Wavelet and Lipschitz exponent based damage identification method for beams using mode shapes

Wa ele and Lipschi z exponen based damage iden i ica ion me hod o beams using mode shapes Abs ac This pape p esen s a damage iden i ica ion me hod o beam- ype s uc u es based on he wa ele analysis. The damage loca ion is p edic ed by a Damage Loca ing Index (DLI) based on a combined iden i ica ion o local maxima om he scalog ams o mode shapes. A new Damage Se e i y Index (DSI) based on he Lipschi z exponen is o mula ed such ha i is no dependen om he damage loca ions, bounda y condi ions, and mode o de , as long as a high enough numbe o measu ing poin s is conside ed. This s udy ocuses on i s applica ion using ela i ely ew measu emen poin s and on he consis ency o he DSI. Resul s om a beam wi h double-no ch damage a e analyzed. The esul s show ha he me hod is applicable wi h a spa se measu emen g id and he DSI Re e ence alues om a s a ic analysis can be e e ed o damage e alua ion using modal displacemen s. The p ac ical limi s o he me hod a e also discussed. Keywo ds: beams, mul iple damage, mode shapes, wa ele ans o m, spa se measu emen s 1. In oduc ion Damage iden i ica ion me hods sol e he p oblem successi ely h ough de ec ion, loca ion, and quan i ica ion. De ec ion and loca ion can be done by analyzing he abno mali y in be ween he pos -damage s uc u al esponse and he e e ence s uc u al esponse, ei he om a s a ic, dynamic, o modal analysis. Quan i ica ion can be sol ed by compa ing a damage s a e wi h e e ence alues ha can be ob ained using nume ical o analy ical models. Ne e heless, he e is always a di e ence be ween a model and a eal s uc u e, which can no be asce ained unless i is iden i ied using measu emen s. In his pape , exis ing o mula ion o a Damage Loca ing Index (DLI) and a Damage Se e i y Index (DSI) [1] based on he Con inuous Wa ele T ans o m (CWT) and Lipschi z exponen a e ede ined and ex ended o he applica ion o damage iden i ica ion om he expe imen al mode shapes o a damaged beam. The pape explo es he applicabili y o he iden i ica ion me hod om h ee pe spec i es: (1) he obus ness and limi a ions o he p oposed DLI on an oscilla ing signal (mode shapes) ob ained om a spa se measu ing g id; (2) he applica ion o he DSI o modal displacemen s and i s independence om bounda y condi ions and damage loca ion; and (3) he po en ial use o he same DSI Re e ence alues om ei he a s a ic o any modal esponse o he e alua ion o he damage. P ep in submi ed o Else ie Decembe 1, 2022 A s uc u al damage is de ined as an al e a ion o he ma e ial o geome y p ope ies o e a local egion, which esul s in a sudden change in he slope o he displacemen ield. A simple example is a no ch on a beam. Al hough his ype o damage is a ely seen in p ac ice, i is commonly used in academic esea ch due o i s simple geome ic o m (easy o be a i icially in oduced and geome ically de ined) and di ec quan i ica ion ela ionship be ween no ch dep h and damage se e i y. In his s udy, he no ch is conside ed always in he open s a e so ha he s uc u al esponse can e lec he e ec o damage. A he loca ion o damage, he induced abno mali y in he displacemen ield (bo h s a ic de lec ions and mode shapes) will esul in a local peak alue o wa ele coe icien s [2–7]. Common me hods o de e mine he loca ion o hese local peaks include he use o a 2D colo map [2, 3, 8] o he use o combined wa ele coe icien s h ough di e en scales [9, 10]. Apa om he exploi a ion o wa ele scales in damage de ec ion, he mode o de is also u ilized in his pape o he de e mina ion o DLI. This pape also add esses he issues on he p edic ion o he damage loca ion due o he oscilla ion o he wa ele unc ion and he consequen shi s o wa ele local peaks caused by he use o a symme ic wa ele unc ion. In he applica ion o CWT o damage de ec ion, he numbe o samples o he disc e e space signal di ec ly a ec s he damage loca ion accu acy. Kuma and Singh [8] compa es he use o 100, 200, and 1000 measu emen poin s o he applica ion o CWT o damage de ec ion and indica es ha he accu acy o he esul s is a ec ed signi ican ly. Hence, nume ical s udies on his opic conside usually a high numbe o measu emen poin s and he expe imen al s udies a e conduc ed wi h measu ing sys ems ha can ob ain a dense measu emen g id, such as lase scanning [7, 11] and pho og amme y measu ing sys em [1, 8, 12–14]. In Table 1, he numbe o samples in some ele an esea ch s udies a e lis ed. Al hough a high densi y o measu emen s can be ob ained expe imen ally, he associa ed expe imen al echniques a e limi ed o small scale s uc u es. Fo la ge scale s uc u es, adi ional senso s, such as accele ome e s and s ain gauges a e mo e common in p ac ice. The numbe o measu emen s o he con en ional senso s is, howe e , much lowe . A o al o 33 measu emen poin s o he dynamic esponse o he es ed beam in his s udy a e acqui ed by using piezoelec ic accele ome e s. E o s o es ima e he damage se e i y ha e led o di e en p oposals o damage indices based on he wa ele coe icien s using s a ic de lec ions [1, 9, 15]. S a ic es ing is simple and e icien . Howe e , i is no always p ac icable o ce ain ypes o s uc u es. Dam- age indices based on modal analysis, can be de eloped using he Lipschi z exponen -wa ele cha ac e is ic [10, 16, 17]. Ne e heless, he damage indices p oposed in he ci ed li e a u e a e dependen on he damage loca ion, mode o de , and bounda y condi ions. In con as , he me hod p oposed in a p e ious s udy by he au ho s [1] no malizes he wa ele local peak coe icien s o he cu a u e a i s loca ion, which leads o a DSI ha is independen om he damage loca ion, mode o de , and he bounda y condi ions. I is a majo ad- an age ha he damage me hod does no equi e he iden i ica ion o ac ual beha io o he bounda y condi ions in p ac ice applica ions. Howe e , when using a limi ed numbe o measu emen poin s, he accu acy o he es ima ion o modal cu a u es is educed. In e po- 2 Table 1: Numbe o samples in he CWT applica ion on Damage Iden i ica ion. Resea ch Analysis ype Displ. ype Beam Leng h Numbe o samples Cao e al. [3] Nume ical Mode shape 500 mm 500 Ghanba i Ma dasi e al. [6] Nume ical S a ic 1000 mm 1001 Expe imen al S a ic 26.5 mm 1080 Deng e al. [7] Nume ical Mode Shape 500 mm 1001 Expe imen al Mode Shape 500 mm 499 Kuma and Singh [8] Nume ical Mode Shape 1000 mm 1000 Expe imen al S a ic 1000 mm 1000 Nigam and Singh [13] Nume ical Mode Shape 1000 mm 4000 Expe imen al S a ic 1000 mm ≈2400 And eaus e al. [9] Expe imen al S a ic 280 mm 312 la ion is one al e na i e way o a i icially inc ease he numbe o measu ing poin s [18–20]. Howe e , when using da a wi h expe imen al noise, on one hand, in e pola ion migh in- oduce ex a unwan ed dis u bance in he da a, which can lead o alse-posi i e e o s in damage de ec ion. On he o he hand, esul s a e highly sensi i e o noise, especially wi h a spa se measu emen g id. The e o e, in his s udy, a polynomial cu e i ing is p oposed o educe he expe imen al noise so no in e pola ion is adop ed, minimizing a i icial in e - ene o he measu emen da a. One should no e ha when using mode shapes, a mode is ob iously non-sensi i e o a damage ha is loca ed a i s nodes (ze o modal ampli ude loca ions). C oss- e e ence o po en ial damage loca ions among di e en modes can esol e his issue. The pape is ou lined as ollows: i s he damage iden i ica ion me hod is p esen ed o- ge he wi h a b ie e iew o he CWT backg ound. The de elopmen o he DLI and DSI a e illus a ed o he es ima ion o damage loca ion and se e i y, espec i ely. A nume - ical example is used o show he pe o mance o he indices wi h bo h dense and spa se measu emen s. Nex , expe imen al esul s o a simply-suppo ed beam wi h wo no ches a e p esen ed o e i y he applicabili y o he me hod wi h eal expe imen al noise pollu ion and a spa se measu ing g id. Finally, he ou pu s o he esea ch a e discussed and conclusions a e d awn. 2. Damage iden i ica ion me hod The p oposed damage iden i ica ion me hod is de eloped based on he heo y p esen ed in Re e ences [1, 21] o s a ic de lec ions. In his sec ion, he heo y is b ie ly e iewed and ex ended o he applica ion wi h mode shapes. He eina e , he undamaged s a e o he s uc u e is deno ed by he Re e ence S a e (S a e R) and he damaged s a e o he s uc u e is deno ed by he Damaged S a e (S a e D). 3 2.1. Damage induced e ec in he displacemen ield Fo a beam s uc u e which is well es ained and subjec ed o some ex e nal s a ic o ces ha gene a e a pu e bending displacemen ield uR(x) and in e nal bending momen dis ibu ion mR(x), being x he longi udinal coo dina e along he beam, he p esence o a concen a ed damage will esul in a change in uR(x). Deno e he pos -damage displacemen ield by uD(x) and le he damage loca ion be x0. The s i ness change a x0 educes he bea ing capaci y o he c oss-sec ion, he eby modi ying he momen dis ibu ion and displacemen ield along he beam. I can be demons a ed [22, 23] ha he damage induced a ia ion o he displacemen ield, ∆u(x) (∆u(x) = uD(x)−uR(x)), is equi alen o he esponse o he damaged beam subjec ed o a pai o sel -equilib a ed bending momen s ha equals mR(x0), applied a x0. This s a e o he beam is named he Inc emen al S a e (S a e I) and he co esponding esponse o he s uc u e is deno ed by uI(x), whe e uI(x) is equal o ∆u(x). The bending bending momen o he damaged beam is deno ed by mD(x). The di e ence be ween he ex e nal bending momen applied in he S a e I a he damage loca ion (mR(x0)) and mD(x0) co esponds o he momen ansmi ed o bo h sides o he damage in he S a e I, which a e ep esen ed by m− I(x0) and m+ I(x0) ( o le and he igh side o he damage, espec i ely): m− I(x0) = m+ I(x0) = mR(x0)−mD(x0). The e o e, he bending momen in he S a e I a he damage loca ion (mI(x0)) is equal o mD(x0). A de ailed explana ion o his decomposi ion scheme can be ound in [22, 23]. Simila ly, in modal analysis, we deno e he undamaged and damaged j h mode shapes by φR,j(x) and φD,j(x), and hei co esponding modal bending momen s by mR,j(x) and mD,j(x), espec i ely. The Inc emen al S a e o he j h mode is desc ibed as he beam subjec ed o a pai o sel -equilib a ed ha monic bending momen s, o an ampli ude equal o he modal bending momen mR,j(x0), ac ing a he damage loca ion wi h a equency equal o he j h na u al equency. The esul ing ope a ional de lec ion φI,j(x) is equal o he mode shape di e ence (φD,j(x)−φR,j(x)), and he esul ing ope a ional bending momen a he damage loca ion (mI,j(x0)) is equal o he modal bending momen a ha loca ion in S a e D (mD,j(x0)), as illus a ed in Figu e 1. F om a p ac ical pe spec i e, i mus be no ed ha , since he damage cha ac e iza ion is based on he change o he modal esponse o he beam, bo h undamaged and damaged modal da a mus be collec ed expe imen ally, as i is illus a ed in Sec ion 4 wi h an expe imen al case s udy. 2.2. Wa ele ans o m and damage indices A wa ele unc ion is a ze o-mean unc ion ha oscilla es wi hin a ce ain span. The Con- inuous Wa ele T ans o m (CWT) o a signal is he con olu ion o a wa ele unc ion and he signal. The span o he wa ele unc ion can be shi ed by he ansla ion pa ame e and s e ched o sh inked by he scale pa ame e sin he CWT: 4 Figu e 1: The Inc emen al S a e o he beam. W ( , s) = 1 √sZ+∞ −∞ (x)ψ∗x− sdx(1) whe e (x) is he signal, ψ∗(x) is he complex conjuga e o he wa ele unc ion, and W is he wa ele coe icien o (x) a posi ion o a ce ain scale s. An impo an ea u e o he wa ele unc ion is i s numbe o anishing momen s. This numbe de ines he highes o de o a polynomial o which he wa ele ans o m would p o ide null alues. I a unc ion (x) can be app oxima ed by a polynomial o o de n,pn(x), and he wa ele unc ion has n+ 1 anishing momen s, hen he CWT o he unc ion would be he same as applying he CWT o he esidual ε(x) be ween he o iginal unc ion (x) and he polynomial app oxima e (Eqs. (2) and (3)): (x) = pn(x) + ε(x) (2) W ( , s) = Wε( , s) (3) In his applica ion, he CWT is applied o each mode shape di e ence φI,j ( (x) = φI,j in Eq. (1), (2), and (3)). Di e en wa ele unc ions exhibi di e en cha ac e is ics in de ec ing singula i y in signals. Mo e de ails o wa ele unc ions and hei applica ion o damage de ec ion can be ound in Re e ences [24–26]. Based on a p e ious s udy [1], he Gaussian wa ele wi h 4 anishing momen s (’Gaus4’) is used in his wo k. When applying CWT wi h ini e leng h signals, he wa ele coe icien s ends o be in ini e a he wo ends o he signal. To adjus his issue, he ini e signal is usually padded a bo h ends. Se e al di e en padding me hods can be ound in Re e ences [27, 28]. In his s udy, he asymme ic padding is used. When using a wa ele unc ion which esul s in local ex eme alues a he discon inui y o he signal in he space-scale domain, he line connec ing hese local ex eme alues ac oss he 5 scales is known as he ”wa ele idge”. In he applica ion o damage de ec ion, he damage loca ions can be iden i ied using he ”wa ele idge”. By u ilizing his p ope y, he p oposed DLI is based on he coun o he appea ance o he wa ele coe icien local ex eme alues a each posi ion: DLIi,j = R X k=1 indexi,j,k (4) indexi,j,k =   1 i W j(xi, sk) is local maximum; −1 i W j(xi, sk) is local minimum; 0 e e ywhe e else. (5) whe e i= 1,2,··· , N,j= 1,2,··· , M, and k= 1,2,··· , R, being Nbe he numbe o measu emen poin s, Mbe he numbe o iden i ied modes, and Rbe he numbe o scales used in he wa ele ans o m. The damage a e loca ed a poin s whe e he absolu e alue o DLI is equal o R. Fo he se e i y quan i ica ion, a di e en p ope y o he wa ele ans o m a he iden i ied discon inui y loca ions is used. A each one o hose loca ions (x0), he wa ele coe icien s o each mode jsa is y: |W j(x0, s)|⩽Asα+1/2,whe e q= 1,2,··· , P (6) whe e Ais a cons an and αis he Lipschi z exponen o he signal a x0. Cons an Ais ela ed o he magni ude o he discon inui y and αis ela ed o he discon inui y o de [1, 16, 17, 29]. Di iding bo h sides o Eq. (6) by he modal bending momen a he damage loca ion in S a e I (which equals he bending momen in S a e D a ha loca ion as s a ed be o e) and aking loga i hms one ge s: log2(W 0 j(x0, s)) ⩽A0+ (α+ 1/2) log2s(7) W 0 j(x0, s) =  W j(x0, s) mD,j(x0) and A0= log2A |mD,j(x0)|(8) being W 0 j(x0, s) he no malized wa ele coe icien s a x0and A0is de ined as he Damage Se e i y Index (DSI) o loca ion x0o mode j. I damage is conside ed as a local physical discon inui y (as ep esen ed adi ionally by a lumped sp ing model), he local singula i y in he esponse o he beam is de ined jus by he damage se e i y and he bending momen ha is ac ing a he damaged sec ion. Conside ing K as he s i ness o he o a ional sp ing ha connec s he wo undamaged pa s o he beam, hen mD,j(x0) = K ∆θI,j, being ∆θ he o a ion discon inui y o he beam ( i s o de de i a i e o he displacemen ield). 6 Based on his assump ion, he discon inui y a x0is p opo ional o he modal bending momen mD,j(x0). Thus, he no maliza ion p ocedu e makes he DSI be only dependen on he damage se e i y and independen om he damage loca ion. Fu he de ails on his issue can be ound in [1]. The DSI can be ob ained o all he a ailable measu emen poin s along he beam. Then, he damage se e i y can be es ima ed a each poin by compa ing he DSI alue wi h a Re e ence Map, as explained in he nex sec ion. 2.3. Re e ence Map o he Damage Se e i y Index The damage quan i ica ion p ocedu e equi es he compa ison o he ob ained DSI on he eal beam wi h e e ence alues o known damage se e i y le els. The e e ence alues can be ob ained nume ically om a ini e elemen model o he beam wi h he same c oss-sec ion and ma e ial p ope ies (he eina e e e ed as he Re e ence Model) by ollowing he p ocedu e desc ibed in he p e ious sec ion. The nume ical esponse (s a ic o modal displacemen s and bending momen dis ibu ions) is ob ained o he undamaged and damaged s a es a he same loca ions as he senso s. F om he esul s ob ained om he Re e ence Model, he Re e ence Map can be plo ed by ep esen ing he DSI alues o di e en damage se e i y le els. The bounda y condi ions o he Re e ence Model do no heo e ically equi e o be consis en wi h he eal beam since he e ec o he damage is locally analyzed a he damage loca ion. This is a p ac ical ad an age because o he unce ain y abou he ac ual beha io o he eal bounda y condi ions. Howe e , a educed numbe o measu ing poin s may a ec he independence o he DSI wi h espec o he bounda y condi ions as well as he damage loca ion. In such si ua ions, he accu acy o he iden i ica ion p ocedu e can be educed by his in luence on he alidi y o he Re e ence Map. All hese issues a e discussed and analyzed om a nume ical illus a ion in he nex sec ion. 3. Nume ical illus a ion A nume ical model o a beam wi h wo no ches is exempli ied o demons a e he p ocedu e o ob ain he p oposed DLI, DSI, and Re e ence Maps. The pe o mance o he iden i ica ion me hod wi h di e en densi ies o measu emen s is also illus a ed. A ini e elemen model is used o he simula ion o an expe imen al es wi h a double damage scena io. A simila model is also used o ob aining he Re e ence Map o he damage quan i ica ion s ep. Fi s , he esul s o a dense measu ing g id a e p esen ed o show he independence o he DSI om he damage loca ions, bounda y condi ions, and mode o de s. Then, a spa se measu ing g id is conside ed o simula ing a p ac ical si ua ion, in which he in luence o he numbe o measu ing poin s, and he limi s o he me hod a e discussed. A ini e elemen model o a beam wi h ec angula c oss-sec ion o dimensions 700(L)× 80(W)×20(H) mm (leng h×wid h×heigh ) is buil wi h 10-node e ahed al quad a ic solid elemen s. The beam is made o s eel (7850 kg/m3densi y, 210 GPa Young’s Modulus, and 0.3 7 Poisson a io). Two no ches o 1 mm wid h and uni o m dep h h oughou he c oss-sec ion a e included in he model a 170 mm and 470 mm wi h di e en dep hs (d) and co esponding ela i e se e i y (ξ=d/H), as lis ed in Table 2. The beam is simply-suppo ed and he i s 5 bending modes (mass-no malized) a e ob ained. Table 2: Damage de ini ion pa ame e s. Damage loca ion (mm) dep h (mm) se e i y (%) D1 170 10 50 D2 470 5 25 The mode shapes co esponding o S a e R, S a e D and S a e I a e shown in Figu e 2. When using he mode shapes o damage de ec ion, a damage may be loca ed a a node o ce ain modes, which he modes would no be sensi i e o ha damage. In he p esen ed example, D2 in Mode 3 and D1 in Mode 4 a e close o a node, in which he iden i ica ion o damage will be mo e sensi i e o e o s. In addi ion, i should be no ed ha , in a mul i-damage scena io o di e en se e i y, he discon inui y induced by he lowe se e e damage may no be pe cep ible in φI, which will complica e he damage loca ing when noise is p esen ed, such as D2 in Mode 2 and Mode 5. 0 200 400 600 -0.5 0 0.5 -0.02 -0.01 0 0.01 0.02 (a) 0 200 400 600 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.05 0 0.05 (b) 0 200 400 600 -0.5 0 0.5 -0.05 0 0.05 (c) 0 200 400 600 -0.5 0 0.5 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 (d) 0 200 400 600 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 (e) Figu e 2: Fi e e e ence mode shapes (φR, black solid lines), damaged mode shapes (φD, black dash-do ed lines), and co esponding inc emen al mode shapes (φI, g ay dashed lines) o he example: (a) Mode 1, (b) Mode 2, (c) Mode 3, (d) Mode 4, (e) Mode 5 ( he e ical dashed lines ma k he damage loca ions). Two se s o nume ical measu emen s o di e en densi ies a e conside ed. One consis s o 176 8 measu emen poin s wi h a 4 mm spacing, conside ed a dense g id example and deno ed by Se -4, and he o he one consis s o 35 measu emen poin s wi h a 20 mm spacing, conside ed a spa se g id example and deno ed by Se -20. 3.1. Dense g id (Se -4) In his case, ou scales a e used o he wa ele ans o m hough a highe numbe could be conside ed o ha la ge numbe o measu ing poin s. Figu e 3 shows ha he wa ele coe icien s a ec ed by he wo damage a e well sepa a ed. The wa ele coe icien s a D1 (mo e se e e) a e highe han hose a D2 (less se e e) as shown in all modes excep in Mode 4 (Figu e 3 (d)), in which D1 is close o a node and consequen ly i s e ec is diminished. A simila phenomenon can be obse ed in Mode 3 o D2 (Figu e 3 (c)) due o i s p oximi y o a node. The wa ele coe icien s o he less se e e damage (D2) a e highe han he mo e se e e one (D1). In case ha he damage is no dis inguishable due o he p esence o expe imen al noise, he de ec ion o such a damage would ely on o he modes. 9 0 200 400 600 -4 -2 0 2 4 6 810-3 -0.01 0 0.01 0.02 (a) 0 200 400 600 -0.02 -0.01 0 0.01 0.02 0.03 0.04 -0.05 0 0.05 (b) 0 200 400 600 -0.04 -0.02 0 0.02 0.04 -0.04 -0.02 0 0.02 0.04 (c) 0 200 400 600 -0.02 -0.01 0 0.01 0.02 0.03 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 (d) 0 200 400 600 -0.15 -0.1 -0.05 0 0.05 0.1 -0.15 -0.1 -0.05 0 0.05 0.1 (e) Figu e 7: The inc emen al mode shapes (g ay solid lines) and hei wa ele coe icien s (black lines) o he example (Se -20): (a) Mode 1, (b) Mode 2, (c) Mode 3, (d) Mode 4, (e) Mode 5 (Scale 1: solid lines; Scale 2: dashed lines; Scale 3: do ed lines; he e ical dashed lines ma k he damage loca ions). 16 0 100 200 300 400 500 600 700 -3 -2 -1 0 1 2 3 (a) 0 100 200 300 400 500 600 700 -3 -2 -1 0 1 2 3 (b) 0 100 200 300 400 500 600 700 -3 -2 -1 0 1 2 3 (c) 0 100 200 300 400 500 600 700 -3 -2 -1 0 1 2 3 (d) 0 100 200 300 400 500 600 700 -3 -2 -1 0 1 2 3 (e) Figu e 8: The DLI o each mode o he example (Se -20): (a) Mode 1, (b) Mode 2, (c) Mode 3, (d) Mode 4, (e) Mode 5 (Scale 1: solid lines; Scale 2: dashed lines; Scale 3: dashed-do ed lines; Scale 4: do ed lines; he e ical dashed lines ma k he damage loca ions). Table 7: P edic ed damage loca ions (mm) o he Se -20 example. Damage Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 D1 190 170 170 170 190 D2 470 470 470 490 470 The alues o he DSI o bo h damage a e included in Table 8. By compa ing he alues wi h hose om Table 5, ob ained o Se -4, one no es ha he DSI alues a e a ec ed by he numbe o measu ing poin s. In addi ion, i can be seen ha he consis ency be ween he di e en modes o each damage (illus a ed by he a iance included in Table 5 and 8) is educed when less numbe o measu ing poin s a e used o he iden i ica ion. Table 8: DSI o he damage o he Se -20 example. Damage Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Va iance D1 −26.0−25.9−25.8−25.4−26.3 0.115 D2 −28.4−28.2−28.9−27.7−29.5 0.473 17 By ollowing he same p ocedu e explained in he p e ious sec ion, he Re e ence Maps om modal analysis ( h ee di e en bounda y condi ions) and s a ic analysis (simply-suppo ed) a e es ablished (Figu e 9). I mus be ecalled ha he Re e ence Map mus be ob ained wi h he same numbe o measu ing poin s om he es s, as he alue o he DSI is a ec ed by his numbe . By compa ing Figu e 9 wi h Figu e 6, i can be clea ly seen ha he consis ency o he Re e ence Maps a e less o Se -20 han o Se -4, especially o Mode 4 and Mode 5. As explained be o e, as mo e oscilla ion akes place in he signal, he accu acy o he wa ele ans o m is mo e a ec ed by he educ ion o he measu ing poin s. 0 10 20 30 40 50 -34 -32 -30 -28 -26 -24 (a) 0 10 20 30 40 50 -34 -32 -30 -28 -26 -24 (b) 0 10 20 30 40 50 -34 -32 -30 -28 -26 -24 (c) Figu e 9: The DSI e e ence maps o di e en bounda y condi ions (Se -20): (a) simply-suppo ed, (b) ixed- ixed, (c) ee- ee. The es ima ed no ch dep hs using Se -20 a e p esen ed in Table 9. The es ima es om Mode 1, Mode 2, and Mode 3 a e be e han hose om Mode 4 and Mode 5 because o he mo e signi ican e ec o he educ ion in he spa ial esolu ion o highe o de modes. Simila o Se -4, a cases whe e damage is close o a node, he es ima e becomes less accu a e han o he s. In gene al, he es ima es o Se -20 a e less accu a e han hose o Se -4. Howe e , conside ing he spa si y o he da a, he damage iden i ica ion should be conside ed success ul. Table 9: Es ima ed damage dep hs (mm) o he Se 2 example. DSI Re . Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Map D1 D2 D1 D2 D1 D2 D1 D2 D1 D2 SSB 9.7 5.0 10.2 5.4 10.2 4.2 10.6 6.0 9.5 3.3 FFB 10.0 5.0 10.3 5.3 10.4 4.1 10.0 5.4 8.4 3.0 FREE 9.7 5.0 10.0 5.2 10.0 4.1 9.5 5.3 8 3.0 STC 10.0 5.3 10.5 5.7 10.7 4.5 11.8 6.6 9.6 3.6 3.3. Discussion o heo e ical and p ac ical issues Th ough he p esen ed nume ical example, i has been shown ha he DSI is no dependen on he damage loca ion, bounda y condi ions, mode o de , and analysis ype s a ic o dy- namic), gi en su icien measu emen poin s. The consis ency be ween he di e en cu es o 18 he p esen ed Re e ence Maps o highe numbe o measu ing poin s (Se -4) demons a es he heo e ical independency o he DSI om bounda y condi ions and mode o de . On he o he hand, he accu acy o he p edic ed damage se e i ies ob ained o di e en damage posi ions ( ha do no co espond o he posi ion used o he Re e ence Maps) demons a e he independency o he DSI om he damage loca ion. By analyzing he esul s o cases Se -4 and Se -20, i is illus a ed how he accu acy is a ec ed by he sampling in e al. When he measu emen poin s a e limi ed, he p oposed DSI is mo e a ec ed by he e ec o di e en bounda y condi ions, mode o de and damage loca ion. As a esul , i pe o ms be e o low o de modes as he numbe o measu ing poin s is smalle . The bounda y condi ions used o build he Re e ence Map do no heo e ically equi e o be consis en wi h he eal si ua ion since he e ec o he damage is locally analyzed a he damage loca ion. This is a p ac ical ad an age because o he unce ain y abou he ac ual beha io o he bounda y condi ions in eal applica ions. Howe e , a educed numbe o measu ing poin s may a ec he independence o he DSI wi h espec o he bounda y con- di ions as well as he damage loca ion. In such si ua ions, he accu acy o he iden i ica ion p ocedu e can be educed by his in luence, as i illus a ed by he esul s om he Se -20 con igu a ion. Rega ding he sensi i i y o he p oposed me hodology, i mus be no ed ha , in noise- ee condi ions, any li le damage could be accu a ely iden i ied as long as a big enough numbe o measu ing poin s is conside ed. The p esen ed nume ical illus a ion shows how he accu acy and sensi i i y o he damage iden i ica ion is a ec ed by he numbe o measu ing poin s. Fo his eason, in p ac ical applica ions, an es ima ion o he smalles damage ha can be be de ec ed can be pe o med h ough a speci ic nume ical sensi i i y analysis conside ing he ac ual numbe o measu ing poin s and he expe imen ally a ailable and eliable mode shapes. 4. Expe imen al case s udy 4.1. Modal es s A s eel beam o he same dimensions o ha om he nume ical model (leng h 700 mm, wid h 80 mm, and heigh 20 mm) is used o examine he p oposed me hod. The beam is simply-suppo ed a wo ends (Figu e 10 (a)). Mode shapes we e iden i ied by pe o ming adi ional impac es s. A o al o 35 posi ions we e labeled on he beam as P1 ( he le suppo ), P2, ..., P35 ( he igh suppo ) wi h an equal 20 mm spacing (Figu e 11). A se o 33 magne ic disks we e used o connec he beam and he piezoelec ic accele ome e s. A o al o 17 accele ome e s, all wi h nominal sensibili y o 100 mV/g we e di ided in o wo se ups in o de o maximize he numbe o measu emen s (33 in o al). One senso was ixed a he impac loca ion (P10) se ing as he e e ence senso (Figu e 10 (b)). The es o he accele ome e s we e used as o ing senso s. Se up 1 consis s o e en posi ions and Se up 2 consis s o odd posi ions, om P2 o P34. Two a i icial no ches, as shown in Figu e 10 (c), we e in oduced as damage a loca ions 170 mm and 470 mm. 19 (a) (b) (c) Figu e 10: The con igu a ion o he impac es (a) he bounda y condi ion; (b) senso s a ound he impac posi ion; (c) damage o 5 mm. Figu e 11: The con igu a ion o he posi ions on he es beam (uni : mm). Fo he modal iden i ica ion, he impac exci a ion was in oduced by an impac hamme o 11 mV/N sensi i i y. The a e aged F equency Response Func ions (FRF) we e ob ained om 10 impac s o each se up and he mass-no malized mode shapes we e iden i ied. The iden i ied mode shapes o he Re e ence S a e a e shown in Figu e 12. They o m he e e ence in o ma ion o he subsequen damage iden i ica ion p ocess. Figu e 12 shows ha he esponse a he suppo s does no co espond o an ideal simply suppo ed-beam. In addi ion, i can be seen ha none o he wo damage a e si ua ed a any node loca ion o any o he iden i ied modes. In addi ion, Figu e 12 shows ha di e en modes ha e dis inc le els o expe imen al noise. The p esence o noise in he mode shapes de ini ely con amina es he damage iden i ica ion esul s. As a esul , noisy mode shapes should be dis ega ded and only eliable modes should be kep o he analysis. In his pape , a p ocedu e based on noise e alua ion is p oposed o selec ing he mos sui able mode shapes. In o de o quan i y and compa e he ela i e noise pollu ion o di e en modes, he noise le el is cha ac e ized h ough a no malized s anda d de ia ion index, ηj(Eqs. (9) and (10)): 20 0 100 200 300 400 500 600 700 0.15 0.2 0.25 0.3 0.35 0.4 0.45 (a) 0 100 200 300 400 500 600 700 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 (b) 0 100 200 300 400 500 600 700 -0.5 0 0.5 (c) 0 100 200 300 400 500 600 700 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 (d) 0 100 200 300 400 500 600 700 -0.5 0 0.5 (e) Figu e 12: The i s 5 e e ence mode shapes o he es beam (φR): (a) Mode 1, (b) Mode 2, (c) Mode 3, (d) Mode 4, (e) Mode 5 ( he e ical dashed lines ma k he damage loca ions). ηj=˜σj max(φj)−min(φj), j = 1,2,3,4,5 (9) ˜σj= u u 1 N−1 N X i=1 (φi,j −¯ φi,j)2, i = 1,2,··· , N (10) whe e φi,j is he i h measu emen o he j h expe imen al mode shape φj,¯ φi,j is he spline i o he mode shape, se ed he e as noise- ee e e ence o noise quan i ica ion, and Nis he numbe o measu emen s. The no malized s anda d de ia ion indices o each mode a e included in Table 10. They show ha Mode 2 and Mode 3 a e signi ican ly less a ec ed by noise han he es o he iden i ied modes. The e o e, in o de o educe he e ec o expe imen al noise in he inal esul s, only Mode 2 and Mode 3 a e conside ed o he applica ion o he p oposed damage de ec ion me hodology. This pape p oposes he applica ion o he p esen ed mode selec ion c i e ion in p ac ical applica ions. Howe e , he de ini ion o a h eshold alue o he no malized s anda d de i- a ion index and subsequen ly he numbe o modes o be selec ed will depend on he quali y o he expe imen al da a. A speci ic analysis is necessa y o each applica ion. 21 Table 10: The noise le el o he iden i ied modes. Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 η0.011 0.0012 0.0032 0.0092 0.0145 4.2. Damage iden i ica ion Two damage scena ios o double damage a e in es iga ed. In Scena io 1 (S1), he wo no ches a e bo h o 5 mm (25% se e i y), and, in Scena io 2 (S2), he wo no ches a e bo h o 10 mm (50% se e i y) a he same loca ions. Figu e 13 shows he damaged mode shapes o Mode 2 and Mode 3 o he wo damage scena ios. I can be seen ha bo h mode shapes a e well iden i ied and he noise le el is low. The damaged mode shapes a e simila o he undamaged ones, which means ha he in luence o damage is no pe cep ible om isual inspec ion. 0 100 200 300 400 500 600 700 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 (a) 0 100 200 300 400 500 600 700 -0.5 0 0.5 (b) 0 100 200 300 400 500 600 700 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 (c) 0 100 200 300 400 500 600 700 -0.5 0 0.5 (d) Figu e 13: The damaged mode shapes o he es beam (φD): (a) S1 Mode 2; (b) S1 Mode 3; (c) S2 Mode 2; (d) S2 Mode 3 ( he e ical dashed lines ma k he damage loca ions). By aking he di e ence o he damaged and undamaged mode shapes, he expe imen al noise e ec becomes disce nible (Figu e 14). The e o e, he mode shapes, φRand φD, a e i ed by splines o diminish he noise e ec . The smoo hed inc emen al mode shapes, ¯ φI, a e ob ained by sub ac ing ¯ φR om ¯ φD, wi h ¯ φRand ¯ φDbeing he i ed undamaged and damaged mode shapes, espec i ely. Figu e 14 shows ha he cu e- i ing p ocess educes signi ican ly he noise pollu ion and p ese es he damage induced discon inui ies. 0 100 200 300 400 500 600 700 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 (a) 0 100 200 300 400 500 600 700 -0.02 0 0.02 0.04 (b) 0 100 200 300 400 500 600 700 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 (c) 0 100 200 300 400 500 600 700 -0.05 0 0.05 (d) Figu e 14: The inc emen al damaged mode shapes o he es beam (φI) and hei smoo hing spline i s (¯ φI): (a) S1 Mode 2; (b) S1 Mode 3; (c) S2 Mode 2; (d) S2 Mode 3 ( he e ical dashed lines ma k he damage loca ions). 22 The wa ele analysis is applied o he i ed inc emen al mode shapes ¯ φI. Due o he limi ed numbe o senso s, only 3 scales a e used. The wa ele coe icien s a e shown in Figu e 15 and he co esponding DLI ob ained by applying Eqs. (4) and (5) a e shown in Figu e 16. The po en ial damage loca ions shown in Table 11 a e iden i ied using he c i e ia es ablished in he nume ical example. Bo h D1 and D2 a e iden i ied success ully in all cases. Fo S1 (25% damage), se e al alse posi i e damage loca ions a e p edic ed, while o S2 (50% damage), no alse posi i e loca ions a e iden i ied. Since he alse posi i es can be due o speci ic easons o each mode, and, he e o e, migh no be consis en among hem, i is p oposed o use he bene i o ha ing se e al iden i ied modes by compu ing a Combined DLI (CDLI) o all he modes, so ha he numbe o alse posi i es can be educed. This combina ion can be ca ied ou by adding up he absolu e alues o DLI o each mode ja each loca ion i(Eq. (11)): 100 200 300 400 500 600 -0.01 -0.005 0 0.005 0.01 (a) 100 200 300 400 500 600 -0.02 -0.01 0 0.01 0.02 (b) 100 200 300 400 500 600 -0.04 -0.02 0 0.02 0.04 (c) 100 200 300 400 500 600 -0.05 0 0.05 (d) Figu e 15: The wa ele coe icien s o he i ed inc emen al mode shapes (Scale 1: solid line; Scale 2: dashed line; Scale 3: do ed line; Scale 4: dash-do ed line): (a) S1 Mode 2; (b) S1 Mode 3; (c) S2 Mode 2; (d) S2 Mode 3 ( he e ical dashed lines ma k he damage loca ions). 100 200 300 400 500 600 -3 -2 -1 0 1 2 3 (a) 100 200 300 400 500 600 -3 -2 -1 0 1 2 3 (b) 100 200 300 400 500 600 -3 -2 -1 0 1 2 3 (c) 100 200 300 400 500 600 -3 -2 -1 0 1 2 3 (d) Figu e 16: The Damage Loca ing Index (DLI) o (a) S1 Mode 2; (b) S1 Mode 3; (c) S2 Mode 2; (d) S2 Mode 3 ( he e ical dashed lines ma k he damage loca ions). CDLIi=X j|DLIi,j|, j = 2,3 (11) As shown in Figu e 16, he absolu e alue o he DLI a he damage loca ion is ei he 3 ( he maximum numbe o scales) wi h no DLI o he same sign a i s adjacen loca ions, o 2 23 Table 11: Po en ial damage loca ions o he expe imen al case s udy. Damage Scena io Mode Damage loca ions (mm) 12 70,170,410,490 3 170,470,550 22 170,490 3 170,470 wi h one o he wo adjacen poin s ha ing a DLI o alue 1 o he same sign. The e o e, he po en ial damage is de e mined a loca ions wi h a CDLIiequaled o g ea e han 5. Figu e 17 shows he ob ained CDLI. i mus be ecalled ha wa ele ex eme alues may appea a adjacen loca ions o di e en scales and modes, as i happens o D2 in bo h scena ios, S1 and S2. Hence, loca ions nex o each o he wi h bo h CDLI equaling 3 should also be conside ed damage loca ion p edic ions. As a esul , he p edic ed damage loca ions a e 170 mm, 470 mm, and 550 mm o S1, and a e 170 mm and 470 mm o S2. 100 200 300 400 500 600 0 1 2 3 4 5 6 (a) 100 200 300 400 500 600 0 1 2 3 4 5 6 (b) Figu e 17: The Combined Damage Loca ing Index (CDLI) o (a) S1; (b) S2 ( he e ical dashed lines ma k he damage loca ions; he ho izon al dashed line ma ks he h eshold o po en ial damage loca ions). Once he damage loca ions a e iden i ied, he modal damaged bending momen has o be iden i ied a hose loca ions om he es ima ion o he damaged modal cu a u e, as p e- sen ed in he nume ical illus a ion. In o de o educe he ins abili y in he cu a u e es ima es, a polynomial i is applied. The o de o he polynomial can be selec ed by check- ing he accu acy o he app oach. When using eal expe imen al noisy mode shapes ins ead o nume ical ones, he polynomial i is p e e ed a he han he spline i . The eason is ha he polynomial ends o app oach smoo hly he global end whe eas he spline i keep pa ially he noisy i egula i y, leading o highe ins abili y in he cu a u e es ima ion. Mo eo e , he polynomial i allows he analy ical e alua ion o he cu a u es and no nu- me ical di e en ia ion is equi ed. Figu e 18 shows he polynomial i s o he expe imen al damaged mode shapes. F om he es ima es o he modal bending momen a he p edic ed damage loca ions, he no malized wa ele coe icien s and he DSI can be ob ained. Since he bounda y condi ions 24 100 200 300 400 500 600 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 (a) 100 200 300 400 500 600 -0.5 0 0.5 (b) 100 200 300 400 500 600 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 (c) 100 200 300 400 500 600 -0.5 0 0.5 (d) Figu e 18: The damaged mode shapes o he es beam (φD) and hei polynomial i s (ˆ φD): (a) S1 Mode 2; (b) S1 Mode 3; (c) S2 Mode 2; (d) S2 Mode 3 ( he e ical dashed lines ma k he damage loca ions). o he ac ual s uc u e is none o he h ee pe ec ypes applied in he nume ical models, and he spa si y o he measu emen s makes ha he DSI migh be a ec ed by he de ini ion o he bounda y condi ions, i is ecommended o use DSI Re e ence Maps o di e en bounda y ypes o he se e i y assessmen . The es ima ed no ch dep hs using he DSI Re e ence Maps om Figu e 9 a e shown in Figu e 19. In S1, bo h o he damage a e unde es ima ed om Mode 2, and D1 is o e es ima ed while D2 is unde es ima ed om Mode 3. The absolu e e o o he es ima e o D1 is be ween −1.5 mm and 1.5 mm, and i is be ween −1.2 mm and −0.8 mm o D2. In S2, bo h Mode 2 and Mode 3 unde es ima e bo h damage. The p edic ion e o o D1 is om −2 mm o −0.5 mm, and o D2 is om −3.0 mm o −0.5 mm. 0 100 200 300 400 500 600 700 0 1 2 3 4 5 6 (a) 0 100 200 300 400 500 600 700 0 1 2 3 4 5 6 (b) 0 100 200 300 400 500 600 700 0 2 4 6 8 10 (c) 0 100 200 300 400 500 600 700 0 2 4 6 8 10 (d) Figu e 19: The es ima ed damage dep h om Re e ence Maps o simply-suppo ed (SSB), ixed- ixed (FFB), ee- ee (FREE) modal analysis and s a ic (STC) analysis: (a) S1 Mode 2; (b) S1 Mode 3; (c) S2 Mode 2; (d) S2 Mode 3. The expe imen al esul s show ha he p oposed me hod can success ully iden i ied he loca ions o a double-damage si ua ion, and es ima e he se e i y o a ce ain accu acy le el. Howe e , wi h he condi ions o he es , e.g. he numbe o measu emen s and he use o wo senso se ups, he expe imen al case s udy se es as an illus a ion o he p ac ical limi s o he applica ion o he p oposed me hodology. 25