Structural Damage Identification Based on Substructure Sensitivity and l 1 Sparse Regularization

Transcription

1 Cvl Stuctual Health Montong Wokshop (CSHM-4) - Poste 1 Stuctual Damage Identfcaton Based on Substuctue Senstvty and l 1 Spase Regulazaton Yuequan BAO *, Shume ZHOU *, Hu LI * *, **, Jnpng OU * School of Cvl Engneeng, Habn Insttute of echnology, Habn, 19, Chna ** School of Cvl and Hydaulc Engneeng, Dalan Unvesty of echnology, Dalan, 11624, Chna Abstact. Spasty constants ae now vey popula to egulaze nvese poblems n the feld of appled mathematcs. Stuctual damage dentfcaton s a typcal nvese poblem of stuctual dynamcs and Stuctual damage s a spatal spase phenomenon,.e., stuctual damage occus, only pat of elements o substuctues ae damaged. In ths pape, a stuctual damage dentfcaton method based on the substuctue-based senstvty analyss and the spase constants egulazaton s poposed. Substuctue senstvty analyss, the establshment of stuctual damage stffness paamete vaaton and change of modal paametes of lnea equatons between the measued degees of feedom s lmted, the equatons fo a mobd equaton. he ntoducton of stuctual damage spasty condtons, to mnmze the l1 nom optmzaton soluton. he numecal example of the 2 bay-tuss stuctue wth consdeng measuement nose, ncomplete of measuements and mult-damage cases ae caed out. he effects of numbe senso and layout to the dentfcaton esults ae also nvestgated. he esults ndcated that the damage locatons and extents can be effectvely dentfed by the poposed method. Addtonally, the senso locaton can be andom aangement, whch has geat sgnfcance to the senso placement of the actual stuctual health montong because obust stuctual damage dentfcaton also can be obtaned even a few of senso ae falue. 1 Intoducton Stuctual damage detecton s a coe ssue of stuctual health montong, whch has been nvestgated long tmes and a lot of methods has been poposed [1].hese methods can be dvded n accodance wth the need fo the pont of vew of the fnte element model, model and non-model-based damage dentfcaton methods. he non-model based damage dentfcaton method fstly dentfy the modal paametes o dectly extact damage featues fom the esponse data, and then constuct the senstve damage ndexes fo dentfy the damage of stuctue. Such as fequency-based methods whch constuct the sutable ndcato fo damage dentfcaton though stuctual fequency change. he mode shapes based methods detect the damage by the changes of mode shapes befoe and afte stuctual damage, whch ncludng the MAC, mode cuvatue, stffness method and flexblty, esdual foce, modal stan enegy methods. he othes methods ncludng wavelet damage featue extacton method [2] o HH (Hlbet Huang anslaton) method [3] ae all the non-model based damage dentfcaton methods. he fnte element (FE) model has an mpotant ole n the damage dentfcaton of stuctue. he model based methods used the modal paametes o damage chaactestc Lcense:

2 paametes extacted fom stuctual vbaton esponse data combng wth the FE model to dentfy the stuctue paametes, and then poceed to the damage dentfcaton, such as the Bayesan damage detecton methods [4]. Fo these methods, the most mpotant s dentfcaton of system paametes by optmzaton poblems. Recently, spasty constants ae now vey popula to egulaze nvese poblems n the feld of appled mathematcs. Fo example, the compessve sensng (CS) technque [-8], whch the coe dea s a hgh-dmensonal sgnal s compessble o spasty n a tansfom doman, then you can use a tansfom base elevant pojected onto a low dmensonal space of the sgnal of the measuement matx, and then by solvng an optmzaton poblem to econstuct the ognal sgnal fom the small amount of pojecton wth ovewhelm pobablty. Essentally, the CS s to fnd the spase solutons of undedetemned equaton. he nvestgaton of CS theoy also put fowad the development of spase egulazaton of nvese poblem, whch povde a new way fo the solvng of nvese poblem whch has spasty n tme doman, spatal doman o some base tansfomaton. Stuctual damage dentfcaton s a typcal nvese poblem of stuctual dynamcs and Stuctual damage s a spatal spase phenomenon. In ths pape, a stuctual damage dentfcaton method based on the substuctue-based senstvty analyss and the spase constants egulazaton s poposed. 2 Substuctue Senstvty coeffcents analyss Senstvty coeffcents analyss fo vbaton paametes s a matue technque. Stffness, mass and dampng changes often esult n changes n dffeent dagnostc paametes, ncludng natual fequences and mode shapes. he changes n stffness coeffcents fo a FE model ae used to epesent damage n ths study, and the fst ode senstvty coeffcents fo the natual fequences and the mode shapes to the damage of substuctue ae deved. Fo a stuctual system wth N degees of feedom, the natual fequences, 1, 2,..., N and mode shapes, φ 1, 2,..., N, can be detemned wth a FE analyss. he equlbum equaton fo undamped stuctual vbaton equaton s 2 K Μφ (1) whee M and K ae mass and stffness matces; and φ ae th ( 1, 2,, N ) fequency and mode shape, whee φ s the nomalzed to be unt-mass mode shapes,.e. φ Mφ 1. he stffness matx can be epesent as, N (2) K 1 K K 1 whee s the substuctue stffness contbuton to the global stffness matx whch could come fom a FE model of undamaged stuctue; 1, 2,..., N s stffness damage paamete and N s the numbe of substuctue. 2.1 Senstvty coeffcents fo natual fequency he fst ode senstvty coeffcents fo natual fequency wth espect to the paamete can be calculated by devatve of Eq. (1) wth espect to as K 2 M 2 φ (3) 2 M φ KM

3 Both sdes of ths equaton ae pemultpled by φ. By utlzng the known elatons ncludng φ Mφ I fo the unt-mass mode shapes, Eq. (1) and that M s ndependent of (theefoe M ), t can be shown that senstvty coeffcent of the th natual fequency n tems of can be deved as 1 K (4) φ φ 2 Note that K 1 N 1 K N K 1 K K 1 hus, the senstvty coeffcents of the natual fequences can be ewtten as 1 S φk φ Senstvty coeffcents fo mode shapes he devaton of fst ode senstvty coeffcents fo mode shapes can efe the pape of Zhao and DeWolf [9]. In the pape, the devaton of mode shapes s espect to stffness damage paamete. As epesented n Zhao and DeWolf [9], the senstvty coeffcents of th mode shape can be defned as follows: N φ (7) S lφ l whee L s undetemned coeffcent that epesents weght of the l th mode shape n the senstvty coeffcent. wo possbltes exst fo l : (1) When s, whee s s subscpt of the coeffcent of, s ae the weghts of all mode shapes except the th mode shape used fo the ght sde of Eq. (7). Eq. (3) s pemultpled by φ s on both sdes. Snce mode shapes ae othogonal, t can be shown that φsmφ, f s ; φ Mφ I ; 2 s s φsk sφsm ; and M. Eq. (1) can be then be wtten as n (8) 2 2 φ K φ φ M φ l1 s s s l l l1 Fo unt-mass nomalzed mode shapes, Eq. (13) can be ewtten as 2 2 φ K φ (9) Solvng fo s s s s 1 s φ 2 2 sk φ s (2) When s, s s the weghts of the th mode shape used fo the ght sde of (2). Fo the unt-mass nomalzed mode shapes t s φ Mφ I (11) akng devatve of Eq. (11) wth espect to esults n () (6) (1)

4 φ φ Mφ φm Consdeng the symmety popety of a mass matx φ φ Mφ φm and usng Eq. (7), Eq. (12) can be ewtten as n 2lφMφ l l1 (12) (13) (14) Based on the othogonal popetes of the mode shapes and Eq. (14), t can be shown that (1) hus the senstvty coeffcents fo the mode shapes can be summazed as follows: 1 (16) φ 2 2 sk φ s S s s 3 Damage dentfcaton appoach he stuctue s assumed to behave lnealy befoe and afte the occuence of damage. he elaton between the changes of modal paametes γ [ ω, φ] and substuctue stffness damage paametes θ can be expessed as γ Sθ (17) whee θ 1, 1,, N s stffness damage paametes vecto whch epesents the damage extent of the substuctues; S s the senstvty matx. S S, S (18) whees and S ae the senstvty matx of the fequency and mode shape, whch contans senstvty coeffcents vecto of the th 1, 2,..., N fequency and mode shapes. S S, S,, 1 S 2 n, S S, S,, 1 S 2 n (19) he changes of modal paametes γ ncludng changes of fequency and mode shapes caused by damage of stuctues s γ ω, φ (2) whee ω and φ ae ω 1, 2,, n,,,, φ φ1 φ2 φn (21) Supposng only small numbe of substuctues ae damaged, theefoe, only small numbe of the elements of vecto θ 1, 1,, N ae nonzeo, whch s spasty. Consdeng the lmtaton of sensos and ncompleton of measued modal paametes, m numbe of sensos ae used and N numbe of modals ae obtaned. Wth consdeng the unavodable measuement nose, the Eq. (17) s changed as γˆ Sθ ˆ e (22)

5 whee e s the eo caused by measuement nose; γˆ s the actual measued changes of modal paametes. γˆ ωˆ, φ ˆ (23) whee ωˆ and φˆ ae ωˆ ˆ ˆ ˆ 1, 2,, n, ˆ,,, φ Γφ1 Γφ2 Γφn (24) mn whee the Γ s a andom samplng opeato compsed of zeos and unts and maps the th whole theoetcal mode shapes vecto φ to the obseved cable foces φˆ Γφ. Fo n 1 convenence, ntoducng a vecto δ R wth elements j 1 f the jth degee of feedom s obseved and j f the jth degee of feedom s not nstalled wth a senso, t can be eadly known that ΓΓdag δ. he andomness of matx Γ means that the selecton of test ponts to be obseved s andom. Eq. (21) s ll-condtoned lke many nvese poblems. adtonally, least squaes method s used to solve ths type of poblem, and sngula value decomposton s used to fnd the pseudo-nvese. In Eq. (21), the vecto ˆθ s known to be spase. heefoe, accodng to CS theoy, the soluton ˆθ can be obtaned by solvng the followng optmzaton poblem. mn θ 且 Sθ ˆ γ ˆ (2) 1 whee s an uppe bound of the eo whch satsfes e. hs s the l 2 1 optmzaton poblem, whee the l 1 -tem enfoces the spasty of the epesentaton. An unconstaned fom of ths objectve s θˆ ag mn Sθ ˆ γˆ θ (26) whee s the Lagange multple and dentfed as a egulazaton paamete. hs objectve functon has been used n a numbe of spase sgnal epesentaton woks [1]. he l 2 -tem makes the esdual Ap ˆ k fˆ k small, whle the l 1 -tem enfoces the spasty of the epesentaton. he paamete contols the tade-off between the spasty of the spectum and esdual nom. Eq. (26) s effcently solvable wth the nteo pont solves [1] and the convex optmzaton package CVX (avalable at s used. A pope Lagange multple s mpotant to obtan meanngful solutons to Eq. (26) and can be calculated by empcal o teatve methods [11-12] Numecal Examples he 2-bay gd tuss stuctue fo numecal example s shown n Fgue 1, of whch the total length, wdth and heght ae 8.,.8 and.6m, espectvely. It conssts of 312 membes wth 18 nodes and all the membes ae steel bas. he connectons of the membes ae bolt-node balls whch ae the half-gdty connectons. hee ae two suppots at the ends of the stuctue: a hnge suppot at the ght end and a vetcal olle suppot at the left.

6 Fg. 1. FE model of 2 bay tuss stuctue Consdeng the lmted test ponts n eal applcaton, 38 acceleometes ae placed on the node of bottom chod of stuctue as shown n Fg. 1. he damage s smulated by deceasng the Young s modulus of elements. hee damage cases ae consdeed. Damage case 1 s that the element No. 23 damaged 1%. Damage case 2 has two elements of No. 17 and 236 ae damaged wth 1%, espectvely. Damage case 3 has fve elements of No. 26, 6, 17, 19 and 26 ae damaged wth %, %, 1%, 1% and 1% espectvely. Moe explctly, the estmated modal paamete set ψˆ n was constucted as: ψˆ ψ(1 ) (27) whee ψ was the exact modal paamete set obtaned fom modal analyss, was a nomally dstbuted andom numbe wth zeo mean and a vaance of 1., was the nose level n tems of pecentage, whch the values ae 1%, % and 1% n ths example. 4.1 Damage dentfcaton esults Damage dentfcaton esults of Case 1 ae shown n Fg. 2, whee Fg. 2(a) s the actual damage and Fgs. 2(b-d) ae the dentfed esults wth consdeng nose 1%, % and 1%, espectvely. Fg. 2 shows that the damage locaton and extent ae dentfed well fo Case 1 even wth 1% nose (a) (b) (c) (d) Fg. 2. Damage dentfcaton esults of Case 1: (a) actual damage; (b) damage dentfcaton esults wth 1% nose; (c) damage dentfcaton esults wth % nose; (d) damage dentfcaton esults wth 1% nose; Damage case 2 has fve damaged elements, whch the dentfcaton esults ae shown n Fg. 3. Fg. 3(a) s the actual damage of stuctue whch shows the elements no. 26, 6, 17, 19 and 26ae damaged %, %, 1%, 1% and 1%, espectvely. Fg. 3(b-d) shows the dentfcaton esults wth 1%, % and 1% nose, espectvely. Fg. 3 shows that fo the multple damage cases, the poposed appoach also can dentfy the damage

7 accuately. Compang wth the cases of small amount of elements damaged such as damage case 1 and case 2, the damage cases 3 ae moe senstve to the nose as esults shown n Fg. 3(d), howeve, whch s also can be acceptable (a) (b) (c) (d) Fg. 3. Damage dentfcaton esults of Case 3: (a) actual damage; (b) damage dentfcaton esults wth 1% nose; (c) damage dentfcaton esults wth % nose; (d) damage dentfcaton esults wth 1% nose; 4.2 Effects of senso numbe o nvestgate the effects of senso numbe on damage dentfcaton esults, the senso numbe fom 8,1,...,38ae consdeed. Wth dffeence senso numbe, the dentfcaton eo s calculated by ˆ θ θ (28) θ whee ˆθ s the dentfed stffness damage coeffcents; θ s the actual stffness damage coeffcents. he damage dentfcaton esults ae shown n Fg. 4, whch shows that wth the nceasng of senso numbe, the damage dentfcaton accuacy s nceased and ths effect s moe sgnfcant to the multple damage cases. Wth same senso numbe, the nceasng of nose also wll ncease the damage dentfcaton accuacy. Eo (%) % nose % nose 1% nose Eo (%) % nose % nose 1% nose Senso numbe Senso numbe (a) (b) Fg. 4. Relatonshp between dentfcaton eo and senso numbe: (a) damage case 1; (b) damage case Effects of senso locaton Accodng to the CS, the selecton of locaton of test ponts can be andom. o show the effects of senso locaton, 24 sensos wth andomly placed ae consdeed as shown n Fg..

8 Wth 24 test ponts, the damage dentfcaton esults of the thee damage cases wth 1% nose ae shown n Fg. 6, whch shows the damage also can be well dentfed. hs futhe llustates the obust esults of the poposed appoach. Random placement of sensos has emakable advantages n actual applcaton of SHM systems. Compang wth the tadtonal seno optmal placement algothms, andomly placed s much moe easly and convenent mplement. Othewse, n actual SHM systems, the complex sevce envonment wll cause the falue of sensos and ths phenomenon s andom. If one o small numbe of sensos ae faled, the obust damage dentfcaton esults also can be obtaned Fg. Locatons of 24 test ponts (a) (b) Fg. 6. Damage dentfcaton esults wth 24 test ponts fo thee damage cases wth 1% nose; (a) dentfcaton esults of damage case 1 wth 1% nose; (b) dentfcaton esults of damage case 2 wth 1% nose Conclusons A stuctual damage detecton appoach by combng CS theoy and substuctue senstvty s poposed n the pape. he appoach establshed the elatonshp between the substuctue damage extent and the measued fequency vbaton vayng. Because the lmtaton of measuement degee of feedom of stuctue and ncompleton of modal paametes, the establshed lnea algebac equaton s ll-condtonal. Based on the spase phenomena of stuctue damage,.e., the damage extent vecto to be solved s spase, the exact solutons can be obtaned by CS theoy. Numecal example of the 2-bay tuss stuctue s caed out to vald the damage dentfcaton ablty of the poposed appoach wth consdeng measuement nose. he poposed appoach can dentfy mult-damages of stuctue ncludng damage locatons and extents wth a small numbe of sensos, ncomplete modal paametes and measuement nose. he senso numbe wll affect the damage dentfcaton accuacy, wth the nceasng of senso numbe, the damage dentfcaton accuacy s nceased and ths effect s moe sgnfcant to the multple damage cases. Random placement of sensos s equed n the appoach, whch has emakable advantages n actual applcaton of SHM systems. If one o small numbe of sensos ae faled, the obust of damage dentfcaton esults also can be obtaned.

9 Acknowledgement hs eseach s suppoted by NSFC (Gant No. 189), and Mnsty of Scence and echnology (Gant No. 211BAK2B2), and Chna Postdoctoal Scence Specal Foundaton (Gant No ). Refeences [1] Sohn H., Faa C.R., Hemez F.M., Shunk D.D., Stnemates D.W., Nadle B.R., A Revew of Stuctual Health Montong Lteatue: , LANL Repot, LA MS, 23. [2] Yen, G.G., Ln and K.C. (2), Wavelet packet featue extacton fo vbaton montong, IEEE. Ind. Electon., 47(3), [3] J.N. Yang, Y. Le, S. Ln, N. Huang. Hlbet-Huang Based Appoach fo Stuctual Damage Detecton. ASCE Jounal of Engneeng Mechancs. 24: 8-9. [4] Vank, M.W., Beck, J.L. and Au, S.K. (2), Bayesan pobablstc appoach to stuctual health montong, J. Eng. Mech., ASCE, 126(7), [] Candès EJ. Compessve Samplng. Poceedngs of the Intenatonal Congess of Mathematcans, Madd, Span; 26, pp [6] Donoho D. Compessed Sensng. IEEE ans Infom heoy 26; 2(4): [7] Baanuk RG. Compessve sensng. IEEE Sgnal Pocess Mag 27; 24(4), [8] Baanuk RG. Moe s less: Sgnal pocessng and the data deluge. Scence 211; 331: [9] Zhao J., DeWolf J.., Senstvty study fo vbatonal paametes used n damage detecton, Jounal of Stuctual Engneeng, 12(4), [1] Boyd SP, Vandenbeghe L. Convex Optmzaton, Cambdge Unv. Pess, Cambdge, U K; 24. [11] Yao H, Gestoft P, Sheae PM. Mecklenbauke C. Compessve sensng of the ohoku-ok Mw 9. eathquake: Fequency-dependent uptue modes, Geophys Res Lett 211; 38, L231 [12] Maloutov DM, Müjdat C, Wllsky AS. A spase sgnal econstucton pespectve fo souce localzaton wth senso aays. IEEE ans Sgnal Pocess 2; 3,