A Proposal of Mode Shape Estimation Method Using Pseudo-Modal Response : Applied to Steel Bridge in Building

Transcription

1 A Proposal of Mode Shape Estmaton Method Usng Pseudo-Modal Response : Appled to Steel Brdge n Buldng More nfo about ths artcle: Doyoung Km 1, Hak Bo Shm 2, Hyo Seon Park 1 1 Department of Archtectural Engneerng, Yonse Unversty. Seoul, KOREA 2 Lotte Research & Development Insttute Seoul, KOREA dydy8422@hanmal.net Key words: System Identfcaton, Modal Parameter, Mode Shape, Modal Responses, Steel Brdge Abstract Studes of system dentfcaton (SI)-based structural health montorng (SHM) are beng actvely conducted to ensure structural safety. Recently, many SI technques have been developed usng the output-only SI paradgm for estmatng the modal parameters. The features of these output-only SI methods are obtaned usng frequency doman decomposton (FDD) and stochastc subspace dentfcaton (SSI), both of whch nvolve the use of algorthms based on orthogonal decomposton, such as sngular value decomposton (SVD). However, the SVD leads to a hgh level of computatonal complexty to estmate modal parameters. Ths paper proposes a technque to estmate the mode shape at lower computatonal cost. Ths technque shows pseudo-modal responses (PMR) through a bandpass flter and suggests a tme-hstory modal assurance crteron (MAC). Fnally, the mode shape s estmated from the PMR and the tme hstory MAC. Expermental tests of the vbraton measurement were performed, and the results of mode shape and computaton tme between a representatve SI method and the proposed method were compared. 1 INTRODUCTION In a varety of felds, ncludng cvl and archtectural engneerng as well as mechancal and aerospace engneerng, structural health montorng (SHM) and damage detecton technques have been actvely developed over the past several decades for evaluaton of the safety of structures[1-4]. The process of estmatng structural modal parameters s referred to as system dentfcaton (SI)[5]. Recently, SI technques have been rapdly developed based on the output-only SI paradgm[6-8]. The modal parameters from SI methods, such as natural frequency, mode shape and modal dampng rato, present structural dynamc propertes that condensng the tme hstory vbraton measurement data. The features of these SI methods are obtaned usng frequency doman decomposton (FDD)[9] and stochastc subspace dentfcaton (SSI)[10]; both of these approaches nvolve the use of algorthms based on orthogonal decomposton, such as sngular value decomposton (SVD), because the applcatons of the algorthms have been valdated n varous felds[11,12]. Therefore, orthogonal decomposton algorthms are used n one part of the SI methods to dentfy the modal parameters of structures from vbraton measurements

2 based on successfully appled examples. Despte the broad use of SVD for estmatng modal parameters, the SVD leads to a hgh level of computatonal complexty when performng SI[13]. SVD starts wth bdagonalzng the target matrx [14]. Next, the bdagonalzed matrces are decomposed nto sngular values and vectors. Modfed procedures of decomposton and new algorthms are proposed to reduce the computatonal cost. For target matrces that are square matrces, several studes[15,16] developed SVD wth less computaton tme usng the bsecton method, one of the optmzaton algorthms. The computaton tme s such a crtcal ssue when estmatng the modal parameters because both FDD and SSI (one of the most popular SI methods lately) nvolve teratve SVDs. As the buldng structures contnue to become hgher, larger and more complex, the computatonal complexty and the tme requred to perform SI wll be very mportant ssues. As a result, SI methods for estmatng the modal parameters must be consdered to reduce computatonal costs. Ths paper proposes a method for vsually estmatng the mode shape usng a band-pass flter and the hstorcal responses of the structure. Ths proposed method does not requre the use of the SVD algorthm, whch causes hgh computatonal complexty. As a result, the proposed method requres a reduced amount of computaton tme compared to the conventonal methods. Expermental tests of a steel brdge n a buldng are performed to verfy ths proposed method. Fnally, the results nvolvng the mode shape and the computaton tme requred for the representatve of SI method, the FDD, and the proposed method are compared. 2 METHOD FOR ESTIMATION 2.1 Target frequency The most mportant aspect of SHM s performng vbraton measurements of a structure. The measured responses of structure at a specfc tme t are presented as u () t. s the number of the sensor locatons, rangng from 1 to M ; M s the total number of sensors to measure. To set the target frequency of the bandpass flter, measured responses are transformed nto the frequency doman through the use of a fast Fourer transform (FFT) usng the measured responses. U ( ) FFT[ u ( t)], 1 to M (1) U ( ) s the frequency doman response of the structure at the th sensor locaton. The peaks of the frequency doman response curve are set as the target frequency of the bandpass flter. 2.2 Pseudo-modal response There are varous flter types currently n use, e.g., Butterworth, Chebyshev and ellptc. The Butterworth type flter has the advantages of constant gan level on bandwdth compared to the other flters. Thus, the Butterworth flter s used n ths proposed method. The measured responses can be dvded nto separate responses of each mode through the use of a bandpass flter. In ths paper, the separate responses obtaned usng a flter are called the pseudo-modal response (PMR). The bandpass flter s desgned wth an approprate range (half of bandwdth) and order based on the target frequency. The desgned transfer functon (TF) of the flter and the frequency doman response (expressed as H ( ) and U ( ), 2

3 respectvely) are used to produce the composte functon. H ( ) s the TF that has a target frequency equal to the th peak frequency of the frequency doman response. Fnally, nverse Fourer transform (IFT) of the composte functon comprsed of the TF and the frequency doman response are calculated to obtan the PMR at each mode and sensor locaton. PMR ( t) IFT[ H ( ) U ( )], 1 to M, 1 to N (2) where PMR () t denotes the pseudo-modal response of the th sensor locaton and th mode at the specfc tme t. M s the maxmum number of sensors, and N s the number of nterestng modes. One of the practcal decsons regardng how to set the number of nterestng modes was the use of the sum of the effectve modal mass ratos up to 90%, based on a prevous analytcal model. 2.3 Components of mode shape A sngle column matrx s formed by combnng the pseudo-modal responses of a specfc mode. In ths paper, ths sngle column matrx s called the components of mode shape (CoMS) because of ts smlarty between the mode shape and the combned pseudo-modal responses. CoMS can be expressed as follows: c 1 M ( t) PMR ( t),, PMR ( t) T (3) The letter c n CoMS s used to dstngush the term from mode shape. CoMS at tme t, that s c () t, presents the global movements of the structure related to the th mode. Thus, the th mode shape s observed through CoMS at some specfc tme steps. 2.4 Tme-hstory Modal Assurance Crteron Ths paper presents a proposed method of dentfyng the mode shape from CoMS. Assumng that CoMS n every tme step s the mode shape, the modal assurance crteron (MAC) can be calculated n every tme step. Practcally, the MAC value s used to analyze the correlaton between mode shapes. However, the tme-hstory MAC value can be obtaned usng the tme-hstory CoMS nstead of the mode shape by assumpton. If the MAC value between the a th and b th mode s close to 1, then the mode shapes exhbt hgh correlaton. In contrast, f the MAC value s close to 0, then the mode shapes are found to exhbt low correlaton and are nearly orthogonal to each other. MAC ab, () t 2 a T b c ( t) c ( t) a T a b T b c ( t) c ( t) c ( t) c ( t) N a 1 ab, ( ) ( ) SUMAC t MAC t a 2 b 1 SUMAC () t s tme-hstory summaton of the MAC values at tme t, and N s the number of nterestng modes. A smaller value of SUMAC () t ndcates that CoMS of each mode satsfes orthogonalty as a prncple vector. Thus, the proposed method regards the CoMS at a specfc tme that mnmzes SUMAC as a part of the mode shape. (4) (5) 3

4 2.5 Mode shape The tme related to the k th smaller SUMAC () t that s defned as the mnmum tme (MTme) s expressed as t mn k. For example, the frst MTme s t mn 1, and the second one s t mn 2. The mode shape can be estmated usng a group of CoMS values at MTme. K mn c ( tk ) k 1 denotes the th unnormalzed mode shape, and K s a certan rato of the number of used CoMS to the number of measured data samples. The estmated mode shapes reflect the ampltude of the CoMS. Therefore, f CoMS at MTme exhbts good orthogonalty and low ampltude, t wll be a bad proporton of mode shape; f not, then CoMS affects a great proporton of the mode shape because of hgh ampltude. Fnally, the obtaned unnormalzed mode shape can be normalzed by usng one of a varety of approaches. The proposed method for estmatng the mode shape does not requre SVD, an approach that ntroduces a hgh level of computatonal complexty and s broadly used n prevous SI technques. Thus, t s expected that the computaton tme wll be reduced when estmatng mode shape usng the proposed method. (6) 3 EXPERIMENTAL APPLICATION 3.1 Expermental set-up The target structure s a steel brdge n a buldng; the dmensons of the brdge are 16 m n length and 4 m n wdth. The target structure s used as a passage. Two grders of 16 m n length were constructed, and three beams were connected wth almost pnned onts. Steel deck plates were located above the grders, and concrete slabs that are 15 cm n thckness were placed on the deck. Three pezoelectrc accelerometers were attached at 1/4, 2/4, 3/4 of the target structure to measure the vertcal vbraton. Durng operatonal modal analyss (OMA), an mportant assumpton s that the loads on the structure are regarded as followng whte nose of Gaussan dstrbuton. To satsfy ths assumpton, the ambent vbraton of structure should be measured. Thus, the measurement was performed wthout any artfcal exctaton. The measurng tme was approxmately 128 seconds at a samplng rate of 128 Hz. The measured data sample equals to The accelerometers have resoluton of g and a frequency range of 0.5 to 1000 Hz. 3.2 Target frequency FFT was conducted usng data from the measurement of ambent vbraton. The natural frequency was approxmately estmated by observng the FFT curve, as shown n below. The results establsh the target frequency of flter accordng to the peaks. ACC1, 2 and 3 are located at 1/4, 2/4 and 3/4 of the brdge length. The target frequences obtaned of 7.1 Hz, 19.3 Hz and 24.4 Hz are related to the frst, the second and the thrd mode, respectvely. The narrow peaks are not consdered as structural modes. 4

5 3.3 Components of mode shape Fgure 1: Results of FFT. The CoMS could be obtaned usng equatons (2) and (3). The bandpass flter was desgned as Butterworth type. The flter s of 5th order and has a range of 1.0 Hz (half of bandwdth). The fltered data known as the pseudo-modal response are combned to make the vector, CoMS. Fgure 2: CoMS of each mode. Fgure 2 shows the three dmensonal plots of the CoMS of each mode durng 1 second. The smlarty between the mode shape and the CoMS can be confrmed vsually n Fgure Tme-hstory MAC The tme-hstory MAC value and SUMAC were calculated usng equatons (4) and (5). For ths expermental test, the number of sensors or the number of nterestng modes can be three. Thus, the SUMAC sets the range of 0 to 3 as gong from the mnmum to the maxmum values. 5

6 Fgure 3: Tme-hstory MAC value. The blue lne presents the tme-hstory MAC, and the black markers present the MTme expressed n equaton (6). The number of MTme, K, was 81,.e., 0.5% of the total number of measured data samples of Mode shape The unnormalzed mode shape can fnally be estmated usng equaton (6). The CoMs of 81 contrbutng mode vectors have measurable ampltudes. Although the selected CoMS have a lower SUMAC, f ts ampltude s also small, then the contrbuton of the selected CoMS for estmatng the mode vector s small. Thus, the proposed method consders both the orthogonalty and the ampltude of the CoMS. Fgure 4: Mode shape. The mode shapes of the proposed method and of the prevous SI method, FDD, were compared. The normalzaton was performed n the manner that has the maxmum value of the mode vector be equal to 1. The frst mode shapes showed good agreement among each other; however, the mode shapes of hgher order dd not exhbt good agreement among each 6

7 other. As a result, the MAC values were calculated to check the orthogonalty, as descrbed below. FDD Proposed 1st 2nd 3rd 1st 2nd 3rd 1st nd rd Table 1: Comparson of MAC value FDD was performed under the followng condtons: 8192 NFFT, Hannng type wndow and 50% averagng rato. The mode shape from FDD showed a hgh correlaton between the frst and the thrd modes, n contrast to the proposed method. In addton, orthogonalty of the mode shapes from FDD was not exactly satsfed. However, such a result s typcal for FDD because the responses are mxed and affected by contguous mode. Thus, the proposed method was confrmed to be able to separate the prncpal vector of the structural responses wth good orthogonalty and to overcome the weakness of FDD. 3.6 Computaton tme Data length (NFFT) FDD Proposed Computaton tme (sec.) Table 2: Computaton tme for estmatng mode shape Fnally, the computaton tmes for the dfferent methods were calculated and compared. The proposed method was confrmed to reduce the computaton tme requred to estmate the mode shape. Practcally, to dentfy a hgh-rse buldng or large spatal structures, long-term measurements and tests have been performed. However, the more the measurement s conducted over a long perod, the more estmaton of the mode shape makes the computaton complcated. The reason why real-tme SI has problems s the ncrements of the computaton tme due to the long measurng tme. Therefore, f the proposed method n ths paper s appled to hgher and larger structures wth long measurement tmes, t wll be more useful to conduct SHM and SI and even to perform damage detecton. 4 CONCLUSIONS In ths paper, the method for vsually estmatng the mode shape was proposed and appled to the expermental test to overcome the hgh computatonal complexty of SVD, the method that s used frequently n the prevous SI methods. The proposed method was appled to a steel brdge n a buldng, and the measurement of the ambent vbraton was used to perform 7

8 OMA. Both the computaton tme and the mode shape for the prevous SI method, FDD, and the proposed method are compared. The mode shapes of low order showed good agreement; n contrast, the hgher order mode shapes showed poor agreement. However, the modal orthogonalty from the proposed method was found to be exactly satsfed, n contrast to the result of FDD. In addton, the proposed method was confrmed to reduce the computaton tme requred to estmate the mode shape. Therefore, f the proposed method were used, the real-tme SHM or SI, even damage detecton, usng the mode shape are apparently possble to perform, wth reduced computaton tme. 5 ACKNOWLEDGEMENT Ths work was supported by the Natonal Research Foundaton (NRF) grant funded by the Korea government (Mnstry of Educaton, Scence and Technology, MEST) (No ). REFERENCES [1] P.C. Chang, A. Flatau, S.C. Lu, Revew paper: Health montorng of cvl nfrastructure, Structural Health Montorng, 12(3), , [2] J. Ko, Y.Q. N, Technology developments n structural health montorng of large-scale brdges, Engneerng Structures, 27, , [3] H.S. Park, H.M. Lee, A new approach for health montorng of structures: Terrestral laser scannng, Computer-Aded Cvl and Infrastructure Engneerng, 20, , [4] C. Sbarufatt, A. Manes, M. Gglo, Applcaton of sensor technologes for local and dstrbuted structural health montorng, Structural Control and Health Montorng, 21, , [5] E.P. Carden, P. Fannng, Vbraton based condton montorng: A revew, Structural Health Montorng, 3, , [6] R. Brncker, L. Zhang, P. Andersen, Modal dentfcaton of output-only systems usng frequency doman decomposton, Smart Materals and Structures, 10, , [7] W.X. Ren, Z.H. Zong, Output-only modal parameter dentfcaton of cvl engneerng structures, Structural Engneerng and Mechancs, 17, , 2004 [8] S. Nagaraaah, B. Basu, Output only modal dentfcaton and structural damage detecton usng tme frequency & wavelet technques, Earthquake Engneerng and Engneerng Vbraton, 30, , [9] R. Brncker, L. Zhang, P. Andersen, Modal dentfcaton from ambent responses usng frequency doman decomposton, Proceedngs of the 18 th Internatonal Modal Analyss Conference,

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