 1984 2.388 35.900 0.067 0.103 1.551 0.067 0.104 0.278 0.373

PCBS, peak computational base shear; PTFD, peak top floor displacement; PTFA, peak top floor acceleration.

denote entire and substructured simulations, respectively; m is the number of time

steps; and l is the time step index.

Table 4.2 shows the RMS differences of the interface force, top floor displacements

and accelerations along with the peak responses and their ratios. It can be observed

that the RMS differences of the interface force are relatively small compared with

their peaks, and their ratios are kept within 0.09 except for the El Centro record.

Similarly, the RMS differences of the top floor displacement are also within a small

range except for the El Centro record. On the other hand, the RMS differences of

the top floor acceleration show larger ratios with respect to their peaks than those of

interface force and the top floor displacement.

The results show that the accuracy of the proposed substructure shake table meth-

ods varies with the input ground motion. This performance variation is considered due

to the effect of actuator dynamics on force tracking in the different frequency contents

of the ground motion as well as error propagation over the simulation time; each input

86


ground motion has unique frequency contents and the actuator control performance

including magnitude and phase varies with frequency. It should be mentioned that

further performance variation and degradation are expected in experimental investi-

gation due to control issues in shake table testing, measurement errors, time delay of

data communication, structural modeling errors and so forth. However, judging from

the above assessment, the overall performance of the proposed substructure method is

quite comparable to the conventional simulation of entire structure. While it cannot

be concluded for general cases from the limited simulation conditions, the results here

demonstrated a potential of the proposed substructure method as a means to assess

the structural responses due to earthquake ground motions.

PERFORMANCE UNDER RANDOM EXCITATION

The earthquake simulations and their assessment in the previous section provide

insight to the proposed method in the time domain. To further investigate the in-

fluence of actuator dynamics, performance of the substructure method is evaluated

in the frequency domain in this section. The same 5-story structure separated at

the second floor is subjected to Gaussian band-limited white noise with a frequency

range from 0.1 to 50Hz. To include the effect of the process in experimental imple-

mentation, the simulation is performed using numerical time step integration as in the

earthquake ground motion. Using spectral techniques, frequency response functions

from the ground acceleration to the structural response are obtained.

Figure 4.7 shows the frequency response functions from ground acceleration to

87


the top floor acceleration under the following cases: substructured systems with no

filtering and ideal actuator model; with filtering and ideal actuator model; with fil-

tering and realistic actuator model. Filtering here refers to the process to ensure

a feasible reference displacement for the actuator in Figure 4.2. The ideal actuator

model means unit magnitude and zero phase for the whole frequency range. These

frequency response functions are compared with the frequency response function of

the entire structure.

As shown in the figure, the substructure with no filtering and ideal actuator model

produced an identical frequency response function to the entire structure. With the

filtering process and ideal actuator model, the frequency response function shows al-

most identical characteristics to the entire structure except around the first natural

frequency. On the contrary, the frequency response function with filtering and re-

alistic actuator model exhibits differences with the entire structure around certain

frequencies: the first mode peak response is reduced while the second mode peak

is amplified. The response around 8.5 Hz is de-amplified. These results show that

the actuator dynamics has an influence on the response of the substructured system.

Thus, actuator dynamics is expected to play a significant role in the experimental

implementation of the substructure method.

Next, the influence of actuator dynamics at different separation floors is evalu-

ated. Figure 4.8 compares the frequency response functions of substructured systems

with separation floor 1 and separation floor 2 to the entire structure. These frequency

88


0 5 10 1510

−2

10−1

100

101

Frequency (Hz)

Magnitude

Entire

Sub. No Filt, Ideal Act

Sub. Filt, Ideal Act

Sub. Filt, Act

0 5 10 15−200

−150

−100

−50

0

50

100

150

200

Frequency (Hz)

Phase (

Deg)

(a)

(b)

Figure 4.7: Comparison of top floor acceleration frequency response functions forsubstructured systems with no filtering and ideal actuator model, filtering and idealactuator model, filtering and realistic actuator model to entire structure: (a) magni-tude and (b) phase.

response functions include filtering and a realistic actuator model. As shown in the

figure, the separation floor does not have much influence in the lower frequency range

including the first mode. However, in the higher frequency range, the separation floor

has significant impact on both magnitude and phase characteristics of the substruc-

tured systems. These simulation results show that the separation floor is also one of

the influential factors of the substructure method.

89


0 5 10 1510

−2

10−1

100

101

Frequency (Hz)

Magnitude

Entire

Sub. Separation Floor 1

Sub. Separation Floor 2

0 5 10 15−200

−150

−100

−50

0

50

100

150

200

Frequency (Hz)

Phase (

Deg)

(a)

(b)

Figure 4.8: Comparison of top floor acceleration frequency response functions for sub-structured systems with separation floor 1 and separation floor 2 to entire structure:(a) magnitude and (b) phase.

Although it cannot be generalized from the limited simulations presented above,

the substructure method can serve as a promising alternative testing method in cer-

tain situations. Performance of the substructure method is highly influenced by a

number of factors including actuator dynamics and choice of separation floor. To

better understand these factors, identify other influential factors that are not dis-

cussed in this paper, as well as design specification for experimental validation of the

90


method, further studies are needed.

4.2 Interface Compatibility using a Force

Controlled Actuator

In this section, another possible implementation of substructure shake table test-

ing for the evaluation of the lower floors in a tall structure is discussed. Here the

experimental force is directly applied to the experimental substructure through a

force controlled actuator. A schematic of the experimental setup is shown in Figure

4.9. More information on force control of hydraulic actuators can be found in Nakata

(2012). The equations of motion for this implementation are identical to those pre-

sented earlier, Equations 4.2-4.4, except that fm is replaced by the the force from the

force controlled actuator, fe.

This section presents the actuator control scheme and numerical simulations to

evaluate the effectiveness of this implementation.

4.2.1 Actuator Control Scheme

Since substructure shake table testing relies heavily on hydraulic actuators for

the performance of the experimental substructure, the problem of actuator control

must be properly addressed. This study utilizes a centralized control scheme to

91


Figure 4.9: Experimental substructure using a force controlled actuator to apply thecomputational base shear.

handle coupling between the actuators. In this technique, the shake table has a

single independent controller while the force actuator has two controllers. The block

diagram of the substructure shake table test method, including the actuator control

systems is shown in Figure 4.10.

Due to coupling between force controlled actuators, Nakata and Krug (2013) a

centralized control approach is used for this implementation to control both the shake

table and the force controlled actuator. Figure 4.10, introduces additional variables

where: the Conv block converts the ground acceleration to an equivalent ground

displacement for the shake table, xg is the reference ground displacement; ed and

ef are the tracking errors between the shake table and force controlled actuators

respectively; Cf and CD are the controllers for the force controlled actuator; the

shake table is assumed to have a PID controller and ud and uf are the voltages sent

to each of the actuator’s servo-valves.

92


Figure 4.10: Block diagram of substructure shake table test method including actu-ator control system.

Since both the shake table actuator and the external force actuator are connected

through the experimental substructure, there will be coupling between both actuators.

However, since shake tables are typically controlled through displacement feedback,

the effect of the force actuator is negligible and the shake table can be controlled

independently. The same cannot be said for the force-controlled actuator, Nakata

and Krug (2013). Therefore, in this control technique, the force actuator has two

controllers: one for reference tracking and disturbance rejection, Cf , and one to

eliminate the effect of the shake table dynamics on the force controlled actuator, CD.

In terms of actuator control, the experimental substructure can be viewed as a

93


two-input two-output system:

xg e

fe

=

HxgudHxguf

HfeudHfeuf

ud

uf

= HEXP

ud

uf

(4.14)

and the controllers are determined based on the relations in Equation 4.14. The shake

table PID controller can be tuned solely from the relation Hxgud. Once the shake table

controller is fixed, the force feedback controller, Cf , can be designed solely using a

loop shaping, Nakata (2012), approach on Hfeuf. If the shake table is completely

independent of the force-controlled actuator then Hxguf= 0 and a straightforward

choice of CD will decouple the force-controlled actuator from the shake table. Using

the previous assumption the decoupling controller can be formed as:

CD = −Hfeud

Hfeuf

· (PID) (4.15)

This choice of the decoupling controller will reduce the effect of shake table on the

force-controlled actuator (complete decoupling is achieved only if Hxguf= 0). This

control technique, assumes that the shake table is uninfluenced by the force actuator.

However in reality some coupling may exist, but as long as the coupling is very small

relative to the other relationships in Equation 4.14 this control technique is still valid.

Once the controllers are defined, the system is ready to run substructure shake

table testing. However due to the influence of actuator delays on the stability of

94


RTHS, additional measures may be needed to remove the delay from the force con-

trolled actuator. Actuator delay compensation techniques have been well established

and the appropriate compensation algorithm should be chosen based on the specific

constraints of the individual actuator. It should be mentioned that delay compensa-

tion is not needed for the shake table since the ground motion is pre-defined and the

experimental system drives the RTHS.

4.2.2 Numerical Case Study

In this section, a numerical case study is performed to investigate the capabilities

of substructure shake table testing including centralized actuator control. Matlab

Simulink is used to simulate both the substructure shake table test and the refer-

ence entire structure. In this study, the response of a 4-story linear-shear structure

subjected to ground motion is investigated. A substructure shake table test of the

4-story structure is completed to investigate the capabilities of the test method. For

the substructured system, the first floor is experimental while the upper three floors

are computational (refer to Figure 4.1 with n = 4 and i = 1).

Realistic parameter values are selected that are compatible with the size of the

shake table at Johns Hopkins. The parameters and models for the shake table and

force controlled actuators are selected in accordance with Nakata (2012). For sim-

plicity, each story of the structure has the same physical properties: m = 70kg,

k = 5 · 104N/m and c = 187Ns/m. With these parameter choices the first vibration

95


Figure 4.11: Performance of experimental setup during step input tests: a.) shaketable displacement; b.) force from second actuator.

mode of the entire structure is 1.48 Hz with a damping ratio of 1.75%.

Before the results of substructure table testing are discussed, the performance of

the experimental substructure is investigated. The performance of the experimental

setup depends on a few criteria, namely: the ability of each actuator to accurately

track its reference signal with little effects of coupling and the time delay of the force-

controlled actuator. The actuator controllers were designed based on the methodology

from the previous section. Results from step input simulations are used to evaluate

the effectiveness of the centralized control strategy. First a step displacement is sent

to the shake table while the force actuator has a zero force reference, then the shake

table has a constant displacement reference while the force actuator receives a step

force input. The results from this simulation are shown in Figure 4.11.

The results from this simulation show that the choice of controllers yields very

good performance from the experimental setup. The shake table is able to follow

the reference displacement with no influence from the force actuator Figure 4.11a.

96


The force-controlled actuator is also able to track the reference step force with only

a small influence from the shake table. The force actuator has a 4N response to the

shake table step, which quickly dies out. It is worth noting the same simulation was

investigated without the decoupling controller and in that case the force actuator had

a 425N response to the shake table step. Thus the decoupling controller reduces the

interaction between the shake table and force actuator by approximately 99%.

While both actuators are able to successfully track their reference inputs indepen-

dently, the force actuator has a relatively large time delay of about 12.5ms. This time

delay is too large for implementation in RTHS. To reduce this delay, the reference

force is passed through a delay compensator block before being sent to the force actu-

ator. The delay compensation algorithm used here is an inverse based compensation

consistent with the discussion in Section 3.6.2. Where the reference signal is sent

through a pseudo-inverse model of the closed loop force actuator. With the addition

of the delay compensation algorithm, the force actuator time delay is brought down

to 4ms, which is suitable for implementation in the substructure shake table test.

To evaluate the substructure shake table test method, a simulation is performed

with the entire RTHS system implemented. The ground motion record used for this

evaluation is the 1995 Kobe ground motion, with the peak ground acceleration scaled

to 0.2 g. A plot of the acceleration tracking performance of the shake table is shown in

Figure 4.12. As shown in Figure 4.12, the shake table reproduces the reference ground

acceleration within a reasonable degree of accuracy. The shake table exhibits a small

97


Figure 4.12: Shake table acceleration during Kobe simulation: a.) entire record; b.)zoomed-in view.

time delay however shows little to no influence from the force controlled actuator.

The performance of the force controlled actuator during the simulation is shown in

Figure 4.13.

The force-controlled actuator is shown to accurately replicate the desired force

from the computational substructure through out the simulation. As shown in Figure

4.13b, a large amount of the actuator time delay is removed by using the corrected

magenta line as the command to the actuator. Although the measured force still

lags behind the true reference, it is acceptable and the simulation was stable. It is

also worth noting that there is no influence from the shake table dynamics and the

addition of the decoupling controller was successful.

Figures 4.12 and 4.13 indicate that the centralized control strategy allows for a

stable simulation and acceptable performance from both the shake table and the force

actuator. Next the accuracy of the structural performance is discussed. To evaluate

98


Figure 4.13: Experimental force during Kobe simulation: a.) entire record; b.)zoomed-in view.

the effectiveness of the substructure shake table test, the substructured response is

compared to a simulation of the entire 4-story structure. To ensure a fair comparison

of results, the input ground motion for the entire structure is the produced accelera-

tion of the shake table during the substructured simulation. A comparison of the 4th

floor absolute acceleration from both simulations is shown in Figure 4.14. A com-

parison of the top floor accelerations from both substructured and entire simulation

shows that the substructure shake table test was able to accurately reproduce the

response of the reference entire structure, Figure 4.14. Figure 4.14a indicates that

the substructured system has almost identical vibration characteristics as the entire

structure. However during the free vibration portion of the simulation, the response

of the substructured system decays quicker than the entire structure. This obser-

vation indicates that the substructure system has slightly more damping than the

entire structure. These observations are again confirmed through a frequency domain

99


Figure 4.14: 4th floor absolute acceleration during Kobe simulations: a.) time histo-ries; b.) Fourier Transform of time histories.

comparison, Figure 4.14b. Here both responses have almost identical characteris-

tics except at the first natural frequency of the structure, where the substructured

response has smaller magnitude due to the larger damping ratio.

Overall the substructure shake table simulation performed exceptionally and was

able to recreate the response of the entire structure within a reasonable tolerance.

The RMS error between the top floor accelerations of both simulations was only

13%. While the results presented in this study are limited to a single simulation, the

simulation data suggests that the substructure shake table test method presented in

this paper could serve as a viable alternative to full-scale shake table tests.

100



This chapter introduced a substructure shake table test method where the lower

stories of a building are tested on a shake table and the upper stories are analyzed com-

putationally. Two possible approaches were examined to impose the computational

forces on the experimental substructure. The first technique incorporates controlled

masses that generate an inertial force equivalent to the computational base shear

and the second technique utilized force controlled actuators to directly impose the

computational base shear.

Numerical simulations were completed to evaluate the performance of each im-

plementation method. The technique using controlled masses was able to accurately

replicate the response of the reference entire structure during a suite of earthquake

simulations. While results for the force controlled implementation were only shown

for a single earthquake, this method was used to accurately simulate the response

of the structure to other earthquakes as well. The research presented in this chap-

ter presented viable techniques to allow researchers to experimentally evaluate the

performance of structural members in the lower portion of a tall building during an

earthquake.

101

Chapter 5

Acceleration Feedback Control of

Shake Tables

The two methods of substructure shake table testing presented in Chapters 3 and

4 require accurate acceleration control of shake tables. While some work has been

completed in this field, acceleration control remains a challenging problem. Thus

most substructure shake table testing implementations use displacement controlled

tables. However improved accuracy can be achieved if acceleration controlled shake

tables are used in substructure shake table testing implementations.

To overcome the limitations introduced by displacement controlled shake tables

this chapter develops a novel acceleration control strategy for shake tables that does

not require any displacement feedback. The contents of this chapter were previously

published in Stehman and Nakata (2013)

102

CHAPTER 5. ACCELERATION FEEDBACK CONTROL OF SHAKE TABLES

5.1 Acceleration Feedback Control with

Force Stabilization

This section presents a new approach for acceleration tracking of shake tables

that adopts direct acceleration feedback control without displacement feedback. Im-

plementation using direct acceleration control of shake tables is inherently unstable

since zero acceleration of the table platform does not necessarily imply that the table

is motionless (i.e. table motion with constant velocity moves with zero acceleration).

Thus due to issues of table drift, implementation of acceleration feedback for shake

tables is fairly limited. In this study, force feedback control is incorporated into the

actuator control system to provide stability for preventing table drift.

5.1.1 Control Architecture

The goal of the control system in shake table testing is to reproduce reference

accelerations. Figure 5.1 shows the block diagram of the proposed acceleration control

strategy used to meet the above goal.

The control system consists of two parallel feedback loops: an acceleration control

loop and a force control loop. In the acceleration control loop, the reference accel-

eration, ar, is first pre-filtered by the pre-gain, Pa, to obtain the modified reference

acceleration, ar. This technique is used to compensate the dynamics of the closed

loop controller. This modified reference acceleration is then sent to the acceleration

103


!"

+!"

Acc. FB.

Controller

Force FB.

Controller Force Pre-Gain

Acc. Pre-Gain

+

Shake Table/Structure Dynamics

Pa

Pf

Ca

C f

Hau

H fu

ar

ar

fru f

ua

u

fm

am

Figure 5.1: Block diagram of proposed acceleration control strategy.

feedback control loop. The acceleration feedback controller, Ca, generates the valve

command, ua, from the acceleration error between the modified reference and the

measured accelerations. In the same way, the reference acceleration is forwarded to

the force control loop that contains the force pre-gain, Pf , and the force feedback

controller, Cf . The reference force, fr, is the converted signal from the reference ac-

celeration and is sent to the force feedback control loop. The force feedback controller,

Cf , generates the valve command, uf , from the force error between the reference and

the measured forces.

The total valve command is the sum of the acceleration and force valve commands.

As shown in the block diagram, acceleration and force measurements of the table are

used as feedback in the servo control loops; in the figure, transfer functions, Hau,

and Hfu, denote the open-loop dynamics of the shake table from the valve command

to the measured acceleration and force, respectively. It should be noted that the

104


control strategy proposed here does not utilize displacement measurement, and thus

it is fundamentally different from the conventional shake table controllers that rely

on displacement feedback.

The closed-loop transfer functions from the reference acceleration to the measured

acceleration and force can be expressed as:

Hamar =amar

=CaPa + CfPf

1 + CaHau + CfHfu

Hau (5.1)

Hfmar =fmar

=CaPa + CfPf

1 + CaHau + CfHfu

Hfu (5.2)

It can be seen that all of the controller terms, namely Pa, Pf , Ca and Cf , affect the

closed-loop acceleration and force transfer functions. In this study, those controller

terms are designed based on the above transfer functions: acceleration tracking per-

formance is evaluated from the acceleration transfer function while stability is judged

from the force transfer function. Therefore, those controller terms need to be designed

for each test setup.

5.1.2 Hardware Requirements

The acceleration control strategy developed in this study also has certain require-

ments in terms of hardware. The first requirement of the proposed control strategy is

a restoring force member between the shake table and the fixed base. The restoring

force member such as springs provide forces proportional to the absolute position of

105


Figure 5.2: Schematic of a uni-axial shake table setup for the proposed accelerationcontrol strategy.

the table, allowing force from the restoring member to serve as a reference to the

table position. To measure the force from the member, a force transducer such as a

load cell has to be installed in series with the actuator rod and table platform.

Figure 5.2 shows a schematic of a uni-axial shake table test setup to illustrate

the requirements for the proposed combined acceleration control strategy. The spring

attached between the shake table and the fixed base provides the restoring force

that is proportional to the position of the table. It should be mentioned that under

dynamic loading the measurement from the load cell includes inertial forces of the

table and test structure; base shear from the test structure; and restoring force from

the spring. For acceleration feedback, accelerometers have to be mounted on the

shake table. It should be also clarified that because the displacement signal is not

used in the proposed control strategy, linear variable differential transducers (LVDTs)

are not required in the test setup.

106


5.2 Experimental Setup

To demonstrate an application of the proposed acceleration control strategy, shake

table tests are conducted using the uni-axial shake table in the Smart Structures and

Hybrid Testing (SSHT) laboratory at the Johns Hopkins University. Figure 5.3a

shows the uni-axial shake table with a three-story test structure.

The shake table has a 1.2m x 1.2m (110kg) aluminum platform driven by a Shore

Western hydraulic actuator (Model: 911D). The actuator has a dynamic load capacity

of 27kN and a maximum stroke limit of 7.6cm. An MTS 252 series dynamic servo

valve is used to control the fluid flow through the actuator chambers. The table

platform is capable of moving a maximum payload of 0.5 ton at an acceleration of 3.8

g. To meet the hardware requirements discussed in Section 2.2, an accelerometer, a

22.5kN dynamic load cell and four compression springs connecting the table platform

to the floor were installed. Together the springs act as a single 15.24 kN/m restoring

member; 2 of the springs are shown in Figure 5.3b. The overall spring stiffness

was chosen such that static drifting of the table resulted in reasonable force levels

measured through the load cell.

The test structure is 2m tall with individual floor masses of 60kg and a total mass

(including support connection) of approximately 225kg. Because the total mass of

the structure is more than double the mass of the table platform, the structure has a

high influence on the dynamics of the actuator than the table platform. Thus, the test

setup here is subjected to a high influence of control-structure interaction, and is used

107


Table 5.1: Dynamic characteristics of three story test structure.

Natural Frequency, Hz Damping Ratio, %Vibration Mode 1 2.04 2.17Vibration Mode 2 13.5 0.41Vibration Mode 3 37.9 0.37

to demonstrate the capabilities of the proposed control strategy for such challenging

conditions. Dynamics characteristics of the three-story structure are listed in Table

5.1.

5.3 Experimental Investigation of the Pro-

posed Acceleration Control Strategy

The proposed acceleration feedback control strategy is experimentally investigated

using the shake table with the three-story structure described in the previous section.

First, the open loop dynamics of the shake table system are experimentally obtained

and modeled using system identification techniques. Then, based on the system mod-

els, acceleration and force feedback controllers along with pre-gains are designed. An

experimental investigation is performed using a series of earthquake ground motions

as the reference to the shake table. This section presents details of the system models,

controller design and experimental results.

108


Figure 5.3: The shake table in the SSHT lab at Johns Hopkins: (a) shake table withthree-story structure; (b) view of restoring springs.

109


5.3.1 Experimental Modeling of the Shake Table

System

Open loop dynamics of the shake table system are determined using experimental

system identification techniques for the design of the controllers. Relationships of

interest here are those from the valve command to measured acceleration and to

measured force. Those open loop relationships are the primary plants, Hau and Hfu,

in the proposed acceleration feedback control strategy, as shown in Figure 5.1.

Figure 5.4 shows the experimentally obtained open loop relationships: valve com-

mand to measured acceleration (Figure 5.4a) and valve command to measured force

(Figure 5.4b). The valve to acceleration relationship has small magnitude at low

frequencies. The magnitude begins to increase with frequency. Then a pole-zero

pair appears around 13Hz, which corresponds to the second natural frequency of the

structure, influence from the first natural frequency is not apparent. Around 25Hz, a

peak is present due to vibrations of support connections after which the magnitude

begins to decrease. Another pole-zero pair occurs around 38Hz, which is the third

natural frequency of the three-story structure. On the other hand, the valve to force

relationship exhibits peaks and valleys within 5Hz. The first zero appears at 1.45

Hz, which corresponds to the natural frequency of the table with springs. The first

peak and the second valley are a pole-zero pair around 2.5 Hz, which is close to the

first natural frequency of the structure. Beyond 5 Hz, the general trend in the valve

110


0 10 20 30 4010

−2

10−1

100

Frequency (Hz)

Magnitude

Experimental

Analytical

0 10 20 30 4010

0

101

102

103

104

105

Frequency (Hz)

Magnitude

Experimental

Analytical

(a) (b)

Figure 5.4: Open loop dynamics for shake table system: (a) valve command to mea-sured acceleration; (b) valve command to measured force.

to force relationship is similar to the valve to acceleration relationship except for the

level of influence of the third mode of the structure around 38 Hz. It can be clearly

seen that the dynamics of the test structure influences the open loop dynamics of the

shake table system, indicating a high level of control-structure interaction in the test

setup.

Analytical models of the relationships are obtained using curve fitting techniques

and are also plotted in Figure 5.4. The valve to acceleration relationship and the valve

to force relationship are captured by 8th order and 9th order rational polynomial

functions, respectively. The mathematical representations of the analytical models

are presented in Table 5.2. As shown in Figure 5.4, the analytical models capture the

experimental relationships with reasonable accuracy throughout the entire frequency

range.

111


Table 5.2: Analytical representations of the open loop shake table dynamics.

Hau = 5.553·106s5+1.444·107s4+3.561·1011s3+3.251·1011s2+2.288·1015ss8+4.774·102s7+1.499·105s6+4.403·107s5+6.946·109s4+1.030·1012s3+1.019·1014s2+5.216·1015s+4.195·1017

Hfu = 1.054·1010s6+4.850·1010s5+8.500·1013s4+3.3541014s3+2.449·1016s2+4.818·1016s+7.782·1017s9+8.416·102s8+1.859·105s7+3.144·107s6+4.369·109s5+1.878·1011s4+2.230·1013s3+4.525·1013s2+3.359·1015s

5.3.2 Design of the Feedback Controllers and Pre-

Gains

A controller design of the proposed acceleration feedback control strategy is de-

veloped employing analytical models of the open loop dynamics of the shake table

system. In this study, the acceleration feedback loop including pre-gain and feedback

controller are designed to provide acceleration tracking of the shake table system while

the force feedback loop is designed to provide stability to prevent table drift. Because

acceleration tracking is the main goal of this study, the acceleration feedback con-

troller, Ca, is considered first. A loop shaping design methodology is employed for the

design of the acceleration feedback controller. Loop shaping is a frequency-domain

approach where the product of the controller and the plant, referred to as the loop

transfer function, is formed to have desirable frequency characteristics, Doyle and

Stein (1981). In this study, the acceleration feedback controller is designed to com-

pensate the dynamics of the valve to acceleration relationship in the entire frequency

range of interest.

Figure 5.5a shows the frequency domain characteristics of the designed accelera-

tion feedback controller along with the valve to acceleration relationship. The math-

ematical representation of the designed acceleration feedback controller is expressed

112


0 10 20 30 4010

−4

10−2

100

Frequency (Hz)

Ma

gn

itu

de

Hau

Ca

0 10 20 30 40

10−2

100

Frequency (Hz)

Ma

gn

itu

de

P

a=1

Pa=2.75

0 10 20 30 40

10−5

100

Frequency (Hz)

Ma

gn

itu

de

Hfu

Cf

0 10 20 30 4010

−4

10−2

100

Frequency (Hz)

Ma

gn

itu

de

(a)

(c)

(b)

(d)

Figure 5.5: Controller design for the proposed acceleration control strategy: (a) accel-eration feedback controller; (b) acceleration closed-loop frequency response function;(c) force feedback controller; (d) force closed-loop frequency response function.

as:

Ca = 30s6+1.272·103s5+2.730·106s4+8.190·107s3+6.218·1010s2+5.263·1011s+3.107·1014s7+1.876·102s6+7.236·104s5+1.202·107s4+9.200·108s3+8.167·1010s2+3.197·1012s+3.213·1013 (5.3)

As seen in the figure, the acceleration feedback controller is essentially the reciprocal

of the valve to acceleration relationship.

The acceleration closed-loop frequency response function from the choice of Ca in

Equation 5.3 is plotted in Figure 5.5b where the pre-gain Pa is set to 1. While it

contains certain desirable characteristics (i.e. flat plateau and decaying magnitude in

113


higher frequencies), its magnitude is too low (approximately 0.36 between 1.5 and 10

Hz). To raise the magnitude to unity, the pre-gain is set to 2.75. The acceleration

control loop with the pre-gain of 2.75 is also plotted in Figure 5.5b. The figure

shows that the selection of Pa(=2.75) and Ca provides desirable acceleration control

performance as well as robustness in high frequencies. It should be noted that while

the acceleration control loop in Figure 5.5b exhibits good performance, it does not

ensure stability for table drift.

To have stability against table drift, the force feedback controller is incorporated

as discussed in Section 5.1.1. However, the impact of the force feedback loop should be

minimized to maintain the acceleration tracking performance. To meet the stability

criterion, the force feedback controller is designed such that force closed-loop transfer

function contains unit magnitude around 0Hz. To ensure small impact of the force

feedback loop on the acceleration performance, the pre-gain for the force controller,

Pf , is set to 0, and the force feedback controller is designed to have sufficiently small

magnitude at higher frequencies in the force closed-loop transfer function; the pre-

gain of zero converts reference acceleration into a zero static reference force. To have

a smoother transition from low to high frequencies in the force closed-loop transfer

function, poles and zeros are placed in the force feedback controller to compensate

the dynamics of the valve to force relationship.

Figure 5.5c and 5.5d show the frequency response function of the designed force

feedback controller and corresponding force closed-loop frequency response function,

114


respectively. The designed force feedback controller is expressed as:

Cf = 3.6·10−1s6+1.498·101s5+1.175·104s4+1.206·105s3+6.634·107s2+6.858·107s+1.008·1010s7+1.006·103s6+3.214·105s5+4.322·107s4+2.538·109s3+6.216·1010s2+4.086·1011s+2.676·1012 (5.4)

As shown in Figure 5d, the force closed-loop frequency response function has unit

magnitude around 0Hz and smaller magnitude at higher frequencies while compen-

sating the dynamics of the valve to force relationship.

Finally, the overall performance and stability of the proposed acceleration con-

trol strategy using the choices of Ca, Pa, Cf and Pf are examined. Figure 5.6a

shows the closed-loop acceleration to force relationship for stability assessment. The

acceleration-to-force relationship has finite magnitude in the neighborhood of 0Hz,

indicating the stability of the shake table system to prevent table drift. Then, the

overall acceleration performance of the proposed control strategy with the incorpo-

ration of the force feedback loop is shown in Figure 5.6b. As shown in the figure,

the designed force feedback loop hardly influences the overall acceleration transfer

function. It can be seen from these figures that the designed acceleration control

strategy provides acceleration control performance while gaining stability to prevent

table drift.

115


0 10 20 30 4010

−2

10−1

100

Frequency (Hz)

Magnitude

With Force Feedback

Without Force Feedback

0 10 20 30 4010

1

102

103

104

Frequency (Hz)

Magnitude

(a) (b)

Figure 5.6: Closed-loop frequency response functions of shake table system using theproposed acceleration control strategy: (a) reference acceleration to measured forcemagnitude; (b) reference acceleration to measured acceleration magnitude.

5.3.3 Experimental Validation of the Proposed Ac-

celeration Control Strategy

The proposed acceleration control strategy with the designed feedback controllers

and pre-gains is experimentally validated using a series of earthquake ground mo-

tions. Figure 5.7(a and b) show the time histories of the reference acceleration and

the measured table accelerations from the JMA record of the 1995 Kobe earthquake.

For comparison, the measured table acceleration using a conventional displacement

control strategy with PID controller is also plotted. Overall both acceleration and

displacement control strategies show reasonable agreement with the reference accel-

eration (see Figure 5.7a). However, when inspected closely (see Figure 5.7b), the

acceleration control strategy exhibited smaller amounts of discrepancy than the dis-

placement control strategy. The two control strategies are also compared in the

frequency domain as in Figure 5.7c. The figure reveals that while the performance

116


1 2 3 4 5 6

−0.5

0

0.5

Time (s)

Accele

ration (

g)

Referecence

Acceleration Control

Displacement Control

2.5 2.6 2.7 2.8 2.9 3

−0.5

0

0.5

Time (s)

Accele

ration (

g)

0 10 20 30 4010

−4

10−3

10−2

10−1

Frequency (Hz)

Fourier

Am

plit

ude

(a)

(b) (c)

Figure 5.7: Results with the 1995 Kobe ground motion as the reference accelera-tion: (a) shake table acceleration tracking comparison; (b) close up view of tableaccelerations; (c) frequency domain comparison of table accelerations.

of both acceleration and displacement control strategies are similar, the acceleration

control strategy has less high-frequency errors.

The measured force time history from the acceleration control strategy is shown

in Figure 5.8. While the force varies during the intense part of the acceleration time

history, it returns to zero afterwards. Because the force control loop regulates static

forces, the table platform is stable and table drift is not observed. In addition to

the performance and stability assessment, the displacement time histories from both

acceleration and displacement control strategies are compared in Figure 5.9. It is

interesting to see from the figure that the table displacements are quite different even

though acceleration time histories from both control strategies are similar (see Fig-

ure 5.9). This observation indicates that two unique displacement time histories can

117


2 4 6 8 10 12

−2000

−1000

0

1000

2000

Time (s)

Fo

rce

(N

)

Figure 5.8: Measured table force from the acceleration control strategy using theKobe reference acceleration.

produce almost identical acceleration time histories. However, from the experimen-

tal results here, the actual displacement experienced during this earthquake cannot

be fully identified. It may also be inferred that deducing the displacement from a

given acceleration time history that is often done in conventional displacement con-

trol strategies may not result in the true ground displacements of the earthquake.

Further performance evaluations are conducted using a series of earthquake ground

motions. A summary of the performance evaluations is discussed using Root Mean

Squared (RMS) percentage errors between reference and measured table accelerations.

Table 5.3 presents errors from both acceleration and displacement control strategies.

All reference accelerations are scaled to a peak ground acceleration of 0.5g. In all

of the tests, acceleration control strategy produces less RMS errors than displace-

ment control strategy. The average RMS error in acceleration control strategy is

118


1 2 3 4 5 6−20

−10

0

10

20

Time (s)

Dis

pla

ce

me

nt

(mm

)



Figure 5.9: Comparison of measured table displacements with the Kobe referenceacceleration.

9.12% whereas the average RMS error in displacement control strategy is 11.29%.

Furthermore, the variation of the RMS errors in the acceleration control strategy is

more consistent than the displacement control strategy where the RMS errors range

from 9.63% to 13.62%. These test results prove that the acceleration control strategy

developed in this study provides more accurate acceleration tracking than the conven-

tional displacement control strategy, showing a 19.2% improvement. In earthquake

engineering emphasis is placed on the peak response of structure, for this reason a

ratio is taken between the peak measured shake table acceleration and reference ac-

celeration, these results are shown in Table 5.3 for each earthquake. The acceleration

control strategy was able to reproduce the peak ground accelerations more accurately

than the displacement control strategy. The acceleration control strategy under am-

plified the peak ground acceleration by 7% while the displacement tended to over

119


Table 5.3: Errors between measured and reference shake table accelerations.

Acceleration Control Displacement ControlRMSE (%) Peak Ratio RSME (%) Peak Ratio

Chi Chi, 1999 08.80 0.98 10.48 1.18Coalinga, 1983 09.14 1.03 10.43 1.39Imperial Valley, 1940 08.49 0.90 11.15 1.15Kobe, 1995 08.48 0.89 09.63 1.12Landers, 1992 08.43 1.04 12.84 1.12Loma Prieta, 1989 08.47 0.79 09.93 1.17Morgan Hill, 1984 11.01 0.95 12.55 1.35Northridge, 1994 10.35 0.79 13.62 1.30Taiwan, 1999 08.94 1.00 10.94 1.14Average 09.12 0.93 11.29 1.24

amplify the peak ground acceleration by 24% on average.

5.4 Impact of Input Acceleration Errors

in Shake Table Tests on Structural Re-

sponse

Uncertainties are inherent in structural response under dynamic loading. Common

sources of uncertainties are material properties, construction qualities, deterioration,

excitation disturbances, environmental effects, etc. While some of the uncertainties

are difficult to measure and often treated as stochastic processes, some can be quan-

titatively assessed. One of such uncertainties is the accuracy of the input ground

motion in shake table testing.

This section investigates the impact of acceleration errors in shake table control

120


(difference between reference and measured accelerations of the shake table) and their

affect on the measured structural response. The investigation is carried out using

measured structural responses from the previous shake table tests. The response

from an analytical model of the three-story structure is used as the reference for the

experimental response.

The top floor acceleration responses of the structure during the 1995 Kobe ground

motion are shown in Figure 5.10. Figure 5.10a shows that while structural responses

are mostly captured using inputs from both acceleration and displacement control

strategies, differences can be seen in the magnitude of the responses; the structural

response in the displacement-controlled test shows larger error in the magnitude. In

addition to the difference in magnitude, different frequency contents can be observed

in the close-up view (see Figure 5.10b); the structural response in displacement-

controlled test exhibits more highly frequency contents that are not present in the

reference. Above observations can be further verified through the frequency domain

comparison in Figure 5.10c. Responses from both tests capture the first vibration

mode well. However, higher frequency responses, in particular second and third

vibration modes, are heavily amplified by the displacement-controlled test. Larger

structural response errors in the displacement-controlled test here are considered to

be a consequence of larger errors in input ground motion discussed previously.

Next, the impact of input ground motion errors on the structural response from the

series of earthquake ground motions is evaluated. Table 5.4 presents the RMS errors

121


1 2 3 4 5 6

−1

−0.5

0

0.5

1

Time (s)

Acce

lera

tio

n (

g)

Reference



2.5 2.6 2.7 2.8 2.9 3

−1

−0.5

0

0.5

1

Time (s)

Acce

lera

tio

n (

g)

0 10 20 30 4010

−4

10−3

10−2

10−1

100

Frequency (Hz)

Fo

urie

r A

mp

litu

de

(a)

(b) (c)

Figure 5.10: Comparison of structural responses with the 1995 Kobe ground mo-tion: (a) top floor structural acceleration comparison; (b) close up view of structuralaccelerations; (c) frequency domain comparison of top floor structural accelerations.

for the top floor accelerations from both the acceleration and displacement-controlled

tests. As expected, RMS errors in structural response from acceleration-controlled

tests are smaller than those from displacement-controlled tests. It is interesting to

note that the largest structural response errors did not occur in the test with the

largest input errors. The average RMS error in the structural response is 32.5% for

the acceleration-controlled tests and 53.0% for the displacement-controlled tests. The

improved accuracy is also verified by analyzing the peak structural acceleration during

each earthquake. The displacement control strategy over amplified the structural

response by 200% while the acceleration control strategy only over amplified the

peak structural response by 18% on average. The variability in the the structural

response errors shown in Table 5.4 is a result of the varying amount of amplification

122


Table 5.4: Errors between measured and reference top floor structural accelerations.

Acceleration Control Displacement ControlRMSE (%) Peak Ratio RSME(%) Peak Ratio

Chi Chi, 1999 30.31 1.24 37.26 1.70Coalinga, 1983 30.98 0.93 43.29 1.32Imperial Valley, 1940 62.34 1.24 105.61 3.89Kobe, 1995 14.83 1.19 20.92 1.68Landers, 1992 46.22 1.00 94.62 1.86Loma Prieta, 1989 15.49 1.19 18.02 1.35Morgan Hill, 1984 35.15 0.93 47.66 1.57Northridge, 1994 44.84 1.63 89.05 3.37Taiwan, 1999 12.64 1.23 20.83 1.75Average 32.53 1.18 53.03 2.05

of the second vibration mode occurring during different shake table tests.

From these comparisons of structural responses and the previous discussion about

the input errors, the following are observed. Errors in structural responses (output

uncertainties) are highly influenced by errors in input ground motions (input uncer-

tainties). Acceleration-controlled tests produce smaller input ground motion errors

than displacement-controlled tests, and in turn provide smaller errors in structural

response.


This study introduced a shake table control strategy that employs direct accel-

eration feedback control without need for displacement feedback. The proposed ac-

celeration control strategy incorporates force feedback for stability to prevent table

drift. The acceleration control strategy was experimentally validated using a series

123


of earthquake ground motions.

Experimental results showed that the acceleration control strategy produced more

accurate table accelerations than the conventional displacement control strategy. This

improved control performance resulted in fewer errors in structural responses, reduc-

ing output uncertainties in shake table tests. Thus, the acceleration control strategy

was proven to be more accurate than conventional displacement control strategies.

It was also found that errors in the input ground motion have an impact on the

errors in the structural response. However, the errors in the structural response are

not simply proportional to the errors in input ground motion. Further studies can

address the relationship between input and output uncertainties in shake table testing.

124

Chapter 6

Conclusions and Future Work

The work presented in this dissertation introduced novel techniques to enhance the

current capabilities of shake tables. The research can be categorized into two fields

of research: substructure shake table testing and shake table control. This research

focused on using shake tables in conjunction with real-time hybrid simulation (RTHS)

techniques to enable a wider variety of experimental testing situations. While this

dissertation focused on applications using shake tables, the concepts, techniques and

methodologies can be applied to many other types of RTHS.

6.1 Conclusions

Advancements in experimental testing technologies are needed to ensure mean-

ingful results are obtained from earthquake engineering studies. In this capacity, the

125

CHAPTER 6. CONCLUSIONS AND FUTURE WORK

research community needs to recognize the limitations in existing methods and ex-

plore new techniques to enable more accurate test setups and ultimately more useful

studies. This dissertation introduced novel techniques to improve the way shake ta-

bles are used in earthquake engineering research. Substructure shake table testing

was discussed which allows researchers to test a tall building with only a portion of

the building being tested on the shake table. Since substructure shake table testing

is a challenging problem, the challenges and limitations of the methods were investi-

gated and techniques were developed to enable stable and accurate simulations. Also,

an acceleration feedback control strategy for shake tables was verified that focused on

acceleration tracking of shake tables and stepped away from traditional displacement-

based techniques.

While earthquake engineering is not a new field, there is still a significant amount

of research required to develop and validate new research concepts. The experimental

earthquake engineering community needs to take full advantage of the technology and

research that other fields are producing. Incorporation of novel research from electri-

cal, mechanical and control systems engineering will further advance the capabilities

of earthquake engineering research facilities.

126


6.2 Future Work

While the research discussed in this dissertation provided significant advances in

shake table testing methods, as with all experimental research, future research will

continue to improve upon the work presented herein. The possible research works

that can build off the work in this dissertation are separated into two categories: near

term goals and long term goals.

6.2.1 Near Term

The future works in this section are direct expansions and continuations of the

work presented through out the dissertation.

• Experimental implementations of the controlled mass system needs to be further

investigated. A logical investigation would consist of:

1. Developing robust and accurate controllers for the controlled mass system

alone.

2. Implementation of substructure shake table testing where the controlled

mass system is the only experimental portion to test the capabilities of the

controlled mass system.

3. Including the lower stories and controlled mass system as the experimental

substructure with the upper stories analyzed computationally.

127


4. Complete shake table tests of the entire structure to experimentally vali-

date the substructure shake table test method.

• A further study of shake tables with force controlled actuators using mixed force

displacement control is also needed.

1. Stable and accurate control of the force controlled actuator should be en-

sured while the shake table is fixed.

2. A multivariable control scheme should be developed to allow independent

control of the shake table displacement and auxiliary actuator force.

3. Investigation of delay compensation techniques for force controlled actua-

tors is expected.

4. Implementation of substructure shake table testing with force controlled

actuators.

5. Complete shake table tests of the entire structure to experimentally vali-

date the substructure shake table test method.

• Further investigations of the acceleration feedback control strategy presented in

Chapter 5 will continue to advance the capabilities of shake tables.

1. Numerical simulations and studies will allow researchers to investigate the

promise and limitations of the strategy.

2. The strategy should be tested on a bi-axial shake table.

128


3. This strategy needs to be implemented and verified for shake table tests

where the structural dynamics has significant influence on the shake table.

• The substructure shake table methods in Chapters 3 and 4 need to be investi-

gated for extreme loading conditions that results in nonlinear response of the

test structures.

1. For the methods in Chapter 3, nonlinearities should be added in the com-

putational simulation of the lower stories.

2. For the methods in Chapter 4, the shake table and force control scheme

should be tested for structures that produce nonlinear response.

6.2.2 Long Term

Some extensions of this dissertation are expected to require significant effort and

are viewed as long term goals for the research field. These research goals focus on

various applications of the methods presented in this dissertation to further broaden

the research applications.

• The controlled mass system concept can be expanded to create more complex

loading situations. This can be achieved by using an array of controlled mass

systems which could be used to generate much larger forces than a single system

as well as impose moments.

129


• The concept of force controlled actuators has significant promise in the field.

Further research topics include more complex loading conditions utilizing mul-

tiple force controlled actuators. Mixed variable control needs to be explored to

investigate setups that utilize actuators with different feedback controllers (dis-

placement, force, acceleration, etc...). Also the application of force controlled

testing for static loading would be extremely useful to the field.

• Further developments will continue to display the merits of the acceleration

feedback control strategy. To make the idea more accessible to researchers,

other ways to stabilize the system against table drift should be investigated.

The concept should also be explored for more advanced shake table systems

which can include vertical and rotational motion.

• While this dissertation focused on advancing the experimental portion of earth-

quake engineering, advances in computational efforts are also needed to further

the research field.

While the preceding list is not exhaustive, it contains logical extensions of the

work presented in this dissertation. This research serves as a stepping stone for

future research in the area of shake table testing and other RTHS methods alike.

130

Appendix A: Experimental

Investigations of Lower Story


While only numerical simulations were studied in Chapter 4, the author investi-

gated preliminary experimental implementations of the controlled mass system and

force controlled actuators as well. Additional challenges were experienced in the ex-

perimental investigations that limited the performance and stability of the controlled

mass system and force controlled actuators.

Challenges in the controlled mass system were due to the introduction of high

frequency vibrations when reference forces were sent to the controlled mass system.

While the controlled mass system was stable, the force tracking results were poor.

With the discrepancies between the reference and measured forces, the controlled

mass system could not be incorporated into substructure shake table testing. It is

expected that improved accuracy can be achieved if further studies are performed

131

APPENDIX A

and more robust control systems are investigated.

Experimental investigations of force controlled actuators in addition to the shake

table also proved challenging. While the numerical results showed accuracy and

stability of the mixed force displacement control scheme, decoupling control of the

shake table displacement and auxiliary actuator force was experimentally challenging.

When the control method presented in Section 4.2.1 was implemented experimen-

tally, the auxiliary actuator was unstable and the system had to be shut down. The

limitation in the proposed control scheme is expected to be due to the inability to

accurately model the coupling between the shake table and force controlled actuator,

Hfeud. Furthermore, since an accurate model of the system coupling was not achieved,

the decoupling controller, CD, was ineffective and led to the instability. Stable im-

plementations were achieved when independent displacement and force controllers

were used. However since the independent approach does not address the coupling

between both actuators, performance was limited to only extremely low frequency

command signals. In future implementations the author recommends a more robust

control strategy which accounts for uncertainties in the system modeling.

132

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Vita

Matthew Joseph James Stehman began his academic

career at Ursinus College were he earned a Bachelor of Sci-

ence degree in Mathematics with minors in Physics and

Computer Science. While at Ursinus he was inducted

into both national and international honors societies cor-

responding to his respective disciplines of study: Kappu

Mu Epsilon (Mathematics), Sigma Pi Sigma (Physics)

and Upsilon Pi Epsilon (Computer Science). His research

on analytical modeling of suspension bridge oscillations earned him departmental

honors in Mathematics.

Matthew continued his academic career at the Johns Hopkins University as the

Saiful and Lopa Islam Fellow, where he obtained a Ph.D from Department of Civil

Engineering. En route to his Ph.D Matthew also earned a Master of Science in

Engineering degree from the Department of Mechanical Engineering with a focus in

Dynamics and Control. His doctoral research focused on innovative techniques using

142

VITA

shake tables (earthquake simulators) in the experimental investigation of structural

response during earthquakes. Matthew’s work is recognized both domestically and

internationally, including publications in well regarded scientific journals.

143

advances in shake table control and …...fully dynamic environment with base excitation of the test...

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