Structural health monitoring using ultrasonic guided-waves and the degree of health index

senso s A icle S uc u al Heal h Moni o ing Using Ul asonic Guided-Wa es and he Deg ee o Heal h Index Se gio Can e o-Chinchilla 1, Ge a do A angu en 2,* , José Manuel Royo 3, Manuel Chiachío 4,5 , Josu E xaniz 2and And ea Cal o-Echenique 3   Ci a ion: Can e o-Chinchilla, S.; A angu en, G.; Royo, J.M.; Chiachío, M.; E xaniz, J.; Cal o-Echenique, A. S uc u al Heal h Moni o ing Using Ul asonic Guided-Wa es and he Deg ee o Heal h Index. Senso s 2021, 21, 993. h ps://doi.o g/10.3390/ s21030993 Academic Edi o : Theodo e E. Ma ikas Recei ed: 1 Janua y 2021 Accep ed: 27 Janua y 2021 Published: 2 Feb ua y 2021 Publishe ’s No e: MDPI s ays neu- al wi h ega d o ju isdic ional clai- ms in published maps and ins i u io- nal a ilia ions. Copy igh : © 2021 by he au ho s. Li- censee MDPI, Basel, Swi ze land. This a icle is an open access a icle dis ibu ed unde he e ms and con- di ions o he C ea i e Commons A - ibu ion (CC BY) license (h ps:// c ea i ecommons.o g/licenses/by/ 4.0/). 1Depa men o Mechanical Enginee ing, Uni e si y o B is ol, B is ol BS8 1TR, UK; se gio.can e [email p o ec ed] 2Elec onic Design G oup, Uni e si y o he Basque Coun y (UPV/EHU), 48013 Bilbao, Spain; [email p o ec ed] 3Techonological Ins i u e o A agón (ITAINNOVA), Ma ía de Luna 8, 50018 Za agoza, Spain; [email p o ec ed] (J.M.R.); [email p o ec ed] (A.C.-E.) 4Depa men o S uc u al Mechanics & Hyd aulics Enginee ing, Uni e si y o G anada, 18001 G anada, Spain; mchiachio@ug .es 5Andalusian Resea ch Ins i u e o Da a Science & Compu a ional In elligence, Uni e si y o G anada, 18001 G anada, Spain *Co espondence: ge a do.a angu [email p o ec ed] Abs ac : This pape p oposes a new damage index named deg ee o heal h (DoH) o e icien ly ackle s uc u al damage moni o ing in eal- ime. As a key con ibu ion, he p oposed index elies on a pa e n ma ching me hodology ha measu es he ime-o - ligh misma ch o sequen ial ul asonic guided-wa e measu emen s using uzzy logic undamen als. The ul asonic signals a e gene a ed using he ansmission beam o ming echnique wi h a phased-a ay o piezoelec ic ansduce s. The acquisi ion is ca ied ou by wo phased-a ays o compa e he in luence o pulse-echo and pi ch-ca ch modes in he damage assessmen . The p oposed moni o ing app oach is illus a ed in a a igue es o an aluminum shee wi h an ini ial no ch. As an addi ional no el y, he p oposed pa e n ma ching me hodology uses he da a s emming om he ansmission beam o ming echnique o s uc u al heal h moni o ing. The esul s demons a e he e iciency and obus ness o he p oposed amewo k in p o iding a quali a i e and quan i a i e assessmen o a igue c ack damage. Keywo ds: s uc u al heal h moni o ing; ul asonic guided-wa es; a igue damage de ec ion; ans- mission beam o ming; deg ee o heal h index 1. In oduc ion C ack ini ia ion and p opaga ion a e he main d i ing o ces o damage in many enginee ing ma e ials [ 1 ]. In he absence o co ec i e ac ions, o example, dec easing he load le els o pe o ming co ec i e main enance ac ions, a c ack may p opaga e up o he ca as ophic ailu e o a s uc u al componen . A wide ange o non-des uc i e es ing (NDT) echniques ha e been p oposed o suppo ope a ion and main enance decision making [ 2 – 4 ]. Visual inspec ion, adiog aphy, o ul asound [ 5 , 6 ] a e some examples o hese NDT echniques; hese a e ega ded as highly eliable bu also ime consuming ( equi ing he in e up ion o se ice) and depen- den on he s uc u e [ 7 ]. Al e na i ely, s uc u al heal h moni o ing (SHM) echniques enable a con inuous on-boa d moni o ing o he s uc u al heal h acili a ing he condi ion- based main enance o he s uc u e [ 8 ]. Some SHM echniques a e especially sui ed o s uc u es made o conduc i e ma e ials such as me als (e.g., eddy cu en s [ 9 ]). O he s, howe e , allow a mo e gene ic applica ion o s uc u es made o any kind o iso opic o aniso opic ma e ials. Fo ins ance, s ain gauges and ibe B agg g a ings [ 10 ] a e ypically used o moni o s ain de o ma ion in s uc u es. Howe e , hei a ea o co e age is ela i ely low, hus po en ially missing small de ec s unless a dense ne wo k o senso s Senso s 2021,21, 993. h ps://doi.o g/10.3390/s21030993 h ps://www.mdpi.com/jou nal/senso s Senso s 2021,21, 993 2 o 17 is used. The acous ic emission echnique has shown e iciency in a igue damage de ec ion and localiza ion [ 11 ], al hough i is known as a passi e me hod ha equi es cap u ing he ins an o he damage g ow h in o de o measu e damage- ela ed in o ma ion. Al e - na i ely, ul asonic guided-wa es [ 12 , 13 ] o e come mos o he e e ed d awbacks and can be e icien ly used as a non-in usi e and on-boa d SHM echnique [ 14 ], due o hei abili y o ac i ely explo e la ge a eas wi h a ela i ely small a enua ion [ 15 ]. These bene- icial p ope ies o ul asonic guided-wa es ha e a ac ed he a en ion o sa e y-c i ical indus ies such he ae ospace o e he las ew decades [ 16 , 17 ]. The a o emen ioned SHM echnique uses piezoelec ic wa e ac i e senso s (PWAS) [ 18 ] pe manen ly a ached o he s uc u e o gene a e and acqui e ul asonic guided-wa es [ 19 , 20 ], which ha e o be u he p ocessed o in e damage- ela ed in o ma ion. Se e al signal p ocessing and damage in e ence s a egies can be applied o ex ac damage in o ma ion ou o he ul asonic da a. A numbe o hem a e based on some o ms o in e se p oblems ha in e ele an damage pa ame e s om he compa ison be ween expe imen al and compu a ionally simula ed ul asonic da a [ 21 ]. Fo example, he localiza ion o damage using ul asonic guided-wa es has been add essed by se e al au- ho s using unce ain y quan i ica ion amewo ks and ime-o - ligh (ToF) models [ 22 , 23 ]. The quan i ica ion o damage has also been add essed using ul asonic measu emen s in he con ex o a p obabilis ic Bayesian in e se p oblem in [ 24 , 25 ]. An al e na i e app oach o ul asound-based damage in e ence is he ime- e e sal me hod [ 26 ], which makes i possible o ocus ul asonic guided-wa es owa ds a de ec in he s uc u e wi hou he need o a baseline. This echnique o e comes he limi a ions s emming om bo h he wa e dispe sion and he unknown ma e ial de o ma ions by ocusing he signal [ 26 ]. Mos o hese me hods a e baseline ee, hence needing no compa ison be ween undamaged and non-p is ine s a es. Howe e , he la e is a he cos o employing a signi ican amoun o physically-g ounded model e alua ions, which bounds i s applicabili y in eal-li e eal- ime enginee ing scena ios no only o i s compu a ional cos bu also o i s model and implemen a ion complexi y. Al e na i ely, he use o baseline-based me hods o damage de ec ion migh ende he equi ed e iciency o complex s uc u es. These a e ypically model- ee, and a s aigh o wa d compa ison be ween signal ea u es ob ained in he p is ine s a e and he subsequen damage s a es is he only in o ma ion equi ed. In his sense, a numbe o me hods ha e been p oposed in he li e a u e. One such example is he econs uc ion algo i hm o p obabilis ic inspec ion o de ec s (RAPID) [ 27 ] and u he a ia ions o he same [28]. This me hod is able o localize de ec s by using signal di e ence coe icien s o senso pai s. Also, in [ 27 ], he au ho s p oposed he use o signal co ela ion coe icien s o de ec and moni o damage e olu ion in an ae ospace panel. O he examples o he econs uc ion o damage using a baseline om an undamaged s a e a e he embedded- ul asonics s uc u al ada [ 29 ] and he delay-and-sum imaging algo i hm [ 30 ]. These echniques use he di e ence be ween signals (acqui ed using a ecei e beam o ming mode) in heal hy and damaged condi ions o ob ain only he signa u es s emming om he de ec [ 31 – 33 ]. Mo e ecen ly, an unsupe ised ea u e-ex ac ion me hod o online damage de ec ion was p oposed in [ 34 ], whe eby a subse o he ul asonic signals ca ying he majo i y o he ene gy con en was used. The baseline app oach was also used in [35], whe e he e ec i eness o he shea -ho izon al guided-wa e mode in moni o ing damage was in es iga ed. Besides, he change in he ToF o he signal peaks due o he p esence o debonding in a composi e pla e was demons a ed in [ 36 ]. This g oup o app oaches enables he in e ence o a lowe deg ee o damage- ela ed in o ma ion compa ed o he a o emen ioned in e se p oblems, and hey app oach eal-wo ld enginee ing scena ios wi h a highe e iciency and model simplici y. Ne e heless, hey s ill equi e a signi ican compu a ional e o , which poses an impo an limi a ion o on-boa d SHM applica ions. The e o e, he e is s ill a need o p o ide compu a ionally e icien pos -p ocessing me hods o ul asonic guided-wa es o eal- ime damage assessmen . Senso s 2021,21, 993 3 o 17 To o e come his limi a ion, his pape p oposes he use o a no el pa e n ma ching pos -p ocessing me hod o he de ec ion and moni o ing o a igue damage in iso opic ma e ials based on signal ea u es s emming om each cha ac e is ic poin (CP) o he signal. These poin s co espond o he peak ampli udes (maximum and minimum) o he acqui ed signal. The ToF o he peaks a e chosen as he base o compa ison o de ec and moni o damage in a s uc u e [ 37 ]. Mo e speci ically, a se o hese poin s acqui ed in he undamaged s a e a e u he used o build a apezoidal unc ion simila ly o a membe ship unc ion o uzzy se [ 38 , 39 ]. Addi ional measu emen s in non-p is ine s a es a e e alua ed in he p oposed unc ion, whe eby a deg ee o heal h (DoH) o he s uc u e is p o ided as ou pu . The easibili y and e iciency o he p oposed algo i hm (based on [ 40 ]) a e demons a ed in a eal a igue es o an aluminum pla e. The ul asonic ansmission beam o ming echnique wi h a linea phased-a ay o six PWAS is adop ed o moni o (i.e., exci e and ecei e ul asonic guided-wa es) he s uc u e wi h an enhanced powe e e y 1000 loading cycles a he same load le el. Ano he symme ic phased-a ay (which only ecei es ul asonic signals) is used o suppo he esul s ob ained om he o me one. The expe imen al esul s show ha he p oposed me hodology is able o de ec a igue damage a an ea ly s age (i.e., he c ack onse ) and o moni o he c ack g ow h in an e ec i e and e icien manne . Fu he mo e, he ease o implemen a ion o his me hodology makes i po en ially applicable o indus ial en i onmen s in o de o pe o m eal- ime SHM. The emainde o he pape is o ganized as ollows—Sec ion 2desc ibes he p oposed pos -p ocessing me hod based on he compa ison o he ToF o he signal peaks; Sec ion 3 shows expe imen al se up as well as he esul s ob ained om his me hod in he a igue expe imen ; Sec ion 4discusses he po en ial impac o he p oposed me hodology on ul asonic guided-wa e based SHM; inally, Sec ion 5p o ides concluding ema ks and sugges s u u e wo ks. 2. Me hodology The p oposed de ec ion and moni o ing me hodology based on a no el damage in- dex is p esen ed in his sec ion. The index, e e ed o he e as DoH, elies on ul asonic guided-wa e da a aken om he SHM o s uc u al panels. Mo e speci ically, he DoH uses baseline ul asonic da a acqui ed when he s uc u e is in p is ine s a e as a basis o compa ison o u he measu emen s when he s uc u e is po en ially damaged. To en- hance he compu a ional e iciency o his echnique, only he ToF o he signal peaks abo e a use -speci ied ampli ude h eshold (named as A ) a e used as ep esen a i e in o ma ion o he aw ul asonic da a, as shown in Figu e 1a. The peaks a e ob ained applying a sliding window o he ul asonic signal. No e ha he wid h o his window is chosen o be p opo ional o he pe iod o he signal, ha is, he in e se o he cen al equency. The maximum and minimum poin s wi hin hese mo ing windows a e selec ed, hence ob aining he signal peaks. Fu he mo e, o pa ially add ess he i educible unce ain y o hese measu emen s and add obus ness o he damage index, an a bi a y amoun o epea ed signals a e acqui ed leading o a se o peaks wi h small di e ences in bo h ime and ampli ude (see Figu e 1b). These a ia ions a e assumed o be an indica o o he measu emen (o alea o y) unce ain y [ 41 ], which suppo s he p e ious obus ness claim o he p oposed damage index. Simila ly o a uzzy se , a apezoidal unc ion is c ea ed a ound he se o ToF poin s acqui ed in p is ine s a e (see Figu e 1c). This unc ion makes i possible o assess he s uc u al heal h by analyzing he ToF misma ch be ween he CPs (see Figu e 1b o e e ence) acqui ed in bo h he deg aded s a e and he p is ine s a e. No e ha as his unc ion gi es deg ee o membe ship alues wi hin he in e al [ 0, 1 ] , he p oposed damage index will ca y quali a i e in o ma ion abou he s uc u al deg ada ion. Senso s 2021,21, 993 4 o 17 2A Time Ampli ude (a) Selec ion o maximum and minimum peaks Time Ampli ude ( b ) Dispe sion o consecu- i e measu emen s Aj iBj iCj iDj i µ=1 µ=0 bj icj i `j i={Cj i,Dj i} Lj i={Bj i,Cj i} `j i={Aj i,Bj i} Time Deg ee o membe ship µ Time Ampli ude (c) T apezoidal unc ion Figu e 1. Panel ( a ): Selec ion o cha ac e is ic poin s (CPs) abo e he h eshold alue A . Panel ( b ): Illus a ion o ToF dispe sion due o epea ed measu emen s. Panel ( c ): T apezoidal unc ion used o e alua e he ToF misma ch based on he epea ed measu emen s o one CP (blue ci cles). Ma hema ically, he e alua ion wi hin he p e iously es ablished apezoidal unc ion (o se ) o CP j i , namely he i - h CP o he j - h signal, i= 1, . . . , N , j= 1, . . . , m , will p o ide he deg ee o membe ship µj i∈[ 0, 1 ] o such a poin o he se , which is de ined as ollows: µj i=                      0 ToFj i≤Aj i ToFj i−Aj i `j i Aj i<ToFj i<Bj i 1Bj i≤ToFj i≤Cj i Dj i−ToFj i `j i Cj i<ToFj i<Dj i 0Dj i≤ToFj i, (1) whe e ToF j i is he ToF o CP j i and `j i={Aj i , Bj i}={Cj i , Dj i} is he ToF in e al be ween Aj i and Bj i , which esul s o be he same as he in e al be ween Cj i and Dj i . The p oposed membe ship unc ion as well as hei ela ed pa ame e s a e illus a ed in Figu e 1c. No e ha he in e al Lj i={Bj i , Cj i} is la ge han he one ob ained by measu ing he dispe sion o he CPs (i.e., {bj i , cj i} ) as shown in Figu e 1c. This is o accoun o u he measu emen unce ain y ha is no cap u ed by he ini ial epea ed measu emen s. No e also he in e als `j iand Lj ineed o be sui ably de ined by he modele in he p oposed app oach. Senso s 2021,21, 993 5 o 17 Thus, when assessing damage h ough an acqui ed guided-wa e, his me hod e alu- a es i CP j i alls wi hin Lj i , hen µj i= 1 meaning ha he s uc u e is unal e ed. Al e na i ely, i he s uc u e su e s a pe manen damage, he ul asonic guided-wa es will be a ec ed by his damage h ough sligh ToF misma ches [ 42 ]. In his case, he CP j i is likely o all wi hin he in e al `j i due o an ad ance o delay o he acqui ed signal, which in u n leads o µj i∈[ 0, 1 ] . Las ly, i CP j i alls ou o he g ea e in e al {Aj i , Dj i} , hen µj i= 0 and i is assumed ha he s uc u e may ha e su e ed a se e e damage o signi ican modi ica ion. The assessmen o a deg ee o membe ship µj i∈[ 0, 1 ) may be an indica ion o s uc- u al damage, bu i can also occu due o en i onmen al o nume ical noise. To o e come his issue, e e y CP j i is assessed and hei deg ees o membe ship µj i a e ob ained, he global deg ee o membe ship o he j - h ul asonic signal (i.e., M j , wi h M being he capi al le e o µ) is ob ained as ollows: Mj=1−g{µj i}N i=1=1−min{µj i}N i=1,j=1, . . . , m. (2) No e ha he unc ion g(·) , conse a i ely chosen in his pape as he minimum o he deg ees o membe ship, migh be adop ed di e en ly such as he weigh ing and segmen a ion o he ToF o he ampli ude [ 43 ]. No e also ha in Equa ion (2) he unc ion g(µj i) has been sub ac ed om he uni y, consequen ly, M j can be iewed as a damage index which inc eases as he de ec becomes mo e se e e, and ice e sa. Mo eo e , when a se o ul asonic signals a e a ailable and pos -p ocessed by his me hod, an a ay (o ma ix) o M j alues a he han a single alue can be ob ained. The use o addi ional M j alues ul ima ely leads o a mo e eliable moni o ing since he in luence o senso mal unc ioning in he moni o ing decision g ea ly dec eases. In his case, he esul ing a ay o ma ix is e e ed he e o as he DoH ma ix o a s uc u e, wi h alues om 0 (maximum heal h) o 1 (maximum deg ada ion). A schema ic wo k low o he p oposed me hodology is shown in Figu e 2. Ul asonic da a PWAS senso s PWAS senso s Signal Peaks Figu e 1a T apezoidal unc ion Figu e 1b S uc u e in p is ine s a e Signal Peaks E al μij - Eq. (1) E al Mj - Eq. (2) S uc u e in ope a ion Damage index (DoH) SHM Decision Measu emen s Figu e 2. Schema ic wo k low o he me hodology di ided be ween he da a acquisi ion o ul asonic da a and i s pos -p ocessing depending on he ac ual s uc u al s a e (i.e., p is ine o in ope a ion). 3. Case S udy The p oposed DoH damage index (Sec ion 2) o damage de ec ion and moni o ing is illus a ed in his sec ion. To his end, a a igue es on an aluminum pla e wi h a cen al no ch has been ca ied ou and i s s uc u al heal h is moni o ed by means o he DoH Senso s 2021,21, 993 6 o 17 using ul asonic guided-wa es exci ed and ecei ed by PWAS and a SHM ul asound sys em (SHMUS). 3.1. Fa igue Tes ing Con igu a ion The a igue es has been pe o med using a middle ension [M(T)] specimen wi h a cen e ed c ack and loaded in ension using a posi i e loading a io. The es specimen o dimensions 245 mm × 500 mm × 1 mm has been ex ac ed om a 1003x503 QQA250/5 ‘O’ 2024 ae onau ic g ade aluminum shee using a wa e je cu ing p ocedu e. The no ch has been cen e ed wi h espec o he es sample cen e line and machined using a lase cu ing p ocedu e, wi h he inal dimensions speci ied in Figu e 3a. The o al leng h o he machined no ch is 22.5 mm app oxima ely. (a) Aluminum specimen and no ch geome y ( b ) Ac ual a igue es ing se up Figu e 3. Panel ( a ): Schema ic o he specimen and no ch geome y, along wi h he posi ion o he piezoelec ic wa e ac i e senso s (PWAS) a ays. Panel ( b ): Pic u e o M(T) aluminum specimen wi h wo pe manen ly a ached phased-a ays moun ed on he a igue es ing machine. An Ins on 8850 se o-hyd aulic a igue es ing machine has been used o conduc he a igue expe imen . A a igue p e-c acking p ocedu e has been applied o de elop a esh and s aigh c ack on o mi iga e he e ec o he machined s a ed no ch [ 44 ]. The c ack onse and g ow h is eco ded h ough a digi al s ill came a. A ension-con olled a igue es has been pe o med wi h a s ess a io ∆R= 0.1 and maximum load ampli ude Pmax = 10 kN, which co esponds o he 60% o he aluminum yield s eng h. The es was pe o med up o 100,000 cycles wi h a cycling equency o 20 Hz. The eade is e e ed o Figu e 3 o u he in o ma ion abou he expe imen al se -up. No e ha he uppe and lowe 50 mm bands o he specimen a e used o clamp he pla e in o he a igue es ing machine. 3.2. Ul asonic Guided-Wa e Based Tes s The de ec ion and moni o ing o he c ack onse and g ow h has been ca ied ou using wo linea phased-a ays o six PWAS, consis ing o six e enly spaced PWAS ha a e linea ly placed in he s uc u e [ 19 ]. Piezoelec ic ce amic disc ansduce s o 7 mm diame e and 0.5 mm hickness ha e been used in his a igue es . These a e used as ansmi e - ecei e (T i ) and ecei e (S i ) a ays o he gene a ion and ecep ion o he ul asonic guided-wa es, espec i ely. No e ha he ansmi e - ecei e a ay (T i ) wo ks in pulse-echo mode by simul aneously gene a ing and acqui ing ul asonic signals, while Senso s 2021,21, 993 7 o 17 he ecei e a ay (S i ) unc ions as a senso only, also known as pi ch-ca ch mode. I is impo an o no e ha he PWAS T 5 mal unc ioned du ing he a igue es and he e o e i s associa ed da a ha e no been conside ed o SHM pu poses. These PWAS ha e a adial mode o ib a ion and a esonan equency cen e ed a 300 kHz. They ha e been e enly spaced wi h a sepa a ion o 10 mm and symme ically placed wi h ega ds o he sample cen e line, as obse ed in Figu e 3, and bonded o he aluminum su ace using 3M ™ Sco ch-Weld ™ DP490 epoxy adhesi e. The piezoelec ic ansduce s a e managed by a SHMUS, which is a cus om-buil compac elec onic de ice o ul asonic guided-wa e based SHM. I includes 12 a bi a y wa e o m gene a o s and 12 acquisi ion sys ems ( e e o [ 45 ] o u he de ails o he SHMUS). The SHMUS is con olled by USB using a ailo -made con ol and p ocessing so wa e ins alled in a lap op. The schema ic o he expe imen al se -up o damage moni o ing is shown in Figu e 4. The exci a ion signals a e sinusoids o 4 cycles, wi h 300 kHz equency (in acco dance o he esonan equency o he chosen PWAS) and 45 ol s peak o peak ampli ude. The ul asonic signals a e acqui ed using a sampling equency o 60 MHz and 12 bi s o esolu ion. SHMUS 12 Signal gene a o s 12 Signal acqui e s Lap op Con ol & p ocessing so wa e Con igu a ion pa ame e s Ul asonic da a Fa igue es ing machine PWAS Aluminum specimen Figu e 4. Schema ic o he ul asonic guided-wa e based es s. Fu he mo e, he ansmission beam o ming echnique [ 46 ] has been adop ed o he moni o ing o he specimen du ing he a igue es . Using his echnique, synch onously delayed signals we e applied o he T i PWAS ( e e o Figu e 3a), so ha he wa e on s can be cons uc i ely summed a a p e-es ablished di ec ion c ea ing a main wa e beam [ 47 ]. The ul asonic es s ha e been ca ied ou by s ee ing he main beam a di e en di ec ions, ha is, om ξ1= 0 ◦ o ξ37 = 180 ◦ wi h an inc emen o ∆ξ= 5 ◦ , which sweeps he en i e moni o ing a ea. No e ha he ul asonic signals a e acqui ed by bo h he Tiand Si phased-a ays o PWAS, hence ecei ing wo se s o 37 × 6 signals (i.e., 37 angles and 6 PWAS o each phased-a ay). P og ammed inspec ions du ing he a igue expe imen ha e been ca ied ou e - e y 1000 cycles. Fo each indi idual inspec ion, he a igue es has been s opped a he minimum load le el and an ul asonic inspec ion has been pe o med using he ansmis- sion beam o ming echnique. The ecei ed ul asonic signals (a ailable in [ 48 ]) by he SHMUS ha e been de-noised using a bandpass il e cen e ed a he equency o exci a ion (i.e., 300 kHz). This allows a ange o equencies a ound he equency o exci a ion o pass, while a enua ing equencies ou o he scope. In his case he passband is de ined in he in e al [250 kHz, 350 kHz], while he s opbands, ha is, a enua ed equency anges, a e de ined wi hin [0 kHz, 200 kHz] and [400 kHz, ∞ ) wi h a minimum a enua ion o − 60 dB. Figu e 5shows he ep esen a ion o his bandpass il e in bo h ime and equency domains. Senso s 2021,21, 993 8 o 17 0 10 20 30 −4 −2 0 2 4 Time [µs] Ampli ude [×10−3] (a) Fil e in ime domain 0 200 400 600 −100 −50 0 S opband S opband Passband F equency [kHz] Magni ude [dB] (b) Fil e in equency domain Figu e 5. Time (panel (a)) and equency (panel (b)) domain ep esen a ions o he bandpass il e . 3.3. Damage Moni o ing Resul s The ul asonic da a acqui ed in p is ine s a e a he minimum load le el (i.e., 1.0 kN) ha e been used o build he apezoidal unc ions ( e e o Equa ion (1) ). The pa ame e s used o c ea e hese se s a e: (1) ampli ude h eshold A = 15% o he maximum ampli ude o he signal in he ime window conside ed o he analysis; (2) `j i= 10% o he pe iod o he exci a ion signal; and (3) Lj i= 10% o he same pe iod in addi ion o he andom dispe sion measu ed om he 10 measu emen s. These alues a e selec ed so ha measu emen s aken du ing he p is ine s a e do no show any alse damage indica ions. Mo eo e , `j i and Lj i a e se so ha he measu emen unce ain y is conside ed while a oiding ha he apezoidal unc ions o wo adjacen peaks o e lap. No e ha he e alua ion o a new signal in he p e ious unc ions has been ca ied ou using a smalle ampli ude h eshold o A = 10%, so ha a con ol loop is c ea ed o he ampli ude in an analogous manne o a hys e esis con olle [49]. Figu e 6a–d show he DoH ma ix o he signals acqui ed a di e en a igue cycles, namely 1000, 20,000, 50,000, and 100,000. No e ha wo DoH ma ices a e p o ided o each da ase , he one on he op is o he pulse-echo a ay (using T i PWAS) and he one on he bo om is o he pi ch-ca ch a ay (using S i PWAS). As is e iden om he esul s, a clea deg ada ion o he s uc u e is app ecia ed by he inc ease o he DoH damage index when inc easing he numbe o cycles. No wi hs anding, wo king wi h he DoH ma ices may be limi ed in p ac ice due o hei complex in e p e a ion. Hence he mean alue o M j , j= 1, . . . , m (i.e., all he alues o he DoH ma ices), is used he e as a simpli ied s a is ic o he SHM da a e olu ion, as shown in Figu e 7a. Resul s also show ha a highe damage index alue is ob ained using he da a acqui ed in he S i senso s (g ey line) compa ed o he T i senso s (black line). Fu he mo e, a compa ison o he e olu ion o bo h he DoH index and a igue c ack leng h is also depic ed in Figu e 7a. The measu emen s o he c ack leng h a e ob ained by digi izing di e en poin s in he c ack pa h, as shown in Figu e 7b. A ema kable ag eemen in he end o bo h damage indica o s is obse ed h oughou he du a ion o he a igue es . No e also ha he ea ly damage is de ec ed using he wo phased-a ays a ound 10,000 cycles, al hough he S i senso s a e able o p o ide an ea lie indica ion o damage (i.e., e en be o e i is isible by op ical means). No e ha o he s a is ics ha illus a e he damage in o ma ion ob ained om he beam o ming es s can be used. In pa icula , he adop ion o he mean o he DoH ma ix has shown accu acy in moni o ing o he deg ada ion o he s uc u e. Howe e , his s a is ic shows no indica ion o he dispe sion o he cells o he DoH ma ices and wo ks as a il e o he e olu ion da a. To p o ide such dispe sion in o ma ion, he e olu ion o he indi idual damage indices M j o he a ay o senso s T i when ocusing a 60 ◦ and 120 ◦ a e shown in Figu e 8a,b, espec i ely. The equi alen da a o he uppe phased-a ay wi h S i Senso s 2021,21, 993 9 o 17 senso s a e shown in Figu e 8c,d. These cu es depic he e olu ion o he 60 ◦ column o he DoH ma ices, equi alen o he ones shown in Figu e 6. ξ1ξ2ξ3ξ4ξ5ξ6ξ7ξ8ξ9ξ10 ξ11 ξ12 ξ13 ξ14 ξ15 ξ16 ξ17 ξ18 ξ19 ξ20 ξ21 ξ22 ξ23 ξ24 ξ25 ξ26 ξ27 ξ28 ξ29 ξ30 ξ31 ξ32 ξ33 ξ34 ξ35 ξ36 ξ37 T1 T2 T3 T4 T6 S1 S2 S3 S4 S5 S6 0 0.2 0.40.60.8 1 (a) DoH ma ix a 1000 cycles ξ1ξ2ξ3ξ4ξ5ξ6ξ7ξ8ξ9ξ10 ξ11 ξ12 ξ13 ξ14 ξ15 ξ16 ξ17 ξ18 ξ19 ξ20 ξ21 ξ22 ξ23 ξ24 ξ25 ξ26 ξ27 ξ28 ξ29 ξ30 ξ31 ξ32 ξ33 ξ34 ξ35 ξ36 ξ37 T1 T2 T3 T4 T6 S1 S2 S3 S4 S5 S6 0 0.2 0.40.60.8 1 (b) DoH ma ix a 20,000 cycles ξ1ξ2ξ3ξ4ξ5ξ6ξ7ξ8ξ9ξ10 ξ11 ξ12 ξ13 ξ14 ξ15 ξ16 ξ17 ξ18 ξ19 ξ20 ξ21 ξ22 ξ23 ξ24 ξ25 ξ26 ξ27 ξ28 ξ29 ξ30 ξ31 ξ32 ξ33 ξ34 ξ35 ξ36 ξ37 T1 T2 T3 T4 T6 S1 S2 S3 S4 S5 S6 0 0.2 0.40.60.8 1 (c) DoH ma ix a 50,000 cycles ξ1ξ2ξ3ξ4ξ5ξ6ξ7ξ8ξ9ξ10 ξ11 ξ12 ξ13 ξ14 ξ15 ξ16 ξ17 ξ18 ξ19 ξ20 ξ21 ξ22 ξ23 ξ24 ξ25 ξ26 ξ27 ξ28 ξ29 ξ30 ξ31 ξ32 ξ33 ξ34 ξ35 ξ36 ξ37 T1 T2 T3 T4 T6 S1 S2 S3 S4 S5 S6 0 0.2 0.40.60.8 1 (d) DoH ma ix a 100,000 cycles Figu e 6. 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