The Effects Of Assumption On Subspace Identification Using Simulation And Experiment Data

Transcription

1 Unversty of Central Florda Electronc Theses and Dssertatons Masters Thess (Open Access) The Effects Of Assumpton On Subspace Identfcaton Usng Smulaton And Experment Data 2013 Yoonhwak Km Unversty of Central Florda Fnd smlar works at: Unversty of Central Florda Lbrares Part of the Cvl Engneerng Commons, and the Structural Engneerng Commons STARS Ctaton Km, Yoonhwak, "The Effects Of Assumpton On Subspace Identfcaton Usng Smulaton And Experment Data" (2013). Electronc Theses and Dssertatons Ths Masters Thess (Open Access) s brought to you for free and open access by STARS. It has been accepted for ncluson n Electronc Theses and Dssertatons by an authorzed admnstrator of STARS. For more nformaton, please contact lee.dotson@ucf.edu.

2 THE EFFECTS OF ASSUMPTION ON SUBSPACE IDENTIFICATION METHODS USING SIMULATION AND EXPERIMENTAL DATA by YOONHWAK KIM B.S. Korea Aerospace Unversty, 2004 M.S. Unversty of Southern Calforna, 2006 A thess submtted n partal fulfllment of the requrements for the degree of Master of Scence n the Department of Cvl, Envronmental, and Constructon Engneerng n the College of Engneerng and Computer Scence at the Unversty of Central Florda Orlando, Florda Sprng Term 2013 Major Professor: Hae-Bum Yun

3 2013 Yoonhwak Km

4 ABSTRACT In the modern dynamc engneerng feld, expermental dynamcs s an mportant area of study. Ths area ncludes structural dynamcs, structural control, and structural health montorng. In expermental dynamcs, methods to obtan measured data have seen a great nflux of research efforts to develop an accurate and relable expermental analyss result. A techncal challenge s the procurement of nformatve data that exhbts the desred system nformaton. In many cases, the number of sensors s lmted by cost and dffculty of data archve. Furthermore, some nformatve data has techncal dffculty when measurng nput force and, even f obtanng the desred data were possble, t could nclude a lot of nose n the measurng data. As a result, researchers have developed many analytcal tools wth lmted nformatve data. Subspace dentfcaton method s used one of tools n these achevements. Subspace dentfcaton method ncludes three dfferent approaches: Determnstc Subspace Identfcaton (DSI), Stochastc Subspace Identfcaton (SSI), and Determnstc-Stochastc Subspace Identfcaton (DSSI). The subspace dentfcaton method s wdely used for fast computatonal speed and ts accuracy. Based on the gven nformaton, such as output only, nput/output, and nput/output wth noses, DSI, SSI, and DSSI are dfferently appled under specfc assumptons, whch could affect the analytcal results. The objectve of ths study s to observe the effect of assumptons on subspace dentfcaton wth varous data condtons. Frstly, an analytcal smulaton study s performed usng a sxdegree-of-freedom mass-damper-sprng system whch s created usng MATLAB. Varous condtons of exctaton nsert to the smulaton test model, and ts exctaton and response are

5 analyzed usng the subspace dentfcaton method. For stochastc problems, artfcal nose s contaned to the exctaton and followed the same steps. Through ths smulaton test, the effects of assumpton on subspace dentfcaton are quantfed. Once the effects of the assumptons are studed usng the smulaton model, the subspace dentfcaton method s appled to dynamc response data collected from large-scale 12-story buldngs wth dfferent foundaton types that are tested at Tongj Unversty, Shangha, Chna. Nose effects are verfed usng three dfferent exctaton types. Furthermore, usng the DSSI, whch has the most accurate result, the effect of dfferent foundatons on the superstructure are analyzed. v

6 ACKNOWLEDGMENTS I would lke to express my apprecaton for the gudance, assstance, and support by my advsor, Professor Hae-Bum Yun. As an engneer and a researcher, he serves as a great model. I have learned a lot from hm and have kept n mnd your advce n my lfe. I would also lke to thank my commttee professors, Professor Necat Catbas and Professor Boo Hyun Nam. Ther advce and comments on my research were nvaluable n honng my thess. Ther support was greatly apprecated. I specally thank my frends and colleagues: Aaron Rank, Bryan Paul, Ganesh Sundaresan, and Ayad Abbas. I also thank our Yun lab s members, and vstng scholars: Dr. Hyunk Km and hs famly, and Dr. K Tae Park. Thanks also must go to those whom I have met at Lake Sherwood OPC: Dr. Larry G. Mnnger, Stephen Chong and hs famly, Drew Culter, Matt Bulter, and famly n fath. I must thank my famly for endless love and support for me. My parents, wthout you I never would have gotten to ths pont. Also thanks to my aunt, I can t forget your help. And our new famly member, my fancée, Sarah Km, whenever I feel frustrated, I am remnded that you are wth me. My strength to complete ths task through to the end s from them. v

7 TABLE OF CONTENTS LIST OF FIGURES... v LIST OF TABLES... x 1. INTRODUCTION Motvaton and Objectve Research Approach Scope THEORETICAL BACKGROUND Dynamc Model and System Identfcaton Input and Output Data Equatons Determnstc Subspace Identfcaton (DSI) Stochastc Subspace Identfcaton (SSI) Determnstc Stochastc Subspace Identfcaton (DSSI) SUBSPACE IDENTIFICATION FOR SIMULATION TESTS Smulaton Setup Descrpton of Models Exctaton Informaton Smulaton Test Procedure Smulaton Result Effect of Nose Effect of Tme Duraton Effect of Movng Average SHAKING TABLE SOIL-FOUNDATION-SUPERSTRUCTURE INTERACTION (SFSI) TESTS Expermental Setup Descrpton of Models Exctaton Informaton Measurement Locatons Test Protocols Data Preprocessng Subspace Identfcaton Method for SFSI Test v

8 4.3.1 Input and output for Subspace Identfcaton Method Experment Data Analyss DISCUSSION Layered Sol Foundaton wth Ple and Box foundaton Modal Parameters at Three Dfferent Foundatons Stffness and Dampng Coeffcent Comparson Mode shapes Comparson All Results respect to the Test Order CONCLUSION APPENDIX LIST OF REFERENCES v

9 LIST OF FIGURES Fgure 1.1: Analyss process flow chart... 3 Fgure 2.1: Dynamc model n dscrete system, u(t) s nput, y(t) s output, and v(t) s dsturbance... 5 Fgure 2.2: A lnear tme-nvarant determnstc system... 8 Fgure 2.3: A lnear tme-nvarant determnstc system Fgure 2.4: A lnear tme-nvarant determnstc-stochastc system Fgure 3.1: 6-DOF mass-damper-sprng system Fgure 3.2: Percentage error saturaton accordng to tme duraton. Blue lne s DSI errors, red lne s SSI errors, and black lne s DSSI errors Fgure 4.1: Sketch of flexble contaner used n the test (2001, Tongj Report Chapter 3) Fgure 4.2: Real pcture of sol contaner (2001, Tongj Report Chapter 3) Fgure 4.3: El Centro exctaton X-drecton graph: (a) the Acceleraton data n tme hstory, (b) FFT n frequency doman Fgure 4.4: Shangha exctaton graph: (a) the Acceleraton data n tme hstory, (b) FFT n frequency doman Fgure 4.5: Kobe exctaton X-drecton graph: (a) the Acceleraton data n tme hstory (b) FFT n frequency doman Fgure 4.6: Measurement locaton on the superstructure: ple, box, and fxed foundaton Fgure 4.7: Sample sensor data at the ple foundaton (Shangha exctaton level 5) Fgure 4.8: Sample sensor data at the box foundaton (Shangha exctaton level 5) Fgure 4.9: Sample sensor data at the fxed foundaton (Shangha exctaton level 5) Fgure 4.10: Flowchart of data preprocessng Fgure 4.11: Preprocessed Shangha artfcal wave, exctaton level Fgure 4.12: Input and output settng for n4sd code (Shangha exctaton level 5) Fgure 4.13: Summary of modal frequency at Shangha exctaton at each case: (a) no nose, (b) RMS 20% nose nserted output only, and (c) RMS 20% nose nserted nput/output Fgure 4.14: Summary of modal frequency at El Centro exctaton at each case: (a) no nose, (b) RMS 20% nose nserted output only, and (c) RMS 20% nose nserted nput/output Fgure 4.15: Summary of modal frequency at Kobe exctaton at each case: (a) no nose, (b) RMS 20% nose nserted output only, and (c) RMS 20% nose nserted nput/output Fgure 5.1: Energy at the Shaker and base at ple foundaton v

10 Fgure 5.2: Frequency content n Shangha exctaton level 1 to 6 at Ple foundaton Fgure 5.3: Energy at the Shaker and base at box foundaton Fgure 5.4: Frequency content n Shangha exctaton level 1 to 6 at Box foundaton Fgure 5.5: El Centro Natural Frequency, Dampng Rato, & MAC result respect to the exctaton level Fgure 5.6: Shangha Natural Frequency, Dampng Rato, & MAC result respect to the exctaton level Fgure 5.7: Kobe Natural Frequency, Dampng Rato, & MAC result respect to the exctaton level Fgure 5.8: Change of stffness and dampng coeffcent n El Centro Fgure 5.9: Change of stffness and dampng coeffcent n Shangha Fgure 5.10: Change of stffness and dampng coeffcent n Kobe Fgure 5.11: Mode shape n El Centro exctaton, (a) Ple and Box (b) Fx and Box Fgure 5.12: Mode shape n Shangha exctaton, (a) Ple and Box (b) Fx and Box Fgure 5.13: Mode shape n Kobe exctaton, (a) Ple and Box (b) Fx and Box Fgure 5.14: Summary of Modal frequency and dampng rato/stffness and dampng coeffcent wth test order (a) 1 st mode, (b) 2 nd mode, (c) 3 rd mode, (d) 4 th mode, (e) 5 th mode, and (f) 6 th mode x

11 LIST OF TABLES Table 3.1: Mechancal Property of Smulaton Test Model Table 3.2: Natural Frequency and Dampng Rato of Smulaton Test Model Table 3.3: Smulaton Test Procedure Table 3.4: The Nose Effect on the Modal Parameters (no nose) Table 3.5: The Nose Effect on the Modal Parameters (RMS 20%) Table 3.6: The Nose Effect on the Modal Parameters (RMS 40%) Table 3.7: The Tme Duraton Effect on Modal Parameters (1 second: 3 cycles) Table 3.8: The Tme Duraton Effect on Modal Parameters (3 second:9 cycles) Table 3.9: The Movng Average Effect on Modal Parameters (0.5%) Table 3.10: The Movng Average Effect on Modal Parameters (1%) Table 3.11: The Movng Average Effect on Modal Parameters (5%) Table 4.1: Man Performance Parameters of Vbraton Platform Table 4.2: Test Schedule. Exctaton type (Exctaton name + Exctaton level) Table 4.3: Modal Parameters of Shangha Artfcal Exctaton level 2 at Fxed Foundaton (no nose) Table 4.4: Modal Parameters of Shangha Artfcal Exctaton level 2 at Fxed Foundaton (output nose only, RMS 20%) Table 4.5: Modal Parameters of Shangha Artfcal Exctaton level 2 at Fxed Foundaton (nput/output nose, RMS 20%) Table 4.6: Modal Parameters of El Centro Exctaton level 2 at Fxed Foundaton (no nose) Table 4.7: Modal Parameters of El Centro Exctaton level 2 at Fxed Foundaton (output nose only, RMS 20%) Table 4.8: Modal Parameters of El Centro Exctaton level 2 at Fxed Foundaton (nput/output nose, RMS 20%) Table 4.9: Modal Parameters of Kobe Exctaton level 2 at Fxed Foundaton (no nose) Table 4.10: Modal Parameters of Kobe Exctaton level 2 at Fxed Foundaton (nose output only, RMS 20%) Table 4.11: Modal Parameters of Kobe Exctaton level 2 at Fxed Foundaton (nput/output, RMS 20%) Table 5.1: Quantfcaton of energy dsspaton at ple and box foundaton x

12 1. INTRODUCTION 1.1 Motvaton and Objectve In the modern engneerng feld, expermental dynamcs s an mportant area of study. Ths area ncludes structural dynamcs, structural control, and structural health montorng. In expermental dynamcs, methods to obtan measured data have seen a great nflux of research efforts to develop an accurate and relable expermental analyss result. A techncal challenge s the procurement of nformatve data that exhbts the desred system nformaton. In many cases, the number of sensors s lmted by cost and dffculty of data archve. Furthermore, some nformatve data has techncal dffculty when measurng nput force and, even f obtanng the desred data were possble, t could nclude a lot of nose n the measurng data. As a result, researchers have developed many analytcal tools wth lmted nformatve data. In order to overcome ths dffculty, numerous output-only system dentfcaton methods have been proposed. Generally, Ibrahm Tme Doman (ITD) and Egensystem Realzaton Algorthm (ERA) have been wdely used n tme doman methods (Ibrahm, 1978, J.-N.Juang and R.S.Pappa, 1985). Recently, Stochastc Subspace Identfcaton (SSI) has emerged as a powerful output-only dentfcaton method (Jacobsen, 2009). Because ths method s based on the projecton algorthm, there s no teratve process, and t s relatvely fast and accurate. Ths SSI serves as one method under an umbrella of subspace dentfcaton methods. Other types of subspace dentfcaton methods nclude Determnstc Subspace Identfcaton (DSI) and Determnstc-Stochastc Subspace Identfcaton (DSSI). 1

13 Consequently, subspace dentfcaton methods are used n many areas; however, they are based on mportant assumptons. Based on the varous data condtons, DSI, SSI, and DSSI are dfferently appled under specfc assumptons, whch could affect ther analytcal results. Therefore, the effects of assumptons of DSI, SSI, and DSSI are quantfed and compared to each other through an extensve smulaton test. Furthermore, based on the concluson of the smulaton test, the effects of assumpton have been verfed through a large-scale expermental data analyss. 1.2 Research Approach Ths study wll contan two dstnct sectons: a smulaton test analyss and a large-scale experment data analyss. The smulaton test analyss has been completed to show the dfference under dfferent condtons of assumpton. Frstly, a sx-degree-of-freedom system wll be created usng MATLAB. From ths smulaton test, artfcal sets of exctaton force, respondng acceleraton, velocty, and dsplacement wll be made. In order to ensure the stochastc condton, known amounts of uncertanty wll be added. Selected assumptons wll be volated ntentonally and the data wll be analyzed by DSI, SSI, and DSSI. The result wll be compared to the known result and the effects of the assumpton wll be concluded. For the experment data analyss, a large-scale experment data set wll be analyzed n a smlar manner. The dynamc response data were tested at Tongj Unversty, Shangha, Chna. Snce the experment data s also determnstc, known amounts of uncertanty wll be added. 2

14 From the concluson of smulaton tests, the effect of assumptons wll be verfed by usng experment data. The entre process s outlned n Fgure 1.1 Fgure 1.1: Analyss process flow chart 1.3 Scope Chapter 2 of the paper s an ntroducton to the necessary theoretcal background on ths study, ncludng: system dentfcaton, DSI, SSI, and DSSI. Chapter 3 represents a smulaton test model and analyss result under varous assumptons on subspace dentfcaton. Chapter 4 verfed the smulaton test concluson usng an experment data from Tongj Unversty. Chapter 5 dscusses the effect of dfferent foundaton on superstructure usng DSSI. Chapter 6 3

15 summarzes the major fndngs of ths study. Chapter 7 puts explanatons about nose effect on DSI. 4

16 2. THEORETICAL BACKGROUND In ths chapter, dynamc model and system dentfcaton s brefly explaned. Furthermore, subspace system dentfcaton s explaned as a man analyss method of ths study. In partcular, determnstc subspace dentfcaton (DSI) that requres nput and output, stochastc subspace dentfcaton (SSI) that requres output only, and determnstc-stochastc subspace dentfcaton (DSSI) that requres nput and output are ntroduced. Ths chapter s ntended to serve as a theoretcal foundaton for Chapters 3 and 4 where the subspace dentfcaton methods are used. The man reference book s Subspace Identfcaton for Lnear Systems (Overschee and Moor, 1996). 2.1 Dynamc Model and System Identfcaton Fgure 2.1: Dynamc model n dscrete system, u(t) s nput, y(t) s output, and v(t) s dsturbance Fgure 2.1 shows a dynamc model n general. In order to know the dynamc model, nput u(t) and output v(t) are mportant nformaton (v(t) s a dsturbance of the system). Usng measured (nput and) output, analyzed and bult the dynamc model s called system dentfcaton. A mass-damper-sprng dynamc model can be expressed as follow: 5

17 (2.1) where M s mass, C s dampng coeffcent, and K s stffness. f(t) s an nput force of the system and become output. Related to the Fgure 2.1, the nput, u(t), s same as f(t) and the output, v(t), s same as. The system nformaton of Equaton 2.1 s M, C, and K. Therefore, system dentfcaton of ths equaton s to know the M, C, and K. As explaned before, there are lots of system dentfcaton methods. In ths study, subspace dentfcaton methods are used and N4sd (MATLAB code) s used for analyss. From the next subsecton, the subspace dentfcaton s explaned. Before the explanaton, there are mportant assumptons n ths method. 1. The dynamc system should be a lnear system. 2. The nose n SSI and DSSI s zero mean, statonary whte nose and uncorrelated wth the nput u(t). 3. The number of measured data n SSI goes to nfnty, and the data s statonary state. 2.2 Input and Output Data Equatons The output block Hankel matrx,, can be constructed usng measured acceleraton data: 6

18 j j j j j j y y y y y y y y y y y y y y y y y y y y y y y y ( ) ( ) (2.2) j j j j j j y y y y y y y y y y y y y y y y y y y y y y y y ( ) ( ) (2.3) where: 1. s the number of block rows that s a user-defned ndex. Snce each block row contans (number of nputs/measurement channel) rows, the conssts of total rows. 2. s the number of columns, n any case, should be larger than. In the stochastc models, s assumed for statstcal reasons 3. The upper part of the matrx, from 1 block row to block row, s defned past and the lower part of the matrx, from block row and block row, s defned future. The subscrpts p ndcates the past and f ndcates the future n Equaton 2.2. Ths block Hankel matrx shows all the response data rearranged wth tme shft, the total shft s whch s the same number of block row.

19 4. ndcates the one block row contaned and ndcates the one block row removed. Ths block Hankel matrx also contans all the response, but the border row for dstngushed future and past s dfferent. Equaton 2.3 shows the detal. 5. The other nput block Hankel matrces are constructed same method as Equaton 2.2 and Equaton 2.3. The state matrx s defned X X X X (2.4) 1 j 2 j Determnstc Subspace Identfcaton (DSI) Fgure 2.2: A lnear tme-nvarant determnstc system In the state space equaton, the determnstc system of order n can be expressed: (2.5) 8

20 where s the system matrx, s the nput matrx, s the output matrx and s the drect feedthrough term. ndcates the states, not the system nput, u s the system nput, and s the system response. The ndcates the number of nputs and means the number of outputs. Snce matrx has a dynamcal characterstc of system and the goal of ths dentfcaton method s to fnd ths matrx. The extended observablty matrx,, s defned as (2.6) [ ] where the subscrpt denotes the number of block rows and s greater than the number of mode, n. The reversed extended controllablty matrx,, s defned as: ) (2.7) The lower block trangular Toepltz matrx,, s defned as: D CB CBA 2 CA B CA 0 D CB 3 B CA 0 0 D 4 B D (2.8) The Equaton 2.5 can be rewrtten as: 9

21 (2.9) (2.10) (2.11) when the orthogonal to the nput matrx,, multples to the Equaton 2.10, t becomes (2.12) [ ][ ] (2.13) (2.14) where s the projecton matrx, ndcates the pseudo-nverse and s the block Hankel matrces consstng of nputs and outputs as: U Y P P (2.15) The left hand sde of Equaton 2.12, only nput and output data s requred. Usng a sngular value decomposton (SVD) of Equaton The extended observablty,, can be obtaned (2.16) 10

22 where s a untary matrx, s a rectangular dagonal matrx, and s a untary matrx (* s a conjugate transpose). The hat, ˆ, on the observablty matrx and the states matrx ndcate the calculated result and the subscrpt 0 on the states matrx ndcates tme lag zero n Equaton 2.4. The tme lag one n the states matrx means one block row of the projecton matrx,, removed at the top and one block row of the observablty matrx,, removed at the bottom. All the states can be expressed usng ths smlarty usng remove blocks and SVD. Therefore, the matrces A, B, C, and D can be solved from Equaton 2.17 Xˆ Y 1 Aˆ Cˆ Bˆ Xˆ Dˆ U (2.17) 2.4 Stochastc Subspace Identfcaton (SSI) Fgure 2.3: A lnear tme-nvarant determnstc system In the state space equaton, the stochastc system of order n can be expressed: (2.18) 11

23 12 where w(t) represents the process nose that causes from the modelng naccuraces and v(t) s called the measurement nose from the sensor naccuraces. It s assumed that w(t) and v(t) are zero mean whte nose vector sequences, ndependent of the state X. The reversed extended stochastc controllablty matrx,, s defned as: G AG G A G A 1 2 (2.19) where G s the state and output covarance matrx (2.20) The block Toepltz matrx,, s constructed from the stochastc output covarance matrces as: (2.21) where the stochastc output covarance matrx (2.22) The projecton matrx,, n the stochastc system s defned as:

24 (2.23) (2.24) In Equaton 2.23, the projecton matrx,, can be calculated from the gven output data. As the Equaton 2.16, usng the sngular value decomposton the observablty matrx,, can be obtaned. In Equaton 2.24, denotes the matrx wthout the last (number of outputs) rows. can be also calculated from the gven output data. Therefore, can be obtaned (2.25) From the output data only, and can be calculated. Xˆ Y 1 A Xˆ C w v (2.26) In order to obtan the G matrx, n Equaton 2.19 should be obtaned frst (2.27) The G matrx s placed at the last columns of. Lastly,. 13

25 2.5 Determnstc Stochastc Subspace Identfcaton (DSSI) Fgure 2.4: A lnear tme-nvarant determnstc-stochastc system In the state space equaton, the determnstc-stochastc system of order n can be expressed: (2.28) where,,, and. s the states, u s the system nput, and s the system response. The ndcates a number of nputs and ndcates a number of outputs. Equaton 2.28 s the combned result of the two prevous explaned theores. The nput-output equatons for the combned system can be defned as follow: (2.29) (2.30) (2.31) 14

26 15 where the upper-scrpt, d, ndcates determnstc and s ndcates stochastc. The oblque projecton,, s defned as (2.32) and the sngular value decomposton (SVD) of the oblque projecton s T T T V S U V V S U U (2.33) Lke Equaton 2.14 and Equaton 2.16, the extended observablty,, can be obtaned. Therefore, the state, can be calculated as (2.34) Fnally, the matrces, A, B, C, and D can be solved from U X D C B A Y X 1 ˆ ˆ ˆ ˆ ˆ ˆ v w (2.35) where the process nose, and the measurement nose,, are uncorrelated wth the nput, and are not dentcally zero.

27 3. SUBSPACE IDENTIFICATION FOR SIMULATION TESTS The theoretcal background for ths study was ntroduced n Chapter 2. A sx-degree-offreedom (6-DOF) mass-damper-sprng system model was created usng MATLAB and tested wth vared exctatons to observe the effect of assumpton on subspace dentfcaton methods. In smulaton tests, the nformaton about the system model was known, so the results of the subspace dentfcaton methods could be compared. 3.1 Smulaton Setup The purpose of ths smulaton test s to observe the effect of assumptons on subspace dentfcaton methods. The prevous chapter addresses the mportant presumptons for the subspace dentfcaton; the system should be a lnear system, the number of data s nfnte and the data s n a statonary state, and the nose s whte Gaussan nose and uncorrelated wth the nput. Among the mportant assumptons, selected assumpton s volated ntentonally. The selected assumpton wll ntroduce n smulaton test procedure Descrpton of Models A sx-story mass-damper-sprng buldng model was created wth MATLAB usng the modal superposton method for the smulaton test (Fgure 3.1). The heght of the model s 0.96 m and every floor s mass, damper, and stffness have same value (materal propertes of the smulaton model s summarzed n Table 3.1). An nput force was appled at the 6 th floor and 16

28 each mass dsplacement, velocty, and acceleraton has been calculated. The man equaton of moton for ths smulaton s (3.1) where F(t) s the nput force, and x,, and are the response of the nput force. M, C, and K are the mass, dampng coeffcent, and stffness, respectvely. M, C, and K are a matrx, and the values are shown n Table 3.1. Fgure 3.1: 6-DOF mass-damper-sprng system 17

29 Table 3.1: Mechancal Property of Smulaton Test Model Property Value Mass Dampng coeffcent Stffness kg Ns/m N/m Table 3.2: Natural Frequency and Dampng Rato of Smulaton Test Model Mode Number Natural Frequency Dampng Rato 1st Hz nd Hz rd Hz th Hz th Hz th Hz The natural frequency and dampng rato of the smulaton model are shown n Table 3.2. Seven ponts, sx lumped masses and the basement pont, make sx dfferent mode shapes, so the number of modes s 6. Therefore, 6 modes of natural frequency and dampng rato are represented as the result. These values n Table 3.2 are the reference result of ths smulaton test. 18

30 3.1.2 Exctaton Informaton In ths smulaton test, a generated exctaton becomes the nput force of the system. In order to observe the effect of varous factors, ths exctaton has to be vared. Whte Gaussan exctaton s the reference exctaton, and accordng to the smulaton schedule the exctaton s volated. The samplng frequency s 200 Hz, and the tme perod of exctaton s 100 seconds. Ths tme perod ncludes approxmately 300 cycles of the 1 st mode of system s natural frequency that s Hz. For a far comparson the root-mean-square (RMS) of each exctaton equalzes to 100 of all exctatons n the smulaton tests. The equaton of RMS s as follow: (3.2) where, n s the number of data. Each exctaton has two cases: one s the no nose case for the determnstc method and the other s the added nose case (RMS 20%) for the stochastc method. The nose artfcally nserts on ts nput exctaton and output acceleratons. Every exctaton has these two cases to observe the dfferent results of DSI, SSI, and DSSI. Therefore, the result represents to two cases: no nose case and nose case. In ths smulaton test, the results of the subspace dentfcaton methods correspond to modal parameters; natural frequency, dampng rato, and mode shape. However, the dampng rato s too small to compare to the value, and the mode shape s only used for checkng the mode order wth the natural frequency. The natural frequency s the comparng factor n ths smulaton test and all calculated results are compared to the reference result n the Table

31 3.1.3 Smulaton Test Procedure A systematc seres of smulaton tests were conducted. The mportant assumptons of subspace dentfcaton are a crucal part of the smulaton test procedure. The frst smulaton test s conducted by whte Gaussan exctaton wth 100 second tme duraton. Ths smulaton test tells the accuracy of DSI, SSI, and DSSI n determnstc condton. The accuracy s calculated by the percentage errors (Equaton 3.3). (3.3) where s the obtaned result from each step s smulaton test, and s the reference result from the smulaton settng. Then, one of the assumptons of subspace dentfcaton methods s volated accordng to the procedure n Table 3.3. As explaned before, the smulaton system s a 6-DOF mass-damper-sprng model, and t can be obtaned by 6 modes. Therefore, the number of modes of ths smulaton system s 6. If there s no nose-nfluenced mode n the condton, 6 modes can be detected. However, f nose affects the mode, or f there exsts an absent mode, the mode place leaves as blank. Ths means the method, whch has nose-nfluenced and absent modes, has less accuracy than a method that can detect 6 modes. 20

32 Table 3.3: Smulaton Test Procedure Order Assumpton Exctaton type Tme Nose duraton (sec) RMS(%) 1 No Volaton Whte Gaussan Nose effect Whte Gaussan Nose effect Whte Gaussan Tme duraton Whte Gaussan Tme duraton Whte Gaussan Tme duraton Whte Gaussan Saturated Smulaton Result Accordng to the smulaton test procedure, subspace dentfcaton methods are used. All the natural frequency results are represented as percentage errors to be compared each other n a far manner. The effect of nose, tme duraton, and statonarty of exctaton s consdered to the factors of the smulaton test Effect of Nose The frst exctaton s whte Gaussan exctaton wth 100 second tme duraton. Ths determnstc case fulflled the assumpton of the subspace dentfcaton methods; whte Gaussan, statonarty, and nfnte exctaton tme duraton. Then, for the stochastc case, the whte Gaussan exctaton and respondng acceleratons are contamnated wth the RMS 20% nose. From comparng these two cases, the nose effect can be checked. Furthermore, n order to observe the effect of the nose magntude, the RMS of nose s ncreased to 40%. Table 3.4, 3.5, and 3.6 represent the result of subspace dentfcaton methods at each condton. 21

33 In the determnstc case, DSI and DSSI have exactly the same result compared to the reference result, and SSI has a 0.17% error average. In the nose-corrupted case wth RMS 20%, DSI, SSI, and DSSI have averages of 0.05%, 0.20%, and 0.05% errors, respectvely. Furthermore, for the nose case wth RMS 40%, DSI, SSI, and DSSI have averages of 0.25%, 0.14%, and 0.05% errors, respectvely. Errors n the all cases are below 1%, even for the RMS 40% nosed case. From ths result, f the assumpton s satsfed, the subspace dentfcaton methods are very accurate. Furthermore, added measurement nose does not affect the subspace dentfcaton methods. 22

34 Mode No. Table 3.4: The Nose Effect on the Modal Parameters (no nose) Modal Frequency Dampng Rato DSI SSI DSSI DSI SSI DSSI Freq(Hz) Error(%) Freq(Hz) Error(%) Freq(Hz) Error(%) Pct(%) Error(%) Pct(%) Error(%) Pct(%) Error(%) 1 st % % % 0.394% % 0.316% % 0.394% % 2 nd % % % 1.160% % 1.119% % 1.160% % 3 rd % % % 1.859% % 1.826% % 1.859% % 4 th % % % 2.449% % 2.646% % 2.449% % 5 th % % % 2.897% % 2.687% % 2.897% % 6 th % % % 3.177% % 3.136% % 3.177% % Mode No. MAC DSI SSI DSSI Pct(%) Pct(%) Pct(%) 1 st % % % 2 nd % 99.98% % 3 rd % 99.99% % 4 th % % % 5 th % 99.99% % 6 th % 99.91% % 23

35 Mode No. Table 3.5: The Nose Effect on the Modal Parameters (RMS 20%) Modal Frequency Dampng Rato DSI SSI DSSI DSI SSI DSSI Freq(Hz) Error(%) Freq(Hz) Error(%) Freq(Hz) Error(%) Pct(%) Error(%) Pct(%) Error(%) Pct(%) Error(%) 1 st % % % 0.333% % 0.888% % 0.395% % 2 nd % % % 1.130% % 1.183% % 1.156% % 3 rd % % % 1.838% % 1.886% % 1.876% % 4 th % % % 2.703% % 2.461% % 2.467% % 5 th % % % 2.680% % 2.912% % 2.923% % 6 th % % % 3.111% % 3.106% % 3.136% % Mode No. MAC DSI SSI DSSI Pct(%) Pct(%) Pct(%) 1 st % 99.98% % 2 nd 99.98% % % 3 rd % % % 4 th 99.99% % % 5 th 99.98% 99.99% 99.99% 6 th 99.93% 99.97% 99.98% 24

36 Mode No. Table 3.6: The Nose Effect on the Modal Parameters (RMS 40%) Modal Frequency Dampng Rato DSI SSI DSSI DSI SSI DSSI Freq(Hz) Error(%) Freq(Hz) Error(%) Freq(Hz) Error(%) Pct(%) Error(%) Pct(%) Error(%) Pct(%) Error(%) 1 st % % % 0.320% 33.01% 0.525% % 0.391% 0.92% 2 nd % % % 1.110% % 0.978% -4.35% 1.100% 5.21% 3 rd % % % 1.838% -7.53% 1.719% -1.09% 1.853% 0.32% 4 th % % % 2.749% 0.81% 2.469% 12.23% 2.482% -1.33% 5 th % % % 2.704% -2.57% 2.823% -6.67% 2.971% -2.57% 6 th % % % 3.224% -3.07% 3.079% -2.87% 3.074% 3.23% Mode No. MAC DSI SSI DSSI Pct(%) Pct(%) Pct(%) 1 st 99.97% % % 2 nd % 99.98% % 3 rd 99.97% % 99.99% 4 th 99.98% 99.99% 99.99% 5 th 99.98% 99.98% 99.99% 6 th 99.80% 99.93% 99.90% 25

37 3.2.2 Effect of Tme Duraton 1) Whte Gaussan exctaton wth extremely short tme duraton (3 cycles) The tme duraton s assumed nfnte long; however, nfnte data s not possble n realty. In ths smulaton, 100 seconds that contaned approxmately 300 cycles of the 1 st mode of natural frequency s used nstead of nfnte tme. The extremely short case s desgned to volate ths tme duraton assumpton; one second tme duraton of whte Gaussan nose s used as the exctaton. Ths tme duraton has approxmately 3 cycles of the 1 st mode of natural frequency. Table 3.7 represents the result of subspace dentfcaton method. DSI and DSSI stll have exactly the same value as the reference result; however, SSI has an average of 2.32% error and the error s centralzed at the low frequency. Ths means that DSI and DSSI are not nfluenced by the tme duraton, but SSI s nfluenced. In order to check the effect of tme duraton, next subsecton s conducted wth longer tme duraton. 26

38 Mode No. Table 3.7: The Tme Duraton Effect on Modal Parameters (1 second: 3 cycles) Modal Frequency Dampng Rato DSI SSI DSSI DSI SSI DSSI Freq(Hz) Error(%) Freq(Hz) Error(%) Freq(Hz) Error(%) Pct(%) Error(%) Pct(%) Error(%) Pct(%) Error(%) 1 st % % % 0.394% % 7.618% % 0.394% % 2 nd % % % 1.160% % 3.346% % 1.160% % 3 rd % % % 1.859% % 1.503% % 1.859% % 4 th % % % 2.449% % 1.972% % 2.449% % 5 th % % % 2.897% % 1.351% 39.29% 2.897% % 6 th % % % 3.177% % 3.451% % 3.177% % Mode No. MAC DSI SSI DSSI Pct(%) Pct(%) Pct(%) 1 st % 97.18% % 2 nd % 99.75% % 3 rd % 98.79% % 4 th % 99.72% % 5 th % 99.97% % 6 th % 99.56% % 27

39 2) Whte Gaussan exctaton wth short tme duraton (9 cycles) In order to check the effect of tme duraton, 3 second tme duraton of whte Gaussan exctaton s used as an exctaton. Ths tme duraton has approxmately 9 cycles of the 1 st mode of natural frequency of the system. Although ths tme duraton s 3 tmes longer than the prevous condton, t s stll a short tme duraton compared to the assumpton. Table 3.8 represents the result of subspace dentfcaton methods at each condton. DSI and DSSI stll have no errors, and SSI has an average of 1.5% errors n each mode. Compared to the 3 cycle tme duraton case, the 9 cycle tme duraton case detects more modes and t has better accuracy. However, because of the assumpton, SSI has stll errors. 28

40 Mode No. Table 3.8: The Tme Duraton Effect on Modal Parameters (3 second:9 cycles) Modal Frequency Dampng Rato DSI SSI DSSI DSI SSI DSSI Freq(Hz) Error(%) Freq(Hz) Error(%) Freq(Hz) Error(%) Pct(%) Error(%) Pct(%) Error(%) Pct(%) Error(%) 1 st % % % 0.394% % 1.029% % 0.394% % 2 nd % % % 1.160% % 1.900% % 1.160% % 3 rd % % % 1.859% % 2.611% % 1.859% % 4 th % % % 2.449% % 3.369% % 2.449% % 5 th % % % 2.897% % 1.819% 87.51% 2.897% % 6 th % % % 3.177% % 2.159% % 3.177% % Mode No. MAC DSI SSI DSSI Pct(%) Pct(%) Pct(%) 1 st % 99.40% % 2 nd % 99.98% % 3 rd % 98.98% % 4 th % 99.92% % 5 th % 97.22% % 6 th % 96.73% % 29

41 3) Convergent Tme n DSI, SSI, and DSSI The prevous two cases do not have enough tme length to obtan acceptable natural frequency. When the acceptable natural frequency has below 1% average error, n the nose case, DSI, SSI, and DSSI have a convergent tme n Fgure 3.2. In order to fnd the convergent tme, the data s measured every 2 second from 1 to 20 second and measured every 20 seconds from 20 to 100 seconds. The convergent tme of SSI s 5 second. The 1 st mode of modal frequency s approxmately 3 Hz, and ths means that 1 second contans about 3 cycles of 1 st mode of modal frequency. Therefore, the 5 seconds n SSI mean that t needs 15 cycles of 1 st mode of modal frequency to obtan the acceptable result. No nose case Fgure 3.2: Percentage error saturaton accordng to tme duraton. Blue lne s DSI errors, red lne s SSI errors, and black lne s DSSI errors. 30

42 3.2.3 Effect of Movng Average In order to allevate nose effect, the movng average s the most common method n dgtal sgnal process. In expermental modal analyss, the movng average s used for the nose reducton. However, snce the movng average makes the orgnal data smooth, t s not good for the frequency pont of vew. Then, how does t affect the result of subspace dentfcaton method f the data s used wth the movng average. To observe the effect of movng average on subspace dentfcaton method, whte Gaussan exctaton nserted RMS 20% s used and the 0.5%, 1%, and 5% movng average are conducted. Table 3.9, 10, and 11 ndcate the 0.5%, 1%, and 5% movng average effect on modal parameters, respectvely. In 0.5% movng average case, DSI and SSI detect 5 modes and DSSI detects all modes. All modal frequences are below 1% errors. In 1% movng average case, DSI and SSI detect 5 modes and DSSI detects all modes. The magntude of modal frequences becomes a lttle bgger, however the average of reman modes s stll below 1% errors. In 5% movng average case, DSI, SSI, and DSSI detect 5 modes. The average of modal frequences of reman modes s stll below 1% errors. That means the movng average affects hgh mode of modal parameters. 31

43 Mode No. Table 3.9: The Movng Average Effect on Modal Parameters (0.5%) Modal Frequency Dampng Rato DSI SSI DSSI DSI SSI DSSI Freq(Hz) Error(%) Freq(Hz) Error(%) Freq(Hz) Error(%) Pct(%) Error(%) Pct(%) Error(%) Pct(%) Error(%) 1 st % % % 0.19% 43.99% 0.216% 63.68% 0.370% 180.2% 2 nd % % % 1.00% 156.4% 0.600% 54.32% 1.116% 187.4% 3 rd % % % 1.58% 153.5% 4.023% 546.5% 2.487% 299.6% 4 th % % % 2.56% 212.7% 3.043% 271.2% 2.855% 248.2% 5 th % % % 2.91% 199.6% 3.574% 268.5% 4.500% 358.7% 6 th NAN NAN NAN NAN % NAN NAN NAN NAN 4.119% 287.2% Mode No. MAC DSI SSI DSSI Pct(%) Pct(%) Pct(%) 1 st 99.99% 99.99% % 2 nd 99.99% 99.98% % 3 rd 99.66% 99.67% 99.98% 4 th 99.37% 99.35% 99.84% 5 th 96.61% 94.98% 98.80% 6 th NAN NAN 84.84% 32

44 Mode No. Table 3.10: The Movng Average Effect on Modal Parameters (1%) Modal Frequency Dampng Rato DSI SSI DSSI DSI SSI DSSI Freq(Hz) Error(%) Freq(Hz) Error(%) Freq(Hz) Error(%) Pct(%) Error(%) Pct(%) Error(%) Pct(%) Error(%) 1 st % % % 0.196% 47.73% 4.988% % 0.477% % 2 nd % % % 1.326% % 1.007% % 1.119% % 3 rd % % % 1.900% % 2.174% % 2.154% % 4 th % % % 2.950% % 2.492% % 2.860% % 5 th % % % 3.003% % 2.970% % 3.865% % 6 th NAN NAN NAN NAN % NAN NAN NAN NAN 4.568% % Mode No. MAC DSI SSI DSSI Pct(%) Pct(%) Pct(%) 1 st 99.96% 99.00% 99.99% 2 nd 99.92% 99.97% 99.99% 3 rd 99.90% 99.78% 99.97% 4 th 99.31% 98.84% 99.70% 5 th 77.66% 91.64% 98.14% 6 th NAN NAN 97.91% 33

45 Mode No. Table 3.11: The Movng Average Effect on Modal Parameters (5%) Modal Frequency Dampng Rato DSI SSI DSSI DSI SSI DSSI Freq(Hz) Error(%) Freq(Hz) Error(%) Freq(Hz) Error(%) Pct(%) Error(%) Pct(%) Error(%) Pct(%) Error(%) 1 st % % % 2.710% % 0.538% % 0.437% % 2 nd % % % 2.565% % 1.273% % 1.229% % 3 rd % % % 1.771% % 1.699% % 2.265% % 4 th % % % 2.341% % 2.282% % 2.494% % 5 th % % % 4.827% % 2.882% % 2.915% % 6 th NAN NAN NAN NAN NAN NAN NAN NAN NAN NAN NAN NAN Mode No. MAC DSI SSI DSSI Pct(%) Pct(%) Pct(%) 1 st 99.88% % % 2 nd 99.34% 99.96% 99.87% 3 rd 99.76% 99.91% 99.61% 4 th 97.14% 99.31% 95.80% 5 th 89.40% 92.74% 80.65% 6 th NAN NAN NAN 34

46 In summary, smulaton tests are conducted n order to observe the effect of assumptons on subspace dentfcaton methods. From the smulaton tests, the followng fndngs are observed: 1) Nose does not affect the subspace dentfcaton methods. 2) Tme duraton n SSI affects the subspace dentfcaton methods. 3) DSI does not have nose terms, however the result s smlar to the results of DSSI and SSI. 4) DSSI s the most accurate method among three approaches. 35

47 4. SHAKING TABLE SOIL-FOUNDATION-SUPERSTRUCTURE INTERACTION (SFSI) TESTS In the Chapter 2, the theoretcal approach of subspace dentfcaton methods has been ntroduced. Through the smulaton test, the effects of assumpton have been analyzed n the prevous Chapter 3. In ths chapter, based on the prevous smulaton test result, a large scale sol foundaton superstructure nteracton expermental data s analyzed at the smlar manner. 4.1 Expermental Setup The large scale sol-foundaton-superstructure nteracton (SFSI) experment was desgned to understand the effects of buldng-foundaton systems. Because tradtonal structural desgn and analyss technques have generally assumed that a sol-foundaton-structure system responds wth a fxed-base (Stewart and Mylonaks, 2010). In order to obtan the purpose, the whole experment was desgned, bult, and tested by Tongj Unversty (Bo, 2002; L, 2004; Lu, 2004; Lu, 2005) Descrpton of Models Three dentcal 1:10 scaled 12-story cast-n-place renforced concrete frame buldng models wth ple, box, fxed foundaton were expermentally tested n the State Key Laboratory for Dsaster Reducton n Cvl Engneerng at Tongj Unversty under varous sesmc loadng condton. The buldng models wth the ple and box, respectvely, constructed on three-layers of sol (slty clay, powder sand sol and sandy sol) contaned n a cylndrcal soft contaner wth 36

48 3000-mm dameter made of 5-mm thckness rubber membrane wth renforcement outsde the contaner to reduce the wave reflecton between the model and the contaner boundary (box effect), durng dynamc tests. Both the model and contaner were placed on a strong hydraulc shake table for un (X-drecton) and b-drectonal (X and Z-drecton) dynamc tests. For fxed foundaton, the buldng model was rgdly attached to the shake table drectly wthout the sol contaner. Fgure 4.1: Sketch of flexble contaner used n the test (2001, Tongj Report Chapter 3) 37

49 (a) Front vew of test contaner (b) Vertcal vew of test contaner Fgure 4.2: Real pcture of sol contaner (2001, Tongj Report Chapter 3) The earthquake smulaton vbraton platform conssts of shakng-table, foundaton, pump pressure, dstrbuton system, vbrator and data acquston system. The man performance parameters of vbraton platform are represented n Table 4.1. Table 4.1: Man Performance Parameters of Vbraton Platform Shakng table dmenson Maxmum loadng weght Vbraton drecton 25 ton X, Y, Z (3-dmenson, 6 DOF) Maxmum acceleraton X: 1.2g, Y: 0.8g, Z: 0.7g Frequency range of operaton 0.1 ~50 Hz 38

50 4.1.2 Exctaton Informaton Three dfferent types of exctatons, El Centro earthquake record (EL), the Shangha artfcal wave record (SH) and the Kobe earthquake record (KB) were appled for the test. The El Centro earthquake occurred on May 18 th, 1940 n the Imperal Valley n Southern Calforna near the nternatonal border of the Unted States and Mexco. The man earthquake took nne lves and caused property damage estmated at $6 mllon. The magntude of frst shock was 6.9 (Stover and Coffman, 1993). Ths exctaton s wdely used as a classc record n structural test and sesmc experment (Jaewook, 2000, Beng Arsoy, 2010, Reyolando, 2010). In ths experment, the man strong shock lasts about 5 second durng the 10 second whole exctaton. The scaled X-drecton (N-S component) tme hstory of the El Centro earthquake record acceleraton and ts Fast Fourer Transform (FFT) are represented n Fgure 4.3 (a) Tme hstory (b) Frequency doman Fgure 4.3: El Centro exctaton X-drecton graph: (a) the Acceleraton data n tme hstory, (b) FFT n frequency doman 39

51 The Shangha artfcal wave was artfcally desgned exctaton that had a broadband frequency. Because of ths rch frequency characterstc, some studes used the Shangha artfcal wave for comparson purpose (Xao, 2011). In ths experment, the man strong shock lasts about 10 second durng the 16 second whole exctaton. It s the longest exctaton among the three exctatons. The scaled tme hstory of Shangha artfcal wave acceleraton and ts FFT are represented n Fgure 4.4 (a) Tme hstory (b) Frequency doman Fgure 4.4: Shangha exctaton graph: (a) the Acceleraton data n tme hstory, (b) FFT n frequency doman The Kobe earthquake or Great Hanshn earthquake occurred on Jan 17 th, 1995 n the southern part of Hyogo Prefecture, Japan. Ths was Japan s worst earthquake n the 20 th century. It caused property damage estmated about $200 bllon. The magntude was 6.8 (Anshel, 1995). The scaled X-drecton (N-S component) tme hstory of Kobe earthquake record acceleraton and ts FFT are represented n Fgure

52 (a) Tme hstory (b) Frequency doman Fgure 4.5: Kobe exctaton X-drecton graph: (a) the Acceleraton data n tme hstory (b) FFT n frequency doman Measurement Locatons Accelerometers, pore pressure gauges, sol pressure gauges and stran gauges were nstalled to measure the dynamc response of the superstructure, foundaton and sol. However, eght accelerometers were only used to obtan dynamc response of the shakng table and the superstructure. A0 was mounted on the shakng table and t was measured by drect exctaton. Includng A1 at the base and A7 on the top of the superstructure, the sensors were nstalled on every other floors of the superstructure. Fgure 4.6 shows the measurement locaton A0 to A7. Fgure 4.7, 4.8, and 4.9 show the stored acceleraton data and ther FFT at the each sensor from A0 to A7 (from top to bottom, respectvely). Fgure 4.7 presents the stored data at the ple foundaton, Fgure 4.8 presents the stored data at the box foundaton, and Fgure 4.9 presents the stored data at the fxed foundaton. In fxed foundaton case, snce there s no sol foundaton, the acceleraton data at A0 and A1 are same. 41

53 (a) Ple foundaton (b) Box foundaton (c) Fxed foundaton Fgure 4.6: Measurement locaton on the superstructure: ple, box, and fxed foundaton 42

54 Fgure 4.7: Sample sensor data at the ple foundaton (Shangha exctaton level 5) 43

55 Fgure 4.8: Sample sensor data at the box foundaton (Shangha exctaton level 5) 44

56 3 Fgure 4.9: Sample sensor data at the fxed foundaton (Shangha exctaton level 5) 45

57 Table 4.2: Test Schedule. Exctaton type (Exctaton name + Exctaton level) Test order Exctaton type Peak acceleraton (G) 1, 2, 3 EL1, SH1, KB , 5, 6 EL2, SH2, KB , 8, 9 EL3, SH3, KB , 11, 12 EL4, SH4, KB , 14, 15 EL5, SH5, KB , 17, 18 EL6, SH6, KB , 20, 21 EL7, SH7, KB Test Protocols A comprehensve seres of tests were conducted n three dfferent earthquake scenaros at seven dfferent exctaton magntudes. The three base exctaton records used were El Centro earthquake record (EL), Shangha artfcal wave (SH) and Kobe earthquake record (KB). The peak acceleraton of the exctatons was used to the scale of the model. The levels of the exctaton were categorzed nto seven levels from 1 to 7: rangng from 0.093G to 0.931G. Test order correlated to exctaton type and level. The loadng schedule ndcated n Table 4.2. Each foundaton test model was appled to the same test protocol. For example, SH1 means that the second test procedure wth peak acceleraton G under Shangha artfcal exctaton wave. KB6 means that the eghteenth test procedure wth peak acceleraton G under Kobe exctaton wave. In ths study, exctatons level 1 to 6 were only used because the fxed foundaton model had been serously damaged after exctaton level 6 loaded to the superstructure. 46

58 4.2 Data Preprocessng The accuracy of system dentfcaton method depends on the amount of data used n the dentfcaton process (Moaven, 2007). In ths study, data preprocessng s appled to nput and output data ncludng detrendng, wndowng, zero-phase dstorton, and low-pass flter. Lastly, numercal ntegraton has been appled to check the correspondng veloctes and dsplacements. More detal procedure s as follow: 1. The DC and up to the 6 th order polynomals trend were subtracted from the raw acceleratons, and a 5% cosne-taper wndow was appled to the acceleraton tme hstores to prevent spectrum leakage. 2. A zero-phase dstorton whch was prevent to phase dstorton and low-pass flter wth cutoff frequency of 120Hz were desgned and appled to the acceleraton tme hstory (samplng frequency s 250 Hz). 3. Standard numercal ntegraton procedures were used to acheve the correspondng velocty dsplacement tme hstory. Fgure 4.10: Flowchart of data preprocessng 47

59 (a) Acceleraton Data (G) (b) Velocty Data (cm/s) (c) Dsplacement Data (cm) Fgure 4.11: Preprocessed Shangha artfcal wave, exctaton level Subspace Identfcaton Method for SFSI Test The stored and preprocessed acceleraton data set from the descrbed experment was analyzed usng the subspace dentfcaton methods. An nput-output format for the subspace dentfcaton method was explaned. In order to observe the effects of assumpton n subspace dentfcaton, several analyses were conducted. The natural frequency usng DSI, SSI, and DSSI was used to compare the methods. 48

60 4.3.1 Input and output for Subspace Identfcaton Method The equaton of base exctaton of the superstructure can be expressed as (4.1) where M s the mass matrx, D s the dampng matrx, and K s the stffness matrx. The M, D and K matrx are matrces and s unt column vector, and s the number of degrees of freedom of the system. It s assumed that the system s lnear and the system parameters are avalable for ths analyss. s absolute acceleraton obtaned from the sensors, s the base acceleraton vector, and s the relatve acceleraton matrx wth respect to the base, where. Equaton 4.1 can be rewrtten as (4.2) The base exctaton nput n ths system s and the outputs are. In Fgure 4.6, the s obtaned from the A1 and s obtaned from the A2 to A7. Therefore, the nput and the output are known values from the measured data. Fgure 4.12 shows the nput,, and the relatve acceleraton,, that s used to the output n the analyss. The base exctaton,, s stored by the sensor A1 and the absolute acceleraton,, s stored by the sensor A2 n Fgure 4.6. In the same manner, the absolute acceleraton are stored by the sensor A3, A4, A5, A6, and A7, respectvely. 49

61 (a) The nput, (b) The relatve acceleraton, (c) The relatve acceleraton, (d) The relatve acceleraton, (e) The relatve acceleraton, (f) The relatve acceleraton, (g) The relatve acceleraton, Fgure 4.12: Input and output settng for n4sd code (Shangha exctaton level 5) 50