Study on Vibration Response Reduction of Bladed Disk by Use of Asymmetric Vane Spacing (Study on Response Reduction of Mistuned Bladed Disk)

Transcription

1 Intenatonal Jounal of Gas ubne, Populson and Powe Systems Febuay 0, Volume 4, Numbe Study on Vbaton Response Reducton of Bladed Dsk by Use of Asymmetc Vane Spacng (Study on Response Reducton of Mstuned Bladed Dsk) Yasutomo Kaneko, Masak Ohta, Kazush Mo and Hohau Ohyama Depatment of Mechancal and Systems Engneeng Faculty of Scence and echnology, Ryukoku Unvesty -5 Yokotan, Ohe, Seta, Ohtsu, Sga 50-94, JAPAN Mtsubsh Heavy Industes, Ltd. ABSRAC It s well known that asymmetc vane spacng can esult n deceased levels of the exctaton at specfc fequences. In ths pape, the esonant esponse educton of mstuned bladed dsks due to asymmetc vane spacng s studed theoetcally fo the most pobable asymmetc vane, n whch the vane count of the uppe and lowe half s slghtly dffeent. Fst, a method fo pedctng the maxmum ampltude of the mstuned bladed dsk fo the asymmetc vane spacng s poposed. Second, a paametc study s caed out usng Monte Calo smulaton to clafy the vbaton esponse chaactestcs of the mstuned bladed dsk fo the asymmetc vane spacng. Fom these esults, t s concluded that asymmetc vane spacng s effectve fo educton of esonant ampltudes of a mstuned bladed dsk f a mult-esonance phenomenon does not appea. INRDUCION In a mult-stage tubo-machney, the nteacton between the vane and the blade geneates the exctaton foce on the blade, whch comes fom the wake of the upsteam vane o the potental feld of the upsteam/downsteam vane. he fundamental fequency of the exctaton foce due to the nteacton between the vane and the blade s the oto speed multpled by the vane count, and f the natual fequency of the blade s concdent wth the fequency of the exctaton foce, the esonant stess of the blade may become vey lage and may cause a blade falue due to HCF. Besdes the tadtonal methods to educe the esonant stess, t s well known that the asymmetc vane spacng can esult n deceased levels of the exctaton foce at specfc fequences [, ]. It seems that asymmetc vane spacng s a vey effectve method fo contollng the blade esponse fo vaable speed engnes whee the axal oto span cannot be lengthened. As fo a tuned bladed dsk, whee dynamc popetes of ndvdual blades on a dsk s the same, the educton effect of the esonant stess due to asymmetc vane spacng has been studed theoetcally and expementally [3, 4]. Howeve, ealstc bladed dsks, although desgned mostly to be cyclcally symmetc, ae mstuned,.e. dynamc popetes of ndvdual blades ae dffeent. In spte that such mpefectons of mstuned bladed dsks ae usually small, t s well known that the loss of cyclc symmety popetes of the bladed dsk by mstunng can dastcally change the foced esponse levels. hee ae numeous papes studyng the mstunng effect. In the ealy papes [5, 6], the foced esponse of mstuned bladed dsk fo symmetc vane spacng has been studed Pesented at Intenatonal Gas ubne Congess 0 Osaka, Novembe 3-8, Osaka, Japan, IGC Revew completed Febuay 7, 0 by use of smple spng-mass model. Moden methods of the numecal analyss ae descbed n the ecent papes [7-9]. Howeve, as fa as authos know, few studes have been caed out on the educton effect of the foced esponse of the mstuned bladed dsk due to asymmetc vane spacng. In ths pape, the esonant esponse educton of mstuned bladed dsks due to asymmetc vane spacng s studed theoetcally fo the most pobable asymmetc vane, n whch the vane count of the uppe and lowe half s slghtly dffeent. Fst, a method fo pedctng the maxmum ampltude of the mstuned bladed dsk fo the asymmetc vane spacng s poposed. Second, a paametc study s caed out usng Monte Calo smulaton to clafy the vbaton esponse chaactestcs of the mstuned bladed dsk fo the asymmetc vane spacng. Fom these esults, t s concluded that asymmetc vane spacng s effectve fo educton of esonant ampltudes of a mstuned bladed dsk. And except fo the mult-esonance phenomena, the maxmum esponse of the mstuned bladed dsk to asymmetc vane spacng can be oughly pedcted by the maxmum esponse of the mstuned bladed dsk to symmetc vane spacng multpled by the esponse educton effect fo fee-standng blade wthout ccumfeental couplng [3]. ANALYSIS MEHOD As fo the asymmetc vane spacng, usually the whole vanes n the stage ae equally dvded nto N s segments, and the followng technques ae used to make asymmetc vane spacng. () Vane count n N s segments s slghtly alteed (Vane count method). () Vane count n N s segments s equal but the vane ptch between segments s changed by shftng segments ccumfeentally wth espect to each othe (angental offset method). (3) All vane ptches ae alteed at andom (Random ptch method). Although these methods ae well known, t seems that only the vane count method, whee the whole vanes ae dvded nto two segments (N s =, uppe half and lowe half) and the vane count n the uppe and lowe half s slghtly alteed, s most pactcal and has been appled to actual engnes, takng the cost of manufactung and mantenance, dsadvantageous effect on the exctaton foce of the lowe engne odes, and the stablty of the flow such as suge nto consdeaton. heefoe, n ths pape, the analytcal method fo pedctng the blade vbaton esponse of the asymmetc vane spacng by use of vane count method (N s = ) s manly descbed, but the same pocedue can be used fo othe methods. Blade-Vane Inteacton Foce fo Asymmetc Vane Spacng When the vane counts n the uppe and lowe half ae N and N, espectvely, two knds of blade-vane nteacton analyses wth 3D Copyght 0 Gas ubne Socety of Japan 35

2 CFD ae fst caed out fo the symmetc vane spacng, whee the whole vane counts ae N and N. hen, the pessue fluctuaton on the blade suface s obtaned n the fom of the tme-hstoy wave by connectng both esults. Fnally, the magntude and phase of the pessue fluctuaton on the blade suface coespondng to each hamonc component (engne ode) ae calculated by Foue analyss of the tme-hstoy wave as expessed by Eq.(). a0 F ( t) cos sn a H Ht bh Ht H a0 P H cos( t H ) H Whee, {F(t)} denotes the exctaton foce on the nodes of the blade suface caused by the blade-vane nteacton, the otaton speed of the oto, H the hamonc (engne ode) of the exctaton foce, =H the exctaton fequency. In ths study, the vbaton esponse to each engne ode s calculated sepaately by the modal analyss method descbed n the next secton, and the steady esponse to the asymmetc vane spacng s obtaned by summng up the calculated esults. he dect cuent component of the exctaton foce n Eq.() s neglected n the vbaton esponse analyss, because t has no nfluence on the vbaton esponse. Vbaton Response Analyss of Mstuned Bladed Dsk by Modal Analyss A bladed dsk wth contnuous ng type stuctue, whee the blades ae contnuously coupled at shouds by blade untwst due to centfugal foce when the blades otate at hgh speed, s modeled by the equvalent spg-mass model n Fg.. Heeafte, the bladed dsk wth contnuous ng type stuctue s efeed to as ISB (Integal Shoud Blade). In Fg., m, k and c ae the equvalent mass, stffness and dampng coeffcent of the blade, espectvely, whle m and k 3 ae the equvalent mass and stffness of the dsk, espectvely. k s the couplng stffness of the shoud, and k 4 s the couplng stffness of the dsk n the ccumfeental decton. he vaable wth supescpt means that the values fo ndvdual blades ae slghtly dffeent. In the analyss of the FSB (fee-standng blade) stuctue wthout shoud and the analyss of the GB (gouped blades) stuctue whee a few blades ae connected by shoud, the same equvalent model can be used, f the couplng stffness k s set to be zeo between blades o goups. When the bladed dsk n Fg. s otatng unde the exctaton foce of Eq.(), the equaton of the moton of the bladed dsk s expessed by Eq.(). he dampng tem s dopped n Eq.(), because the modal dampng s ntoduced nto the equaton of the moton late. Shoud Blade Dsk M x P x Q x Q x f Rotaton Fg. Equvalent spng mass model of bladed dsk () () Whee, M, P, In Eq.(3), {x } denotes the dsplacement vecto of -th blade, and {x 0 } = {x N } and {x N+ } = {x } ae satsfed, whee N s the blade count of the whole bladed dsk. {f e } s the vecto of the modal exctaton foce on the blade, whch can be calculated by the blade-vane nteacton foce {P H } n Eq.() and the vbaton mode calculated by FEA (Fnte Element Analyss). denotes the angula fequency of the exctaton foce, and s the phase angle caused by the otaton of the bladed dsk, whch can be expessed by Eq.(4). Assumng the soluton of Eq.() by Eq.(5), and applyng the conventonal modal analyss method to Eq.(), the fequency esponse to the exctaton engne ode H s obtaned as Eq.(6). Whee, m 0 Q, x x x, k 0 0 k4 H H, H ( ), H N jt x x x x X e X 0 m N N F H H ( ) H ( ) F fe fee fee fee (7) In Eq.(6), { } and ae the egenvecto and the natual fequency of the bladed dsk obtaned fom the egenvalue equaton, n whch the exctaton foce n Eq.() s set to be zeo. k denotes the modal stffness, and the modal dampng ato, whee the subscpt s the ode of the vbaton modes. When the mstuned bladed dsk s otatng though symmetc vane spacng, the vbaton esponse chaactestcs s descbed n Fg.(a), whee the domnant engne ode of the exctaton foce s just one, H. heefoe, the vbaton esponse of the bladed dsk can be calculated dectly by Eq.(6). On the othe hand, when the mstuned bladed dsk s otatng though asymmetc vane spacng, the vbaton esponse chaactestcs s epesented by Fg.(b), whee the bladed dsk s excted by a few domnant engne odes. heefoe, fst, the vbaton esponse to each engne ode s calculated by Eq.(6), and then, the vbaton esponse to the asymmetc spacng s obtaned by summng up the esults. In bef, the mstuned bladed dsk otatng though asymmetc vane spacng s obtaned as follows. () he blade-vane nteacton foce s obtaned by 3D CFD fo a bladed dsk otatng though asymmetc vane spacng, and the ampltude of the exctaton foce coespondng to each engne ode s calculated by Foue analyss. In ths study, fo smplcty, the blade-vane nteacton foce s assumed by Eq.(8), and s expanded to Foue sees. k ( j ) k k k f 0 j j j N k j( t ) j( t ) f f e e e k k k3 k4 (3) (4) (5) (6) 36

3 f fe f e sn(nt ) ( n uppe half ) sn(n t ) ( n lowe half ) f 0 e (8) () Based on the nfomaton obtaned fom FEA esults and expemental data, the modal paametes of the tuned bladed dsk (m, m, k, k, etc.) ae detemned n such a manne that the natual fequency and the logathmc decement of the analyss model become equal to those of an actual bladed dsk fo the nodal damete mode of nteest.. (3) he natual fequency f of the ndvdual blade on a dsk s selected fom the nomal dstbuton of the blade alone fequency, N( f b, f ) ( f b : mean value of blade alone fequency, f : standad devaton), and k s obtaned fom k = m (f ). Mstuned bladed dsk s assembled usng k. (4) Fequency esponse of a mstuned bladed dsk to each engne ode s calculated by Eq.(6), and then the fequency esponse of the bladed dsk otatng though asymmetc vane spacng s obtaned by summng up the esults as shown n Fg.. (5) Afte many epeated calculatons of pocedue (3) and (4) (one hunded tmes n ths study), the calculated esults ae pocessed statstcally. he maxmum ampltude and the standad devaton of the vbaton esponse of the mstuned bladed dsk ae evaluated (Monte Calo smulaton). ANALYSIS RESULS Fequency esponse analyss of a mstuned bladed dsk otatng though symmetc and asymmetc vane spacng (vane count method, N s =) was caed out accodng to pocedue descbed n the pevous secton. Fst, n ode to compae the vbaton esponse chaactestcs of the FSB and ISB stuctue to asymmetc vane spacng, whch ae the typcal bladed dsks used n tubomachney, the fequency esponse analyss was caed out fo two knds of models. he paametes of bladed dsks analyzed ae shown n able. he modal paametes of bladed dsks n able ae detemned fom the nfomaton fom FEA esults and expemental data. Fgue 3 shows the natual fequency of the bladed dsks shown n able. able Paametes of bladed dsk analyzed N m m k k k 3 k 4 ISB H FSB Fg. 3 Natual fequency of tuned bladed dsk (a) Response to symmetc vane (b) Response to asymmetc vane Fg. ypcal esponse of mstuned bladed dsc to symmetc and asymmetc vane H 3 H H Analyss Results of bladed dsk wth FSB stuctue Vbaton esponse of FSB stuctue to symmetc vane spacng. In ode to get the efeence data fo compason, fst, the vbaton esponse analyss of bladed dsks to symmetc vane spacng was caed out. One hunded mstuned bladed dsks wee assembled accodng to the pocedue explaned n the pevous secton, and the fequency esponses of all bladed dsks wee calculated. Pocessng the calculated esults statstcally, the maxmum value and the dstbuton of the esponses wee evaluated. Fgue 4 shows the typcal esult of the fequency esponse analyss of the 0th engne ode fo the FSB stuctue, whee the standad devaton of the blade alone fequency ( f / fb ) s 3 %. In Fg. 4, fequency esponses of all blades on a dsk (80 blades) ae supeposed. Fgue 5 shows the hstogam of the maxmum ampltude fo all blades on one hunded mstuned bladed dsks (80 blades tmes 00 dsks). he odnates of Fg. 4 and the abscssa of Fg.5 ae nomalzed by the maxmum esponse (the esonant ampltude) of the tuned bladed dsk. Fgue 6 shows the elatonshp between the standad devaton of the blade alone fequency and the maxmum and mean ampltude of the mstuned bladed dsks. In Fg.6, the coeffcent of the vaaton of the esponse (the ato of the standad devaton and the mean value) s also plotted. As shown n Fg. 4 to Fg. 6, the FSB stuctue s vey senstve to the mstunng. If the fequences of the blades on a dsk ae slghtly dffeent, the esponses of the blades change dastcally, and the maxmum ampltude eaches nealy twce that of the tuned bladed 37

4 No.ofblades dsk as shown n Fg. 5 and Fg. 6. he maxmum ampltude eaches ts maxmum value n the vcnty of 0.5 % standad devaton of the blade alone fequency, and f the standad devaton of the blade alone fequency becomes geate than ths, the maxmum ampltude begn deceasng as shown n Fg. 6. On the othe hand, the mean ampltude of the mstuned bladed dsk eaches ts maxmum at 0 % standad devaton of blade alone fequency (at the tuned bladed dsk). In othe wods, the mean ampltude of the mstuned bladed dsk s always smalle than that of the tuned bladed dsk. hs s because n the mstuned bladed dsk, lage ampltudes appea only n a few specfc blades, whle n the most othe blades, the esponse becomes smalle than that of the tuned bladed dsk. Fg. 4 Fequency esponse of mstuned bladed dsk ( f = 3 %, H=0) f = 0.5% f = 3% of the mstuned bladed dsks. In Fg.9, the coeffcent of the vaaton of the esponse (the ato of the standad devaton and the mean value) s also plotted. Roughly evaluatng, the vbaton esponse chaactestcs of the mstuned bladed dsk to the exctaton foce of the 7nd engne ode s the nealy same as that to the exctaton foce of the 0th engne ode. In the fequency esponse to the 7nd engne ode, howeve, the maxmum ampltude eaches ts maxmum value n the vcnty of.0 % standad devaton of blade alone fequency, and the maxmum ampltude eaches moe than twce that of the tuned bladed dsk as shown n Fg. 9. he dffeence of the vbaton esponse chaactestcs of the mstuned bladed dsk due to the engne ode of the exctaton foce s closely elated to the chaactestcs of the natual fequency of the bladed dsk shown n Fg. 3 [5, 6]. he featue of the calculated fequency esponse of the mstuned bladed dsk fo symmetc vane spacng s consstent wth those of publshed papes. Fg.7 Fequency esponse of mstuned bladed dsk ( f = 3 %, H=7) f = 0.5% f = 3% Nomalzedampltude Nomalzedampltude Fg.5 Hstogam of blade ampltude (H=0) No.ofblades Nomalzedampltude Nomalzedampltude Fg. 8 Hstogam of blade ampltude (H=7) Meanampltude Max.ampltude Coeffcentofvaaton Fg. 6 Fequency devaton and maxmum ampltude (H=0) Max.ampltude Meanampltude Coeffcentofvaaton he esults of vbaton esponse analyss to the 7nd engne ode ae shown n Fg. 7 to Fg. 9. Fgue 7 shows the typcal esult of the fequency esponse analyss of the 7nd engne ode fo the FSB stuctue, whee the standad devaton of the blade alone fequency ( f / fb ) s 3 %. In Fg. 7, fequency esponses of all blades on a dsk (80 blades) ae supeposed. Fgue 8 shows the hstogam of the maxmum ampltude fo all blades on one hunded mstuned bladed dsks (80 blades tmes 00 dsks). Fgue 9 shows the elatonshp between the standad devaton of the blade alone fequency and the maxmum and mean ampltude Fg.9 Fequency devaton and maxmum ampltude (H=7) Vbaton esponse of the FSB stuctue to asymmetc vane spacng. Fgue 0 shows the esult of the fequency esponse analyss to the asymmetc vane spacng (vane count of the uppe half N =vane count of the lowe half N =9) fo the FSB stuctue, whee the standad devaton of the blade alone fequency ( f / fb ) s 3 %. In Fg. 0, fequency esponses of all blades on a 38

5 dsk (80 blades) ae supeposed. In Fg. 0, the abscssa s the otaton speed, and the odnate s nomalzed by the maxmum esponse (the esonant ampltude) of the tuned bladed dsk to the exctaton foce of the 0th (H=N +N ) engne ode. Fgue shows the hstogam of the maxmum ampltude fo all blades on one hunded mstuned bladed dsks (80 blades tmes 00 dsks). he abscssa of Fg. s also nomalzed by the maxmum esponse (the esonant ampltude) to the exctaton foce of the 0th engne ode of the tuned bladed dsk. Fgue shows the elatonshp between the standad devaton of the blade alone fequency and the maxmum and mean ampltude of the mstuned bladed dsks. In Fg., the coeffcent of the vaaton of the esponse (the ato of the standad devaton and the mean value) s also plotted. Compang Fg. 4 wth Fg. 0, t can be sad that because the bladed dsk otatng though asymmetc vane spacng s excted by exctaton foces caused by a few engne odes, the numbe of the peaks n the fequency esponses s lage than that to symmetc vane spacng, and the esponse chaactestcs become vey complcated. As shown n Fg. and Fg., howeve, the maxmum and mean ampltude of the mstuned bladed dsk s deceased by 40 % to 50 %, compang to those to symmetc vane spacng. hat s, compang Fg. 6 wth Fg., t s sad that although the geneal tendency of the maxmum ampltude, the mean ampltude and the coeffcent vaaton to the fequency devaton s the nealy same fo both of symmetc and asymmetc vane spacng, the absolute value of esponses to asymmetc vane spacng s deceased by 40 % to 50 %. hs s because although the bladed dsk though asymmetc vane spacng s excted by a few engne odes, each exctaton foce s smalle than that caused by symmetc vane spacng. heefoe, even f the esponse to each engne ode s summed up as shown n Fg. (b), the total esponse to asymmetc vane spacng s less than that to symmetc vane spacng, unless the mult-esonances occu. In efeence [3], the esponse educton effect by asymmetc vane spacng fo the FSB stuctue wthout the ccumfeental couplng) s summazed as shown n Fg. 3. In Fg. 3, the esponse educton effect s calculated fo a fee-standng blade, whee the flexblty of the dsk (the couplng of a bladed dsk n the ccumfeental decton) s neglected. he abscssa of Fg. 3, N vane denotes the poduct of the whole vane count (N vane =N +N ) and the logathmc decement of the blade. Because the value of N vane coespondng to Fg. s 0. (N vane =0, logathmc decement =0.0), the esponse educton effect fo the condton of Fg. s estmated to be 0.48 fom Fg.3. Consequently, fo the FSB stuctue whee the ccumfeental couplng s weak and the mult-esonances hadly occu, t s concluded that the maxmum esponse of the mstuned bladed dsk though asymmetc vane spacng (Fg.) can be estmated by the maxmum esponse of the mstuned bladed dsk to symmetc vane spacng (Fg. 6) multpled by the esponse educton effect n Fg. 3 (-R=0.5). No.ofblades Fg. 0 Fequency esponse of mstuned bladed dsk ( f = 3 %, N =, N =9) Nomalzedampltude f = 0.5% Fg. Hstogam of blade ampltude (N =, N =9) Fg. Fequency devaton and maxmum ampltude (N =, N =9) Nomalzedampltude f = 3% Max.ampltude Meanampltude Coeffcentofvaaton ReductoneffectR DampngpaameteN vane Fg. 3 Reducton effect of asymmetc vane [3] Fgue 4 shows the esult of the fequency esponse analyss to the asymmetc vane spacng (vane count of uppe half N =37vane count of lowe half N =35) fo the FSB stuctue, whee the standad devaton of the blade alone fequency ( f / fb ) s 3 %. In Fg. 4, fequency esponses of all blades on a dsk (80 blades) ae supeposed. In Fg. 4, the abscssa s the otaton speed, and the odnate s nomalzed by the maxmum esponse (the esonant ampltude) of the tuned bladed dsk to the exctaton foce of the 7nd (H=N +N ) engne ode. Fgue 5 shows the hstogam of the maxmum ampltude fo all blades on one hunded mstuned bladed dsks (80 blades tmes 00 dsks). he abscssa of Fg. 5 s also nomalzed by the maxmum esponse (the esonant ampltude) to the exctaton foce of the 7nd engne ode of the tuned bladed dsk. Fgue 6 shows the elatonshp between the standad devaton of the blade alone fequency and the maxmum and mean ampltude of the mstuned bladed dsks. In Fg.6, the coeffcent of the vaaton of the esponse (the ato of the standad devaton and the mean value) s also plotted. Roughly evaluatng, the vbaton esponse chaactestcs of 39

6 the mstuned bladed dsk to asymmetc vane spacng of N vane =7 (Fg. 6) s the nealy same as that of N vane =0 (Fg. ). As fo asymmetc vane spacng of N vane =7, the value of N vane n Fg. 3 s 0.7, and theefoe, the esponse educton effect s evaluated to be 0.40 fom Fg.3. Consequently, fo the case of N vane =7, t s concluded that the maxmum esponse of the mstuned bladed dsk though asymmetc vane spacng (Fg.6) can be estmated by the maxmum esponse of the mstuned bladed dsk to symmetc vane spacng (Fg. 9) multpled by the esponse educton effect n Fg. 3 (-R=0.60). devaton of the blade alone fequency ( f / fb ) s 3 %. In Fg. 7, fequency esponses of all blades on a dsk (80 blades) ae supeposed. Fgue 8 shows the hstogam of the maxmum ampltude fo all blades on one hunded mstuned bladed dsks (80 blades tmes 00 dsks). he odnates of Fg. 7 and the abscssa of Fg. 8 ae nomalzed by the maxmum esponse (the esonant ampltude) of the tuned bladed dsk. Fgue 9 shows the elatonshp between the standad devaton of the blade alone fequency and the maxmum and mean ampltude of the mstuned bladed dsks. In Fg. 9, the coeffcent of the vaaton of the esponse (the ato of the standad devaton and the mean value) s also plotted. As shown n Fg. 9, the ISB stuctue s vey nsenstve to the mstunng. If the fequences of the blades on a dsk ae dffeent, the esponses of the blades hadly change. he maxmum esponse of the mstuned bladed dsk becomes 0 % lage than that of the tuned bladed dsk at most n the pactcal ange of the standad devaton of the blade alone fequency. he featue of the calculated fequency esponse of the mstuned bladed dsk wth the ISB stuctue fo symmetc vane spacng s also consstent wth the fact ponted by many publshed papes. No.ofblades Fg. 4 Fequency esponse of mstuned bladed dsk ( f = 3 %, N =37, N =35) f = 0.5% f = 3% Nomalzedampltude Fg. 5 Hstogam of blade ampltude (N =37, N =35) Fg. 6 Fequency devaton and maxmum ampltude (N =37, N =35) Analyss Results of bladed dsk wth ISB stuctue. Response analyss of the ISB stuctue was caed out n the same pocedue as that of the FSB stuctue. Fst, the vbaton esponse analyss of bladed dsks to symmetc vane spacng was caed out to get the efeence data. One hunded mstuned bladed dsks wee assembled, and the fequency esponses of all bladed dsks wee calculated. Pocessng the calculated esults statstcally, the maxmum value and the dstbuton of the esponses wee evaluated. Fgue 7 shows the esult of the fequency esponse analyss of the 7nd engne ode fo the ISB stuctue, whee the standad Nomalzedampltude Meanampltude Max.ampltude Coeffcentofvaaton No.ofblades Fg. 7 Fequency esponse of mstuned bladed dsk (ISB, f = 3 %, H=7) Nomalzedampltude f = 0.5% Fg. 8 Hstogam of blade ampltude (ISB, H=7) Fg. 9 Fequency devaton and maxmum ampltude (ISB. H=7) Nomalzedampltude Max.ampltude Meanampltude Coeffcentofvaaton f = 3% 40

7 No.ofblades Fgue 0 shows the esult of the fequency esponse analyss to the asymmetc vane spacng (vane count of uppe half N =37vane count of lowe half N =35) fo the ISB stuctue, whee the standad devaton of the blade alone fequency ( f / fb ) s 3 %. In Fg. 0, fequency esponses of all blades on a dsk (80 blades) ae supeposed. In Fg. 0, the abscssa s the otaton speed, and the odnate s nomalzed by the maxmum esponse (the esonant ampltude) of the tuned bladed dsk to the exctaton foce of the 7nd (H=N +N ) engne ode. Fgue shows the hstogam of the maxmum ampltude fo all blades on one hunded mstuned bladed dsks (80 blades tmes 00 dsks). he abscssa of Fg. s also nomalzed by the maxmum esponse (the esonant ampltude) to the exctaton foce of the 7nd engne ode of the tuned bladed dsk. Fgue shows the elatonshp between the standad devaton of the blade alone fequency and the maxmum and mean ampltude of the mstuned bladed dsks. In Fg., the coeffcent of the vaaton of the esponse (the ato of the standad devaton and the mean value) s also plotted. Fg. 0 Fequency esponse of mstuned bladed dsk (ISB, f = 3 %, N =37, N =35) Fg. Hstogam of blade ampltude (ISB, N =37, N =35) Nomalzedampltude f = 0.5% Fg. Fequency devaton and maxmum ampltude (ISB. N =37, N =35) Nomalzedampltude Max.ampltude Meanampltude Coeffcentofvaaton f = 3% Fo the ISB stuctue wth lage blade dampng (=0.03), the value of N vane n Fg. 3 s. fo asymmetc spacng of N vane =7, and theefoe, the esponse educton effect s evaluated to be 0. fom Fg. 3. Consequently, fo the ISB stuctue, t s also concluded that the maxmum esponse of the mstuned bladed dsk though asymmetc vane spacng (Fg.) can be estmated by the maxmum esponse of the mstuned bladed dsk to symmetc vane spacng (Fg. 9) multpled by the esponse educton effect n Fg. 3 (-R=0.78). On the othe hand, Fg. 3 shows the elatonshp between the standad devaton of the blade alone fequency and the maxmum and mean ampltude of the mstuned bladed dsks fo symmetc vane spacng (H =0) and asymmetc vane spacng (vane count of the uppe half N =the lowe half N =9), espectvely. As shown n Fg. 3, f the fequences of the blades on a dsk change, the esponses of the blades hadly change. Moeove, the maxmum esponse to asymmetc vane spacng becomes slghtly lage than that to symmetc vane spacng f the fequency devaton s beyond %. hs s because fo the ISB stuctue analyzed, the multesonances happen to appea aound the esonance pont to the 0th engne ode as shown n Campbell dagam of Fg. 4, whee the esonance ponts satsfyng esonance condton [0] ae plotted. heefoe, f the esponses to each engne ode ae summed up, the total ampltude becomes lage than that to symmetc vane spacng. Consequently, t s ponted out that when asymmetc vane spacng s appled fo the ISB stuctue, t s essental to check whethe thee ae mult-esonances wthn opeaton ange. Max.amp. (Asymmetcvane) Meanamp. (Asymmetcvane) Fg. 3 Fequency devaton and maxmum ampltude of ISB fo Symmetc (H=0) and asymmetc (N =, N =9) vane Fg.4 Campbell dagam of st mode famly of ISB Max.amp (SymmetcVane) Meanamp. (Symmetcvane) Hamoncs 4

8 Analyss Results of bladed dsk wth GB stuctue Fgue 5, Fg. 6 and Fg. 7 show the analyss esults of the GB stuctue. he numbe of the blades of the GB stuctue analyzed s 0 (N =0), and the bladed dsk conssts of the 0 goups of blades. he analyss model of ths GB stuctue s the same as that n the efeence []. Fgue 5 shows the natual fequency, and Fg. 6 shows the Campbell dagam of the bladed dsk, n whch the esonance ponts satsfyng the esonant condton [0] ae plotted. Fgue 7 shows the elatonshp between the standad devaton of the blade alone fequency and the maxmum ampltude of the mstuned bladed dsks to the symmetc and asymmetc vane spacng. As shown n Fg. 7, the esponse educton effect of the GB stuctue analyzed s small even fo the tuned bladed dsk, because the mult-esonances happen to appea aound the esonance pont to the 6th engne ode as shown n Fg. 6. In addton, when the standad devaton of the blade alone fequency s lage than 5 %, the maxmum esponse of the mstuned bladed dsk to the asymmetc vane spacng becomes lage than that to the symmetc vane spacng. hs s also caused by the mult-esonances aound the esonance pont to the 6th engne ode. Fg. 5 Natual fequency of tuned gouped bladed dsk Fg. 6 Campbell dagam of tuned gouped bladed dsk stmodefamly stmodefamly H=40 ndmodefamly ndmodefamly SymmetcvaneH=6 H=30 3dmodefamly Multesonance egon AsymmetcVaneN =,N =4 H=0 Fg. 7 Maxmum esponse of bladed dsk wth GB stuctue CONCLUSION In ths study, usng the equvalent spng mass model of the bladed dsk, the esonant esponse analyss of the mstuned bladed dsk was caed out, and the esonant esponse educton due to asymmetc vane spacng was studed wth Monte Calo method. Fom these esults, t s concluded that fo the mstuned bladed dsk, the asymmetc vane spacng s effectve to educe the esonant esponse, and the maxmum esponse of the mstuned bladed dsk otatng though asymmetc vane spacng can be oughly estmated by the maxmum esponse of the mstuned bladed dsk to symmetc vane spacng multpled by the esponse educton effect fo fee-standng blade wthout the ccumfeental couplng, unless the mult-esonances happen to appea. On the othe hand, f the mult-esonances occu, t s possble that the asymmetc vane spacng nceases the esonant esponse level. heefoe, t s ponted out that when asymmetc vane spacng s appled fo the ISB o GB stuctue, n whch the mult-esonances could appea due to stong couplng of the bladed dsk n the ccumfeental decton, t s essental to check whethe thee ae mult-esonances wthn opeaton ange. REFERENCES [] Kemp, R. H., Hschbeg, M. H. and Mogan, W. C., heoetcal and expemental analyss of the educton of oto blade vbaton n tubomachney though the use of modfed stato vane spacng, NACA techncal note, 4373 (958), pp [] Clak, J. P., Aggawala, A. S., Velons, M. A., Gacek, R. E., Magge, S. S. and Pce, F. R., Usng CFD to educe esonant stesses on a sngle-stage, hgh-pessue tubne blade, ASME G , (00). [3] Kaneko, Y. et al., Reducton of vbatoy stess of compesso blade by use of asymmetc vane spacng (n Japanese), ansactons of the Japan Socety of Mechancal Engneeng, Sees C, Vol.7, No.7 (005), pp [4] Kaneko, Y. et al., Reducton of esonant stess of tubne blade by use of asymmetc vane spacng (n Japanese), ansactons of the Japan Socety of Mechancal Engneeng, Sees C, Vol.7, No.70 (006), pp [5] Kaneko, Y. et al., Vbaton esponse analyss of mstuned bladed dsk (n Japanese), ansactons of the Japan Socety of Mechancal Engneeng, Sees C, Vol.58, No.547 (99), pp [6] S.. We and C. Pee, Localzaton phenomena n mstuned assembles wth cyclc symmety pat : Fee vbatons, ansactons of ASME, J. of Vbaton, Acoustcs, Stess, and Relablty n Desgn, Vol. 0, (988), pp [7] Petov, E.P., Sanltuk, K. Y. and Ewns, D. J., A new method fo dynamc analyss of mstuned bladed dscs based on exact elatonshp between tuned and mstuned systems, ansactons of ASME, J. of Engneeng fo Gas ubnes and Powe, Vol., (00), pp [8] Bladh, R. and Pee, C., Dynamc esponse pedcton fo a mstuned ndustal tubomachney oto usng educed-ode modelng, ansactons of ASME, J. of Engneeng fo Gas ubnes and Powe, Vol.4, (00), pp [9] Fene, D. M. and Gffn J. H., A fundamental model of mstunng fo a sngle famly of modes, ansactons of ASME, J. of Engneeng fo Gas ubnes and Powe, Vol.4, (00), pp [0] Namua, K., he vbaton chaactestcs of steam tubne blades : nd epot (n Japanese), ansactons of the Japan Socety of Mechancal Engneeng, Sees C, Vol.5, No.474 (986), pp [] Kaneko, Y., Vbaton chaactestcs and mstunng effect of gouped blades (n Japanese), Poceedngs of JSME, No (995), pp