ACCOMMODATING SYNCHRONIZATION-INDUCED ERRORS IN OPERATIONAL MODAL ANALYSIS

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1 ACCOMMODATING SYNCHRONIZATION-INDUCED ERRORS IN OPERATIONAL MODAL ANALYSIS Kosas DRAGOS, Triantafyllos MAKARIOS, Ioanna KARETSOU 3, George D. MANOLIS 4, Kay SMARSLY 5 ABSTRACT Operational odal analysis (OMA) is frequently eployed in structural health onitoring (SHM) for obtaining inforation on the dynaic properties, e.g. natural frequencies and ode shapes, of onitored structures. To extract accurate estiates of ode shapes, absolute synchronization between sets of acceleration response data collected fro the onitored structure is required. Discrepancies between data sets typically occur in wireless SHM systes, where the synchronization of wireless sensor nodes is not intrinsically anaged by a centralized server. Moreover, in cable-based SHM systes, if ore than one data acquisition unit is used, absolute synchronization between sets of acceleration response data is not guaranteed. This paper presents a ethod for detecting and accoodating synchronization-induced errors in OMA. While several synchronization scheas propose the offline reoval of tie lags between sets of acceleration response data at the output stage of OMA, the synchronization ethod presented here perfors the detection and accoodation of the tie lags at an interediate stage of OMA. Tie lags are detected using estiates of the (expected) ode shapes of the onitored structure to define the relationship between the phase angles of different sets of acceleration response data at odal peaks of the respective Fourier spectra. The synchronization ethod is validated through field tests on a pedestrian overpass bridge, showcasing the ability of the ethod to yield accurate estiates of ode shapes. Keywords: Structural health onitoring; abient vibrations; pedestrian bridges; operational odal analysis; dynaic response. INTRODUCTION Syste identification in structural health onitoring (SHM) is associated with obtaining estiates of structural properties. Several syste identification ethods eployed in SHM focus on extracting inforation on the structural dynaic properties of onitored structures, such as experiental odal analysis ethods for estiating natural frequencies and ode shapes (Dragos and Sarsly, 06a). In experiental odal analysis tests, early approaches have suggested artificially exciting the structure and identifying the structural dynaic properties based on the relationship between the input (load) and the output (structural response data) of the tests. However, as the artificial excitation of full-scale structures has proven to be cubersoe and expensive, research has been increasingly focused on developing output-only ethods for identifying structural dynaic properties (Cunha et al., 005). Tered operational odal analysis (OMA), as structures are analyzed under operational conditions without any artificial excitation, the category of output-only ethods has been increasingly adopted in SHM, owing to the low ipleentation cost and the negligible disturbance in the operation of the structures (Zhang et al., 005). Doctoral Candidate, Civil Engineering, Bauhaus University Weiar, Weiar, Gerany, kosas.dragos@uni-weiar.de Asst. Professor, Civil Engineering, Aristotle University, Thessaloniki, Greece, akariostr@civil.auth.gr 3 PhD Candidate, Civil Engineering, Aristotle University, Thessaloniki, Greece, kioanna@civil.auth.gr 4 Professor, Civil Engineering, Aristotle University, Thessaloniki, Greece, gd@civil.auth.gr 5 Professor, Civil Engineering, Bauhaus University Weiar, Weiar, Gerany, kay.sarsly@uni-weiar.de

2 OMA ethods for extracting natural frequencies and ode shapes are applied either directly to the raw structural response data (tie doain ethods) or after transforing the structural response data into the frequency doain via the Fourier transfor (frequency doain ethods). While natural frequencies are detected fro the frequency coponents in the structural response data that correspond to resonant response (odal frequency coponents), ode shapes are extracted fro the phase relationship between the odal frequency coponents. Therefore, to extract accurate estiates of ode shapes, the structural response data sets collected fro the onitored structure ust be synchronized (Krishnautry et al., 008). Although synchronization probles typically affect wireless SHM systes (Rainieri and Fabbrocino, 04), whose decentralized nature renders an inherent global clock synchronization ipossible, cable-based SHM systes ay also be prone to synchronizationinduced errors, particularly if ore than one data acquisition units are used. To ensure the highest possible quality in ode shapes extracted through OMA ethods, synchronization-induced errors need to be detected and appropriately accoodated. Synchronization scheas for wireless sensor networks have priarily been studied in the field of coputer science. Reviews on clock synchronization scheas based, e.g., on clock offset or on clock drifting, have been reported by Youn (03) and Sundararaan et al. (005). Other approaches are based on incorporating GPS transceivers into wireless sensor nodes (Bojko et al., 009) and on transitting beacon signals for establishing a global network tie (Severino et al., 00). With respect to cable-based systes with ultiple data acquisition units, ost research approaches focus on incorporating GPS transceivers into the data acquisition units (Peeters, 0; Magalhães et al., 006). Even though GPS technology serves as an attractive solution to synchronization probles, using GPS transceivers has been proven power-consuing and error-prone in cases of unfavorable weather conditions or poor satellite coverage (Wang and Xu, 00). Finally, approaches towards the offline reoval of synchronization-induced errors at the outcoe stage of OMA have also been reported (Maes et al., 06; Brincker and Brandt, 0). This paper presents a synchronization ethod for detecting and accoodating synchronizationinduced errors at an interediate stage of OMA. The synchronization ethod is based on the relationship between the phases of odal frequency coponents fro different structural response data sets, which are either positively or negatively correlated. By iposing this correlation as a necessary condition, tie lags between the structural response data sets are detected and accoodated. The synchronization ethod proposed in this paper is verified through laboratory tests on a shear frae structure and validated through field tests on a pedestrian bridge. In the first section of this paper, the atheatical background is presented, providing details on the correlation condition between the structural response data sets and on the tie lag detection process. Next, the laboratory tests are described, followed by the validation field tests on the pedestrian bridge. The paper concludes with a suary and an outlook on future research.. A METHOD FOR DETECTING AND ACCOMMODATING SYNCHRONIZATION-INDUCED ERRORS IN OMA In this section, the ethod towards the detection and accoodation of synchronization-induced errors in OMA is described. First, the atheatical background on detecting tie lags through the phase difference between acceleration response data sets is presented, and, next, the steps of the synchronization ethod are illuinated.. Detection of tie lags between two acceleration response data sets For the sake of siplicity, a single-degree-of-freedo oscillator, instruented with two data acquisition units, is considered. Assuing sinusoidal excitation of the oscillator, the two acceleration response data sets collected by the two units, unit d and unit d, are t Y sin t t kδt k t Y t t kδt k y, () y sin, ()

3 y(t) In Equations and, y represents the acceleration response data, Y is the axiu acceleration value for each acceleration response data set, θ is the phase angle, and ω is the natural frequency of the oscillator. If there is no tie delay between set y and set y, θ is equal to θ. However, if there is tie delay between set y and set y, the tie lag τ (Figure ) is obtained by the difference in the phase angles, as shown below: ~ ~ y t Y sin t y t Y sin t, (3) ~ (4) τ Set Set Tie t (s) Figure : Tie lag between two sets of acceleration response data. Fro Equations 3 and 4, it is evident that the phase angles of the acceleration response data sets, which are calculated via the fast Fourier transfor, are required for obtaining the tie lag τ: X N i y e N N n [0, ) N Δt n 0 n N (5) In Equation 5, y is the N-point acceleration response data set transfored into an N-point coplex discrete Fourier series X. The natural frequency of the th eleent in the Fourier series is depicted as ω, and Δt is the tie step at which the acceleration data set is sapled. Drawing fro the theory of signal processing, an estiate of the phase angle φ of the th frequency coponent in the Fourier series ay be derived fro the arguent of X : I X argx arctan t N X Δ (6) Re Equation 6 yields values that lie within the interval [0, π), which, when applying Equation 4, result in a axiu tie lag detectable in a straightforward anner: ax T In Equation 7, T is the natural period of the oscillator. To enable the detection of tie lags longer than T, the trigonoetrical solution of Equation 6 is generalized as follows: (7) 3

4 I X arctan k k (8) Re X gen In Equation 8, ζ is the phase angle derived fro the general trigonoetric solution. By extending Equation 7 accordingly, the generalized expression for estiating the tie lag between two sets of acceleration response data is: T a a a a a a a, a, a (9) The tie lag detection, as shown in Equation 9, is used as a basis for the synchronization ethod, which is described in the next subsection.. Description of the synchronization ethod The tie lag detection described previously is based on the assuption that for absolute synchronization between two acceleration response data sets no phase differences are observed. This assuption is valid for the exaple considered in the previous subsection, where both data sets are obtained fro the sae single degree of freedo (DOF) of the oscillator. However, in practice, for acceleration response data sets obtained fro different degrees of freedo of a ulti-dof structure, the expected phase differences ay differ fro zero and are estiated by the ode shapes. More specifically, considering the frequency coponents of vibration ode r fro two degrees of freedo i and j, the expected phase difference depends on the correlation ρ between the frequency coponents, which, as shown in Dragos and Sarsly (06b), is either positive or negative. As a result, the expected phase difference between DOF i and DOF j is: 0 for Δ r i j i, j (0) r for Moreover, as the dynaic behavior of ulti-dof systes is described by the superposition of vibration odes, the vibration ode for applying Equations 9 and 0 needs to be selected. This selection is based on the degree of participation of each vibration ode in the structural response; for conventional civil engineering structures, the first (fundaental) vibration ode (i.e. the ode with the lowest natural frequency) is doinant. Furtherore, as integer a in Equation 9 is generally unknown, an additional condition is necessary, which is forulated by iposing the expected phase differences of a set of h vibration odes higher than the fundaental ode and by obtaining a through trial and error. Hence, by considering p degrees of freedo, and by setting the acceleration response data set of one DOF as reference, with respect to which the acceleration response data sets of the other degrees of freedo are synchronized, Equation 9 is extended to the phase shift condition, shown in Equation. τ Ta r δ θ () In Equation, τ is the vector holding the tie lags between the acceleration response data sets of p degrees of freedo and the acceleration response data set of the reference DOF ( reference data set ), a is the vector holding the p integers a for applying Equation 9 for each DOF, and φ is the vector holding the estiates of the phase angles fro the p data sets derived fro Equation 6. The estiate of the phase angle for the reference data set is denoted as φ r, and ω and T are the fundaental natural frequency and the corresponding period, respectively. The h-sized vector δ 4

5 represents the estiates of the phase angles of the reference data set for the h higher vibration odes, and φ is the h-sized vector with the corresponding phase angles fro the rest of the acceleration response data sets. Finally, θ is the h-sized vector of the expected phase differences between the reference data set and the rest of the acceleration response data sets for the h higher odes, calculated fro Equation 0. The phase shift condition, shown in Equation, fors the basis of the synchronization ethod. Fro Equation, it is evident that knowledge on the expected phase differences between the reference data set and the rest of the acceleration response data sets is required. This knowledge ay be derived either fro nuerical analysis or fro engineering judgent. Hence, the synchronization ethod is divided into two parts, (i) preliinary analysis, and (ii) ain analysis, depending on whether knowledge on the expected phase differences exists or needs to be obtained. The steps of each part are described below. A detailed description of the synchronization ethod can be found in Dragos et al. (07). Preliinary analysis: An initial estiate of the h + vibration odes (natural frequencies and ode shapes) used for the synchronization ethod is obtained either fro nuerical analysis or fro engineering judgent. The initial estiate of the h + vibration odes is updated by collecting a preliinary set of acceleration response data fro the onitored structure. Vector θ is calculated. Main analysis: Acceleration response data sets are collected fro all p degrees of freedo and transfored into the frequency doain. Vectors φ, φ, and δ and paraeter φ r are calculated fro Equation 6. The phase shift condition (Equation ) is applied, and vector τ is obtained. The acceleration response data sets are shifted according to vector τ. Mode shapes are extracted by applying any OMA ethod to the synchronized data sets. 3. VERIFICATION OF THE SYNCHRONIZATION METHOD THROUGH LABORATORY TESTS The synchronization ethod is verified through laboratory tests on a shear frae structure with known structural dynaic properties. This section first presents the laboratory structure and the experiental setup, then describes the laboratory tests, and concludes with a discussion on the results. 3. Experiental setup The structure used for the laboratory tests is a four-story shear frae structure, as illustrated in Figure. Each story consists of a rectangular aluinu plate, resting on four aluinu coluns with rectangular cross sections of diensions 0. The longer sides of the colun cross sections are aligned with the shorter sides of the aluinu plates. The coluns are fixed at their base into a solid wooden block ensuring quasi-claped restraint conditions. Each story has a vertical clearance of 300, and the plate-to-colun connections are considered fully-fixed. The structure is instruented with a wireless SHM syste without global clock anageent. One wireless sensor node is placed at the center of each story, and the acceleroeter ebedded into each sensor node saples acceleration response data at 5 Hz. The experiental setup is illustrated in Figure. 5

6 Figure : Experiental setup of the laboratory tests with the shear frae structure and the wireless sensor nodes. 3. Description of the tests The sensor nodes are assigned to start sapling once an acceleration threshold of 0. g is exceeded. The synchronization discrepancies between the sensor nodes are caused by non-siultaneous triggering of the sensor nodes. The sequence of sensor node triggering depends on the load profile used for inducing the excitation. To deonstrate the applicability of the proposed synchronization ethod under any excitation conditions, two tests are perfored with different excitation characteristics. In the first test, the excitation is induced by deflecting the top (fourth) story of the structure, while, in the second test, the structure is excited by deflecting the second story. The excitation in both tests is applied along the weak direction of the colun cross section, i.e. along the long side of the plates (300 ). Each acceleration response data set has a length of 048 points. The nuber of higher odes for applying the phase shift condition is set to h = 3, and integer a is set equal to 4. In both tests, the acceleration response data set of the first story is set as reference data set. The ode shapes are extracted by applying frequency doain decoposition (Brincker and Zhang, 0). The expected phase differences are derived fro preliinary tests conducted on the sae structure cobined with nuerical analysis. For the nuerical analysis, the aterial elastic odulus of the aluinu alloy is set to E = 70 GPa, and the aterial density is set to ρ = 700 kg/ 3. The results fro applying the proposed synchronization ethod (noralized to the fourth story) are suarized in Table. Mode nuber Mode (f =.53 Hz) Mode (f = 4.56 Hz) Mode 3 (f = 7.03 Hz) Mode 4 (f = 8.79 Hz) Table. Mode shapes extracted fro both tests. Story Unsynchronized Synchronized Test Test Test Test Expected

7 Mode 4 Mode 4 Mode 3 Mode 3 Mode Mode Mode Mode For coparing the ode shapes before and after applying the synchronization ethod, the odal assurance criterion (MAC) value, which expresses the relationship between two odal vectors (Alleang, 980), is used, as shown in Equation. MAC T, () s q s q T T s s q q In Equation, ψ expresses the vector of ode s and ode q, respectively. A MAC value close to unity shows consistent correspondence between the two odal vectors, whereas a MAC value close to zero indicates no correspondence. By progressively assigning the vectors of the expected ode shapes to ψ q (q =,, 3, 4) and the vectors of the ode shapes extracted fro the two tests to ψ s (s =,, 3, 4), two sets of 4 4 atrices of MAC values (MAC atrices) are obtained. For illustrative purposes, the MAC atrices of test before and after applying the synchronization ethod are plotted in Figure 3. Unsynchronized Synchronized Mode Mode Mode 3 Mode 4 Mode Mode Mode 3 Mode 4 Figure 3: MAC atrices of test before and after applying the synchronization ethod In the MAC atrices, strong correspondence between the odal vectors is expected for s = q, which is expressed by MAC values being close to unity along the diagonals of the atrices, thus indicating that the ode shape extracted fro test is siilar to the corresponding expected ode shape. No correspondence is expected for s q, characterized by non-diagonal values of the atrices being close to zero. Fro Figure 3, it is evident that before applying the synchronization ethod there are significant deviations fro unity for the diagonal values of the MAC atrix as well as several deviations fro zero for the non-diagonal values. Furtherore, after applying the synchronization ethod, both the unity values along the diagonal and the zero non-diagonal values are restored. Finally, the stronger effects observed at higher odes, as copared to the effects of the first two odes, are attributed to the flexibility of the structure; the tie lags between the sensor nodes are in the order of soe tens of illiseconds and, therefore, considerably lower than the periods of the first two odes (T = 0.65 s, T = 0. s), thus not affecting noticeably the respective ode shapes. Figure 4 illustrates the plots of the ode shapes before and after applying the synchronization ethod in test. In accordance with the MAC atrices shown in Figure 3, the effect of the synchronization discrepancies on the shapes of odes 3 and 4 is clearly stronger than the respective effect on odes and. The field tests to validate the applicability of the synchronization ethod with data fro real structures are presented in the next section. 7

8 Mode Mode Mode 3 Mode 4 Unsynchronized Synchronized Expected Figure 4: Mode shapes extracted fro test before and after applying the synchronization ethod. 4. APPLICATION OF THE SYNCHRONIZATION METHOD IN FIELD TESTS ON A PEDESTRIAN BRIDGE For validating the applicability of the synchronization ethod using acceleration response data fro a real structure, a field test is conducted on a pedestrian overpass bridge in Greece. First, the pedestrian bridge and the instruentation are presented. Next, the test is described followed by the discussion of the results. 4. Description of the pedestrian bridge and of the instruentation The pedestrian overpass bridge used for the field test facilitates pedestrian traffic over the Inner Ring Road of Thessaloniki, Greece at the unicipality of Evosos (Figure 5). The bridge coprises a long by 5.35 wide deck, which is supported by two steel girders one at each longitudinal side of the deck. Each girder rests on two reinforced concrete coluns with circular sections of 800 diaeter. The deck is coposed of a corrugated steel trough filled with concrete. The girder-tocolun connections are achieved through spherical base isolation bearings, which act as load interfaces. Moreover, 0 equidistant beas and 0 X-braces, all fixed to the girders, add to the deck stiffness in the transverse direction. Finally, copleentary vertical support to the deck is provided by two skewed arches, each suspended via 6 twisted strand cables fro one girder. Figure 5: (left) Panoraic view of the pedestrian bridge and (right) view of the support syste of the bridge. Two cable-based SHM systes are installed on the bridge deck, each having a separate data acquisition unit, shown in Figure 6. Each SHM syste consists of two acceleroeters easuring in the direction vertical to the bridge deck. The acceleroeters, labeled A, B, C and D, are placed at 8

9 5.35 distances equal to /5 of the total length of the bridge deck, i.e. at 8.6 intervals. The data acquisition units are placed at the center of the deck, each data acquisition unit being interfaced with separate software, thus resulting in non-synchronized operation between the SHM systes. Since no prior knowledge on the dynaic properties of the bridge is available, preliinary analysis is required. Therefore the field test is conducted in two steps, step being the preliinary analysis and step being the ain analysis. SHM syste SHM syste A B C D y z x 4. Step : Preliinary analysis Figure 6: Plan view of the bridge and instruentation layout. Using inforation fro structural drawings of the bridge and fro on-site inspection, a finite eleent odel of the bridge is forulated, which is shown in Figure 7. The finite eleent odeling of the bridge is restricted to the deck, neglecting the two stairways at each end of the deck, which are not onolithically connected to the rest of the structure. The finite eleent odel consists of bea eleents used to odel the girders, the beas, the arches, and the X-braces. In addition, cable eleents represent the suspension cables of the arches. The deck stiffness is incorporated via odification factors to the bending stiffness of the girders and of the beas, and the girder-to-colun partial fixity connections (due to the bearings) is approxiated. The elasticity odulus of the steel structural eleents is taken equal to E s = 0 GPa, and the elasticity odulus of the cables is set equal to E c = 65 GPa. The ass density of steel is set equal to ρ s = 7850 kg/ 3. The finite eleent odel is calibrated using inforation fro a preliinary set of acceleration response data collected fro the structure. Then, odal analysis is perfored using the finite eleent odel to obtain the expected phase differences (vector θ) needed for the ain analysis. Figure 7: Nuerical odel of the bridge. 9

10 Mode 4 Mode 4 Mode 3 Mode 3 Mode Mode Mode Mode 4.3 Step : Main analysis Each acceleration response data set collected fro the four instruented locations of the deck has a size of 3,768 points sapled at a rate of 5 Hz. The bridge is tested under operational conditions, i.e. the excitation is induced by vehicular traffic passing under the bridge and by pedestrian traffic passing over the bridge. Since the SHM systes are cable-based and each SHM syste collects data fro two locations, an overall tie lag between the left-ost and the right-ost pair of instruented locations, i.e. an overall tie lag between the two SHM systes theselves, is expected. For applying the phase shift condition, the first four vibration odes (h + = 4), having been found fro the preliinary analysis to have the highest contribution to the structural dynaic response, are considered. Integer a (Equation 0) is set to 4 for all vibration odes. The acceleration response data set fro location A is set as reference data set. The ode shapes are extracted by applying frequency doain decoposition to the acceleration response data sets, before and after applying the synchronization ethod. The results of the synchronization ethod for the first four vibration odes (noralized to location D), along with the results of the preliinary analysis (expected ode shapes) are suarized in Table, and the respective MAC atrices are illustrated in Figure 8. The ode shapes before and after applying the synchronization ethod are shown in Figure 9. Table.Mode shapes of the bridge extracted fro the field test. Mode nuber Location Unsynchronized Synchronized Expected Mode (f =.4 Hz) Mode (f = 4.8 Hz) Mode 3 (f = 5.78 Hz) Mode 4 (f = 7.58 Hz) A B C D A B C D A B C D A B C D Unsynchronized Synchronized Mode Mode Mode 3 Mode 4 Mode Mode Mode 3 Mode 4 Figure 8: MAC atrices of the ode shapes of the bridge extracted fro the field test. 0

11 Mode Mode Mode 3 Mode 4 Unsynchronized Synchronized Expected Figure 9: Mode shapes of the pedestrian bridge extracted fro the field test. Fro the MAC atrices in Figure 8 and fro the ode shapes in Figure 9, it is evident that the synchronization ethod proposed in this paper is capable of reoving synchronization-induced errors fro the results of OMA and of yielding accurate estiates of ode shapes. Furtherore, the weak correction effect of the synchronization ethod on the fourth ode is attributed to the inadequate excitation of this ode and, therefore, to the low contribution of this ode to the structural dynaic response. 5. SUMMARY AND CONCLUSIONS The identification of structural dynaic properties, such as natural frequencies and ode shapes, through processing of acceleration response data falls into the category of operational odal analysis. Specifically in identifying ode shapes, the absolute synchronization between acceleration response data sets collected fro different locations of a structure is a prerequisite for yielding accurate estiates of the ode shapes. Synchronization discrepancies are typical of wireless structural health onitoring systes, where due to the decentralized nature of wireless sensor networks global clock anageent of wireless sensor nodes is challenging. However, synchronization discrepancies ay also affect cable-based systes, e.g. if ore than one data acquisition units are used. To ensure the highest possible accuracy in extracting ode shapes, synchronization-induced errors ust be detected and accoodated. While ost research efforts, particularly in coputer science, have been focusing on ipleenting clock synchronization protocols, this paper has addressed the issue of reoving synchronization-induced errors at an interediate stage of OMA. The synchronization ethod proposed in this paper is based on iposing the expected relationship between the phase angles of frequency coponents in the structural response corresponding to vibration odes. The expected relationship is obtained fro preliinary analysis of the onitored structure, e.g. via nuerical odeling or fro engineering judgent. The synchronization ethod has been verified through laboratory tests on a shear frae structure with known structural dynaic properties and has been validated through field testing on a pedestrian overpass bridge. Both the verification and the validation tests have showcased the applicability of the synchronization ethod and the ability of the ethod to yield accurate estiates of ode shapes. The effect of the synchronization ethod on the ode shapes has proven to be weak in cases where the period of the vibration ode is significantly longer than the tie lags between the acceleration response data sets and in cases where the vibration odes are inadequately excited. Future research will focus on the applicability boundaries of the synchronization ethod as well as on alternative patterns in the frequency coponents of acceleration response data sets that ay also be used for synchronization purposes. 6. ACKNOWLEDGMENTS The authors gratefully acknowledge the support offered by the Geran Research Foundation (DFG) through the DFG research training group GRK 46 entitled Evaluation of Coupled Nuerical and Experiental Partial Models in Structural Engineering. The authors wish to acknowledge the valuable assistance provided by Dr. Panayiotis Panetsos, project anager at Egnatia Odos S.A., in

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