Fault Diagnosis in Stationary Rotor Systems through Correlation Analysis and Artificial Neural Network

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1 Faul Dago Saoary oor Syem hrough Correlao aly ad rfcal Neural Newor leadre Carlo duardo a ad obo Pederva b a Federal Uvery of Ma Gera (UFMG). Deparme of Mechacal geerg (DMC) aceduard@homal.com b Sae Uvery of Campa (UNICMP). Faculy of Mechacal geerg/deparame of Mechacal Deg robo@fem.ucamp.br brac. h wor pree a mehodology of parameer faul dago roag mechacal yem eced by ubalace ad coloured ochac oe, hrough he correlao aaly baed o he Ljapuov Mar ad he rfcal Neural Newor (NN). he procedure of parameer faul dago decrbed ue oly meaured ae varable. he drec meaureme of he ochac ecao o eceary. Coloured oe ochac ecao modeled ug a dyamcal yem eced by whe oe. he propoed mehod avod he ecey of meaurg he ecao force ad wor o he dplaceme ad velocy gal obaed form dffere po of he roor yem. Ug correlao he oupu varable, poble o derve pecfc relao volvg he phycal parameer of he yem ad he correlao marce of he meaured varable. Faul he roor ca be deeced by moorg he varao of he phycal parameer hrough a comparo of heorecal ad emaed correlao fuco. Some varable are dffcul o be meaured ad he correlao fuco volvg uch varable cao be drecly emaed. rfcal Neural Newor ued a a ool o map uch correlao. he aaly of he reul made wh he applcao of he mehod o a roor yem modeled wh degree of freedom. he good reul of hee eample how he vably of furher ude h area. Keyword:Faul Dago, Correlao aly, Ljapuov Mar quao, rfcal Neural Newor. Iroduco Faul deeco ad dago procee ha bee recevg coderable aeo rece year due o he obvou mporace of h problem. I addo, he creag avalably of powerful compug evrome ad he progre problem-olvg paradgm ha faclaed he applcao of a wde varey of mehod for real-me dago. May of hee mehod were developed dffere feld, uch a corol heory, gal proceg ad arfcal ellgece (I). a reul, he mehod dffer wh repec o he ype of pror owledge hey are baed o ad how h owledge ued for developg a dagoc yem. he hear of faul dago le he model-baed approach, where by a may varable ad yem parameer are ae o accou a poble, order o coruc a dealed mahemacal model of he yem uder obervao. Oce he dyamc behavour of he yem ha bee "adequaely" modelled, hould heorecally be poble o deec faul va aaly of chage he model o pu parameer. Several approache are baed o pary pace equao, dagoc oberver, ad proce defcao (Fra, 99; Gerler, 99; Pao ad Che, 99; Ierma, 984; Pao e al., 989; Ierma, 993, 999; Che e al., 999). Veaaubramaa e al (3) coclude ha hee mehod are baed o he aalycal redudacy here o modellg proce. hey alo roduce he problem of faul dago ad revew ome approache baed o quaave model. Pederva (99) ad Charello (998) propoe a alerave approach o face he problem of parameer emao ad faul deeco aoary dyamc yem. he relao bewee correlao fuco ad phycal parameer are ued ad evdec ha he correlao fuco ed o eady ad coa value. h wor how ha poble o epad he propoed formulao by roducg he relao bewee he Ljapuov Mar ad he rfcal Neural Newor (NN). h procedure of parameer faul dago ue oly meaured ae varable. he correlao fuco volvg oly oupu varable ca be emaed hrough he meaureme of hee varable. Oe ca verfy he compably of he equao applyg he emaed correlao aocao wh he heorecal value of he parameer. quao ha deped o parameer ha have chaged wll pree coecy ad he oher o. I h way, poble o oly o deec chage he yem bu alo o locae hem. umercal eample how ha alo poble o locae he faul parameer, eve whe mechacal yem are eced by ochac force.

2 . Dyamcal Model Coder a roor yem ha ca be decrbed by he followg dffereal mar equao, () P () Q () Su( ) H( ) M () where he marce M, P ad Q are repecvely, he ma, dampg ad ffe marce wh approprae dmeo. he marce S ad H are pu marce of ochac force, ubalace force ad harmoc perurbao. he vecor () he dmeoal vecor of dplaceme ad he do dcae dffereao wh repec o me. he vecor u() repreeg he ochac force codered a a whe oe proce. I eay o raform he above dmeoal equao o a dmeoal ae pace model, () { () M () }, () wh he ae pace equao ca be obaed, () () u() () (3) M Q I M P I ( f, f ) ( f, f ) ( f, f ) ( f, f ) (4) M S ( f, p ) ( f, p ) ( ) ad f, p M H ( f, p ) (5) he quared mar of order f called he yem mar; he mar (, p) he pu mar; ad () a harmoc ecao gve by, () co ( Ω) ( ) Ω (6) I q. (3), f u() o a whe oe may be modelled a he oupu of a mlar dyamc yem (fler). Coderg he dyamc yem () () S w() (7) () he ae vecor -dmeoal. he pu u() of dyamc equao (3) a coloured proce, u () H () (8) u H u a mar wh dmeo (p,). Fgure how he dagram of a dyamc yem eced by coloured oe ( ) w () u( ) FIL Fgure Dyamc yem eced by coloured oe S loc he fgure how bloc he dyamc yem eced by coloured oe, where: loc - modelg coulored oe hrough whe oe; loc - compoo of ecao modeled a he oupu of a mlar dyamc yem (fler); loc 3- repoe of yem mechacal. H u ( ) MCHNICL SYSM ( ) loc loc 3

3 epaded model he obaed a a reul of he compoo of he yem equao ad he fler equao. h ew epaded howed Fg. (), mlar o q. (3), wh ew marce ad, q. (9). ocag equao (7), (8) ad (3) oe ca oba he epaded dyamc equao wh, () () w() () (9). () () L () () (,) ( ), L ad L () ( ), he ew mar of he yem ha he followg rucure M L M L () M. Phycal Model he mechacal yem codered a roor wh a fleble haf coupled o a fed ed elerc moor. ewee he moor ad he dc, here a bearg of ma m fed elacally. he degree of freedom are he four ralao ad wo roao luraed he Fgure. able () how he umercal value ued he mulao. able : Value parameer Parameer Value U m 5 g m g I.5 g.m I polar.5 g.m L L.4 m 9. N/m. N/m c 3. g/ c 37.5 g/ Ω 6 rad/ Fgure - he 6-DOF model he dplaceme are repreeed he vecor, { y y ψ ϕ} (3) he dagoal ma mar M, { m m I m m } M dag (4) I

4 he mar S ha he followg rucure, { } S, (5) wh (,...6) coa value. he mar H, Ω Ω e m e m H, (6) where e he roor ubalace eccercy.. epoe of Dyamcal Syem ced by Coloured Noe he me repoe of he lear yem a uperpoo of he repoe due o he ochac ecao w () ad he repoe caued by ubalace (), () () () () q q w w Ω Ω co (7) wh q ad q repreeg dmeoal amplude vecor. he correlao mar, ( ) ca be defed a Yaglom (96), ( ) ( ) ( ) { } ε. (8) where repree a me lag. Subug he oluo of he ae pace, q.(9), ad wh q.(7) o q. (8), he followg relao ca be obaed for aoary codo (Yaglom, 96), ( ) ( ) ( ) ( ) ww e ψ, (9) where he mar ww ψ he ey mar of he ecao proce w(). he eq. (9) called Ljapuov Mar quao be ued o develop he mehod of faul deeco decrbed h wor. padg he q. (9), [ ] [ ] ψ ww e I I () he ub marce mulplcao of he q. () provde he followg equao, ( ) ( ) ( ) ( ) () quao () -dmeoal ad coa he relao bewee he correlao fuco of he oupu (dplaceme ad veloce) ad he phycal parameer q. (). quao () coa alo he correlao bewee he ubalace force ad he oupu. lhough, ca be furher developed ad mplfed. he correlao mar bewee ubalace ad oupu varable ca be evaluae. From he defo ( ) ( ) ( ) { } ε. ()

5 ( ) ε{ ( ). ( )} (3) Subug he equao (6) ad (7) o () ad (3),ad afer ome mahemacal maupulao we oba, ( ) coω ( q q ) Ω ( q ) q ( ) coω ( q q ) Ω ( q q ) (4) (5) I mea ha uder aoary codo he umerc value of marce ad are coa depedg o he amplude q ad q, he roao peed ad he me lag ( ). he erm coag h correlao he q.() are coa oo. quao () repree he bae for he propoed approach ad coa relao of compably. O ca verfy ome varao o he parameer by checg hee relao. he marce ad he q.() have he followg eral rucure (Pederva, 99), (6) c ΩI p c ΩI p (7) padg q.() o ca oba he followg relao volvg he parameer of he bearg, he haf ad he ubalace: c b (8) eω (9) b (3) c (3) he dce j dcae he correlao bewee he ae varable ad he ae varable j. he equao of compably (8) o (3) have dffere depedece o he parameer. For eample, he q.(8) oly hree parameer appear. I mea ha h equao o eve o he parameer e relaed o he ubalace. I alo eve o parameer ad 33. h propery very ueful, o oly o deerme varao he parameer bu alo o locae where he falure pree. 3. Propoed Mehod I propoed a mehodology for faul dago aoary roor yem hrough correlao aaly ad arfcal eural ewor ug mul-layer percepro o map he correlao fuco volvg varable ha cao be drecly emaed. Mul-layer percepro (MLP) are eural e uually referred o a fuco appromaor. MLP a geeralao of oebla' percepro (958).he fudameal mporace of a eural ewor o oly he way a euro mplemeed bu alo how her ercoeco (more commoly called opology) are made. Oe of he eae form of h opology rece year made of hree layer : oe pu layer (he pu of he ewor) oe hdde layer oe oupu layer (he oupu of he ewor)

6 ll euro from oe layer are coeced o all euro he e layer. Oe way o do h by repreeg he mappg of equao (8) o (3), a lluraed he Fg. 3. pplyg emaed correlao fuco o he pu ad comparg he oupu, h cae, he correlao fuco. 77 a 3 a 77 7 c M a M Fgure 3- he rucure eural for equao (8) ach pu ha a aocaed wegh W. he u compue ome fuco f of he weghed um of pu: f W 77 he weghed um w called he e pu o u, ofe wre e. (3) Noe ha W refer o he wegh from u o u. he fuco f he u' acvao fuco. I he mple cae, f he dey fuco, ad he u' oupu ju e pu. h called a lear u. Srucurally, h e compoed by hree euro pu layer (correpodg he correlao fuco), ffee euro layer hdde (phycal parameer of he yem c... ad radom wegh a... a ) ad a oupu layer (correpodg he correlao fuco aocaed wh he velocy vecor). he correlao aocaed o he roao ad veloce coordeae of he roor had bee dcodered (q. 8 o 3). I effec wa raed by he e. For he archecure of he eural e wa ued he fuco, ad 7 a he pu layer ad 77 a he oupu layer. For he archecure of he eural e wa ued he fuco, ad a he pu layer ad 78 a he oupu layer. For he archecure 3 of he eural e wa ued he fuco ad a he pu layer ad 79 a he oupu layer. For he archecure 4 of he eural e wa ued he fuco, ad 4 5 a he pu layer ad a he oupu layer. 4. eul 7 I order o demorae he applcably of he propoed approach, ome operaoal uao of he dyamc behavour a roag yem had bee mulaed. he repoe of dyamcal yem eced by coloured oe wa aalyed for dffere cu frequece. he Fgure 4 how o he drbuo of he coloured oe-ype ecao geeraed from he whe oe. he udy of he varao of he parameer wh he cuoff-frequecy aocae o parameer varao he reoace rego. Fgure 4- Drbuo of he ecao coloured oe

7 o mulae a falure, he value of he parameer wa chaged abou %. he yem wa mulaed ad he correlao calculaed ad he eural e a decrbed wa raed. he oupu of he e from he faul codo wa compared o oe whou falure. o compare he oupu wa ued a Mea Squared Devao (MSD) bewee he fuco. 4. Sude cae quao (8 o 3) repree he bae for he propoed approach ad coa relao of compably. Oe ca verfy ome varao o he parameer by checg hee relao. able how he ubjecve perceage of faul able : Value parameer Faul codo Faul Decrpo reduco of % reduco of % c 3 reduco of % c 4 reduco of % ubalaced 5 reduco of % ad 6 reduco of % c ad c 7 reduco of % ad ubalaced 8 reduco of % ubalaced ad c 4.. Varao he parameer of he yem ad cuoff frequece of he fler Fgure 5(a)-(d) how he devao of he four equao codered (8) o (3). I codered varao cuoff frequece for he faul codo o 4(able ) Fgure 5 (a)-(d) - Squared Mea Devao of ewor oupu: parameer varao ad cuoff frequecy. Fgure 5(a) how he effec of he varao he parameer. I oberved ha for dffere cuoff frequece of he fler, he eural archecure appromaely pree varao. h fac o foud he oher archecure (equao 9 o 3). I mea ha are o depede o he parameer. Fgure 5(b) how he behavor of he archecure aocaed wh a varao he parameer c. I ereg o oe ha he q.(8) pree a gfca varao whe compared o he behavor of he oher equao. h becaue equao (9) o (3) are o depede o he parameer. c Fgure 5(c) how he behavor of he equao aocaed wh a varao he parameer c. he ame behavor occur, dcag ha he falure aocaed wh q.(3). Oe ca ee ha he equao aocaed wh h parameer pree a grea varao whe compared o he oher, dcag ha he

8 falure aocaed o he parameer of h equao. Fally, Fg. 5(d) lurae he varao he ubalace of he roor. he cocluo are mlar o he cae already commeed. hee cuoff frequecy howed Fg.4 (a)-(d) are drbuo of he ecao coloured oe (Fg. 4). able 3 how h correpodecy. he able 4 how he correpodecy eural archecure. able 3: Correpodecy cuoff frequecy a y Cuoff frequecy f c 5 rad/ f c rad/ 3 f c 85 rad/ 4 f c 5 rad/ 5 f c 6 rad/ able 4: Correpodecy Neural archecure a Neural archecure q. 8 q. 9 3 q. 3 4 q Combg Varao he parameer ad cuoff frequece Fgure 6(a)-(d) how he devao of he four equao codered (8) o (3). I codered dffere cu-off frequece for he faul codo 5 o 8(able ). Fgure 6(a)-(d) - Squared Mea Devao of ewor oupu: parameer varao ad cuoff frequecy Fgure 6(a) how he behavor of he archecure aocaed wh a varao he parameer c ad c. I ereg o oe ha equao (8) ad (3) pree a gfca varao whe compared o he behavor of he oher equao. Fgure 6(b) how he behavor of he archecure aocaed wh a varao he parameer ad. he cocluo are mlar o he cae already commeed. Fgure 6(c) how he behavor of he archecure aocaed wh a varao he parameer ubalace ad c. he equao (9) ad (3) pree a gfca varao whe compared o he behavor of he oher equao. Fally, Fg. 6(d) how he varao he ubalace of he roor ad he parameer c.here he ame behavor occur, dcag ha he falure aocaed wh equao (8 ad 9). Oe ca ee ha he equao aocaed wh h parameer pree a grea varao whe compared o he oher, dcag ha he falure aocaed wh he parameer of h equao Perceual Varao of coloured oe yem oupu Fgure 7(a)-(d) how he devao of he four equao codered, q. (8) o (3). he parameer are codered o be coa. Coloured oe addoed yem oupu (varao he order of up o 5% of value MS of he orgal gal).

9 Fgure 7(a)-(d) - Squared Mea Devao of ewor oupu: varao of coloured oe oupu Oe ca ee ha he equao (8) o (3) do o pree a grea parameer varao, dcag ha he falure aocaed wh he coloured oe added yem oupu of hee equao. I addo, he effec of coloured oe mmed a he oupu of he e for dffere archecure ad oe level. 5. Cocluo I h wor, a mehod of faul moorg of a roag yem wa developed, ug he aocao of Ljapuov Mar quao ad arfcal eural ewor. he model mulaed for he uded cae preeed afacory reul. I he codered mehod, here o ecey o ow he ecao force. However, he owledge of he ype ad he form of he ecao eceary. I poble for a cera operaoal codo of he roag mechacalal yem, o defy ad o locae he faul eacly. 6. cowledgeme he auhor would le o ha CNPq (Proce 463/999-4) for facal uppor. 7. eferece Che, J. ad Pao,. J., 999. obu Model-aed Faul Dago for Dyamc Syem. Kluwer cademc Publher. Charello, G..,998. Faul Deeco Mechacal Syem hrough Correlao aly. PhD he, Sae Uvery of Campa (UNICMP). Fra, P. M. 99. Faul Dago Dyamc Syem ug alycal ad Kowledge-aed edudacy - Survey ad Some New eul. uomaca, vol.6, o 3, pp Gerler, J. 99. alycal edudacy Mehod Faul Deeco ad Iolao - Survey ad Syhe. I: Pleary paper, IFC Safeproce Sympoum, ade-ade, Germay, ep. -3. Ierma, Proce Faul Deeco baed o Modelg ad mao Mehod - urvey. uomaca, pp.387-, vol., 4. Ierma, Faul Dago of Mache va Parameer mao ad Kowledge Proceg - uoral paper. uomaca, vol.9, 4, pp Pao,.J., Fra, P. ad Clar,.,989. Faul Dago Dyamc Syem, Prece Hall, UK. Pao,., J., Che, J. 99. O-le edual compeao obu Faul Dago of Dyamc Syem. I: IFC rfcal Iellgece eal me Corol, Delf, he Neherlad. Proceedg, pp.-7. Pederva,. 99. Parameer Idefcao of Mechacal Syem eced by Sochac Force. PhD. he, Sae Uvery of Campa (UNICMP). oebla, F., 958. he Percepro: Probablc Model for Iformao Sorage ad Orgaao he ra. Pychologcal evew, 65, Veaaubramaa V., egawamy., Y, K., Kavur, S.N., 3. evew of Proce Faul Deeco ad Dago Par I: Quaave Model-baed Mehod Compuer ad Chemcal geerg 7 pp Yaglom,. M. 96. Iroduco o he heory of Saoary adom Fuco, Prece Hall, 96.