Application of support vector machine in health monitoring of plate structures

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1 Appcaton of support vector machne n heath montorng of pate structures *Satsh Satpa 1), Yogesh Khandare ), Sauvk Banerjee 3) and Anrban Guha 4) 1), ), 4) Department of Mechanca Engneerng, Indan Insttute of echnoogy Bombay, Mumba, Inda 3) Department of Cv Engneerng, Indan Insttute of echnoogy Bombay, Mumba, Inda 1) Satsh.satpa@gma.com ABSRAC hs paper demonstrates the use of Support Vector Machne (SVM) for detecton of damage ocaton and ts ntensty n an aumnum pate. weve damage ocatons and nne damage ntenstes have been smuated by reducng thckness of the pate at varous ocatons usng the fnte eement anayss package Abaqus. he frst mode shape data s extracted at varous ponts on the pate and t has been used as nput data for SVM to predct the damage ocatons and ther ntenstes. hs approach does not requre data of the pate n damaged state. In order to make the mode shape data more reastc n nature, Gaussan nose from 30dB to 80dB has been added. he resuts demonstrate that SVM can be used as a too for structura heath montorng wthout usng data of heathy (undamaged) state. 1. INRODUCION Structura Heath Montorng (SHM) s of great mportance n cv, mechanca and aerospace structures for safety purpose and to avod economca oss. he process of mpementng a damage dentfcaton strategy for above mentoned structures s referred to as SHM (Farrar et a., 007). he presence of damage n the structure eads to change n the moda parameters (natura frequency, dampng and stffness), and nterpretng the changes n these parameters one can ensure whether the structure s damaged or ntact. he change n the natura frequency was not suffcent to ocate the damage, hence, there was need to deveop methods based on mode shape data and Frequency Response Functon (FRF) data of the structure (Banerjee et a., 005, 009). he use of SVM for predcton of faut n power systems has been demonstrated by 1) Ph.D. Student ) P.G. Student 3) Assocate Professor 4) Assstant Professor 1631

2 Kumar et a. (011). hey used support vector cassfcaton to predct the damage ocaton. he nputs used for SVM mode are Power and Votage Vaues. Buut et a. (007) demonstrates the damage detecton n cv structure usng SVM cassfer and waveets. hey found that the SVM was a robust cassfer n presence of nose whereas waveet-based compresson gracefuy degrades ts cassfcaton accuracy. he present artce uses vbraton data (mode shape data) for regresson anayss usng SVM n order to ocate damage and ts ntensty n the rectanguar pate. Fgure 1 Damage ocatons Fgure FE mesh & data acquston ponts abe 1 modfed ocaton abes n reference to Fgure 1 Locatons from center of pate Rearranged ocaton Rada dstance (mm) Locaton abe Locaton abe Rada dstance (mm) (as per Fg. 1) (as per Fg. 1) FINIE ELEMEN MODELLING AND ANALYSIS A smpy supported pate of dmensons 500mm x 400mm x 3mm, wth foowng propertes: Young s moduus = 70GPa, Densty = 700 Kg/m3, Posson s rato = 0.3 s consdered. FE modeng and anayss of the pate s carred out n ABAQUS usng 4 163

3 node rectanguar she eement of sze 10mm X 10mm. In Fgure (1) damage ocatons are shown whch are smuated by reducng the thckness from 10% to 50 % of the orgna pate thckness n steps of 5%. However for better understandng of resuts, the pate centre s taken as reference (0, 0) and ocatons are defned as per ther rada dstance from center. he purpose of ths arrangement s to hghght trend of error n damage predcton wth respect to ocaton from the centre of the pate. he arrangement can be expaned from abe OVERVIEW OF SUPPOR VECOR MACHINE FOR REGRESSION A bref formuaton on SVM for regresson anayss gven by Vojsav (001) s presented n ths secton.svm s ntay deveoped for sovng cassfcaton probems, and successfuy apped n regresson probems. he genera formuaton of regresson earnng s carred out as foows. Gven tranng data set for earnng the machne (agorthm), t attempts to earn the nput-output reatonshp f(x). A tranng data set D = {[x (), y ()] n, = 1,, } conssts of pars (x 1, y 1 ), (x, y ),, (x, y ), where the nputs x are n- dmensona vectors x n, and the system responses y are contnuous vaues. Here frst near regresson probem formuaton s consdered and extended to non-near probem. f( x, w) w x b (1) where, x s nput vector, w s weght vector and b s bas term. ypcay regresson anayss s assocated wth approxmatng nput-output reatonshp consderng error of approxmaton. he near oss (error) functon wth nsenstvty zone ntroduced by Vapnk s gven as yf( xw, ) ε 0 y f( x, w) ε y f( x, w) ε otherwse () he near oss (error) functon wth -nsenstvty zone s shown graphcay n the Fgure (4). e Fgure 3 parameters used n (1D) SV regresson x Fgure 4 Loss (error) functon, 1633

4 he vaue gven by the Eq. (1) s predcted one and y s the actua vaue of the system response for gven nput x. he oss or error s equa to zero f the dfference between predcted and actua vaue s ess than tube. Vapnk s -nsenstvty oss functon aows us to set mt or some measure of error whch can be toerated and gven by a sma vaue. If the predcted pont es outsde the tube, then the oss s equa to magntude of the dfference between the predcted vaue and the radus of the tube whch termed as sack varabe and s gven by y f( x, w) ε ζ for data "above" an ε tube (3) y f( x, w) ε ζ * for data "beow" an ε tube (4) as A new emprca rsk s ntroduced n order to perform SVM regresson and s gven R ε emp 1 wb, yw xb 1 ε (5) he objectve of SVM regresson s to mnmze the emprca rsk R ε emp and norm of w wegh vector smutaneousy. hus, man goa s to estmate a near regresson hyperpane f( x, w) w xb by mnmzng 1 R w C yw xb (6) 1 ε Usng expressons for sack varabes the emprca rsk becomes Under the constrants 1 R w C ζ ζ * (7) 1 1 y w x b ε ζ, 1 * w x b y ε ζ, 1 ζ 0, 1 ζ * 0, 1 (8) (9) here are many two parameters whch have to be tuned to get good performance from SVM regresson anayss. he constant C nfuences the trade-off between an approxmaton error and the weght vector norm. Another parameter whch has to choose by the user, that defnes the precson requred n predcton. hs constraned probem s soved by formng prma Lagrangan (L p ) functon, and s gven by 1634

5 * * * 1 * * * * wbζ,,, ζ α, α, β, β w C ζ ζ α y x b ε ζ βζ βζ Lp w w (10) hs prma Lagrangan functon has to be mnmzed wth respect to prma varabes w, b,, and and maxmzed wth respect to,,,. he probem s soved n ts dua form and s gven as foows, Maxmze Subject to * * * 1 * * Ld α, α εα αα αy α α α j α jx x j (11) 1 1, j1 * α α (1) 1 1 * 0 α C, 1 0 α C, 1 (13) If we ook at the dua form of probem t s expressed n terms of Lagrange mutpers and ony. hs standard optmzaton probem can be expressed n a matrx form and gven as: Mnmze Subject to constrants Eqs. (1), (13). Where for near regresson α L 0.5 d α Hα f (14) H x x1 (15) f εy εy εy εy εy εy (16) 1 N 1 he souton of above probem w gve Lagrange mutpers pars. he number of support vector s equa to the nonzero parameters or. After cacuatng Lagrange mutpers the weght vector and bas term s found as foows * 1 w α α x (17) N 1 y x 1 b w (18) he best regresson hyperpane n case of near probem s gven by f( x, w) w x b (19) 1635

6 Whe desgnng SV machnes for non-near regresson anayss frst map the nput vectors x n n to vectors z of a hgher-dmensona feature space F, where ϕ represents a mappng), and sove a near regresson probem n ths feature space. he most mappng (kerne) functons are poynomas and rada bass functons wth Gaussan kernes. he gven optmzaton probem s soved wth change n ony Hessan matrx H and s gven as H G G G G (0) Where G s the correspondng kerne matrx G (, ) and weght vector and bas term s gven by w α * α (1) 1 b y g 1 () and the best non-near regresson functon s gven by g Gw (3) z f x, w Gw b (4) 4. RESULS AND DISCUSSION Snce SVM regresson agorthm gves ony snge output, SVM regresson anayss s carred out n two stages. 4.1 SAGE1: DAMAGE LOCAION PREDICION he damage ocaton predcton was done n two steps. Step 1 nvoved predctng the X coordnate and step nvoved predctng the Y coordnate of the damage ocaton. he tranng nput set used for step 1 s mode shape data for a damage ntenstes and tranng output set was correspondng X coordnate of damage ocatons. est set was the mode shape data whose damage ocaton and ntensty was to be predcted. SVM now predcts X coordnate of damage ocaton n step 1. Step was smar to step one except that the Y coordnate of damage ocaton was predcted. Stage nvoved damage ntensty predcton. In ths stage, mode shape data for a partcuar ocaton was used as nput and damage ntensty at that ocaton was used as output. hs was repeated for a the ocatons. Parameters for SVR are taken as: C=, e=0.0005, Rada Bass functon (RBF) kerne, - nsenstve oss functon, kerne wdth=0.6 for damage ocaton predcton, and C=10, kerne wdth=1 for damage ntensty. he percentage error s cacuated as gven beow. 1636

7 X X Y Y % error maxmum ength of dagona X 100 (1) Frst mode shape data obtaned at a the ponts of the smuated rectanguar pate hghghted n the Fgure s consdered as tranng nput for the SVM, and correspondng damage ocaton and/or ntensty as tranng output. he damage ocaton s represented by the mdpont of the damaged area n order to get snge vaued output for the SVM. abe Error n damage ocaton predcton averaged over damage ocaton of pate nose eve Intensty% no nose 80 db 70dB 60 db 50 db 40 db 30 db % Error no nose 80 db 70dB 60 db 50 db 40 db 30 db % Error no nose 80 db 70 db 60 db 50 db 40 db 30 db Damage ntenstes % Damage ocaton Fgure 5 Error n damage ocaton predcton averaged over damage ntenstes Fgure 6 Error n damage ocaton predcton averaged over damage ocaton he same % errors for consdered nose eve cases now are averaged over damage ntensty and the varaton of % error wth the damage ocatons s tabuated n tabe 3 and s potted n the Fgure 5 averaged over damage ntenstes and averaged over damage ocatons n Fgure 6 respectvey. For no nose case, the % error remans beow % up to damage ocaton 134mm and t suddeny ncreases at the ocatons 1637

8 14mm and 156mm whch are far away from the center of the pate. As we add nose n the data for ow nose eves the same error, whch was up to 134mm n the case of no nose case, now t s at 15mm. he detaed resuts are gven n the tabe. abe 3 Error n damage ocaton predcton averaged over damage ntensty of pate Dstance nose (mm) no nose db db db db db db SAGE: DAMAGE INENSIY PREDICION abe 3 summarzes the error n ntensty predcton by SVM for those ocatons found n the stage 1. For nose eve up to 50dB the % errors are amost same at ow damage ntensty. abe 3 Error n damage ntensty predcton averaged over damage ocaton of pate nose no nose Intensty% 80 db 70dB 60 db 50 db 40 db 30 db he percentage error n ntensty predcton s cacuated as gven beow % error Damage ntensty Damage ntensty Damage ntensty X 100 () 1638

9 % Error no nose 80 db 70 db 60 db 50 db 40 db 30 db Damage ocaton Fgure 7 Error n damage ntensty predcton averaged over damage ocaton for ow nose eve % Error no nose 80 db 70dB 60 db 50 db 40 db 30 db Damage ntenstes n % Fgure 8 Error n damage ntensty predcton averaged over damage ocaton for hgh nose eve abe 4 represents the detaed vaues of % error for dfferent nose eves ncudng no nose case, and t s potted n the Fgure 7 and Fgure 8. For no nose case the % error s hgh for ony two ocatons (14mm and 156mm) but, when we add the nose, t s hgh for three ocatons for nose eves 80dB to 50dB. he error for the case of 40dB nose s acceptabe ony for the ocatons coser to the center of the pate.e. up to 75mm from the center. abe 4 Error n damage ntensty predcton averaged over damage ntensty of pate Dstance nose no nose db db db db db db CONCLUSIONS he SVM has been traned wth vbraton-nduced dspacements coected at 99 ponts for the frst mode shape as nput and damage ntensty or ocaton as output. After tranng, the SVM s abe to predct any damage ntensty or ocaton of the tranng set data wth amost neggbe error. he % error n predcton of damage ocaton and ntensty s ess at the center of the pate and goes on ncreasng away from the center. he predcton capabty of SVM s degraded wth addton of nose n the data. For ow 1639

10 nose eves % error remans amost same as that of no nose case n the data that means SVM can toerate such nose eves wth ess devaton n the errors. REFERENCES Banerjee S, Rcc F, Monaco E, Ma A (009), A wave propagaton and vbraton based approach for damage dentfcaton n structura components. Journa of Sound and Vbraton (3), Buut A, Sng A.K, Shn P, Fountan, Jasso H, Yan L, Egama A (005), Rea-tmenon destructve structura heath montorng usng support vector machnes and waveets. In: SPIE 5770, Advanced Sensor echnooges for Nondestructve Evauaton and Structura Heath Montorng, 180. UC Santa Barbara, Santa Barbara, 13 May 005. Cherkassky V, Ma V (004), Practca seecton of SVM parameters and nose estmaton for SVM regresson.neura Network (17), Farrar CR, Worden K (007), An ntroducton to structura heath montorng. Ph rans R Soc A (365), Kumar SK, Jayabarath, Naveen S (011), Faut dentfcaton and ocaton n dstrbuton systems usng support vector machnes. European Journa Scentfc Research (51), Lee Jong Won, Krkera Goutham. R, Kang Inp, Schuz Mark J and Shanov Vessen.N (006), Structura heath montorng usng contnuous sensors and neura network anayss, Smart Mater.Struct. (15), Ma A, Rcc F, Banerjee S, Shh F (005), A conceptua structura heath montorng system based on vbraton and wave propagaton. Structura Heath Montorng 4(3),

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