Fatigue crack detection based on changes of the structure vibration characteristics


 Agatha Lambert
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1 Fatgue crack detecton based on changes of the structure vbraton characterstcs Rassa D.M. BRANDÃO,, Nacer HAMAOUI, Franços GIRARDIN, Pascal VOAGNER, Therry MAOYER Unv Lyon, INALyon, Laboratore Vbratons Acoustque F696 Vlleurbanne, France ACOEM Group Chemn des Ormeaux, Lmonest Cedex, France Abstract In order to mplement a technque for detectng fatgue cracks n vbratng structures, a test bench was bult n the laboratory and an expermental nvestgaton was conducted. The case of ths study s a smplstc representaton of a bucket wheel excavator from mnng ndustres, desgned to have a fragle structural member actng as a mechancal fuse. No artfcal exctaton s used. The structural member s naturally damaged durng operaton and ts vbraton s montored over tme, untl the crack occurs. Ths paper deals wth spectral and transmssblty matrces, by means of whch two measurements are compared about changes n the frequency content. The emphass s not on the modal parameters themselves, but on comparng the spectral waveforms by means of two metrcs, the relatve error and the lkelhood metrc from a statstcal test. The performance of each feature to detect the changes caused by the presence of cracks n the testbench s llustrated. Introducton tructural health montorng (HM) has become a research focus n almost all engneerng communtes, ncludng the feld of mechancal engneerng. The reason s clear: the mplementaton of an effectve montorng strategy ncreases the ndustres compettveness through a better control of the producton tools health. The prncple conssts n perodcally evaluatng the health state of a machne n order to carry out mantenance operatons only at the most convenent tme. In ths way, HM encompasses local and global strateges of survellance []. Local HM methods nclude vsual nspectons and nondestructve evaluaton technques, such as acoustc emsson, ultrasonc and eddy current. The lmtaton of all these methods comes from the fact that a pror knowledge of the damage zone locaton s requred. In contrast, global vbratonbased methods are an alternatve to overcome ths lmtaton and, n some cases, they can be appled wth natural exctatons. everal works (e.g. [4]) have made crtcal revews on the topc, whch has been developed for more than thrty years. The man dea behnd the vbratonbased HM s that damage changes the stffness, mass and energy dsspaton characterstcs of the structure and therefore ts dynamcal behavour. As stated by Farrar et al. [3], the most fundamental challenge of ths approach s the fact that damage s typcally a local phenomenon and may not sgnfcantly nfluence the lowerfrequency global response of a structure that s typcally measured durng vbraton tests. At the same tme, t s wellknown that envronmental factors, such as temperature and mosture, also produce changes n dynamc characterstcs []. Therefore, makng accurate and repeatable vbraton measurements at a lmted number of locatons on structures often operatng n adverse envronments represents many practcal ssues [3].
2 Hstorcally, the vbratonbased HM has been appled to aeronautcs, cvl structures (brdges and buldngs) and offshore platforms. Our obectve s to work wth ndustral machnery submtted to fatgue cracks. We want to mplement, from measurement data acqured durng operaton, a montorng technque that allows early detecton of the presence of cracks that can compromse the correct operaton of the machne. In ths way, we have desgned an expermental testbench on the laboratory representng smplfed verson of a real machne, on whch fatgue cracks are naturally generated durng operaton and ts vbraton s montored perodcally. The frst part of ths paper presents the spectral approach developed for montorng the ntegrty of structures from ts vbraton sgnals. It s based on trackng the frequency content changes on spectral and transmssblty matrces. Then, the expermental setup s detaled and the results obtaned are llustrated and dscussed. pectral approach for HM Ths secton dscusses the spectral approach developed n order to detect the exstence of structural damage on a machne, through the analyss of ts dynamc behavour. It gves the detals on the spectral quanttes and ther metrcs nvestgated to track the changes on these measured responses.. pectral and transmssblty functons Consder a structure subected to several ndependent sources of exctaton, as llustrated n Fgure. Fgure : tructure wth several nputs (grey arrows) and several measured responses (black arrows). Let x (t) be the th response sgnal measured on ths structure and X (f) ts spectrum at the frequency f, obtaned by a Dscrete Fourer Transform (DFT). The spectral matrx s a 3D matrx that gathers the autoand crossspectra functons of the set of p measured response sgnals. For a gven frequency f, each element of ths matrx s gven by the followng equaton: * ( f ) X ( f ) X ( f ), wth, p, = =,,... () where the subscrpt * represents the complex conugate. The lmtaton of usng spectral matrces n the context of HM s because they are senstve to any frequency changes, not only the ones caused by damage. For ths reason, the use of transmssblty functons, whch do not depend on the exctatons as long as the nput degrees of freedom do not change, has been extensvely nvestgated. The reader s referred to [5] for a revew of lterature on the use of transmssblty concept for damage detecton and localzaton. The transmssblty functons are generally defned as the rato of the spectra of two homogeneous varables (acceleraton response/acceleraton nput), for a sngle exctaton at a gven locaton [5]. From equaton (), we can defne these functons as: ( f ) ( f ), T, ( f ) =, wth, =,,... p (),
3 However, there are few cases n the practce where only one source of exctaton acts on the system of nterest. For ths reason, the transmssblty concept has been generalzed for the case of multple exctatons to a matrx quantty (e.g. see [6]). In ths case, under the assumpton that the structure behavour s lnear, the p measured responses and hence the spectral matrx can be splt nto references and outputs subsets: r, r ( f ) r, o( f ) ( ) ( ) ( ) f = (3) o, r f o, o f where r =,,..., m and o =,,..., (pm) correspond to the r th and o th measured responses consdered respectvely as references and output. Therefore, f the number m of reference responses s at least equal to the number of uncorrelated sources of exctaton, the submatrx r,r (f) s wellcondtoned and the transmssblty matrx can be defned as [7,8]: ( ) T T ( f ) = r ( f ) ( f ) T (4), o where the subscrpt T refers to the transpose. Notce that ths defnton of the transmssblty matrx s equvalent to the H estmator of transfer functons [8]. It represents a lnear system that characterses the structure for any exctatons appled at the gven nput degrees of freedom.. Montorng metrcs Wth the purpose of determnng f a structure s damaged or not, we compare two measurements (at t and at t n >t ) about changes on the frequency content by means of the estmated spectral and transmssblty matrces... The relatve error The frst metrc used to montor the ntegrty of the structure s based on a relatve dfference at each frequency of the analyss band. For smplcty, the spectral and transmssblty matrces are genercally named here as M(f). As descrbed n the equaton below, t corresponds to the sum of squared absolute errors between the spectral quanttes at t k and t, normalzed by the power of the spectral quantty at t. r, r ( f ) M ( f ) EM ( f, ) = (5) t M ( f ) M,, t, where the operator M represents the complex magntude of M and the subscrpts and represents the measured sgnals composng the matrx M(f). Thus, the feature used to track the changes s the rootmeansquare (RM) value of E M, estmated as: f RM E ( ) = ( (, ) ) M EM f nf f = f (6) where n f represents the number of frequences on the analyss band [f f ]. In ths way, the structure wll be consdered damaged for large values of E M or E M RM. 3
4 .. The lkelhood approach The second metrc nvestgated s based on a hypothess test. Let x(t) be a pvarate measured response, such as: x ( t) = [ x ( t) x ( t) L x ( t) ] (7) p Let X(f) be the Dscrete Fourer Transform (DFT) of the measured response. It derves from the central lmt theorem for random varables that the coeffcents of the DFT at a gven frequency f wll have a complex normal dstrbuton wth zero mean and varance equal to the spectral matrx at ths partcular frequency: X ( f ) ~ N (, ( f )) (8) Therefore, the comparson of the measurements at t and t k about changes n the frequency content can be seen as a problem of determnng f the two measurements come from the same probablstc dstrbuton. From equaton (8), ths problem can be encoded n the followng hypothess test: Η Η : alternatve : t ( f ) = t ( f ) ( f ), f ( f ), f (9) In other words, f the hypothess Η s accepted, no statstcally sgnfcant changes occur on the frequency contents of the measurements and, assumng that the measurements ponts and exctatons are the nvarant, the system s consdered healthy. Alternatvely, Η s reected f there are changes on the spectral matrces and, n ths case, the system s consdered damaged. The problem descrbed by the equaton (9) can be tested by means of the lkelhood rato λ ( f ), whch s defned by the rato between the maxmum probabltes under each hypothess. However, when dealng wth complex normal dstrbuton, t s nterestng to use the negatve loglkelhood rato [9], whch s defned n ths case as (see Appendx A for more detals on ths development): ln λ( f ) = Nln N W( f ) = n W( f ) n N = n t t t t ln + n ( f ) n t ( f ) n ( f ) ln ( f ) () where n t and n correspond to the number of averages adopted for the estmaton of the spectral matrces at t and t k respectvely and the operator M represents the determnant of the matrx M. In ths way, the hypothess Η s reected and the structure s consdered damaged f ths metrc reaches a nonneglgble value. For convenence, we have mposed a constant number of averages over the test, such as n t = n = n. In addton, because the determnant of a matrx depends on ts dmensons, the statstcal metrc defned to track changes on the spectral matrces s: ln λ( f ) V ( f, ) = () n. p Here agan, the feature used to track the changes over the expermental test s the RM value of V estmated as n equaton (6). 4
5 3 Expermental setup The case of study s an expermental test bench representng n a smplstc form a bucket wheel excavator from mnng ndustres (Fgure ). It s composed by a structural member sustanng an electrcal motor, the reducton stages and a bucket wheel. The structural member was desgned to have a fragle part wth exchangeable peces (two thn plates screwed at postons 3 and 4), where the crack s expected to appear. The global sze of the model s 8 x 4 x 5 mm Y X Fgure : Pctures of the expermental test bench from dfferent ponts of vews. Durng the tests, the motor drves the bucket wheel through a step (Fgure, bottom left), causng shocks. The motor speed (6 Hz), the dmenson ( mm of heght) and locaton ( mm from the vertcal bucket) of the step were chosen to nect enough vbratons on the structure regardng the fatgue damage. The tests were conducted untl the cracks were vsually detected (at postons and 3). The vbraton of the structural member s montored perodcally (every 5 mnutes) n dfferent ponts by usng accelerometers and stran gauges. Table summarzes the sensors poston and drectons of measurement wth respect to the mark pont of Fgure. Locaton X Y γ 3, γ 3, 3 ensors reference X Y γ, γ, γ X γ, Y γ, γ γ and stran gauges (see Fgure 3) X γ 4, Y γ 4, γ 5 γ 4 Table : ensors locaton and the respectve drectons of measurement. 5
6 Fgure 3: Pcture of the exchangeable plates at the end of one test. It s mportant to menton that the stran gauges were used to stop the test at a convenent tme and, hence, to avod that other components of the test bench get damaged. They were not used for the HM spectral approach developed here, because of ther frequency lmtaton. These sensors are based on dsplacement measurements, so that they cannot measure hgher frequences, whch are known to be more senstve to damagng phenomenon than the lower ones. Because the vbratons to whch the testbench was subected were sgnfcant and appled to all of ts components, other phenomena n addton to the cracks on the exchangeable plates were observed durng the tests. For example, t happened that the screws used to fx the motor to the structural member fractured and that the fxaton of the test bench to the ground was lost. Because of these other phenomena, some varablty was observed between the tests. The advantage of ths expermental strategy s that the fatgue cracks are obtaned naturally (.e. no notch s done a pror). The test bench bult n the laboratory s a smplstc representaton of the real structure, but allows the nvestgaton of ts structural member ageng durng operaton. In addton to ths, no drect measurement of the sources of exctatons s done and the analyses are based on response measure only. The perspectve s to apply ths HM technque to real complex machnes under complex loadng condtons, whch can also be subected to varyng operatonal/envronmental condtons. 4 Applcaton of the HM spectral approach The spectral approach presented n ths paper was appled to the acceleraton sgnals measured on the expermental test bench. We remnd the reader that our obectve s to detect, durng operaton and as soon as possble, the presence of a crack. Therefore, to gve the same mportance to the changes observed through all frequences beng analysed, the frequency resoluton used to the spectral estmates was progressvely ncreased as ndcated on Table. Frequency band [5] Hz ]5] Hz ] 4] Hz ]45] Hz Frequency resoluton Hz Hz 5 Hz Hz Table : Frequency resolutons used to the spectral estmates. For the test used as example to demonstrate the results n the followng, the crack was vsually detected at 85 mnutes. At the end of the test fve mnutes later, both exchangeable plates were cracked. Durng ths test, t was notced that one screw, fxng the motor to the structural member, broke. We do not know when exactly t fractured. Ths screw was changed after the measurement at 55 mnutes. Frstly, the Fgure 4 llustrates the relatve error metrc appled to the spectral matrces estmated durng the example test. The thck arrow marks the tme when the crack was vsually detected. 6
7 Fgure 4: Evoluton of the relatve error metrc for the spectral matrces. The analyss of the mappng above shows that the most mportant changes n the spectral matrces occur after 8 or 85 mnutes at dfferent frequency bands. An excepton s observed between 4 and 55 mnutes, but n ths case, the changes are probably caused by the problem wth the motor screw descrbed before. On the other hand, regardng the RM value estmated n the frequency band [5] Hz (Fgure 4, rght), t s clear that the changes detected n the spectral matrces are more sgnfcant when they are caused by the presence of the crack. By usng ths RM feature to montor the ntegrty of the test bench, the detecton s possble at the same tme as by vsual nspecton at 85 mnutes. econdly, the Fgure 5 shows the evoluton of the statstcal metrc durng the example test. Once agan, the thck arrow marks the tme when the crack was vsually detected. It s nterestng to know that, when ths metrc reaches the unt value, t means that ( f ) 9t ( f ). In contrast, when the metrc based on the relatve error (E (f,t k )) has an unt value, then ( f ) ( f ). t Fgure 5: Evoluton of the statstcal metrc for the spectral matrces. Usng ths statstcal approach, the most sgnfcant changes on the spectral matrces are observed n the low frequences (below to Hz). Also from the analyss Fgure 5, ths metrc seems more robust than the one based on the relatve error, consderng that the changes between 4 and 55 mnutes are no longer detected. In contrast, some less sgnfcant changes can be notced, startng at mnutes, for whch ths metrc reaches the value of.4 and.6 at some frequences. The RM value of ths metrc for the frequency band [5] Hz has an nterestng monotonc evoluton wth ageng, but no abrupt change s clearly notced near to the tme when the crack was vsually detected. It s mportant to remark that nether of the metrcs developed for the spectral matrces crcumvents the maor lmtaton of usng these quanttes n the context of HM. Frequency changes other than those caused by spectral damage can be detected by these two features. Ths statement motvated the nvestgaton of the transmssblty matrces. In ths case of study, all measurements beng under constant operatonal condtons, the choce of reference sgnals was done wth respect to the condton number of the submatrx r,r. It s obvous that the mean source of exctaton s the rotaton of the bucket wheel through the step, so that the sgnal of the 5 accelerometer placed on the bearng ( γ ) should be taken as a reference. But only one reference s not suffcent to characterse the exctatons sources on the test bench, due to ts operatonal nature. Hence, as a result of an teratve procedure based on the condton number of r,r, the sgnals retaned as references to the transmssblty matrx approach were γ, γ and γ. Y X 3 5 7
8 The Fgure 6 below llustrates the evoluton of the relatve error metrc appled to the transmssblty matrces for the example test. Fgure 6: Evoluton of the relatve error metrc for the transmssblty matrces. The fgure above shows mportant frequency changes startng manly at 8 or 85 mnutes. Notce that the changes between 4 and 55 mnutes are also observed n some specfc frequences. Compared to the other metrcs, the changes on the E T feature are more concentrated to some frequency bands and they seem more stable (the value of ths feature s low and approxmatvely constant for most of the example test). Concernng ts RM feature, the evoluton observed n Fgure 6 (rght) s smlar to the RM value of E, and the most mportant changes observed can be assocated wth the presence of the crack. By usng the transmssbltybased RM feature nstead of the one based on the spectral matrces, the detecton can be done fve mnutes earler at 8 mnutes. Fnally, to estmate the performance of the features developed to detect the presence of a fatgue crack, the ReceverOperatngCharacterstc (ROC) curve was used. By ts defnton, the ROC curve shows the tradeoff between the true postve rate and false postve rate as one change the threshold for postvty []. In ths way, f both rates are nearly equals, the detector s consdered random, and a good detector wll be the one wth hgh true postve rate and small false postve rate. For our case of study, the true postves were defned from the tme when the crack was vsually detected. All sx tests done wth the expermental testbench were treated together n order to construct a globalroc curve for each of the developed features (Fgure 7). Fgure 7: Global ROC curves for each of the developed features. The fgure above demonstrates that the three features developed n ths paper have relatvely good performances to detect the presence of a crack on the expermental test bench studed. Among them, the worst detecton performance s obtaned by the RM E feature. However, even f the performances of the two others are close, the reader should recall that, n contrast to the spectral matrces, the transmssblty matrces are not dependent on the exctatons level and type, as long as the nput degrees of freedom reman constant. The ROC curves llustrated n Fgure 7 were constructed under constant operatonal condtons and do not consder the lmtaton of the use of spectral matrces to HM. Under varyng operatonal condtons, we expect the performance of the RM E T feature to surpass the others. 8
9 5 Conclusons Wth the purpose of detectng the presence of fatgue cracks n a structure durng the machne operaton, a spectral approach was developed and ts performance tested on an expermental testbench bult n the laboratory. From vbratons measurements on the structure, whch can be consdered healthy n the begnnng of the tests, the dea conssted n trackng changes on spectral and transmssblty matrces, n order to dentfy the moment when the changes correspond to damage. In ths way, a relatve error measure and, for the spectral matrces, a statstcal metrc were nvestgated. Even f the statstcal metrc demonstrated to mprove the performance of usng the spectral matrces to HM purposes, both features developed to track changes on these matrces have the lmtaton of beng dependent on the exctaton levels. Under varyng operatonal condtons, the best performance s consequently expected for the metrc based on the transmssblty matrces. For all experments performed and for a gven threshold, the RM value of ths metrc presented a true postve rate of 89.7% and a false postve rate of.6%. The perspectve s to apply ths HM technque to real complex machnes by takng proft from the statement made by Leclère et al [8] that, gatherng nformaton from several operatonal condtons, the transmssblty matrces estmates can be mproved through a better charactersaton of the structure for gven exctaton confguraton. Acknowledgments Ths work s supported by the French Natonal Assocaton for Research and Technology (ANRT) and the Brazlan Natonal Councl for centfc and Technologcal Development (CNPq), under the CIFRE Brazl program, and performed wthn the framework of the Labex CeLyA of Unversté de Lyon, operated by the French Natonal Research Agency (ANRLABX6/ ANRIDEX7). References [] B. Gunes and O. Gunes, tructural health montorng and damage assessment  Part I: A crtcal revew of approaches and methods, Int. J. Phys. c., 8(34) (3), pp [].W. Doeblng, C.R. Farrar, M.B. Prme and D.W. hevtz, Damage Identfcaton and Health Montorng of tructural and Mechancal ystems from Changes n Ther Vbraton Characterstcs: A Lterature Revew, Report, Los Alamos Natonal Laboratory, LA37M, 996. [3] C.R. Farrar,.W. Doeblng and D.A. Nx, Vbratonbased structural damage dentfcaton, Phl. Trans. R. oc. Lond, A () (359), pp [4] C.R. Farrar and K. Worden, tructural Health Montorng: a machne learnng perspectve, st edton, John Wley & ons, UK, 3. [5]. Chesné and A. Deraemaeker, Damage localzaton usng transmssblty functons: A crtcal revew, Mechancal ystems and gnal Processng, 38 (3), pp [6] A.M.R. Rbero, J.M.M. lva and N.M.M. Maa, On the generalzaton of the transmssblty concept, Mechancal ystems and gnal Processng, 4() (), pp [7] M. Fontul, A.M.R. Rbero, J.M.M. lva and N.M.M. Maa, Transmssblty matrx n harmonc and random processes, hock and Vbraton, (4), pp [8] Q. Leclère, N.B. Roozen and C. ander, On the use of the Hs estmator for the expermental assessment of transmssblty matrces, Mechancal ystems and gnal Processng, 43 (4), pp [9] K.V. Marda, J.T. Kent and J.M. Bbby, Multvarate analyss, st edton, Academc Press, UK, 979. [] K. HaanTlak, Recever Operatng Characterstc (ROC) Curve Analyss for Medcal Dagnostc Test Evaluaton, Caspan J. Intern. Med., 4() (3), pp [] A.A. Mranda, pectral Factor Model for Tme eres Learnng, PhD Thess, Unversté Lbre de Bruxelles, Belgum,, Chapter, pp
10 [] A.A. Mranda, C. Olsen and G. Bontemp, Fourer spectral factor model for predcton of multdmensonal sgnals, gnal Processng 9 (), pp A Appendx Lkelhood rato test for spectral denstes The development of the lkelhood rato test for spectral denstes presented here s based on the words of Marda et al [9] and Mranda et al [,]. uppose X(f) be a pvarate DFT of measured responses. From equaton (8), the loglkelhood of observng n samples of X around a target frequency f becomes (the dependency wth f s omtted): n n = * l( X) = ln π X X (A.) The lkelhood value l descrbed above represents the probablty that X(f) follows the complex normal dstrbuton of varance (see equaton (8)). Takng advantage of the trace property, ths expresson becomes: l( X) = ln π Φ = n n n = n tr ( X X * Φ) (A.) Notce that Φ s the sample spectral denstes matrx, estmated wth n averages. Let now X t ~ N(, t) and X ~ N(, ) be two pvarate ndependent measurements, whch we want to compare about changes on the frequency content through the hypothess test defned by equaton (9). In ths case, the loglkelhood functon becomes: l n n t t t = ln π n tr ( Φ ) + ln π n tr ( Φ ) (A.3) Under the null hypothess Η, =, so that: t = l = Np ln π N ln tr ( N = n W = n t t Φ t + n n Φ W) (A.4) By consequence, the maxmum value that the lkelhood functon takes under Η wll be for ˆ = N W : l max a W = Np ln π N ln N Np (A.5) On the other hand, under the alternatve hypothess, no constrants are mposed to the expresson of the lkelhood functon and ts maxmum value wll occur for Ŝ = Φ : max a l = Np ln π ( nt ln Φt + nt k ln Φ ) Np (A.6) As a result, the hypothess test from equaton (9) can be tested usng the followng metrc: max max ( la l ) = Np ln N W nt ln Φt nt k ln Φ ln λ = (A.7)
11 That means, Η s reected f ln λ >>, f (the reader should recall that the dependency wth frequency was omtted for smplcty). It s mportant emphasze that Φ corresponds to an estmate of the spectral matrx, so that the equaton (A.7) s equvalent to the equaton to the equaton () presented earler n ths paper.
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