SOME RESULTS IN MODEL BASED TRANSVERSE CRACK IDENTIFICATION IN ROTOR SYSTEMS

Size: px

Start display at page:

Download "SOME RESULTS IN MODEL BASED TRANSVERSE CRACK IDENTIFICATION IN ROTOR SYSTEMS"

Piers Hodge
6 years ago
Views:

1 SOME RESULTS IN MODEL BASED TRANSVERSE CRACK IDENTIFICATION IN ROTOR SYSTEMS Nicolò Bachschmid () Paolo Peacchi () Sylvie Audebert () () Dipartimeto di Meccaica, Politecico di Milao, P.zza Leoardo da Vici, 3 - I-33 Milao, Italy icolo.bachschmid@polimi.it, paolo.peacchi@mecc.polimi.it () Divisio Recherche et Développemet, Dép. Acoustique et Mécaique Vibratoire EDF - Electricité de Frace, aveue du Gééral de Gaulle - 9 Clamart CEDEX, Frace, sylvie.audebert@edf.fr Abstract This paper presets some experimetal results obtaied o the EUROPE (Esamble Utilisat u ROtor Pour Essais) test rig, which was expressly desiged by EDF (Electricité de Frace) for ivestigatig the dyamical behavior of cracked rotors. The results are used for validatig a model based trasverse crack idetificatio method, which was developed durig a Europea commuity fuded research project called MODIAROT (MOdel based DIAgosis of ROTor systems i power plats). The excellet accuracy obtaied i idetifyig positio ad depth a crack proves the effectiveess ad reliability of the proposed method. Keywords: Idetificatio, Crack, Rotordyamics.. INTRODUCTION Propagatig trasverse cracks have bee discovered i the last years (Alliaz, 987) i several rotors of steam turbies or geerators of Europea power plats. Fortuately, as far as the authors kow, they have bee detected before the crack had propagated to a critical depth, that meas before the occurrece of a catastrophic failure. The importace of early detectio of cracks, possibly by meas of a automatic diagostic methodology that uses the iformatios furished by stadard moitorig systems, which geerally aalyze the vibratios measured i correspodece of the bearigs oly, appears obvious from these cosideratios. The dyamical behavior of rotors with trasverse crack has bee studied by may authors (a extesive survey is give i Dimarogoas, 996) ad therefore the symptoms of a cracked rotor are well kow: a chage i x rev., x rev. ad 3x rev vibratio vector is suspect. A chage i vibratio vector meas ot just a icrease or a decrease i amplitude, but also a chage i phase oly with costat amplitude. However, x rev. compoets ca be caused by may other faults (e.g. ubalace, bow, couplig misaligmets) ad x rev. compoets ca be due also to polar stiffess asymmetries (i geerators), to surface geometry errors

2 (joural ovalizatio) ad to o-liear effects i oil film bearigs. These two last causes ca also geerate 3x rev. compoets. It is the extremely importat to have a referece situatio, stored by the moitorig system, i which the behavior of the rotor system without faults ad i similar operatig coditios is aalyzed. The referece situatio would be better represeted by ru-dow trasiet which furishes much more iformatios about its dyamical behavior, rather tha by a steady state coditio at ormal operatig speed. By comparig the the actual behavior durig a ru-dow trasiet with the referece behavior, the chage i vibratios ca be evaluated ad by meas of oe of the automatic diagostic procedures based o fault-symptom matrices or o decisio trees approach, the type of the most probable impedig fault ca be idetified. Oce the type of fault has bee idetified i a shaft lie, also its most probable positio ad its severity (f.i. i the case of a crack, its depth) should be idetified. This is the possible by meas of the least square approach i the frequecy domai, which is described i the followig paragraph.. MODEL BASED IDENTIFICATION As described i (Bachschmid et al. ), assumig a fiite beam elemet model for the rotor, the effect of a crack o the statical ad dyamical behavior of the rotor ca be simulated i the frequecy domai, by applyig to the rotor differet sets of equivalet forces, oe set for each oe of the three harmoic compoets i correspodece of the cracked beam elemet. The problem of the idetificatio of the positio of the crack is the reduced to a exteral force idetificatio procedure, described i (Bachschmid et al. 999). The fial equatios are recalled here below. The differece, betwee the measured vibratio of rotor system that has a fault ad the referece case, represets the vibratioal behavior due to the fault. These vibratios are the used i the idetificatio procedure. By applyig the harmoic balace criteria i the frequecy domai, the differeces δ, betwee the vibratios, which are calculated by meas of suitable models of the system ad of the fault, ad measured vibratios X Bm ca be defied for each harmoic compoet as: δ = α F X () B f where α B is the partitioed iverse of the system dyamical stiffess matrix ad F f is the -th compoet of the fault force vector. Eq () ca be writte for each oe of the differet rotatig speeds which are take ito cosideratio for the idetificatio. Sice the ukow force vector is composed by few forces applied to odes oly of the f.e. model, the ukows are much less tha the umber of the equatios () ad a least square approach ca be used. A relative residual may be defied by the root of the ratio of the squared δ, divided by the sum of the squared measured vibratio amplitudes X Bm : δ r = Bm T [ α F X ] [ α F X ] B f X Bm T Bm X By meas of the hypothesis of localizatio of the fault, the residual is calculated for each possible ode of applicatio of the defect. The set of equivalet forces i the case of a crack ca be reduced to a couple of opposite ad equal momets which have x rev., x rev. ad 3x rev. compoets. B Bm f Bm ()

3 Where the residual reaches its miimum, there is the most probable positio of the crack. It is worth otig that the x rev. vibratio compoets are due both to the breathig mechaism of the crack ad to the local bow which geerally has developed durig the crack propagatio. Therefore, whe o other sources of bow are preset, the x rev. compoet is useful for the localizatio of the crack, but ot for the idetificatio of its depth. The 3x rev. compoet is rather small ad geerally masked by some oise. Ofte this compoet ca be recogized oly whe approachig the resoat coditio at a rotatig speed equal to /3 the rotor s critical speed. The x rev. compoet is therefore the most suitable symptom for detectig positio ad depth of the crack; the highest values are obviously reached durig a ru-dow trasiet whe approachig the resoat coditio at / critical speed. 3. CRACK DEPTH IDENTIFICATION The followig procedure has bee implemeted for the idetificatio of the crack depth. The L.S. idetificatio procedure, described i the previous paragraph, idetifies the crack positio i a particular elemet of the rotor, whose legth is kow from the D f.e.m. ad equal to l. The equivalet momet compoets M (x, x ad 3x rev. compoets) are applied to this elemet. These equivalet bedig momet compoets M, M ad M 3 have bee calculated from the correspodig x, x ad 3x rev. measured vibratioal behavior. The the statical bedig momet M i correspodece of the same elemet, due to the weight ad to bearig aligmet coditios, is calculated from model data. The ratios of the x rev. equivalet bedig momet M to the statical bedig momet M are all depedet o the relative crack depth p oly. This is represeted i Figure for the x, x ad 3x rev. compoet ad expressed by the relatioship i eq. (3). M = M f ( p) I the same figure also the curve M /M for a always ope crack (a slot or otch) is show: i this case the x rev. compoet oly is preset ad x ad 3x rev. compoet are abset. Eq. (3) ca the be used for determiig the crack depth. But, as show i (Bachschmid et al. ), the legth l c of the equivalet, reduced stiffess, beam elemet that simulates the behavior of the cracked beam, is also depedig o the relative crack depth p: l c () = g( p) D The fuctio g(p) is represeted i Figure. Now we have the equivalet bedig momets M which are applied to a elemet with a wrog legth: l istead of l c. It is worth otig that the x rev. measured displacemets are due to the relative rotatio of the cracked elemet extremity odes, which is proportioal to the product M l of the idetified x rev. bedig momet compoet applied to oe elemet of the f.e. model of the rotor, multiplied by its legth. The equivalet bedig momet compoet M, applied to a equivalet cracked beam elemet of legth l c, ca therefore be calculated as: M l = M l (5) c (3) 3

c / 3.5 M / M M / M M 3 / M M slot / M.6.5. M / M.5 l D.3..5. % 5% % 5% % 5% 3% 35% % 5% 5% Crack relative depth Figure.

4 c / 3.5 M / M M / M M 3 / M M slot / M.6.5. M / M.5 l D % 5% % 5% % 5% 3% 35% % 5% 5% Crack relative depth Figure. Bedig momets ratio o the equivalet cracked beam, as a fuctio of crack relative depth for the x rev. compoet. % 5% % 5% % 5% 3% 35% % 5% 5% Crack relative depth Figure. Relatioship betwee the crack relative depth p, the diameter D ad the legth l c of equivalet beam..6.. f( p) g( p) x rev. f( p) g( p) x rev. f( p) g( p) 3x rev. f( p) g( p) x rev. slot f( p ) g( p) % 5% % 5% % 5% 3% 35% % 5% 5% Crack relative depth Figure 3. Fuctio for the calculatio of the crack depth. Figure. -bearig -composed shaft test rig of EDF-Electricité de Frace o rigid foudatio. By assumig that the statical bedig momet M applied to the origial elemet of legth l does ot chage much alog the elemet, the same M ca be cosidered applied to the elemet with equivalet legth l c. Recallig eq. (3) we ca derive: ad usig eq. () we get: M M l f ( (6) = = p) M M l c M l f ( p) g( p) (7) = M D Eq. (7), show i Figure 3 for the x rev. compoets, ca the be used for determiig, from the kow left had side, the relative depth of the crack.. EXPERIMENTAL RESULTS. EUROPE Test Rig Descriptio The EUROPE test rig, show i Figure, is composed by a shaft divided i three parts supported by two equal three lobe oil film bearigs. The omial diameter of the shaft is 7 mm ad the overall legth is 3.5 m. The distace betwee the bearigs is.88 m. The

5 total mass is 5 kg ad the mai iertia disk is of 5 kg. I this cofiguratio the st critical speed is close to 5 rpm. The supportig structure ca be cosidered as rigid i the speed rage 5 rpm. The proximity probes for the measuremets of relative shaft-joural vibratios are close to the bearigs, but ot iside of them, as usually occurs i real machies. Sice the cetral part of the rotor ca be disassembled, several types of crack ca be geerated. The crack, cosidered i this paper to validate the idetificatio procedure, has bee started from a otch ad made grow up to a depth of 33 mm, which correspods to a depth of 7% o a shaft of 7 mm of diameter. The possibility of disassemblig of the cetral part of the rotor has the mai advatage to ot dismout the etire rotor i order to create a crack. However this leads to some difficulties to have a valid referece case. I fact, by cosiderig a ru-dow of the ucracked rotor ad a ru-dow of a cracked oe, small differeces i aligmet might be itroduced whe the cetral part is coupled to the other extremities. Moreover, the cracked part presets usually a permaet bow due to the fatigue solicitatio used to geerate the crack (see also.3).. Referece Situatio Eve if the so-called referece situatio caot actually be cosidered as the true referece situatio of the same rotor i this test rig, due to the reaso previously expressed, evertheless it has bee used to calculate the vibratio differece. The measured referece situatio is reported i Figure 5 to Figure for the x, x ad 3x rev. compoets. From the aalysis of the x rev. compoet i Figure 5 ad Figure 6, it is evidet the critical speed at about 5 rpm ad that the rotor presets a bow which geerates aroud 5 µm at very low speed i bearig. As regards the x rev. compoet i Figure 7 ad Figure 8 the secod critical speed is rather evidet at about / of the critical speed, but also a peak at about rpm that idicates a o liear effect of the oil film ca be recogized. The relatively high value of the x rev. compoet i the secod bearig (about µm) at low speed, which remais the mai compoet over all the speed rage as is also show by the phase tred, idicates a geometrical error of the shaft (joural ovalizatio) i the bearig. The phase differece of 8 betwee horizotal ad vertical compoets is typical for ovalizatio errors. The 3x rev. compoet has very reduced amplitude i both bearigs (about µm ad µm respectively) ad is maily due to some oise, however a smaller peak at about /3 of the critical speed is recogizable. x -5 Bearig, Harm., Referece case -5 x Bearig, Harm., Referece case Figure 5. Referece case: x rev. vibratio compoets for bearig. 6 8 Figure 6. Referece case: x rev. vibratio compoets for bearig. 5

6 [m] x -6 3 Bearig, Harm., Referece case x Bearig, Harm., Referece case Figure 7. Referece case: x rev. vibratio compoets for bearig. 6 8 Figure 8. Referece case: x rev. vibratio compoets for bearig..5 x -6 Bearig, 3 Harm., Referece case.5 5 x -6 Bearig, 3 Harm., Referece case [m] Figure 9. Referece case: 3x rev. vibratio compoets for bearig. 6 8 Figure. Referece case: 3x rev. vibratio compoets for bearig..3 Cracked Rotor The experimetal measures obtaied with a crack of 7% depth of the diameter i the cetral sectio without subtractig the referece case are show i Figure -Figure 6. From the x rev. compoet (Figure ad Figure ) it ca be iferred that the rotor presets a permaet bow which is icreased with respect to the referece rotor. As regards the x rev. compoet, the high amplitude (about µm) of the peak at / critical speed is clearly due to the crack, which produces also a high resoace amplitude (about µm) of the 3x rev. compoet at /3 critical speed. By usig the measured vibratios due to the crack ad subtractig the referece case vibratios, a attempt to idetify the positio ad the depth of the crack has bee carried out. The results i Figure 7 show that the locatio of the crack is precisely idetified by all the three harmoic compoets. Moreover, the x rev. compoet idetifies the positio with a particularly reduced value of the relative residual. This result has bee obtaied by processig the experimetal data i the followig way: first the ubalaces o the disks were idetified, ad the the dyamical behavior due to the ubalaces oly has bee subtracted from measured data i order to obtai the bow iduced vibratios. This leads to a very good agreemet betwee the experimetal ad the simulated behavior for the x rev. compoet as show i Figure 8 ad Figure 9, i which bow ad ubalaces have bee superposed. 6

7 x -5 Bearig, Harm., 7% Crack depth 6 x -5 Bearig, Harm., 7% Crack depth Figure. 7% crack: x rev. vibratio compoets for bearig. 6 8 Figure. 7% crack: x rev. vibratio compoets for bearig. x -5 Bearig, Harm., 7% Crack depth 6 x -5 Bearig, Harm., 7% Crack depth Figure 3. 7% crack: x rev. vibratio compoets for bearig. 6 8 Figure. 7% crack: x rev. vibratio compoets for bearig. x Bearig 3, Harm., 7% Crack depth.5 x -5 Bearig, 3 Harm., 7% Crack depth Figure 5. 7% crack: 3x rev. vibratio compoets for bearig. Figure 6. 7% crack: 3x rev. vibratio compoets for bearig. As regards the x rev. compoet, the relative residual ca be cosidered as good, but the most remarkable result is the exact idetificatio of the depth of the crack. Also the simulated behavior i Figure ad Figure is good. As cocerig the 3x rev. compoet, the relative residual is quite high, but this ca be explaied by cosiderig that this compoet is ormally masked by oise. However, the residual curve presets a well defied miimum i 7

8 correspodece of the crack eve this is ot so evidet i Figure 7 due to the scale. For this compoet, the compariso is reported i Figure ad Figure 3. Crack Crack residual (x, x, 3x comp.) Residual:.8 Modulus:.e+3 x rev. Elemet: Residual:.567 Modulus: 3.e+ x rev. Elemet: 3 Phase:. (+8) Crack depth: 7. % Residual:.976 Modulus: 5.9e+ 3x rev. Elemet: 3 Figure 7. 7% cracked shaft. Relative residuals of the crack idetificatio. 6 x -5 7% Crack depth, simulated vs. experimetal, bearig, harm. Aalytical Aalytical Experimetal (differece) Experimetal (differece) 8 x % Crack depth, simulated vs. experimetal, bearig, harm Aalytical Aalytical Experimetal (differece) Experimetal (differece) 6 8 Figure 8. 7% crack: compariso betwee simulated ad experimetal (differeces) x rev. vibratio compoets for bearig. 6 8 Figure 9. 7% crack: compariso betwee simulated ad experimetal (differeces) x rev. vibratio compoets for bearig. x -5 7% Crack depth, simulated vs. experimetal, bearig, harm. Aalytical Aalytical Experimetal (differece) Experimetal (differece) 6 x -5 7% Crack depth, simulated vs. experimetal, bearig, harm. Aalytical Aalytical Experimetal (differece) Experimetal (differece) Figure. 7% crack: compariso betwee simulated ad experimetal (differeces) x rev. vibratio compoets for bearig. 6 8 Figure. 7% crack: compariso betwee simulated ad experimetal (differeces) x rev. vibratio compoets for bearig. 8

9 x -5 7% Crack depth, simulated vs. experimetal, bearig, 3 harm. Aalytical Aalytical Experimetal (differece) Experimetal (differece) x -5 7% Crack depth, simulated vs. experimetal, bearig, 3 harm. Aalytical.5 Aalytical Experimetal (differece) Experimetal (differece) Figure. 7% crack: compariso betwee simulated ad experimetal (differeces) 3x rev. vibratio compoets for bearig. 6 8 Figure 3. 7% crack: compariso betwee simulated ad experimetal (differeces) 3x rev. vibratio compoets for bearig. 5 CONCLUSIONS These results, alog with others referred to other test rigs or to other crack depths o the EUROPE test rig, validate the method proposed by the authors to idetify the crack. 6 ACKNOWLEDGEMENTS This work is partially fuded by the MURST (Italia Miistry for the Uiversity ad Scietific Research) Cofiaziameto IDENTIFICAZIONE DI MALFUNZIONAMENTI IN SISTEMI MECCANICI for the year REFERENCES Alliaz, 987, ALLIANZ Berichte, r. Nov. 987, ISSN Bachschmid N., Vaia A., Tazi E. ad Peacchi P., 999, Idetificatio ad Simulatio of Faults i Rotor Systems: Experimetal Results, Proc. of EURO DINAME 99 - Dyamic Problems i Mechaics ad Mechatroics, Wisseschaftszetrum Schloß Reiseburg der Uiversität Ulm, -6 July 999, Güzburg, Germay, pp. 3-. Bachschmid, N., Vaia, A. ad Audebert, S.,, A Compariso of Differet Methods for Trasverse Crack Modellig i Rotor Systems, Proc. of ISROMAC-8 Coferece, 6-3 March, Hoolulu, Hawaii, ISBN X, pp Dimarogoas, A.D., 996, Vibratio of cracked structures. A state of the art review, Egieerig Fracture Mechaics, 55, pp

Principle Of Superposition

Principle Of Superposition ecture 5: PREIMINRY CONCEP O RUCUR NYI Priciple Of uperpositio Mathematically, the priciple of superpositio is stated as ( a ) G( a ) G( ) G a a or for a liear structural system, the respose at a give