ADVANCES IN SHAKE TABLE CONTROL AND SUBSTRUCTURE SHAKE TABLE TESTING. Matthew Joseph James Stehman

Transcription

1 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

2 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 reason 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. Developments 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 between the substructures have limited the number of implementations of substructure shake table testing to date; thus in this dissertation, appropriate methods of addressing 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

3 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 methods. Primary Reader: Narutoshi Nakata Secondary Readers: Benjamin Schafer and Judith Mitrani-Reiser iii

4 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 Substructure Techniques for Shaking Table Tests (grant no. CMMI ). iv

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

6 Contents Abstract ii Acknowledgments iv List of Tables xi List of Figures xii 1 Introduction Literature Review Shake Table Testing Real-Time Hybrid Simulation Substructure Shake Table Testing Overview of Dissertation Shake Table Dynamics Dynamics of Shake Tables with Hydraulic Actuators vi

7 CONTENTS 2.2 Acceleration Relationships Summary and Discussion Substructure Shake Table Testing of Upper Stories in Tall Buildings Formulation Equations of Motion Compatibility Requirements Concept of Substructure Shake Table Testing Experimental Setup and Modeling Uni-Axial Shake Table Control and Data Acquisition System Experimental Substructure Computational Substructure Measurement of Base Shear Acceleration Control Performance Issues of Acceleration Control and Control-Structure-Interaction Propagation of Input Acceleration Errors Substructure Shake Table Test System with Error Compensation Numerical Integration for the Computational Substructure State Observer and Kalman Filter Model-Based Actuator Delay Compensation vii

8 CONTENTS Corrector for Errors in Base Shear Induced by Input Acceleration Errors Experimental Results Harmonic Ground Excitation Inputs Earthquake Ground Excitation Input Advanced Model-Based Shake Table Compensation Techniques Feedforward Compensation using Derivatives of Reference Signal IIR Compensation Technique for Significant Control-Structure- Interaction Experimental Investigation of Model-Based Delay Compensation Techniques Summary and Discussion Substructure Shake Table Testing of Lower Stories in Tall Buildings Interface Compatibility using a Controlled Mass Simulation Models Numerical Investigation Interface Compatibility using a Force Controlled Actuator Actuator Control Scheme Numerical Case Study Summary and Discussion viii

9 CONTENTS 5 Acceleration Feedback Control of Shake Tables Acceleration Feedback Control with Force Stabilization Control Architecture Hardware Requirements Experimental Setup Experimental Investigation of the Proposed Acceleration Control Strategy Experimental Modeling of the Shake Table System Design of the Feedback Controllers and Pre-Gains Experimental Validation of the Proposed Acceleration Control Strategy Impact of Input Acceleration Errors in Shake Table Tests on Structural Response Summary and Discussion Conclusions and Future Work Conclusions Future Work Near Term Long Term Appendix A 131 ix

10 CONTENTS References 133 Vita 142 x

11 List of Tables 3.1 Dynamic properties of the entire 10-story RTHS structure Summary of shake table performance from substructure shake table testing using 3 different compensation techniques Properties and dynamic characteristics of the 5-story structure RMS differences for simulation responses under different earthquake simulations Dynamic characteristics of three story test structure Analytical representations of the open loop shake table dynamics Errors between measured and reference shake table accelerations Errors between measured and reference top floor structural accelerations.123 xi

12 List of Figures 1.1 Schematic of a early stage hand powered shake table, CREDIT: Severn (2011) Schematic of the E-Defense shake table system, CREDIT: Ogawa et al. (2001) Early implementation of pseudo-dynamic testing, CREDIT: Nakashima et al. (1992) Implementation of RTHS where the experimental substructure is a single damper, CREDIT: Carrion et al. (2009) Concept of substructure shake table testing including a tuned mass damper, CREDIT: Igarashi et al. (2000) Schematic of a uni-axial shake table with linear structure Block diagram of shake table system including hydraulic actuator, test structure and feedback controller Schematics of substructure shake table testing in comparison with the entire simulation A block diagram for the concept of substructure shake table testing A three-story steel frame structure on the uni-axial shake table at Johns Hopkins University Frequency response curves of the three-story steel experimental substructure Frequency response curves of closed-loop (reference to measured) displacement and acceleration: (a) displacement magnitude; (b) acceleration magnitude; (c) displacement phase; and (d) acceleration phase Frequency response curve and coherence from the table acceleration to measured base shear A block diagram of the substructure shake table test system with compensation techniques for experimental errors xii

13 LIST OF FIGURES 3.8 Acceleration and base shear time histories under 2.0 Hz harmonic ground excitation: (a) the entire acceleration time histories; (b) a zoomed section of the acceleration time histories; (c) the entire base shear time histories; and (d) a zoomed section of the base shear time histories Structural responses under 2.0 Hz harmonic ground excitation: (a), (d), and (g), relative floor displacement at the 10 th, 6 th and 2 nd floor, respectively; (b), (e), and (h), absolute floor displacement at the 10 th, 6 th and 2 nd floor, respectively; and (c), (f), and (i), absolute floor acceleration at the 10 th, 6 th and 2 nd floor, respectively Acceleration and base shear time histories under 6.0 Hz harmonic ground excitation: (a) the entire acceleration time histories; (b) a zoomed section of the acceleration time histories; (c) the entire base shear time histories; and (d) a zoomed section of the base shear time histories Structural responses under 6.0 Hz harmonic ground excitation: (a), (d), and (g), relative floor displacement at the 10 th, 6 th and 2 nd floor, respectively; (b), (e), and (h), absolute floor displacement at the 10 th, 6 th and 2 nd floor, respectively; and (c), (f), and (i), absolute floor acceleration at the 10 th, 6 th and 2 nd floor, respectively Acceleration and base shear time histories under the 1995 Kobe earthquake excitation: (a) the entire acceleration time histories; (b) a zoomed section of the acceleration time histories; (c) the entire base shear time histories; and (d) a zoomed section of the base shear time histories Structural responses under the 1995 Kobe earthquake excitation: (a), (d), (g), (j), and (m), relative displacement at the even floors from top to bottom (10 th to 2 nd ); (b), (e), (h), (k), and (n), absolute displacement at the even floors from top to bottom (10 th to 2 nd ); and (c), (f), (i), (l), and (o), absolute displacement at the even floors from top to bottom (10 th to 2 nd ) Experimental reference to measured frequency response functions for shake table with 3DOF experimental substructure: (a) and (c) displacement tracking magnitude and phase; (b) and (d) acceleration tracking magnitude and phase Absolute shake table response from substructure shake table tests with 3DOF 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

14 LIST OF FIGURES 3.16 Shake table tracking errors from substructure shake table tests with 3DOF experimental substructure subjected to Kobe earthquake record: (a) and (d) displacement and acceleration errors using feedforward compensator; (b) and (e) displacement and acceleration errors using IIR compensator; (c) and (f) displacement and acceleration errors using iterative compensator Schematics of the Entire and substructure systems A block diagram of the substructure shake table test using controlled masses Closed-loop frequency response function of the 911-D actuator with a mass of 45kg: (a) magnitude; (b) phase Displacement comparison for controlled mass system in a simulation using Kobe ground motion Comparison of force achieved by the controlled mass system and target computational base shear in a simulation using the Kobe ground motion Comparison of top floor structural responses: (a) accelerations; (b) velocities; (c) displacements in a simulation using the Kobe ground motion Comparison of top floor acceleration frequency response functions for substructured systems with no filtering and ideal actuator model, filtering and ideal actuator model, filtering and realistic actuator model to entire structure: (a) magnitude and (b) phase Comparison of top floor acceleration frequency response functions for substructured systems with separation floor 1 and separation floor 2 to entire structure: (a) magnitude and (b) phase Experimental substructure using a force controlled actuator to apply the computational base shear Block diagram of substructure shake table test method including actuator control system Performance of experimental setup during step input tests: a.) shake table displacement; b.) force from second actuator Shake table acceleration during Kobe simulation: a.) entire record; b.) zoomed-in view Experimental force during Kobe simulation: a.) entire record; b.) zoomed-in view th floor absolute acceleration during Kobe simulations: a.) time histories; b.) Fourier Transform of time histories Block diagram of proposed acceleration control strategy Schematic of a uni-axial shake table setup for the proposed acceleration control strategy xiv

15 LIST OF FIGURES 5.3 The shake table in the SSHT lab at Johns Hopkins: (a) shake table with three-story structure; (b) view of restoring springs Open loop dynamics for shake table system: (a) valve command to measured acceleration; (b) valve command to measured force Controller design for the proposed acceleration control strategy: (a) acceleration feedback controller; (b) acceleration closed-loop frequency response function; (c) force feedback controller; (d) force closed-loop frequency response function Closed-loop frequency response functions of shake table system using the proposed acceleration control strategy: (a) reference acceleration to measured force magnitude; (b) reference acceleration to measured acceleration magnitude Results with the 1995 Kobe ground motion as the reference acceleration: (a) shake table acceleration tracking comparison; (b) close up view of table accelerations; (c) frequency domain comparison of table accelerations Measured table force from the acceleration control strategy using the Kobe reference acceleration Comparison of measured table displacements with the Kobe reference acceleration Comparison of structural responses with the 1995 Kobe ground motion: (a) top floor structural acceleration comparison; (b) close up view of structural accelerations; (c) frequency domain comparison of top floor structural accelerations xv

16 Chapter 1 Introduction Each year earthquakes cause a significant amount of structural damage, economic turmoil and ultimately human causalities. The United States Geological Survey estimates 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

17 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 research 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

18 CHAPTER 1. INTRODUCTION 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 engineering 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, recent 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 desirable, 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

19 CHAPTER 1. INTRODUCTION 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

20 CHAPTER 1. INTRODUCTION 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 table 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 incorporate substructure techniques; extrapolation of scale effects; interpretation, implication, 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 displacementcontrolled 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 incorporated in commercial shake table controllers to compensate for the dynamics of the 5

21 CHAPTER 1. INTRODUCTION 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, displacement 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 conventional displacement control strategies can be found in literature. Stoten and Gómez (2001) proposed adaptive control using the minimal control synthesis (MCS) algorithm 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 advanced 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

22 CHAPTER 1. INTRODUCTION sitivity analysis showed the effectiveness of the velocity feed-forward term in the magnitude characteristics of the displacement transfer function. Nakata (2010) developed an acceleration trajectory tracking control strategy that combines acceleration 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 Real-Time Hybrid Simulation While shake table testing is the preferred experimental testing technique, size constraints typically limit researchers to the use of scaled experiments. Real-time hybrid simulation (RTHS) is a promising new experimental technique that enables systemslevel performance assessment of structures at the true dynamic loading rate. In RTHS, responses of entire structural systems are simulated combining computational models (computational substructures) and physical tests (experimental substructures); in general, only structural members of which responses are difficult to model are experimentally 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

23 CHAPTER 1. INTRODUCTION 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), Figure 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. Challenges 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

24 CHAPTER 1. INTRODUCTION Figure 1.3: Early implementation of pseudo-dynamic testing, CREDIT: Nakashima et al. (1992). Figure 1.4: Implementation of RTHS where the experimental substructure is a single damper, CREDIT: Carrion et al. (2009). 9

25 CHAPTER 1. INTRODUCTION from such errors may not be dissipated fast enough resulting in an unstable simulation. 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 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 substructure 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)). Substructure 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

26 CHAPTER 1. INTRODUCTION Figure 1.5: Concept of substructure shake table testing including a tuned mass damper, 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 Günay (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

27 CHAPTER 1. INTRODUCTION 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 simulation. 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 performance 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

28 CHAPTER 1. INTRODUCTION 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 techniques 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 control 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 testing of upper stories in tall buildings using a shake table. Compensation techniques 13

29 CHAPTER 1. INTRODUCTION 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 dissertation with general remarks, technical conclusions and openings for future research. 14

30 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

31 CHAPTER 2. SHAKE TABLE DYNAMICS m n c n k n x n m 1 f t c 1 k 1 x 1 mt x t Figure 2.1: Schematic of a uni-axial shake table with linear structure. 2.1 Dynamics of Shake Tables with Hydraulic Actuators Many research topics, from traditional shake table testing to substructure shake table testing, rely on accurate acceleration tracking of shake tables to generate acceptable 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

32 CHAPTER 2. SHAKE TABLE DYNAMICS In Figure 2.2, m i, c i and k i are the i-th floor mass, story damping coefficient and story stiffness of the test structure; x i is the i-th floor absolute displacement; m t is the mass of the table platform; x t is the absolute displacement of the platform and f t 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: m t 0 1 n 0 n 1 M ẍ t ẍ s + c 1 c c T C ẋ t ẋ s + k 1 k kt K x t x s = f t 0 n 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; x s is the vector of structural displacements (i.e.: x s = [x 1... x n ] T ); c and k are the damping and stiffness coupling terms, defined as: c 1 c = 0 n 1 and k k 1 = 0 n 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: m t 0 1 n 0 n 1 M s2 + c 1 c c T C s + k 1 k k T K X t (s) X s (s) = F t (s) 0 n 1 (2.3) 17

33 CHAPTER 2. SHAKE TABLE DYNAMICS Where s is the Laplace variable. Equation 2.3 can be partitioned into a single transfer function matrix acting on the system displacements: H tt fx (s) H st fx (s) Hts fx (s) Hss fx (s) X t (s) X s (s) = F t (s) 0 n 1 (2.4) Where Hfx tt (s) is the transfer function of the table platform; Hts fx (s) and Hst fx (s) are the coupling transfer function matrices (such that: H ts fx (s) = (Hst fx (s))t ) and H ss fx (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: F t (s) = ( Hfx(s) tt H ts fx(s) ( H ss fx(s) ) ) 1 H st fx(s) X t (s) = H ftxt (s)x t (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: X t (s) = 1 H ftx t (s) F t(s) = H xtf t (s)f t (s) (2.6) The previous equations addressed the dynamic relationships for the shake table platform with test structure. The relationships show that the force-displacement transfer function for the table, H xtf t (s), is influenced not only by the properties of the table 18

34 CHAPTER 2. SHAKE TABLE DYNAMICS Shake Table Actuator sa s 2 N 2 (s) A t (s) S a (s) A m (s) X r (s) E(s) C(s) W (s) U (s) V (s) Q(s) H qv (s) 1/k e F t (s) H xt f t (s) Accelerometer N 1 (s) X t (s) S x (s) X m (s) Shake Table Controller Servo Valve Table Platform & Structure LVDT sk l Figure 2.2: Block diagram of shake table system including hydraulic actuator, test structure 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, X r (s) is the reference displacement for the shake table, X m (s) is the measured shake table displacement; E(s) =X r (s) X m (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); H qv (s) is the transfer function of the servo valve from actual command to oil flow through the actuator 19

35 CHAPTER 2. SHAKE TABLE DYNAMICS chambers, Q(s); k e is the force-flow coefficient; k l 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; X t (s) is the true displacement of the shake table; N 1 (s) represents the displacement measurement noise; S x (s) is the transfer function of the displacement transducer (LVDT); A t (s) is the true acceleration of the shake table; N 2 (s) represents the acceleration measurement noise; S a (s) is the transfer function for the shake table accelerometer and A m (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, X t (s), is obtained through a block diagram reduction as: X t (s) = H xtf t (s) k e + (k l + AH xtf t (s)) s H qv(s)v (s) = H xtv(s)v (s) (2.7) The appearance of H xtft (s) in the actuator transfer function was identified by Dyke et al. (1995) as control-structure-interaction, CSI. Because H xtft (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 k e, k l and A. This type of CSI is often seen in shake table testing (Conte and Trombetti, 2000; 20

36 CHAPTER 2. SHAKE TABLE DYNAMICS 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, N 1 (s), which are always present in shake table testing. Including these effects, the measured shake table displacement is represented as: X m (s) = S x(s)h xt v(s)c(s) X 1+S x (s)h xt v(s)c(s) r(s) + S x(s)h xt v(s) W (s) + S x (s) N 1+S x (s)h xt v(s)c(s) 1+S x (s)h xt v(s)c(s) 1(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. Traditionally the reference displacement, X r (s), is obtained from a conversion of the true 21

37 CHAPTER 2. SHAKE TABLE DYNAMICS reference acceleration, A r (s), where the relationship between the two is: X r (s) = T (s)a r (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: A m (s) = s2 S a (s)h xtv(s)c(s)t (s) 1 + S x (s)h A s 2 S a (s)h xtv(s) r(s) + xtv(s)c(s) 1 + S x (s)h W (s) xtv(s)c(s) s2 S a (s)s x (s)h xtv(s)c(s) N 1 (s) + S a (s)n 2 (s) (2.10) 1 + S x (s)h xtv(s)c(s) Equation 2.10 indicates that the measured shake table acceleration is not only influenced by the reference table acceleration but also by the process, displacement and acceleration measurement noises. The middle two terms in Equation 2.10 demonstrate 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 guarantee accurate acceleration tracking of the table in the presence of noise. However 22

38 CHAPTER 2. SHAKE TABLE DYNAMICS 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 table 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 multidegrees of freedom test structure. The equations included the dynamics of the hydraulic 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 hydraulic 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

39 Chapter 3 Substructure Shake Table Testing of Upper Stories in Tall Buildings Some researchers are interested in experimentally testing the response of nonstructural 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

40 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 computationally 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 compatibility requirements, this section presents the underlying dynamics of substructure shake table testing 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: m i ẍ i + R i (d i, d i ) R i+1 (d i+1, d i+1 ) = m i ẍ g (i = 1,..., n and R n+1 = 0) (3.1) where m i is the mass of the i-th floor; x i is the i-th floor relative displacement with respect to the ground; d i is the i-th floor story drift that can be expressed as x i x i 1 ; R i is the i-th floor restoring force including damping; and ẍ g is the ground acceleration. Note that R i is the function of the relative story velocity, ẋ i ẋ i 1, 25

41 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS m n R n x n m e _ ne R e _ ne x e _ ne Computational Substructure m i+1 R i+1 x i+1 f c _ nc m e _1 R e _1 x e _1 m i R i x i m c _ nc R c _ nc x c _ nc x g _ e Shake Table m 1 R 1 x 1 m c _1 R c _1 x c _1 Experimental Substructure x g x g _ c (a) Entire Simulation (b) Substructure Shake Table Hybrid Simulation Figure 3.1: Schematics of substructure shake table testing in comparison with the entire simulation. and the story drift, x i x i 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: n c - story computational substructure and n e -story experimental substructure that represent the lower and upper parts of the entire system, respectively (n = n c + n e ). The equations of motion of the substructure system can be expressed as: m c i ẍ c i + R c i (d c i, d c i ) R c i+1 (d c i+1, d c i+1 )= m c i ẍ g c (i =1,...,n c 1) (3.2) m c i ẍ c i + R c i (d c i, d c i ) f c nc = m c i ẍ g c (i = n c ) (3.3) 26

42 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS m e i ẍ e i + R e i (d e i, d e i ) R e i+1 (d e i+1, d e i+1 ) = m e i ẍ g e (i = 1,..., n e and R e ne+1 = 0) (3.4) where m c i and m e i are the i-th floor mass of the computational and experimental substructures, respectively; x c i and x e i are the i-th floor relative displacement of the computational and experimental substructures, respectively; d c i and d e i are the i-th floor story drift of the computational and experimental substructures, respectively; R c i and R e i are the i-th floor nonlinear restoring forces of the computational and experimental substructures, respectively; and ẍ g c and ẍ g e are the ground acceleration of the computational and experimental substructures, respectively; and f c nc is the interaction force from the experimental substructure at the n c -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 Compatibility Requirements For the substructure system to have the equivalent dynamics as the entire system, 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, m c i = m i and R c i = R i for i = 1,..., n c, and m e i = m i and R e i = R i for i = 1,..., n e. With the above model assumptions, the remaining conditions that have to be satisfied are input compatibility conditions. 27

43 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS 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, ẍ g c = ẍ g (3.5) The computational acceleration compatibility is straightforward since the input ground acceleration in the computational substructure is known in advance and can be directly 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, ẍ g e = ẍ c nc + ẍ g c (3.6) The experimental acceleration compatibility implies that the reference ground acceleration 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, f c nc = R e 1 (d e 1, d e 1 ) (3.7) 28

44 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS 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 ) have to be satisfied at any given instance during the simulation Concept of Substructure Shake Table Testing 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

45 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS x g x g _ c Comp. structure x c _ nc + x g _ c x g _ e Exp. structure f c _ nc R e _1 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 threestory 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 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

46 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS Figure 3.3: A three-story steel frame structure on the uni-axial shake table at Johns Hopkins 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 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 actuator. A general-purpose accelerometer is installed on the table to measure absolute acceleration of the shake table 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

47 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS 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 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) 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 bandlimited 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 functions, indicating the first, second, and third natural frequencies of the experimental 32

48 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS Figure 3.4: substructure. Frequency response curves of the three-story steel experimental 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 Computational Substructure The computational substructure is a linear elastic seven-story shear building with the story mass of 226 kg, floor stiffness of 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

49 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS Table 3.1: Dynamic properties of the entire 10-story RTHS structure Mode Natural Frequencies Damping Ratios 1 st 2.52 Hz 7.76 % 2 nd 6.80 Hz % 3 rd 9.60 Hz % 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: 3 R e 1 = m e i (ẍ e i + ẍ g e ) (3.8) i=1 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

50 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS 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 acceleration of the computational substructure. However, acceleration control of shake tables is extremely difficult mainly due to limitations in displacement control as explained 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 Issues of Acceleration Control and Control- Structure-Interaction While shake tables are designed to produce reference accelerations, primary controllers, inner-loop servo controllers, for actuators are displacement control. In almost all the cases, the inner-loop servo controllers are proportional-integral-derivative (PID) controllers or PID with additional feedbacks (e.g., differential pressure feedback). In practice, a command shaping controller/filter such as the inverse dynamics 35

51 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS compensation techniques is added to cancel out the dynamics of the inner-loop control system, Spencer and Yang (1998). Basically, command shaping is an off-line process that alters the reference displacements to produce the closest possible reference 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

52 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS Figure 3.5: Frequency response curves of closed-loop (reference to measured) displacement 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

53 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS this study. The characteristics of the acceleration FRC in Figure 3.5 indicate that input acceleration errors are present in this control system 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 coherence 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 substructure. 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

54 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS Figure 3.6: Frequency response curve and coherence from the table acceleration to measured base shear. 3.4 Substructure Shake Table Test System with Error Compensation All of the compatibility requirements have to be satisfied during real-time computational 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 errors 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

55 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS x g _ c Comp. structure x c Delay Compensator x p Exp. structure f c _ nc x r x e + ΔR e _1 + R e _1 Force Corrector + _ State Observer R e _1 x m Figure 3.7: A block diagram of the substructure shake table test system with compensation techniques for experimental errors. corrector; state estimator for the experimental substructure; and actuator delay compensation for the shake table. Details of each process are discussed herein 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 computationally. Therefore, equations of motion for the entire structure, Equations , 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

56 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS 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 stiffness 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 x c and the input vector u c are available at the j-th step, the output vector y c at the j-th step and the state vector at the (j + 1)-th step are calculated as: x c [j + 1] = A c x c [j] + B c u c [j] (3.9) y c [j] = C c x c [j] + D c u c [j] (3.10) where A c, B c, C c, and D c are the discrete-time system, input, output and feedthrough matrices of the computational substructure, respectively; The input vector 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

57 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS displacement of the computational substructure. That is, [ u c [j] = ẍ g [j] f c nc [j] ] T (3.11) [ y c [j] = ẍ c nc [j] + ẍ g c [j] x c nc [j] + x g 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 State Observer and Kalman Filter If the top floor absolute displacement, x c nc [j] + x g c [j], is sent to the shake table controller as the reference at the j-th step, the measured table acceleration, ẍ m [j], at this step will not match the reference acceleration ẍ r [j] that is the top floor absolute acceleration, ẍ c nc [j] + ẍ g 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 compensation 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 initial conditions in the delay compensation process is not available in the substructure 42

58 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS 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 u e and output y e in the experiment and the estimated state vector x e at the j-th step, the state vector at the (j+1)-th step can be estimated as: x e [j + 1] = (A e LC e ) x c [j] + (B c LD e )u e [j] + Ly e [j] (3.13) where A e, B e, C e, and D e 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 accuracy of the model for experimental substructure. The measured input and output in the experiment are the shake table acceleration and the base shear, respectively. u e [j] = ẍ g e [j] (3.14) y e [j] = R e 1 [j] (3.15) 43

59 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS Furthermore, the measured output in the experiment can be filtered to reduce the influence of the process and measurement noises as: R e 1 [j] = ȳ e [j] = C e x e [j] + D e u e [j] (3.16) Where R e 1 [j] is the filtered base shear in the experiment that is used in the force correction technique 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: [ û e [0, j] = P 1 y c [j] = ẍ c nc [j] + ẍ g c [j] where P 1 = 1 0 ] (3.17) ˆx e [0, j] = x e [j] (3.18) ˆx c [0, j] = x c [j] (3.19) 44

60 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS 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: ˆx e [k + 1, j] = A eˆx e [k, j] + B e û e [k, j] (3.20) ŷ e [k, j] = C eˆx e [k, j] + D e û e [k, j] (3.21) [ û c [k, j] = ẍ g c [j + k] ŷ e [k, j] ] T (3.22) ˆx c [k + 1, j] = A cˆx c [k, j] + B c û c [k, j] (3.23) ŷ c [k, j] = C cˆx c [k, j] + D c û c [k, j] (3.24) û e [k + 1, j] = P 1 ŷ c [k, j] (3.25) The delay compensation technique repeats the above processes (Equations ) 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, x p, is specified as: [ x p [j] = P 2 ŷ c [h, j] = ˆx c nc [h, j] + x g c [j + h] where P 2 = 0 1 ] (3.26) Where ˆx c nc [h, j] is the top floor relative displacement of the computational substructure at the h-th future step predicted from the current j-th simulation step; and x g c [j + h] is the ground displacement at the computational substructure at the (j + h)-th step. 45

61 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS 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 propagation 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: x c [j + 1] = A e x e [j] + B e ũ e [j] (3.27) ỹ e [j] = C e x e [j] + D e ũ e [j] (3.28) where ũ e [j], ỹ e [j] and x e are the input, output, and state vectors for the error correction process. The input and output in this process can be expressed as: ũ e [j] = ẍ r [j] ẍ m [j] (3.29) ỹ e [j] = R e 1 [j] (3.30) where ẍ r [j] ẍ m [j] is the input acceleration error and R e 1 [j] is the erroneous base shear induced by the input acceleration error. The corrected force is the sum of the 46

62 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS filtered and error-induced base shear as: f c nc [j] = R e 1 [j] + R e 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 implemented in the control system at the Johns Hopkins University. A series of substructure 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 obtained. 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 =

63 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS Figure 3.8: Acceleration and base shear time histories under 2.0 Hz harmonic ground excitation: (a) the entire acceleration time histories; (b) a zoomed section of the acceleration time histories; (c) the entire base shear time histories; and (d) a zoomed section of the base shear time histories 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 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

64 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS 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 vibrations 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 reference 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 technique. 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 harmonic. 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

65 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS 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 2 nd, 6 th, and 10 th floors are shown in the plots. Note that the displacement responses at the 10 th floor (the top floor of the experimental substructure) 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 agreement 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. Because 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

66 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS Relative Displacement (mm) Absolute Displacement (mm) Absolute Acceleration (g) Figure 3.9: Structural responses under 2.0 Hz harmonic ground excitation: (a), (d), and (g), relative floor displacement at the 10 th,6 th and 2 nd floor, respectively; (b), (e), and (h), absolute floor displacement at the 10 th,6 th and 2 nd floor, respectively; and (c), (f), and (i), absolute floor acceleration at the 10 th,6 th and 2 nd floor, respectively. 51

67 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS Figure 3.10: Acceleration and base shear time histories under 6.0 Hz harmonic ground excitation: (a) the entire acceleration time histories; (b) a zoomed section of the acceleration time histories; (c) the entire base shear time histories; and (d) a zoomed section 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 frequency vibration of approximately 30 Hz. Because of the poor acceleration tracking, the input acceleration errors are present at this frequency as expected from the observation 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

68 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS 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 As is the case with the previous simulation with the 2.0 Hz excitation, discrepancies 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 responses is also obtained in this simulation with the 6.0 Hz excitation. It is interesting to see that the 6 th floor accelerations are smaller than those at the 2 nd 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 6 th 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 compensation techniques. The substructure shake table tests are stable and valid, indicating 53

69 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS Relative Displacement (mm) Absolute Displacement (mm) Absolute Acceleration (g) Figure 3.11: Structural responses under 6.0 Hz harmonic ground excitation: (a), (d), and (g), relative floor displacement at the 10 th,6 th and 2 nd floor, respectively; (b), (e), and (h), absolute floor displacement at the 10 th,6 th and 2 nd floor, respectively; and (c), (f), and (i), absolute floor acceleration at the 10 th,6 th and 2 nd floor, respectively. that experimental errors are not propagated through the real-time hybrid simulation processes 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

70 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS Figure 3.12: Acceleration and base shear time histories under the 1995 Kobe earthquake excitation: (a) the entire acceleration time histories; (b) a zoomed section of the acceleration time histories; (c) the entire base shear time histories; and (d) a zoomed 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

71 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS Figure 3.13 shows the structural responses under the Kobe earthquake during the substructure shake table test. Relative displacement, absolute displacement and absolute 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 vibration 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 10 th -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

72 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS Computational Substructure Experimental Substructure Relative Displacement (mm) Absolute Displacement (mm) Absolute Acceleration (g) 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 (10 th to 2 nd ); (b), (e), (h), (k), and (n), absolute displacement at the even floors from top to bottom (10 th to 2 nd ); and (c), (f), (i), (l), and (o), absolute displacement at the even floors from top to bottom (10 th to 2 nd ). 57

73 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS 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 magnitude 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 modifying 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) Feedforward Compensation using Derivatives of Reference Signal The first technique considered in this study is a feedforward compensator developed by Phillips et al. (2013), as an extension from the work by Carrion and Spencer 58

74 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS (2007), to compensate for actuator dynamics and time delay in RTHS for a magnetorheological 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 relationship 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 Ĥx mx r (s). The curve fitting is completed using a transfer function with three poles, such that: Ĥ xmx r (s) = 1 a 3 s 3 + a 2 s 2 + a 1 s + a 0 (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 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, x p, is obtained as a weighted series of derivatives of the true reference displacement, x r = x c nc + x g c : x p [j] = a 3... x r [j] + a 2 ẍ r [j] + a 1 ẋ r [j] + a 0 x r [j] (3.33) 59

75 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS 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) IIR Compensation Technique for Significant Control-Structure-Interaction While the compensation method introduced by Phillips et al. (2013) can compensate for time delays and high frequency actuator dynamics, it cannot address the inertial effects from CSI which often appear in substructure shake table testing, Equation 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, Ĥx mx r (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

76 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS 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: Ĥ xmx r (s) = b ls l + b l 1 s l b 1 s + b 0 a m s m + a m 1 s m a 1 s + a 0 with 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 (Ĥx mx r (s)) 1. The additional poles are thus determined from an r th order high-pass filter, H f (s). That is, the modified reference signal is calculated as: X p (s) = (Ĥxmx r (s)) 1 Hf (s)x r (s) = F (s)x r (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

77 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS and Equation 2.8), of the shake table can now be expressed as: X m (s) = S x(s)h xt v(s)c(s)f (s) 1+S x (s)h xt v(s)c(s) X r(s) + S x(s)h xt v(s) 1+S x (s)h xt v(s)c(s) W (s) + S x (s) 1+S x (s)h xt v(s)c(s) N 1(s) (3.36) From Equation 3.36, it is clear that the IIR compensator only influences the relationship 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 table relationship and that the compensated closed loop system does not amplify high frequency vibration Experimental Investigation of Model-Based Delay Compensation Techniques To test the efficacy of these advanced compensation techniques, the same earthquake simulation that was presented in Section is completed using both compensation 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 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

78 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS Magnitude (m/m) Displacement Tracking Uncompensated Feedforward IIR (a) Frequency (Hz) 200 Magnitude (g/g) Acceleration Tracking (b) Frequency (Hz) 200 Phase (Deg.) Phase (Deg.) (c) Frequency (Hz) (d) Frequency (Hz) Figure 3.14: Experimental reference to measured frequency response functions for shake table with 3DOF experimental substructure: (a) and (c) displacement tracking magnitude and phase; (b) and (d) acceleration tracking magnitude and phase 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 The displacement tracking of the shake table is significantly 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 magnitudes 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

79 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS 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 It is worth noting that the response using iterative compensation (Section 3.4.3) is also available from Figure The closed loop performance using iterative compensation has the same magnitude response as the uncompensated system (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 simulations are shown in Figure 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

80 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS Displacement (m) Numerical RTHS, Feedforward RTHS, IIR RTHS, Iterative x (a) Time (s) (b) Time (s) Acceleration (g) (c) Time (s) (d) Time (s) Figure 3.15: Absolute shake table response from substructure shake table tests with 3DOF 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

81 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS Table 3.2: Summary of shake table performance from substructure shake table testing using 3 different compensation techniques. Feedforward IIR Iterative Compensation 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 tracking results are much more distorted. Results show that all compensation techniques yielded reasonable tracking of the major frequency contents in the reference acceleration, 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

82 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS 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). Undercompensation 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 technique was able to produce more accurate shake table tracking performance during substructure shake table testing when compared to two existing compensation techniques. The increased performance is due to the ability of the IIR compensator to 67

83 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS Feedforward Error IIR Error Iterative Error Reference Displacement (m) Reference Displacement (m) Reference Displacement (m) (a) Measured Displacement (m) (b) Measured Displacement (m) (c) Measured Displacement (m) Reference Acceleration (g) 0 Reference Acceleration (g) 0 Reference Acceleration (g) 0 (d) Measured Acceleration (g) (e) Measured Acceleration (g) (f) Measured Acceleration (g) Figure 3.16: Shake table tracking errors from substructure shake table tests with 3DOF experimental substructure subjected to Kobe earthquake record: (a) and (d) displacement and acceleration errors using feedforward compensator; (b) and (e) displacement and acceleration errors using IIR compensator; (c) and (f) displacement and acceleration errors using iterative compensator. 68

84 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS account for complex shake table CSI. While results from this study are not exhaustive, 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. 3.7 Summary and Discussion 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 acceleration, model-based actuator delay compensation with state observer, and force correction using Kalman filter. The effectiveness of those techniques was experimentally 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

85 CHAPTER 3. TESTING OF UPPER STORIES IN TALL BUILDINGS 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

86 Chapter 4 Substructure Shake Table Testing of Lower Stories in Tall Buildings Another particular interest in the earthquake engineering community is the dynamic 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 structure. 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

87 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 Controlled 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: m j k=1 j ẍ k +R j (x j, ẋ j ) R j+1 (x j+1, ẋ j+1 ) = m j ẍ g (j = 1,..., n and R n+1 = 0) (4.1) Substructured System: m j k=1 j ẍ k + R j (x j, ẋ j ) R j+1 (x j+1, ẋ j+1 ) = m j ẍ g e (j = 1,..., i 1) (4.2) m i k=1 i ẍ k + R i (x i, ẋ i ) f m = m i ẍ g e (j = i) (4.3) 72

88 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS Entire System Substructured System x 1 x i + 1 x i x n m 1 ( ) R 1 x 1,ɺx 1 m n ( ) R n x n,ɺx n m i + 1 ( ) R i+1 x i+1,ɺx i+1 m i R i ( x i,ɺx i ) Experimental Substructure Controlled Mass Shake Table x i x m 1 1 x m f m m m m i ( ) R i ( x i,ɺx i ) R 1 x 1,ɺx 1 ɺɺx g_e!" Computational Substructure x i + 1 x n m i + 1 m n ɺɺx g_c R n ( x n,ɺx n ) R i+1 ( x i+1,ɺx i+1 ) ɺɺx g Figure 4.1: Schematics of the Entire and substructure systems. m j k=1 j ẍ k + R j (x j, ẋ j ) R j+1 (x j+1, ẋ j+1 ) = m j ẍ g c (j = i + 1,..., n and R n+1 = 0) (4.4) where m j is the mass of the j-th floor; x j is the relative displacement of the j-th floor with respect to the (j 1)-th floor; R j (x j, ẋ j ) is the nonlinear restoring force from the j-th floor; ẍ g is the ground acceleration; ẍ g c and ẍ g e are the input acceleration to the computational and experimental substructures, respectively; and f m 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

89 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS (experimental acceleration compatibility), i.e., ẍ g e = ẍ g (4.5) Second, the input acceleration to the computational substructure has to be the absolute acceleration at the i-th floor (computational acceleration compatibility), i.e., i ẍ g c = ẍ g e + ẍ k (4.6) k=1 And last, the force due to the controlled inertial mass at the i-th floor in the experimental 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., f m = R i (x i+1, ẋ i+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 condition. 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

90 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS 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 control 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 substructure to an equivalent inertial force of the mass. From the kinetic relationship, the 75

91 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS equation of motion of the mass can be expressed: m m ( ẍ gc + ẍ k + ẍ m ) = f m (4.8) where m m is the mass of the controlled mass and ẍ m 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 numerical integration of the relative acceleration. Using the trapezoidal rule, the relative displacement can be obtained as follows: ẋ m [n] = ẋ m [n 1] + t 2 (ẍ m[n] + ẍ m [n 1]) (4.9) x m [n] = x m [n 1] + t 2 (ẋ m[n] + ẋ m [n 1]) (4.10) 76

92 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS where ẍ m [n], ẋ m [n] and x m [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. x m fil [z] = H fil [z]x m [z] (4.11) where H fil is the discrete transfer function of the high-pass filter; x m 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

93 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS ( x g ) ɺɺ x g_e =ɺɺ f m( = R i+1 ) Exp. Acc. Compatibility Experimental Substructure Interface Force Compatibility i ɺɺ x g_c =ɺɺ x g_e + ɺɺ x k k=1 Comp. Acc. Compatibility Computational Substructure R i+1 Actuator/ Mass x m _ fil High-pass Filter x m Numerical Integration ɺɺ x m Acceleration Conversion Figure 4.2: A block diagram of the substructure shake table test using controlled masses 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 configuration 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

94 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS Table 4.1: Properties and dynamic characteristics of the 5-story structure. Structural Properties Floor mass 50kg Story stiffnes 97.5KN/m Damping ratio 6.50% Natural frequencies 1st mode 2.00Hz 2nd mode 5.84Hz 3rd mode 9.20Hz 4th mode 11.83Hz 5th 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 substructure: 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

95 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS 10 0 Magnitude (m/m) (a) Frequency (Hz) (b) Frequency (Hz) Phase (Deg) Figure 4.3: Closed-loop frequency response function of the 911-D actuator with a mass 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) = s s s s (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

96 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS numerical simulations to represent the interface force from the computational substructure 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 = 1 and β = 1. The sampling rate of 2 6 the simulation is selected at 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 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

97 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS 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 substructured 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 ). 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

98 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS Direct Int. Disp. Target Disp. Achieved Disp. Displacement (mm) Time (s) Figure 4.4: Displacement comparison for controlled mass system in a simulation using Kobe 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

99 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS Comp. Base Shear Achieved Force Force (N) Time (s) Figure 4.5: Comparison of force achieved by the controlled mass system and target computational 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 = 1 m m (φ E [l] φ S [l]) 2 (4.13) i=1 Where is the RMS difference; φ is the simulation response; subscripts E and S 84

100 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS Acceleration (m/s 2 ) Entire Substructured (a) Time (s) 0.02 Velocity (m/s) (b) Time (s) 2 (c) Displacement (mm) Time (s) Figure 4.6: Comparison of top floor structural responses: (a) accelerations; (b) velocities; (c) displacements in a simulation using the Kobe ground motion. 85

101 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS Table 4.2: RMS differences for simulation responses under different earthquake simulations. Controlled mass response Top floor structural response Force Displacement Acceleration (N) PCBS(N) /PCBS (mm) PTFD(mm) /PTFD (g) PTFA(g) /PTFA Chi Chi, Coalinga, Duzce, El Centro, Imperial Valley, Kobe, Kocaeli, Landers, Loma Prieta, Morgan, 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 methods 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

102 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS 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 investigation 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 influence 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 implementation, 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

103 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS the top floor acceleration under the following cases: substructured systems with no filtering and ideal actuator model; with filtering and ideal actuator model; with filtering 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 almost identical characteristics to the entire structure except around the first natural frequency. On the contrary, the frequency response function with filtering and realistic 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 evaluated. 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

104 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS 10 1 (a) Magnitude Entire Sub. No Filt, Ideal Act Sub. Filt, Ideal Act Sub. Filt, Act Frequency (Hz) Phase (Deg) (b) Frequency (Hz) Figure 4.7: Comparison of top floor acceleration frequency response functions for substructured systems with no filtering and ideal actuator model, filtering and ideal actuator model, filtering and realistic actuator model to entire structure: (a) magnitude 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 substructured systems. These simulation results show that the separation floor is also one of the influential factors of the substructure method. 89

105 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS 10 1 (a) Magnitude Entire Sub. Separation Floor 1 Sub. Separation Floor Frequency (Hz) Phase (Deg) (b) Frequency (Hz) Figure 4.8: Comparison of top floor acceleration frequency response functions for substructured 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 certain 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 discussed in this paper, as well as design specification for experimental validation of the 90

106 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS method, further studies are needed. 4.2 Interface Compatibility using a Force Controlled Actuator In this section, another possible implementation of substructure shake table testing 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 presented earlier, Equations , except that f m is replaced by the the force from the force controlled actuator, f e. This section presents the actuator control scheme and numerical simulations to evaluate the effectiveness of this implementation 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

107 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS Figure 4.9: Experimental substructure using a force controlled actuator to apply the computational 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 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, x g is the reference ground displacement; e d and e f are the tracking errors between the shake table and force controlled actuators respectively; C f and C D are the controllers for the force controlled actuator; the shake table is assumed to have a PID controller and u d and u f are the voltages sent to each of the actuator s servo-valves. 92

108 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS Figure 4.10: Block diagram of substructure shake table test method including actuator 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, C f, and one to eliminate the effect of the shake table dynamics on the force controlled actuator, C D. In terms of actuator control, the experimental substructure can be viewed as a 93

109 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS two-input two-output system: x g e f e = H xgu d H feu d H xgu f H feu f u d u f = H EXP u d u f (4.14) and the controllers are determined based on the relations in Equation The shake table PID controller can be tuned solely from the relation H xgud. Once the shake table controller is fixed, the force feedback controller, C f, can be designed solely using a loop shaping, Nakata (2012), approach on H feuf. If the shake table is completely independent of the force-controlled actuator then H xgu f = 0 and a straightforward choice of C D will decouple the force-controlled actuator from the shake table. Using the previous assumption the decoupling controller can be formed as: C D = H f eu d H feu f (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 H xgu f = 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

110 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS RTHS, additional measures may be needed to remove the delay from the force controlled 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 compensation is not needed for the shake table since the ground motion is pre-defined and the experimental system drives the RTHS 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 reference 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 simplicity, each story of the structure has the same physical properties: m = 70kg, k = N/m and c = 187Ns/m. With these parameter choices the first vibration 95

111 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS Figure 4.11: Performance of experimental setup during step input tests: a.) shake table 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 forcecontrolled 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 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

112 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS 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 independently, 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 actuator. The delay compensation algorithm used here is an inverse based compensation consistent with the discussion in Section 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 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

113 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS 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 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

114 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS 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 acceleration of the shake table during the substructured simulation. A comparison of the 4th floor absolute acceleration from both simulations is shown in Figure A comparison 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 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 observation indicates that the substructure system has slightly more damping than the entire structure. These observations are again confirmed through a frequency domain 99

115 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS Figure 4.14: 4 th floor absolute acceleration during Kobe simulations: a.) time histories; b.) Fourier Transform of time histories. comparison, Figure 4.14b. Here both responses have almost identical characteristics 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

116 CHAPTER 4. TESTING OF LOWER STORIES IN TALL BUILDINGS 4.3 Summary and Discussion 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 computationally. 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 implementation 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 chapter 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

117 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

118 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. Implementation 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 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 acceleration, a r, is first pre-filtered by the pre-gain, P a, to obtain the modified reference acceleration, â r. This technique is used to compensate the dynamics of the closed loop controller. This modified reference acceleration is then sent to the acceleration 103

119 CHAPTER 5. ACCELERATION FEEDBACK CONTROL OF SHAKE TABLES a r Acc. Pre-Gain â P r a Force Pre-Gain!" + Acc. FB. Controller C a Force FB. Controller Shake Table/Structure Dynamics u a H au a m u P f f r!" + C f u f H fu f m Figure 5.1: Block diagram of proposed acceleration control strategy. feedback control loop. The acceleration feedback controller, C a, generates the valve command, u a, 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, P f, and the force feedback controller, C f. The reference force, f r, is the converted signal from the reference acceleration and is sent to the force feedback control loop. The force feedback controller, C f, generates the valve command, u f, 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, H au, and H fu, 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

120 CHAPTER 5. ACCELERATION FEEDBACK CONTROL OF SHAKE TABLES 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: H ama r = a m a r = C a P a + C f P f 1 + C a H au + C f H fu H au (5.1) H fma r = f m a r = C a P a + C f P f 1 + C a H au + C f H fu H fu (5.2) It can be seen that all of the controller terms, namely P a, P f, C a and C f, 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 performance 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 Hardware Requirements The acceleration control strategy developed in this study also has certain requirements 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

121 CHAPTER 5. ACCELERATION FEEDBACK CONTROL OF SHAKE TABLES Figure 5.2: Schematic of a uni-axial shake table setup for the proposed acceleration control 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

122 CHAPTER 5. ACCELERATION FEEDBACK CONTROL OF SHAKE TABLES 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 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

123 CHAPTER 5. ACCELERATION FEEDBACK CONTROL OF SHAKE TABLES Table 5.1: Dynamic characteristics of three story test structure. Natural Frequency, Hz Damping Ratio, % Vibration Mode Vibration Mode Vibration Mode to demonstrate the capabilities of the proposed control strategy for such challenging conditions. Dynamics characteristics of the three-story structure are listed in Table Experimental Investigation of the Proposed 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 models, 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

124 CHAPTER 5. ACCELERATION FEEDBACK CONTROL OF SHAKE TABLES Figure 5.3: The shake table in the SSHT lab at Johns Hopkins: (a) shake table with three-story structure; (b) view of restoring springs. 109

125 CHAPTER 5. ACCELERATION FEEDBACK CONTROL OF SHAKE TABLES 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, H au and H fu, in the proposed acceleration feedback control strategy, as shown in Figure 5.1. Figure 5.4 shows the experimentally obtained open loop relationships: valve command 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

126 CHAPTER 5. ACCELERATION FEEDBACK CONTROL OF SHAKE TABLES Experimental Analytical Magnitude 10 1 Magnitude (a) Experimental 10 2 Analytical Frequency (Hz) (b) Frequency (Hz) Figure 5.4: Open loop dynamics for shake table system: (a) valve command to measured 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

127 CHAPTER 5. ACCELERATION FEEDBACK CONTROL OF SHAKE TABLES Table 5.2: Analytical representations of the open loop shake table dynamics. H au = s s s s s s s s s s s s s H fu = s s s s s s s s s s s s s s s Design of the Feedback Controllers and Pre- Gains A controller design of the proposed acceleration feedback control strategy is developed 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 controller, C a, 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 compensate 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 acceleration feedback controller along with the valve to acceleration relationship. The mathematical representation of the designed acceleration feedback controller is expressed 112

128 CHAPTER 5. ACCELERATION FEEDBACK CONTROL OF SHAKE TABLES Magnitude Magnitude 10 0 P a =1 P a =2.75 H au (a) C a Frequency (Hz) (b) Frequency (Hz) Magnitude 10 0 H fu C f Magnitude (c) Frequency (Hz) (d) Frequency (Hz) Figure 5.5: Controller design for the proposed acceleration control strategy: (a) acceleration feedback controller; (b) acceleration closed-loop frequency response function; (c) force feedback controller; (d) force closed-loop frequency response function. as: C a = 30s s s s s s s s s s s s s (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 C a in Equation 5.3 is plotted in Figure 5.5b where the pre-gain P a is set to 1. While it contains certain desirable characteristics (i.e. flat plateau and decaying magnitude in 113

129 CHAPTER 5. ACCELERATION FEEDBACK CONTROL OF SHAKE TABLES 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 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 P a (=2.75) and C a 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 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, P f, 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 pregain 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

130 CHAPTER 5. ACCELERATION FEEDBACK CONTROL OF SHAKE TABLES respectively. The designed force feedback controller is expressed as: C f = s s s s s s s s s s s s s (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 compensating the dynamics of the valve to force relationship. Finally, the overall performance and stability of the proposed acceleration control strategy using the choices of C a, P a, C f and P f 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 incorporation 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

131 CHAPTER 5. ACCELERATION FEEDBACK CONTROL OF SHAKE TABLES With Force Feedback Without Force Feedback Magnitude 10 2 Magnitude 10 1 (a) Frequency (Hz) (b) Frequency (Hz) Figure 5.6: Closed-loop frequency response functions of shake table system using the proposed acceleration control strategy: (a) reference acceleration to measured force magnitude; (b) reference acceleration to measured acceleration magnitude Experimental Validation of the Proposed Acceleration Control Strategy The proposed acceleration control strategy with the designed feedback controllers and pre-gains is experimentally validated using a series of earthquake ground motions. 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 acceleration (see Figure 5.7a). However, when inspected closely (see Figure 5.7b), the acceleration control strategy exhibited smaller amounts of discrepancy than the displacement 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

132 CHAPTER 5. ACCELERATION FEEDBACK CONTROL OF SHAKE TABLES Acceleration (g) Referecence Acceleration Control Displacement Control (a) Time (s) Fourier Amplitude 10 2 Acceleration (g) (b) Time (s) (c) Frequency (Hz) Figure 5.7: Results with the 1995 Kobe ground motion as the reference acceleration: (a) shake table acceleration tracking comparison; (b) close up view of table accelerations; (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 Figure 5.9). This observation indicates that two unique displacement time histories can 117

133 CHAPTER 5. ACCELERATION FEEDBACK CONTROL OF SHAKE TABLES 2000 Force (N) Time (s) Figure 5.8: Measured table force from the acceleration control strategy using the Kobe reference acceleration. produce almost identical acceleration time histories. However, from the experimental 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 control 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 displacement control strategy. The average RMS error in acceleration control strategy is 118

134 CHAPTER 5. ACCELERATION FEEDBACK CONTROL OF SHAKE TABLES Displacement (mm) Acceleration Control Displacement Control Time (s) Figure 5.9: Comparison of measured table displacements with the Kobe reference acceleration. 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 conventional 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 acceleration, 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 amplified the peak ground acceleration by 7% while the displacement tended to over 119

135 CHAPTER 5. ACCELERATION FEEDBACK CONTROL OF SHAKE TABLES Table 5.3: Errors between measured and reference shake table accelerations. Acceleration Control Displacement Control RMSE (%) Peak Ratio RSME (%) Peak Ratio Chi Chi, Coalinga, Imperial Valley, Kobe, Landers, Loma Prieta, Morgan Hill, Northridge, Taiwan, Average amplify the peak ground acceleration by 24% on average. 5.4 Impact of Input Acceleration Errors in Shake Table Tests on Structural Response 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 quantitatively 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

136 CHAPTER 5. ACCELERATION FEEDBACK CONTROL OF SHAKE TABLES (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 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 displacementcontrolled 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

137 CHAPTER 5. ACCELERATION FEEDBACK CONTROL OF SHAKE TABLES 1 Reference Acceleration Control Displacement Control 10 0 Acceleration (g) (a) Time (s) Fourier Amplitude 10 2 Acceleration (g) (b) Time (s) (c) Frequency (Hz) Figure 5.10: Comparison of structural responses with the 1995 Kobe ground motion: (a) top floor structural acceleration comparison; (b) close up view of structural accelerations; (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

138 CHAPTER 5. ACCELERATION FEEDBACK CONTROL OF SHAKE TABLES Table 5.4: Errors between measured and reference top floor structural accelerations. Acceleration Control Displacement Control RMSE (%) Peak Ratio RSME(%) Peak Ratio Chi Chi, Coalinga, Imperial Valley, Kobe, Landers, Loma Prieta, Morgan Hill, Northridge, Taiwan, Average 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 uncertainties). Acceleration-controlled tests produce smaller input ground motion errors than displacement-controlled tests, and in turn provide smaller errors in structural response. 5.5 Summary and Discussion This study introduced a shake table control strategy that employs direct acceleration feedback control without need for displacement feedback. The proposed acceleration control strategy incorporates force feedback for stability to prevent table drift. The acceleration control strategy was experimentally validated using a series 123

139 CHAPTER 5. ACCELERATION FEEDBACK CONTROL OF SHAKE TABLES 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, reducing 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

140 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 meaningful results are obtained from earthquake engineering studies. In this capacity, the 125

141 CHAPTER 6. CONCLUSIONS AND FUTURE WORK research community needs to recognize the limitations in existing methods and explore new techniques to enable more accurate test setups and ultimately more useful studies. This dissertation introduced novel techniques to improve the way shake tables 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 investigated 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 displacementbased 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 electrical, mechanical and control systems engineering will further advance the capabilities of earthquake engineering research facilities. 126

142 CHAPTER 6. CONCLUSIONS AND FUTURE WORK 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 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

143 CHAPTER 6. CONCLUSIONS AND FUTURE WORK 4. Complete shake table tests of the entire structure to experimentally validate 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 ensured 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 actuators is expected. 4. Implementation of substructure shake table testing with force controlled actuators. 5. Complete shake table tests of the entire structure to experimentally validate 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

144 CHAPTER 6. CONCLUSIONS AND FUTURE WORK 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 investigated 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 computational 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 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

145 CHAPTER 6. CONCLUSIONS AND FUTURE WORK The concept of force controlled actuators has significant promise in the field. Further research topics include more complex loading conditions utilizing multiple force controlled actuators. Mixed variable control needs to be explored to investigate setups that utilize actuators with different feedback controllers (displacement, 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 earthquake 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

146 Appendix A: Experimental Investigations of Lower Story Substructure Shake Table Testing While only numerical simulations were studied in Chapter 4, the author investigated preliminary experimental implementations of the controlled mass system and force controlled actuators as well. Additional challenges were experienced in the experimental 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

147 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 was implemented experimentally, 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, H feud. Furthermore, since an accurate model of the system coupling was not achieved, the decoupling controller, C D, was ineffective and led to the instability. Stable implementations 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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157 Vita Matthew Joseph James Stehman began his academic career at Ursinus College were he earned a Bachelor of Science degree in Mathematics with minors in Physics and Computer Science. While at Ursinus he was inducted into both national and international honors societies corresponding 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