Damage Detection using Stochastic Subspace Identification

Transcription

1 Daage Detection using Stochastic Subspace Identiication S. H. Si 1 and B. F. Spencer, Jr. 2 1 Departent o Civil and Environental Engineering, University o Illinois at Urbana-Chapaign, Urbana, IL 6181, USA 2 Departent o Civil and Environental Engineering, University o Illinois at Urbana-Chapaign, Urbana, IL 6181, USA ssi2@illinois.edu, bs@illinois.edu ABSRAC Vibration-based daage detection is coonly used in structural health onitoring. Such vibration-based approaches typically conduct odal analysis using easured dynaic responses to extract odal inoration that can be used to deine daage sensitive indices. hus, the daage detection result is highly dependent on the identiied odal properties. As such, identiication ethods play a ey role; however, it s ipact on daage detection has not been ully investigated. Stochastic Subspace Identiication (SSI) has the potential to iprove the daage detection capability. SSI can reliably estiate odal properties due to ultiple reerences and Canonical Variate Algorith (CVA) weighting. his paper investigates daage detection using SSI/CVA. he perorance in ters o odal properties and daage detection results o three coonly used output-only identiication ethods is copared: (1) Natural Excitation echnique in conjunction with Eigensyste Realization Algorith (NEx/ERA), (2) Frequency Doain Decoposition (FDD), and (3) SSI/CVA. A nuerical exaple is presented to deonstrate the eicacy o the proposed SSI/CVA approach. INRODUCION SHM is oten reerred to as a process o ipleenting a daage detection strategy or aerospace, echanical, and civil engineering inrastructure (Sohn 23); however, a broader deinition can include a spectru o applications that such as design veriication, assessent o structural state ater catastrophic events, control o the construction process, and assisting with building and bridge aintenance. In the civil engineering ield, SHM has becoe a proinent tool to address probles associated with deteriorating civil inrastructure. For instance, the collapse o the Silver Bridge between West Virginia and Ohio in 1967 subsequently initiated the US governent to develop and ipleent the National Bridge Inspection Standards (NBIS) (Sall et al. 1999). In addition, the Korean governent has been exercising bridge onitoring since the collapse o the Sungsu Bridge 1 Doctoral candidate 2 Nathan M. and Anne M. Newar Endowed Chair o Civil Engineering

2 over the Han River in 1994 resulted in 32 casualties. As such, assessing daage at an early stage and retroitting or repairing structures in a tiely anner is o paraount iportance or public saety as well as to reduce aintenance costs. Daage detection is one o the central objectives o SHM. In bridge inspections conducted biannually by the Federal Highway Adinistration (FHWA) in the United States, visual inspection and siple tap testing are coon practice or daage assessent (Chance et al. 23). However, one o the coon eatures o all bridge ailures, including the I-35 bridge collapse in Minneapolis, is that they have all undergone and passed regular inspection, leading to the conclusion that the current inspection approach is insuicient. Indeed, daage is oten luring in locations that are not accessible. he need or ore accurate daage identiication has driven the research counity to pursue various approaches or daage detection. One proising approach is daage detection based on the dynaically easured lexibility atrix. Unlie the stiness atrix, the lexibility atrix is insensitive to higher requency odes that are generally diicult to deterine ro easured data (Gao et al. 24). his eature has ocused research eorts on lexibility-based, rather than the stiness-based, daage detection approaches. For exaple, Pandey and Biswas (1994) used changes in the easured lexibility atrix, deonstrating that daage locations could be ound ro the irst two lower odes. Bernal (22) presented the Daage Locating Vector (DLV) ethod that utilizes changes in the lexibility atrix due to daage. he DLV ethod can ind daage with only a liited nuber o DOFs in which the lexibility atrix is deined, i the DOFs are in the proxiity o the localized daage. he DLV ethod was expanded to the Stochastic DLV (SDLV) ethod or output-only cases (Bernal 26), and urther generalized to tae advantage o the transer unction atrix that can be thought as a lexibility atrix extended to the requency doain (Bernal 27a; Bernal 27b). In these lexibility-based daage detection ethods, accurate estiation o odal properties, which is subsequently used in deterination o the lexibility atrix, is critical. Because the input orce is diicult to easure in civil engineering structures, output-only odal identiication is oten required. he Natural Excitation echnique (NEx) (Jaes et al. 1992) in conjunction with the Eigensyste Realization Algorith (ERA) (Juang and Pappa 1985) has been otenties eployed to obtain odal paraeters ro the abient vibration or daage detection (Gao and Spencer 22; Duan et al. 25; Nagayaa and Spencer 27). Covariance-driven Stochastic Subspace Identiication (SSI) has the potential to iprove the odal identiication results. In particular, the covariance-driven SSI with the Canonical Variate Algorith (CVA) weighting noralizes the natural odes in ters o energy; thus, less excited odes can be better identiied (Heran and Van Der Auwarer 1999). Because this study is ainly ocused on the covariance-driven, rather than the datadriven, SSI ethod, the covariance-driven SSI will be reerred to siply as SSI in the reainder o this docuent. In this paper, the perorance o SSI/CVA in daage detection is investigated. he DLV ethod is selected to copare daage detection perorance ro the coonly used output-only odal identiication ethods; (1) NEx/ERA, (2) Frequency Doain Decoposition (FDD) (Brincer et al. 21), and (3) SSI/CVA. A nuerical exaple is presented to deonstrate the eicacy o the proposed SSI/CVA approach. BACKGROUND For copleteness, the DLV ethod and SSI are briely reviewed. DLV Method he lexibility atrices o a linear structure beore and ater daage are deterined ro the easured data and are denoted as F u and F d, respectively. Assue the load vectors L that produce identical displaceents at the sensor locations beore and ater daage exist and can be written as

3 ( F F ) L= F L= (1) u d hese vectors, L, are tered the daage locating vectors (DLVs). he DLVs constitute a set o loads that induce no stress in the daaged eleents. Excluding the trivial case where F Δ =, L is in the null space o F Δ and can be deterined using the singular value decoposition. Once deterined, each o the DLVs can be applied to a nuerical odel o the undaaged structure, and the stress in each eleent calculated and noralized as ollows: Δ j ij j = and j = abs ax ( ) (2) i= 1 ax ( i) where j is the noralized accuulated stress, ij is the stress in the j th eleent induced by the i th DLV, and is the nuber o DLVs. I the noralized stress in an eleent is zero, then this eleent is a daage candidate. However in practice, the cuulative stress ay not be exactly zero due to intrinsic uncertainties such as easureent noise or odel error. hus, a threshold value needs to be selected to deterine the daaged eleents. I the cuulative stress o an eleent is less than the selected threshold, then the eleent is considered as a daage candidate. his nonzero error stress at the daaged eleent caused by uncertainties is required to be sall or reliable daage detection. Bernal (26) extended the DLV ethod to accoodate the output-only case (i.e., the input excitation is not easured), resulting in the Stochastic DLV (SDLV) ethod. In this approach, an alternative atrix that spans the sae null space as F Δ in Equation (1) is used to deterine the DLVs. he stochastic DLV ethod has been extended, ipleented, and experientally veriied on a networ o sart sensors (Nagayaa and Spencer 27). Stochastic Subspace Identiication Consider a discrete tie stochastic state space odel: x + = Ax + w y = Cx + v 1 (3) where w and v are uncorrelated zero ean white noise vectors, x is the n by 1 state vector, and y is the by 1 output vector. he correlation unction R o the output sequence can be expressed as: = = (4) 1 R E y lyl CA + G where G = E x+ 1y. he bloc Hanel atrix H p, q with p bloc rows and q bloc coluns, coposed o the correlation unction R, can be decoposed as: H R1 R q = = OC p, q p q Rp R p + q 1 (5)

4 where C CA = = p 1 CA q 1 Op and Cq G AG A G O p and C q are the observability and extended controllability atrices, respectively. he weighted bloc Hanel atrix calculated by pre- and post-ultiplying weighting atrices W 1 and W 1 becoes: S V WH W WO CW U U USV pq, 2 = 1 p q 2 = [ 1 2] = V2 (6) One possible solution to Equation (6) is Op = W U S (7) hen, the syste atrices A and C can be readily obtained using Equation (5) with the nown observability atrix O p in Equation (7). he syste atrix A can be calculated by solving the ollowing equation. where O O = O A p 1 p 1 CA C 2 CA CA p 1 = and O p 1 = p 1 p 2 CA CA (8) he syste atrix C is siply the irst rows o O p. Depending on the weighting atrices, SSI is called Balanced Realization (BR) i no weighting atrices are eployed, or Canonical Variate Algorith (CVA) i the weighting atrices W 1 and W 1 are deined as: where + ( ) and ( ) 1 1 W = L W = L 1 2 R R R R p 1 q R = = L L and R = = L L R R R R p 1 q 1 (9) COMPARAIVE ANALYSIS FOR DAMAGE DEECION o veriy the eicacy o SSI/CVA in daage detection, output-only odal identiication ethods, NEx/ERA, FDD, and SSI/CVA, are copared in the DLV ethod. he 53 DOF planar truss shown in Figure 1 is considered in this nuerical study. he truss consists o 53 eleents o the identical

5 sectional and aterial properties as suarized in able 1. Daage is siulated by 4% stiness reduction in eleent 5, 31, and 44. A band-liited white noise, ranging ro to 19 Hz that encopasses the irst our odes o the truss, is applied vertically to node 6. ranser unctions between the input excitation and the horizontal and vertical accelerations at 5 are shown in Figure 2. Note that the pea o the 3 rd ode is uch saller than the others. A total o 12 horizontal and vertical accelerations at node 5, 6, 7, 19, 2, and 21 in the 4 th and 5 th bays are easured. 5% RMS noise is added to all easureents, and each siulation is repeated 1 ties to handle the uncertainty due to the noise. Figure DOF planar truss able 1. Sectional and aterial properties cross sectional area ( 2 ) Moent o inertia ( 4 ) Young s odulus (Pa) Mass density (g/ 3 ) Magnitude Magnitude Frequency (Hz) Frequency (Hz) (a) Horizontal (b) Vertical Figure 2. ranser unctions between the input excitation and the acceleration at node 5. With the acceleration easureents, odal properties are estiated using NEx/ERA, FDD, and SSI/CVA. In data processing, FF points o 248, 5% overlap, and 2 averages o spectral windows are used. Accuracy in the natural requency and ode shape are exained as well as daage detection results. Natural Frequency Error o the estiated natural requency is deined as: E ( est ext ) = (1) ext where est and ext are the estiated and exact natural requencies o the th ode, respectively. E = 1, 2,3, and 4. As can be seen in the Figure 3 shows the ean and standard deviation o ( )

6 igure, SSI/CVA estiates the natural requencies consistently with low error and deviation. μ E (%) Mode 1 Mode 2 Mode 3 Mode 4.1 NEx/ERA FDD SSI/CVA (a) Mean o E E (%) Mode 1 Mode 2 Mode 3 Mode 4.2 NEx/ERA FDD SSI/CVA E (b) Standard deviation o Figure 3. Error in the estiated natural requencies. Mode Shape Error o the estiated ode shape is deined as: 2 φest φ ext E φ = arccos (11) φ 2 2 est φext where φ est and φ ext are the estiated and exact ode shapes, respectively. he error E φ can be interpreted as the angle between two vectors, φ est and φ ext. In Figure 4, SSI/CVA is ound to have lower error and deviation than NEx/ERA and FDD. Note that the 3 rd ode that has saller energy is better identiied by SSI/CVA. Noralized Accuulated Stress he noralized accuulated stresses at daaged eleents, daage indices in the DLV ethod, are shown in Figure 5 or the three daage cases: daaged eleents 5, 31, and 44. In Figure 5, SSI/CVA has the sallest noralized accuulated stresses or all daage cases. Note that the agnitude o the stress represents error in daage detection results because the stress should be identically zero when no error is involved in the DLV ethod. Because the odal paraeters are ore accurately and reliably identiied by SSI/CVA, the resulting noralized accuulated stress (i.e., error) is lower than those ro other odal identiication ethods. Due to the low error level, the

7 probability o the alse negative detection, a ailure to identiy the existing daage, can be reduced..4.3 Mode 1 Mode 2 Mode 3 Mode 4 μ Eθ.2.1 NEx/ERA FDD SSI/CVA (a) Mean o E φ E (%) Mode 1 Mode 2 Mode 3 Mode 4 NEx/ERA FDD SSI/CVA E φ (b) Standard deviation o Figure 4. Error in the estiated ode shapes. Noralized accuulated stress NEx/ERA FDD SSI/CVA Daage Eleent 5 Daage Eleent 31 Daage Eleent 44 Figure 5. Mean o the noralized accuulated stress. CONCLUSION Daage detection using SSI/CVA was investigated. Nuerical siulation is conducted using the truss odel to veriy the proposed approach. hree coonly used output-only odal identiication ethods, NEx/ERA, FDD, and SSI/CVA, were eployed to estiate odal paraeters ro the siulated accelerations. he lexibility atrix constructed ro the odal paraeters was then used

8 in the DLV ethod to copare the perorance o the odal identiication ethods. Fro the siulation results, SSI/CVA has been shown to iprove the daage detection perorance. REFERENCES Bernal, D. (22). Load vectors or daage localization. J. Engineering Mechanics, ASCE, 128(1), Bernal, D. (26). Flexibility-based daage localization ro stochastic realization results. J. Engineering Mechanics, ASCE, 132(6), Bernal, D. (27a). Daage localization ro the null space o changes in the transer atrix. AIAA Journal, 45(2), Bernal, D. (27b). Daage localization in systes with unnown inputs. Key Engineering Materials, 347, Brincer, R., Zhang, L., and Anderson, P. (21). Modal identiication o output-only systes using requency doain decoposition. Sart Materials and Structures, 1(3), pp Chance, J., olinson, G. R. and Worden, K. (1994). A sipliied approach to the nuerical and experiental odeling o the dynaics o a craced bea. Proc. 12th International Modal Analysis Conerence, Duan, Z., Yan, G., Ou, J., and Spencer Jr., B. F. (25), Daage localization in abient vibration by constructing proportional lexibility atrix, Journal o Sound and Vibration, 284, pp Gao, Y. and Spencer Jr., B. F. (22), "Daage Localization under Abient Vibration Using Changes in Flexibility," Journal o Earthquae Engineering and Earthquae Vibration, 1(1), pp Gao, Y., Spencer Jr., B. F., and Bernal, D. (24). Experiental veriication o the daage locating vector ethod. Proc., 1st Int. Worshop on Advanced Sart Materials and Sart Structures echnology, Honolulu, Hawaii, January Herans, L. and Van Der Auweraer, H. (1999) Modal testing and analysis o structures under operational conditions: Industrial applications. Mechanical Systes and Signal Processing, 13(2), Juang, J. N. and Pappa, R. S. (1985). An Eigensyste realization algorith or odal paraeter identiication and odel reduction. Journal o Guidance Control and Dynaics, vol. 8, pp Jaes, G. H., Carne,. G., Lauer, J. P. and Nord, A. R. (1992). Modal testing using natural excitation. Proc. 1 th International Modal Analysis Conerence, San Diego, CA. Nagayaa,. and Spencer Jr., B. F. (27). Structural Health Monitoring Using Sart Sensors. Newar Structural Engineering Laboratory (NSEL) Report Series, No. 1, University o Illinois at Urbana-Chapaign, Urbana, Illinois ( Sohn, H., Farrar, C. R., Heez, F. M., Shun, D. D., Stineates, D. W. and Nadler B. R. (23). A review o structural health onitoring literature: Los Alaos National Laboratory Report, LA MS. Sall, E. P., Philbin,., Fraher, M., and Roac, G. P. (1999). he current status o bridge anageent syste ipleentation in the United States. 8 th International Bridge Manageent Conerence, Denver, Colorado.