Vibration Analysis Of a Self-Excited Elastic Beam

Transcription

1 Vbrto lyss Of Self-Excted Elstc Bem M.. Brró-Mez Deprtmeto de Mterles Uversdd utóom Metropolt zcpotzlco v. S Pblo 80, ol. Reyos-Tmulps.P. 000, Méxco, D.F., MÉXIO bmm@correo.zc.um.mx BSTRT The vbrto behvor d the eergy exchge mog the orml modes of clmped-free self-excted elstc bem re lyzed ths wor. To model ths d of bem, the dmpg term of v der Pol osclltor s drectly dded to the equto of ler elstc bem, yeldg sgle oler prtl dfferetl equto. To solve ths equto, spectrl method s employed. Three vbrto modes re cosdered the lyss, d the vlues of the self-exctg costt re vred order to cover from ler to oler vbrto behvor. Multple frequeces of the oler bem re determed through the power spectrl desty of the bem free-ed tme seres. Gve tht ths reltvely smple model mmcs t lest qulttve wy some ey ssues of the flud-structure problem, t could be potetlly useful for ftgue studes d vbrto lyss of rottg bldes turbomchery. Keywords: Bem vbrto, flud-structure problem, modl tercto, self-excted bem, spectrl method, turbe blde vbrto, v der Pol osclltor. RESUMEN E este trbo se lz el comportmeto bo vbrcó y el tercmbo de eergí etre los modos ormles de u brr elástc utoexctd co u extremo fo y el otro lbre. Pr modelr est clse de brr se le greg drectmete el térmo de mortgumeto de u oscldor v der Pol l ecucó de u brr elástc lel, obteédose u sol ecucó dferecl prcl. Pr resolver est ecucó se us el método espectrl. E el álss se cosder tres modos de vbrcó, y los vlores de l costte de uto-exctcó se vrí modo de cubrr u comportmeto l vbrcó desde lel hst o lel. Ls múltples frecuecs de l brr o lel se determ medte el espectro de potecs de ls seres de tempo del extremo lbre. Ddo que este modelo reltvmete smple reproduce, l meos culttvmete, lguos spectos clve del problem fludo-estructur, puede ser poteclmete útl pr estudos de ftg y álss de l vbrcó de álbes rottoros e turbomqur.. Itroducto Vbrto of turbe bldes turbomchery my cuse flure by ftgue [,]. Expermetl lyss of blde vbrto s very dffcult becuse sesors re trusve d terfere wth mche operto. lso, ocotct mesuremet techques hve bee proposed; however, frequetly, they re expesve [3]. O the other hd, umercl studes of blde vbrto requre lot of computtol power gve tht the fludstructure problem must be solved [4-6]. From forml pot of vew, the flud-structure problem requres the umercl soluto of the Nver- Stoes equtos coupled wth the equtos of elstcty for the structure usg Drect Numercl Smulto d Fte Elemet techques, respectvely; however, both pproches c be very expesve from computtol pot of vew [7,8]. The, smplfed model whch smultes the flud-structure tercto of turbe blde, but t the sme tme ot be very computtolly demdg, s requred. Frequetly turbe bldes re represeted the lterture s elstc clmped-free bems, both umercl [9-4] d expermetl [5] studes. Self-exctto c be te to ccout by troducg oler dmpg term the clsscl ler bem equto to smulte the fludstructure tercto turbe bldes [3]. Ths s ustfed by fct tht the turl frequecy of the self-excted osclltor s proportol to the free strem velocty d the structure moto tercts Jourl of ppled Reserch d Techology 7

2 Vbrto lyss Of Self Excted Elstc Bem, M.. Brró Mez, 7 39 wth the osclltor [6]. The v der Pol osclltor s oe tht hs bee used the lterture: [7] t s suggested tht v der Pol type osclltor c be used to represet the tme-vryg forces o cylder due to vortex sheddg. I [8] ths osclltor s used for the study of eroelstc system possessg lmt cycle osclltos. lso, [9] v der Pol-bsed model tht cptures my of the fetures see expermetl results of elstclly mouted cylder exposed to wd s troduced. Sometmes, self-exctto fludstructure problems s cosdered by troducg ddtol v der Pol equto, whch s med the lterture s we-osclltor model [6,0]. Both the bem equto d the we-osclltor model re coupled through the ccelerto term d they re smulteously solved usg umercl techques,.e. fte dffereces [6] d collocto []. Ufortutely, we-osclltor models re best suted for log flexble sleder structures exposed to vortex duced vbrto, d therefore they hve bee mly ppled the oce egeerg feld structures such s moorg cbles d ol rsers [6,]. I ths wor the trset d log-term vbrto behvor d the eergy exchge mog modes of clmped-free self-excted elstc bem re lyzed. v der Pol self-exctg term s drectly dded to the equto of ler bem to represet through sgle prtl dfferetl equto the flud-structure tercto problem d model olertes of turbe blde. The prmeter tht dctes the stregth of the dmpg s the self-exctto costt,. By mes of the spectrl method, the prtl dfferetl equto s trsformed to set of coupled oler ordry dfferetl equtos whch re umerclly solved. Gve tht hgh vbrto mpltudes ssocted to low frequeces re sgfct for flure by ftgue [3], to smplfy the computtol ts d the lyss of the modl tercto, ust three vbrto modes re cosdered. Vlues of re vred from 0 to 0 order to study from ler to oler the vbrto behvor of the self-excted bem. For Є[0,0.33], sgle vbrto frequecy s preset d the bem rems vbrtg the tlly excted mode,.e. o eergy trsfer betwee modes rses d the bem behves ler fsho. For >0.33 multple frequeces re preset d the vbrto behvor s oler. I ths cse the power spectrl desty bsed o the bcwrd Dscrete Fourer Trsform of the bem free-ed tme seres s employed to determe the domt frequecy.. Mthemtcl model Formlly, tclg the flud-structure problem requres the umercl soluto of the Nver- Stoes equtos [6-8]. Ths procedure results expesve from computtol pot of vew; therefore, ew less computtolly expesve models to study flud-structure tercto turbe blde d elstc bem vbrto problems re eeded. To smplfy the vbrto lyss, turbe bldes re frequetly modeled the lterture s elstc clmped-free bems umercl [9-4] d expermetl [5] studes. The requred fludstructure tercto s te to ccout by troducg self-exctg fucto, such s the gve by the v der Pol osclltor, the clsscl ler bem equto. Ths s ustfed by fct tht the turl frequecy of the self-excted osclltor s proportol to the free strem velocty d the structure moto tercts wth the osclltor [6]. Besdes, v der Pol type osclltor c be used to represet the tme-vryg forces o cylder due to vortex sheddg [7]. The trsverse moto of elstc bem of homogeeous secto d propertes s govered by Equto [4]: wth 4 y 4 x EI c y 0 t () () where x s the dstce from the root, y(x,t) s the trsverse dsplcemet, t s tme, d E, I, ρ d c re the Youg's modulus, the momet of ert, the mss desty per ut volume, d the cross sectol re of the bem, respectvely. Defg the o-dmesol vrbles X=x/L, Y=y/L, τ= 8 Vol.8 No. ugust 00

3 Vbrto lyss Of Self Excted Elstc Bem, M.. Brró Mez, 7 39 t/l, where L s the bem legth, the followg s obted: 4 Y Y 0 4 X (3) The v der Pol equto s well ow secod order ordry dfferetl equto used 98 by B. v der Pol d J. v der Mr [5,6] to represet, by frst tme, o-electrcl oler osclltors. s tme hs elpsed, t hs bee wdely used my dscples to model y d of relxto osclltor. I ths wor the followg verso of the v der Pol equto s proposed: 3. Boudry codtos clmped-free bem, self-excted or ot, s bem tht s oly supported o oe of ts eds whle the other ed s free. Mthemtclly, for ths d of bem the boudry codtos re s follows [4]. For the clmped ed, X=0: Y(0) 0 (6) Y(0) 0 X d for the free ed, X=L: (7) Y Y Y Y 0 (4) Y(L) X 0 (8) where s the self-exctg costt, whch s prmeter tht mesures the stregth of the oler dmpg [7]. Besdes, self-susted osclltos re provded by the term Y -. 3 Y(L) 3 X 0 (9) I ths wor the oler term (Y -) of the v der Pol equto s drectly dded to the ler o-dmesol bem equto to model the selfsusted osclltos d clude the fludstructure tercto. s stted bove, ths s ustfed by the fct tht the v der Pol osclltor hs bee used the lterture for log tme to represet the tme-vryg forces o cylder due to vortex sheddg [7,9]. The, by ddg the oler dmpg d the self-sustg terms of Eq. (4) to Eq. (3), sgle prtl dfferetl equto rses: 4 X Y 4 Y Y Y 0 (5) Eq. (5) supposedly rets two ey ssues of the flud-structure problem: the oler flud-structure tercto d the elstc respose of the structure. These ssues re resposble of the complex vbrto behvor of rottg bldes turbomchery. 4. Spectrl soluto To determe the turl frequeces d orml vbrto modes of the self-excted bem, the spectrl method s employed. Ths method s bsed o the decomposto of prtl dfferetl equto to fte summto of two seres of orthogol fuctos [8]. The, the dmesoless trsverse dsplcemet Y(τ,X) of the bem c be expressed terms of the turl vbrto modes s [9],X Y ( ) (X) (0) where (τ) s tme depedet coeffcet d Φ (X) s the th orml vbrto mode of the bem. The ssumpto of orthogolty requres tht [4] for. L, dx 0 () 0 Jourl of ppled Reserch d Techology 9

4 Vbrto lyss Of Self Excted Elstc Bem, M.. Brró Mez, 7 39 Vol.8 No. ugust (7) X X X X X cosh sh cos s ) ( 4 3 () where β s the egevlue correspodg to the th mode, d,, 3 d 4 re costts ssocted to the th mode, whose vlues re determed from the boudry codtos. Moreover, t c be verfed tht Eq. () stsfes the ortogholty codto gve by Eq. () [4]. The followg reltoshp rses whe Eq. () s four tmes derved respect to X: (X) X) ( 4 '''' (3) where Φ'''' s the fourth dervtve of Φ respect to X.Substtutg Eq. (0) to Eq. (5) gves ' '' '''' 0 (4) where ' d '' re the frst d the secod tme dervtves of, respectvely, d s the umber of vbrto modes cosdered. Expdg d rerrgg the bove equto yelds '''' ' '' 0 (5) Substtutg Eq. (3) to Eq. (5), tg er product gst Φ d rerrgg t s obted: 0 f 3 ' '' (6) Eq. (6) represets set of coupled secod order oler ordry dfferetl equtos. oeffcets of such equtos re gve by,,,, (8) soluto for Φ(X) of Eq.(0) for the oler bem terms of combto of ses, coses, hyperbolc ses d hyperbolc coses fuctos c be ssumed of the form [4]

5 Vbrto lyss Of Self Excted Elstc Bem, M.. Brró Mez, , (9) ' f m m, m, (0) m 5. Egevlues d egefrequeces Egevlues β correspodg to the th mode c be determed by umerclly solvg the followg expresso [4] L cosh L 0 cos () d the gulr egefrequeces ω ssocted to the correspodg egevlues re estmted from [4] () Substtutg Eq. () to the boudry codtos X=0 gve by Eqs. (6-7) gves 3 4 (3) (4) osderg the boudry codtos X=L defed by Eqs. (8-9), the followg expresso for s obted (5) 6. Results d dscusso To smplfy the computtol ts d the lyss of the modl tercto, ust three vbrto modes re cosdered Eq. (0). From more prctcl pot of vew, ths s ustfed by the fct tht the frst modes hve low frequeces d hgh vbrto mpltudes, fctors whch re sgfct flure by ftgue of turbe bldes [3]. Eq. () s umerclly solved for the egevlues β usg the frst order Newto-Rphso procedure. Egefrequeces ω ssocted to the egevlues re determed from Eq. (). Tble shows the vlues of the egevlues d ther correspodg egefrequeces for the frst three vbrto modes of the self-excted bem ssumg the dmesoless vlue L=. By fxg = d clcultg from Eqs. (5) d (6), vlues of 3 d 4 re esly determed from Eqs. (3) d (4). I Tble the vlues of the bove costts re show. The three turl vbrto modes re clculted from Eq. () usg the vlues of β d from Tbles d Tble, respectvely. The tme depedet coeffcets of Eq. (0) re obted by umerclly solvg Eq. (6) through the clsscl Fourth Order Ruge- Kutt method usg dmesoless tme step of τ =0-6. where e L s sh L cos L L s L (6) th Mode Β F, Hz Tble. Egevlues d egefrequeces of the cosdered vbrto modes. Jourl of ppled Reserch d Techology 3

6 Vbrto lyss Of Self Excted Elstc Bem, M.. Brró Mez, 7 39 th Mode Tble. Numercl vlues of the costts of Eq. (). Oce (τ) d Φ (X) re ow, the bem dmesoless trsverse dsplcemet Y(τ,X) s determed from Eq. (0). dmesoless tegrto tme of 000 s cosdered for the logterm rus gve tht the umercl smultos showed tht ths tme the vbrto ptter of the bem rems stble. trcg pot s locted t the free-ed of the bem correspodg to X=L=. Through power spectrl desty lyss bsed o the bcwrd Dscrete Fourer Trsform, sgles or multple frequeces of the free-ed tme seres re determed. 6. Ler bem For =0, oe hs the clsscl d well-ow ler bem whose vbrto modes re ucoupled d o eergy trsfer s preset mog turl vbrto modes. The self-excted bem rems vbrtg, trset or stedy stte, the mode tlly excted. For Є[0,0.7], the self-excted bem stll behves s ler oe; however, for Є[08,0.33], some eergy trsfer mog modes s observed whe the frst mode s excted. Ths trsfer s demostrted through the presece of two frequeces of the free-ed tme seres of Fg. for =0.8, where t c be pprected tht the bem vbrto cossts of hgh frequecy compoet rdg over low frequecy compoet. Besdes, Fg. shows strogly modulted tercto betwee wdely spced modes,.e. the frst d the thrd modes. Ths behvor hs bee reported ctul bems wth exterl exctto [3]. The domce of frequecy of 0.56 Hz belogg to the frst mode becomes evdet the power spectrum of the bove tme seres show Fg., where smll pe of 9.8 Hz, whch correspods to the thrd vbrto mode, s preset. Ths d of power spectrum s reported ctul stem turbe bldes []. I spte of the presece of ddtol frequecy, for the self-excted bem the log-term domt frequecy correspods to the tlly excted mode,.e. the frst oe. Fgure. Tme seres for =0.8. The frst mode s tlly excted. 3 Vol.8 No. ugust 00

7 Vbrto lyss Of Self Excted Elstc Bem, M.. Brró Mez, 7 39 Fgure. Power spectrum correspodg to the tme seres of Fg.. Whe the secod or the thrd modes re excted for Є[0.8,0.33], the self-excted bem rems vbrtg the correspodg mode, d therefore the bem behves ler fsho. 6. Noler bem Trsfer of eergy mog vbrto modes s chrcterstc of oler bems [3]. For vlues of the self-exctg costt beyod crtcl vlue, the cosdered oler self-excted bem vbrtes complex wy. Ths behvor becomes ppret the form of vbrto ttrctors wth multple frequeces. Numercl results show tht the domt frequecy of prtculr ttrctor depeds o the vlue of the self-exctg costt d o the tlly excted mode. Due to the eergy trsfer mog modes, the log-term domt frequecy does ot lwys correspod to the mode tlly excted. I Fg. 3 the domt frequeces of the selfexcted bem t log tmes for Є[0,0] whe ech vbrto mode s seprtely excted re show. Smlrly to the cse of smll vlues of the selfexctg costt, Fg. 3 shows tht the oler self-excted bem resembles ler oe whe the secod d the thrd modes re dvdully excted. O the cotrry, rch oler log-term behvor s exhbted by the self-excted bem whe ust the frst mode s excted. The followg results d dscusso correspod to ths cse. Fgure 3. Bem domt frequeces for log tmes s fucto of the self-exctg costt. Excted vbrto modes: thc le, frst mode; dotted le, secod mode; dshed le, thrd mode. Jourl of ppled Reserch d Techology 33

8 Vbrto lyss Of Self Excted Elstc Bem, M.. Brró Mez, 7 39 For Є[0.34,5.76], eergy s trsferred from the frst mode to the thrd oe, d the ttrctor hs sgle frequecy of 9.8 Hz. The free-ed tme seres d the power spectrl desty lyss dcte tht the bove frequecy s sgle oe; therefore, o other frequecy ppers for the foremetoed rge of vlues of. Fg. 3 shows tht for Є[5.77,6.], the ttrctor frequecy correspods to the frequecy of the secod mode,.e Hz. Ths mples tht eergy s trsferred from the frst to the secod mode. very complex log tme oler behvor s exhbted by the self-excted bem for Є[6.,0]. Fg. 3 dctes tht for ths rge of vlues of the domt frequecy of the ttrctor correspods to the frequecy of the frst mode; however, closer loo revels qusperodc behvor d the presece of multple frequeces, s c be observed Fgs. 4 d 5, whch show the free-ed tme seres d the power spectrum for =0, respectvely. Power spectrum of Fg. 5 uvels the exstece of severl frequeces for =0, beg the domt oe tht correspodg to the frst vbrto mode. The foremetoed complex vbrto behvor detected ths wor for oler self-excted bems s corroborted, qulttvely t lest, by the fct tht ctul turbe d compressor bldes exhbt multmodl tercto, the presece of multple vbrto frequeces d power spectrl destes smlr to tht show Fg. 5 [30-3]. Fgure 4. Tme seres for =0. The frst mode s tlly excted. 34 Vol.8 No. ugust 00

9 Vbrto lyss Of Self Excted Elstc Bem, M.. Brró Mez, 7 39 Fgure 5. Power spectrum correspodg to Fg Eergy trsfer mog modes The trsfer of eergy mog vbrto modes occurs oler bems, such s the selfexcted bem proposed ths wor. The trsto mog modes s observed the free-ed tme seres of Fgs. 6 d 7 for =3 d =6, respectvely, whch ust the frst mode s excted. I both cses ths trsto occurs from hgh mpltude-low frequecy mode to low mpltudehgh frequecy oe. For =3, roud 5 dmesoless tme uts re requred to complete the trsto, wheres 38 tme uts re requred for =6. Ths suggests tht, t lest for the two cses cosdered, dymcs s fster s the vlue of the self-exctg costt s cresed. Expermetl evdece of the eergy trsfer mog modes, deduced from the tme seres of Fgs. 6 d 7, hs bee reported bems wth exterl exctto [3,33]. Fgure 6. Trsto mog modes for =3. The frst mode s tlly excted. Jourl of ppled Reserch d Techology 35

10 Vbrto lyss Of Self Excted Elstc Bem, M.. Brró Mez, 7 39 Fgure 7. Trsto mog modes for =6. The frst mode s tlly excted. 7. oclusos model of clmpled-free elstc bem wth drectly dded v der Pol self-exctg term to smulte the vbrto behvor of turbe blde s proposed. Regos of ler d oler vbrto behvor of the bem re foud terms of vlues of the self-exctg costt. I the oler rego, multple frequeces the log term vbrto behvor re detected. Trsfer of eergy from low frequecy mode to hgh frequecy oe s foud. The dymcs of the bove trsto s fster s the vlue of the self-exctg costt s cresed. From qulttve pot of vew, the greemet betwee the obted results d expermetl results reported the lterture for ctul bems d turbe bldes s good; therefore, t c be cocluded tht the proposed model s potetlly useful for ftgue studes d vbrto lyss of bldes turbomchery. Refereces [] Y.L. Lu, R..K. Leug, d R.M.. So. Vortexduced vbrto effect o ftgue lfe estmte of turbe bldes. Jourl of Soud d Vbrto, Vol. 307, 006, pp [] Z. Mzur, R. Grc-Illescs, J. gurre-romo, d N. Perez-Rodrguez. Stem turbe blde flure lyss. Egeerg Flure lyss, Vol. 5, 008, pp [3] P. Beuseroy d R. Legelle. Notrusve turbomche blde vbrto mesuremet system. Mechcl Systems d Sgl Processg, Vol., 007, pp [4] Y.Y. Jg, S. Yoshmur, R. Im, H. Ktsur, T. Yoshd, d. Kto. Qutttve evluto of flowduced structurl vbrto d ose turbomchery by full-scle wely coupled smulto. Jourl of Fluds d Structures, Vol. 3, 007, pp [5].G. Rodrguez, E. Egusquz, d I.F. Stos. Frequeces the vbrto duced by the rotor sttor tercto cetrfugl pump turbe. SME Jourl of Fluds Egeerg, Vol. 9, 007, pp [6] R. Volette, E. de Lgre, d J. Szydlowsy. omputto of vortex-duced vbrtos of log structures usg we osclltor model: omprso wth DNS d expermets. omputers d Structures, Vol. 85, 007, pp [7] K.B. Sugset d.m. Lrse. Drect umercl smulto d expermetl vestgto o suppresso of vortex duced vbrtos of crculr cylders by rdl wter ets. Flow, Turbulece d ombusto, Vol. 7, 003, pp [8] E. Gulmeu d P. Queutey. Numercl smulto of vortex-duced vbrto of crculr cylder wth low 36 Vol.8 No. ugust 00

11 Vbrto lyss Of Self Excted Elstc Bem, M.. Brró Mez, 7 39 mss-dmpg turbulet flow. Jourl of Flud d Structures, Vol. 9, 004, pp [9] G.S. Psreo d Y.S. Vorobev. Issues of smulto of turbomche blde vbrto. Stregth of Mterls, Vol. 3, 000, pp [0] J. L, S.T. Le, d Z. e. Numercl lyss of dymc behvor of strem turbe blde group. Fte Elemets lyss d Desg, Vol. 35, 000, pp [] G. Dmtrds, I.B. rrgto, J.R. Wrght, d J.E. opper. Blde-tp tmmg mesuremet of sychroous vbrtos of rottg blded ssembles. Mechcl Systems d Sgl Processg, Vol. 6, 00, pp [] S. Kumr, N. Roy, d R. Ggul. Motorg low cycle ftgue dmge turbe blde usg vbrto chrcterstcs. Mechcl Systems d Sgl Processg, Vol., 007, pp [3] M.. Brro d M. Se. Sychrozto of coupled self-excted elstc bems. Jourl of Soud d Vbrto, Vol. 34, 009, pp [4] P. Bseg d G. ruso. Optmzto of ssve vbrto cotrol scheme ctg o blded rotor usg homogezed model. Structurl d Multdscplry Optmzto, Vol. 39, 009, pp [5] J.S. Ro d. Sldh. Turbomche blde dmpg. Jourl of Soud d Vbrto, Vol. 6, 003, pp [6] R.D. Gbb d H. Beroy. overvew of modelg d expermets of vortex-duced vbrto of crculr cylders. Jourl of Soud d Vbrto, Vol. 8, 005, pp [7] R.E.D. Bshop d.y. Hss. The lft d drg forces o crculr cylder flowg flud. Proceedgs of the Royl Socety Seres, Vol. 77, 963, pp [8] Y.S. Lee,.F. Vs, L.. Bergm, d D.M. McFrld. Suppresso of lmt cycle osclltos the v der Pol osclltor by mes of pssve o-ler eergy ss. Structurl otrol d Helth Motorg, Vol. 3, 006, pp [9] R.T. Hrtle d I.G. urre. Lft-osclltor model of vortex duced vbrto. Jourl of the Egeerg Mechcs, Vol. 96, 970, pp [0].H.K. Wllmso d R. Govrdh. bref revew of recet results vortex-duced vbrtos. Jourl of Wd Egeerg, Vol. 96, 008, pp [] D. Lucor d M.S. Trtfyllou. Prmetrc study of two degree-of-freedom cylder subect to vortexduced vbrtos. Jourl of Flud d Structures, Vol. 4, 008, pp [] M.L. Fcchett, E. de Lgre, d F. Bolley. ouplg of structures d we osclltors vortexduced vbrtos. Jourl of Fluds d Structures, Vol. 9, 004, pp [3].H. Nyfeh. Noler Iterctos. Wley, New Yor, NY, 000. [4] K.L. Grff. Wve Moto Elstc Solds. Oxford Uversty Press, Lodo, UK, 975. [5] B. v der Pol d J. v der Mr. The hertbet cosdered s relxto oscllto, d electrcl model of the hert. Phlosophcl Mgze, Vol. 6, 98, pp [6] J. Grsm, F. Verhulst, d S. Shh. The Lypuov expoets of the v der Pol osclltor. Mthemtcl Methods ppled Sceces, Vol. 8, 005, pp [7] D.W. Stort d R.H. Rd. Dymcs of two strogly coupled relxto osclltors. SIM Jourl o ppled Mthemtcs, Vol. 46, 986, pp [8] J.S. Hesthve, S. Gottleb, d D. Gottleb. Spectrl Methods for Tme-Depedet Problems. mbrdge Uversty Press, mbrdge, UK, 007. [9].N. Yme, R. Tchouuego, d P. Wofo. No-ler dymcs of elstc bem uder movg lods. Jourl of Soud d Vbrto, Vol. 73, 004, pp [30] M. Zels d G. Zller. Nocotct vbrto mesuremets o compressor rotor bldes. Mesuremet Scece d Techology, Vol., 000, pp [3] M.L.J. Verhees. Expermetl Modl lyss of Turbe Blde. Treshp Report, Techsche Uverstet Edhove, Edhove, The Netherlds, 004. [3] E.D. ohe. Vbrto Detecto Turbomchery Usg No-cotctg Sesors. Mster's Thess, Msschusetts Isttute of Techology, mbrdge, M, 006. Jourl of ppled Reserch d Techology 37

12 Vbrto lyss Of Self Excted Elstc Bem, M.. Brró Mez, 7 39 [33] P. Mltr d.h. Nyfeh. O the trsfer of eergy betwee wdely spced modes structures. Noler Dymcs, Vol. 3, 003, pp cowledgmets Ths wor ws prtlly doe whle the uthor ws o sbbtcl leve from the Uversdd utóom Metropolt zcpotzlco (UM) d vstg the Uversty of Notre Dme (UND), Notre Dme, IN, US. He grtefully cowledges both the fcl support from the Progrm de poyo Estcs Sbátcs de Ivestgcó of the UM d the gudce of Professor Mhr Se from the Deprtmet of erospce d:.mechcl.egeerg.of.und. 38 Vol.8 No. ugust 00

13 Vbrto lyss Of Self Excted Elstc Bem, M.. Brró Mez, 7 39 uthor s Bogrphy Mguel Ágel BRRÓN-MEZ hemcl-metllurgcl Egeer from Uversdd Ncol utóom de Méxco. Mster of Scece chemcl egeerg from Uversdd utóom Metropolt Iztplp, Mexco. PhD mterls scece from Isttuto Poltécco Ncol, Mexco. Posdoctorl fellow t Uversdd Poltécc de tluñ, Brcelo, Sp. Sbbtcl fellow t Uversty of Notre Dme, Notre Dme, IN, US. Member of Sstem Ncol de Ivestgdores (Ntol System of Reserchers), Mexco. urretly, professor t Deprtmeto de Mterles of Uversdd utóom Metropolt zcpotzlco, Mexco. Jourl of ppled Reserch d Techology 39