Frequencies in the Vibration Induced by the Rotor Stator Interaction in a Centrifugal Pump Turbine

Transcription

1 C. G. Rodrguez 1 Department of Mechancal Engneerng, Unversty of Concepcon, Caslla 160-C, Concepcon, Chle e-mal: crstan.rodrguez@udec.cl E. Egusquza Center of Industral Dagnostcs and Flud Dynamcs, Techncal Unversty of Catalona, Avenda Dagonal 647, Barcelona, Span e-mal: egusquza@mf.upc.edu I. F. Santos Department of Mechancal Engneerng, Techncal Unversty of Denmark, DK-2800 Kongs Lyngby, Denmark e-mal: fs@mek.dtu.dk Frequences n the Vbraton Induced by the Rotor Stator Interacton n a Centrfugal Pump Turbne The hghest vbraton levels n large pump turbnes are, n general, orgnated n the rotor stator nteracton (RSI). Ths vbraton has specfc characterstcs that can be clearly observed n the frequency doman: harmoncs of the movng blade passng frequency and a partcular relatonshp among ther ampltudes. It s valuable for the desgn and condton montorng to count on these characterstcs. A CFD model s an approprate tool to determne the force and ts characterstcs. However, t s tme consumng and needs hghly qualfed human resources whle usually these results are needed mmedately and n stu. Then, t s useful to determne these characterstcs n a smple, quck, and accurate method. At present, the most sutable method ndcates a large amount of possble harmoncs to appear, wthout ndcatng the relatve mportance of them. Ths paper carres out a theoretcal analyss to predct and explan n a qualtatve way these frequences and ampltudes. The theoretcal analyss ncorporates the number of blades, the number of gude vanes, the RSI nonunform flud force, and the sequence of nteracton. Ths analyss s compared wth the method currently n use, and both methods are appled to a practcal case. The theoretcal analyss gves a resultng force over the pump turbne, whch corresponds well to the measured behavor of a pump turbne n terms of ts frequences and the relatonshp between ther ampltudes. A correctve acton s proposed as a result of the analyss and after t s carred out n one of the unts, the vbraton levels are reduced. The vbraton nduced by the RSI s predcted consderng the sequence of nteracton and dfferent ampltudes n the nteractons between the same movng blade and dfferent statonary blades, gvng a dfferent and orgnal nterpretaton about the source of the vbraton characterstcs. A successful correctve acton s proposed as a consequence of ths new nterpretaton. DOI: / Keywords: rotor stator nteracton (RSI), frequency content, pump turbne, vbraton analyss 1 Introducton The current trend of ncreasng the power concentraton n large hydraulc pump turbnes over 75 MW ether for new power plants or durng the upgrade of exstng ones together wth the trend of ncreasng the operatng range to allow hgher regulaton capactes have lead to a rse n the pressure and flud veloctes, a reducton n the thckness of the mpeller-runner, and extreme off-desgn operatng condtons. As a consequence, the vbraton levels have ncreased and the fatgue problems have become more common. Generally, n large pump turbnes, the hghest vbraton levels are orgnated n the rotor stator nteracton RSI Ohura et al. 1, Tanaka 2, Egusquza et al. 3,4, Egusquza 5 Fsher et al. 6, and Mateos 7. Ths vbraton has specfc characterstcs that can be clearly observed n the frequency doman: harmoncs of the movng blade passng frequency at a partcular relatonshp among ther ampltudes. Ths relatonshp depends on the desgn and the operaton of the pump turbne. The ampltudes of these harmoncs are sgnfcantly ncreased when the machne s operated under specfc condtons. Ths rse n the ampltudes s not equal for all the harmoncs and some are more affected than others Mateos 7. Ths rse s a consequence of the new requrements 1 Formerly at the Center of Industral Dagnostcs and Flud Dynamcs, Techncal Unversty of Catalona. Contrbuted by the Fluds Engneerng Dvson of ASME for publcaton n the JOURNAL OF FLUIDS ENGINEERING. Manuscrpt receved February 9, 2006; fnal manuscrpt receved May 26, Revew conducted by Akra Goto. and trends. At present, ths behavor lmts the power output of new desgns and condtons the operaton of nstalled machnes; for these reasons, t s mportant to dentfy whch harmoncs have sgnfcant ampltude and are prone to be ncreased and how to avod them. An mportant dffculty for desgners s to avod the harmoncs wth sgnfcant ampltudes to concde wth a natural frequency; ths condton can produce an accelerated fatgue, an excessve wear, or even a catastrophc falure Egusquza 5. For condton montorng, t s mportant to know the source of the vbraton; ths nformaton allows the correcton of possble bad practces or to detect faults n the machne. For both desgners and condton analysts, t s useful to count on a tool that dentfes the sgnfcant harmoncs and the source of that vbraton. Ths nformaton s usually needed mmedately and/or n stu. Mateos 7, Tsukamoto et al. 8, Sh and Tsukamoto 9, Wang and Tsukamoto 10, and Zhang and Tsukamoto 11 predct the RSI pressure pulsaton wth a reasonable accuracy usng CFD. Recently, Zhang and Tsukamoto 11 also determned the resultng hydrodynamc force on the mpeller. The smulatons referred above consder one 7,9 11 or two 8 combnatons of a number of statonary and movng blades, whch s known to be an mportant aspect to determne the sgnfcant harmoncs as can be notced by comparng the two cases studed by Tsukamoto et al. 8. These smulatons are tme consumng and needs hghly qualfed human resources. They are also a complex challenge due to the geometry, the mesh, and especally the boundary condtons. From 1428 / Vol. 129, NOVEMBER 2007 Copyrght 2007 by ASME Transactons of the ASME

2 Table 1 Expected frequences and dameter modes N h q=9 2 q=2 3 q=5 q=11 4 q=4 q=12 5 q=3 6 q=6 q= q=4 q=8 9 q=1 the expermental pont of vew, the efforts have been focused on quantfyng the nfluence of some constructon and operatng practces as the angles of the blades Ohash et al. 12 and Ino and Kasa 13, the flow, the rotatonal speed and the dstance between the statonary and movng blades Ohash et al. 12 and Arndt et al. 14,15, and the combnaton of number of statonary and movng blades Guo and Okamoto 16 and Guo and Maruta 17 consder three dfferent combnatons. These studes have shown that the harmoncs wth hgher ampltude depend manly on the number of statonary and movng blades, on the dstance between the statonary and movng blades, and on the operatng condtons. From the prevous studes, Zhang and Tsukamoto 11 and Ohash et al. 12 work wth the resultant hydrodynamc force over the mpeller-runner, whch s the result of all the RSI pressure pulsatons actng around the permeter of the mpeller-runner. These studes gve an understandng of the flud dynamc prncples of the RSI and ts resultng force over the mpeller-runner. Wth respect to the frequences, they ndcate that the pressure pulsaton and the hydrodynamc force have harmoncs of the movng blade passng frequency, but they do not gve gudelnes to determne a pror the most sgnfcant ones and/or the relatonshp among ther ampltudes. At present, the method used to determne these vbraton characterstcs s the one developed by Kubota et al. 18. It proposes the equaton hz V ±N=q, where h and q are postve ntegers, and Z V are the number of movng and statonary blades, and N s the excted dameter mode. The excted dameter mode s a fxed pattern of vbraton rotatng n the same or opposte drecton of the mpeller-runner. The drecton s gven by the sgn used to satsfy the equaton, a postve value means forward drecton, and a negatve one, backward drecton. When the equaton s satsfed, the value of q ndcates whch harmonc of the movng blade passng frequency f B s excted. As an example, let us consder =7 and Z V =16. The equaton s satsfed for several values, some of them are lsted n Table 1. For example, for N=1 and h =1, the equaton takes 1 16±1=q 7, whch cannot be satsfed. For N=2 and h=1, the equaton take 1 16±2=q 7, whch s satsfed f the sgn s negatve and q s 2, then the expected harmonc s the second one, wth a backward excted dameter Mode 2. Kubota et al. ndcate wth Table 1 a possble vbraton at f B q=1, 2f B q=2,..., and others all the values of q. Ths procedure gves a large amount of possble dameter modes and harmoncs of f B to appear. It s ndcated that the most probable harmoncs to appear are correspondng to the lower dameter modes. Consderng the values n Table 1 for ths partcular case, t s expected that the hghest component n the spectrum wll be the 9th harmonc, then the 2nd, followed by the 5th, the 11th, the 4th, and so on. In summary, the study of Kubota et al. lmts the frequences to harmoncs of the movng blade passng frequency, and assumes dameter modes that are prone to be excted. The excted dameter mode wll depend on whch s the hghest harmonc of the vbraton. These results have been wdely used by several authors as Tanaka 2, Guo and Okamoto 16, Guo and Maruta 17, and Franke et al. 19. In ths paper, the approach of Kubota et al. s complemented wth geometrcal and tmng consderatons that allow us to go further wth the equaton. The clue s the sequence of nteracton. Some other mportant characterstcs of the pump turbne are ncorporated, such as the dstance between the statonary and movng blades. Then, the proposed procedure s appled to a pump turbne, showng that the source of the vbraton can be detected. 2 Theoretcal Analyss In order to characterze the resultng force over the mpellerrunner related to the RSI, let us consder an mpeller-runner wth blades movng blades and a spral case that has Z V gude vanes statonary blades. Any tme a blade passes n front of a gude vane, there s an nteracton that produces a flud force over the blade and the gude vane. Let us assume, for the moment, that when the nth blade passes n front of the mth gude vane, t receves the same flud force as passng n front of the next m +1 th gude vane and so on. After an entre revoluton, the nth blade receves Z V tmes the same force. If the rotatng speed s constant, then the blade takes 2 / s to complete an entre revoluton, recevng a repettve force wth perod T V =2 / Z V. Ths perodc force can be descrbed by a Fourer seres as follows: f n = F,n sn Z V t,n 1 Here F s the ampltude of the force, the phase, and the subscrpts and n ndcate that the force or phase corresponds to the th harmonc of Z V at the nth blade. Equaton 1 ncorporates the acton of the nonunform flow between two dfferent pars of gude vanes and consders that ths nonunform flow s the same between the dfferent pars of gude vanes. The force n Eq. 1 s movng together wth the nth blade and s referred to the rotatng system of reference, as s shown n Fg. 1. It can be observed n Fg. 1 that f n n the x axs of the nertal system of reference f n,x s gven by f n,x = f n cos t + 0,n 2 where 0,n s the ntal angular poston of the nth blade: 0,n = n Here 0 s the ntal angular poston of the runner. Then, takng Eqs. 1 3 nto account, f n,x = 1 F,n sn Z V +1 t,n + 0,n 2 + sn Z V 1 t,n 0,n 4 Journal of Fluds Engneerng NOVEMBER 2007, Vol. 129 / 1429

3 Fg. 1 Scheme of the forces over the runner at the reference ntal tme t=0 Fg. 2 Tme between nteractons The resultant force over the runner n the nertal system of reference s obtaned addng the forces over the blades as follows: f x = f n,x = 1 2 F,n sn Z V +1 t,n + 0,n + sn Z V 1 t,n 0,n 5 Usng trgonometrc denttes n Eq. 5 and arrangng the terms, f x = 1 2 F,n sn Z V +1 t + 0 cos,n n 2 + cos Z V +1 t + 0 sn,n n cos,n + n 2 + F,n sn Z V 1 t 0 cos Z V 1 t 0 sn,n + n 2 Kubota et al. 18 obtan a smlar result at ths pont, but they do not go further wth the equaton. To go further n Eq. 6, ts necessary to consder the sequence of nteracton. Durng an entre revoluton, each of the blades nteracts wth all the Z V gude vanes, so that there are a total of Z V nteractons separated n tme by t RSI = 2 7 Z V Fgure 2 schematcally shows the tme sequence for blade-gude vane nteractons. Fgure 2 a shows the mpeller-runner rotatng and ts correspondng nteractons that are shown n a tme lne n Fg. 2 b. At the begnnng, t=0, Blade 1 nteracts wth Gude Vane 1. After t RSI, Blade p nteracts wth Gude Vane s, etc. After nteractons n 1/Z V turns, Blade 1 nteracts agan wth a 6 gude vane. The nteractons begn at dfferent tmes for dfferent blades and ths lag produces a phase dfference dependng on the correspondng frequency Z V n Eq. 6. Fgure 3 shows the tme lag at Z V and 2Z V, respectvely. The phase dfference correspondng to the tme lag between two consecutve nteractons Blade 1 and then Blade p s constant for the same harmonc value of and s gven by,p,1 = t RSI Z V = 2 8 Equaton 8 gves the phase dfference between two consecutve nteractons but does not consder the order of nteracton. The order of nteracton depends on the combnaton of and Z V and as an example, Tables 2 4 summarze the order for combnatons of =2,3,4, and Z V =5, respectvely. These tables are lsted for an entre revoluton. At the end of each row, there s the nteracton of Blade 1 and Gude Vane 1, where all the sequence starts agan. Ths order s graphcally shown n Fgs. 4 and 5 for the nertal and rotatng system of reference, respectvely. Fg. 3 Phase for a sgnal of frequency Z V Table 2 Interacton order for Z V =5 and =2 Blade Gude vane Table 3 Interacton order for Z V =5 and =3 Blade Gude vane Table 4 Interacton order for Z V =5 and =4 Blade Gude vane / Vol. 129, NOVEMBER 2007 Transactons of the ASME

4 Fg. 4 Blade and gude vane nteracton n the nertal system of reference for a combnaton of fve gude vanes and a 2, b 3, and c 4 blades It can be observed that the poston of nteracton moves wth respect to the nertal system of reference n dfferent drectons around the stator: clockwse n Fgs. 4 a and 4 c and counterclockwse n Fg. 4 b. Ths s also observed wth respect to the rotatng system of reference where the drectons of moton are alternatve not rotatng n Fg. 5 a, counterclockwse n Fg. 5 b, and clockwse n Fg. 5 c. In both systems of reference, the nteracton poston moves a fxed nteger number of blades or gude vanes. For example, n Fg. 5 c, the poston moves three blades from Blade 1 to Blade 4, after that t moves other three blades to reach Blade 3. As a consequence, the values of,1,,2,...,,zb are separated by the same angle that s an entre multple s s=3 n Fg. 5 c of the phase lag between two consecutve nteractons n Eq. 8 :,2,1 = s 2 9 After all the blades have nteracted, the angle n Eq. 9 completes entre crcular angular dsplacements. Now, t s necessary to calculate the value of s. Fgure 6 shows a schematc of the frst four nteractons for =4 and Z V =5 the case shown n Fgs. 4 c and 5 c. The entre permeter where the nteractons occur s drawn n a bar dvded by all the Z V nteractons experenced durng one revoluton. The frst nteracton corresponds to Blade 1 and Gude Vane 1. After t RSI Blade 4 nteracts wth Gude Vane 5. After 3 t RSI Blade 2 nteracts wth Gude Vane 3, n ths case, s=3. The quantty of tme ntervals t RSI for the nteracton of the nth blade and the mth gude vane can be determned from Fg. 6: s n,m = m 1 n 1 Z V 10 It can be notced that s 1,1 =0 and that the followng nteracton of Blade 2 wth Gude Vane m s gven by Fg. 6 blades s 2,m = m 1 Z V 11 The frst gude vane that nteracts wth Blade 2 can be determned as follows: m 1st = Z V where the operator rounds the correspondng value to the upper nteger number. It follows that s n Eq. 9 s s 2,m1st = Z V Z V 13 From Eq. 6, t can be observed that gven a fxed harmonc, the value of the terms depends on the behavor of the phase,n n2 /. Ths behavor can be deduced from Eq. 9 : The addton of cosnes and snes of,n n2 / s zero unless,n+1 n+1 2 /,n n2 / =z2, wth,2..., 1 beng z an nteger. Ths condton has to be satsfed to obtan a nonzero term at the correspondng frequency Z V ±1 of Eq. 6. Ths condton can be graphcally shown n Fg. 7 for the case of =5. The dfference between two consecutve angles n the summaton n Eq. 6 s,n+1,n 2 /. Observng Eq. 8, t can be notced that ths value s a multple of 2 /. For the case presented n Fg. 7, the angles advance s=2,3,4,and 5 entre multples of 2 / and the only condton where the cosne or sne summaton s not equal to zero s when the phase dfference s 5 2 /5, an nteger multple of 2. Takng the explanaton gven above nto account, Eq. 5 can be smplfed usng the followng derved condton: Tme between the nteractons of two consecutve sn Z V +1 t,n + 0,n =0 f s 1 j sn Z V 1 t,n 0,n =0 f s +1 j Fnally, the exctaton obtaned s 14 Fg. 5 Blade and gude vane nteracton n the rotatng system of reference for a combnaton of fve gude vanes and a 2, b 3, and c 4 blades f x = 1 2 F,n sn Z V ±1 t,n ± 0,n wth s 1=j 15 Equaton 15 gves the frequences n the resultant force over the runner when the nteractons between the blades and dfferent pars of gude vanes are equal. In practce, ths condton cannot be satsfed and there are dfferences between the ampltude of the nteractons of one blade and two dfferent gude vanes Mateos Journal of Fluds Engneerng NOVEMBER 2007, Vol. 129 / 1431

5 The modulated force obtaned multplyng Eqs. 1 and 16 ncorporates the nonunform flow effect of Eq. 1 together wth the change n the ampltude dependng on the crcumferental poston. Ths modulated force ncludes the behavor documented n the references cted by ths artcle. Usng the same procedure as the one used for the no modulated force, the exctaton obtaned s f x = 1 2 F sn Z V ±1 t,n ± 0 + wth s 1=j A k,n F,n sn Z V ± k ±1 t k,n ± 0,n wth s k 1=j 17 It should be observed that f there s no modulaton and the exctatons have the same ampltude n all the permeter of the runner, then A k,n =0 and the frequences are at Z V ±1 t. The frequences gven by Z V ±k±1 t appear only when the values of A k,n 0 and are a consequence of the dfferent ampltudes of the nteractons around the permeter of the runner. The frequences n Eq. 17 are resumed n Eq. 18, whch ncludes the frequences j that are measured n hydraulc turbomachnery. Fg. 7 Graphcal explanaton of the zero ampltude components 7. These dfferences can be nterpreted as the consequence of two man effects, shown n Fg. 8, whch are related to the constructon and mountng of the mpeller-runner: because of the cut water and because of the statc eccentrcty of the runner. In the frst case, the cut water has dfferent desgn and workng condtons, presentng a pressure dfference that s hgher than n any other par of gude vanes. In the second case, the dstance between the blades and the gude vanes vares; ths dstance s an mportant factor that affects the ampltude of the pressure pulsaton: the hgher the dstance, the more unform the flow and the exctaton ampltude s reduced Arndt et al. 14,15. In both cases, the ampltude of the forces depends on whch gude vane s nteractng wth the blade; ths s equvalent to modulate n ampltude the exctaton over the nth blade n Eq. 1 wth a sgnal a of perod T=2 / : a =1+ k A k,n sn k t k,n 16 Fg. 8 Nonunform dstrbuton of the RSI. Cut water and statc eccentrcty. Z V ± k ±1=j wth s k 1=j 18 It should be notced that Eq. 18 s smlar to the equaton gven by Kubota et al. 18. The dfference les on the value of k that n ths analyss represents the pattern of the forces between dfferent pars of gude vanes around the permeter of the runner, whle Kubota et al. used the value of ±k±1 to defne a dameter mode. Other mportant dfference n Eq. 18 s that the value of s has also to be satsfed; ths restrcton ncorporates the sequence of nteracton and reduces the amount of possble harmoncs to appear. To llustrate the dfference between both methods, let us consder the same case of =7 and Z V =16 used n the Introducton Table 1. The frst step s to determne the values of n the equatons above. Ths value ndcates the harmoncs presented n the radal force over a blade. Ohura et al. 1, Tanaka 2, and Ohash et al. 12 show that the stress of a rotatng pont n the mpeller-runner s found to be manly a snusodal wave at the gude vane passng frequency: Z V. If we assume ths, then n the above equatons take the sngle value of 1. After ths assumpton, the force n Eq. 1 can be modulated by the sgnal n Eq. 16. For a 7-16 combnaton, the value of s s equal to 5. If there s no modulaton, and all the ampltudes are equal, then the condton n Eq. 18 takes the followng values: =j 7. Ths cannot be satsfed, then ths condton should not show any harmonc. For the eccentrcty and the cut water, the forces are modulated as shown n Fg. 9. Consderng a statc eccentrcty, the ampltude modulaton s a sngle snusodal at the rotatng speed Rodrguez 20. In that case, k=1, and the condton n Eq. 18 takes =j 7; ths s satsfed for 5+1+1=7 where j =1, then the correspondng frequency s =2 7 and the excted harmonc s the 2. For the cut water, the modulatng sgnal ncludes harmoncs of the rotatng speed, whch s equvalent to modulate the sgnal by a step functon. The forces n Fg. 9 are appled to the blades consderng the correspondng tme lag between dfferent blades. Equaton 6 n combnaton wth Eq. 16 s numercally solved and the obtaned frequences n the resultng force are shown n Fg. 10 for both cases / Vol. 129, NOVEMBER 2007 Transactons of the ASME

6 Table 5 Expected frequences Harmonc Ampltude n the nteractons Unform Eccentrcty Cut water 1 24% 2 100% 32% 3 22% 4 7% 5 3% 6 3% Others 9% Fg. 9 Exctaton force over a blade consderng a an eccentrcty and b the cut water The results shown n Fg. 10 correspond to the ampltude n the harmonc of the resultng force dvded by the ampltude of the no modulated force over a blade; ths result s shown as a functon of the level of modulaton A 0,n /F 1,n. In the eccentrcty, the resultng force Fg. 10 a s a sngle snusodal at 2 as predcted wth the theoretcal analyss. For the cut water, the relaton between the ampltude of the harmoncs of s fxed and corresponds to a 24% for the frst harmonc, 32% for the second, 22% for the thrd, 7% for the fourth, 3% for the ffth, 3% for the sxth, and 9% for the rest of the harmoncs. The harmonc wth hgher ampltude s n both cases 2. Notce that for the same level of modulaton, the rse n the ampltude of the resultng force s hgher for the eccentrcty. For nstance, for a level of modulaton 0.5, the ampltude n the second harmonc of the resultng force s 0.60 tmes the ampltude of the force over a blade n the eccentrcty, whle ths level of modulaton n the cut-water effect produces a force 0.17 tmes the ampltude of the force over a blade. The prevous analyss leads to the followng: If there s only one component at 2, then the source s the eccentrcty. If there are several harmoncs where the relatonshp between ther ampltudes s the one shown n Fg. 10 b, then the source s an excessve pressure pulsaton at the cut water. The predcton of ths method s summarzed n Table 5. 3 Comparson Wth a Pump-Turbne Applcaton A pump turbne wth a combnaton of =7 and Z V =16 s studed n order to corroborate the accuracy n the predctons of both methods. The machne nomnal power s 96 MW operatng as a pump and 105 MW operatng as a turbne. The rotatng speed s 600 rpm. In pump nomnal operatng condtons, the flow rate s 25 m 3 /s and the net head s 379 m. In turbne nomnal operatng condtons, the flow rate s 32 m 3 /s and the net head s 376 m. The predcton for the frequences n the force orgnated n the RSI for a combnaton of =7 and Z V =16 s exposed n Table 1 for the approach of Kubota et al. and n Table 5 for the analyss of ths artcle. Both methods predct the force actng over the mpeller-runner. Especally n a prototype, ths force s dffcult to measure; for that reason, t s preferred to measure vbraton. The force actng over the mpeller-runner produces a vbraton n the machne. Ths vbraton s not equal to the force and s modfed by the response of the machne. The vbraton s a consequence of the force n combnaton wth the response of the machne. To dscrmnate f the resultng vbraton characterstcs are due to the RSI force or to the response of the system, t s necessary to determne the response between the poston where the force s appled and the measurng poston. The response analyss of Rodrguez 20 ncludng start-up and run-down analyss, expermental modal analyss, and the consderaton of the added mass effect of water Rodrguez et al. 21 and Lang et al. 22 ndcates that there s no resonance or antresonance near the frst fve harmoncs of the blade passng frequency. Ths allows us to compare the ampltudes of the vbraton wth those of the force. Takng these consderatons nto account, the vbraton should at least nclude Fg. 10 Frequency content of the resultant exctaton consderng a an eccentrcty and b the cut water Journal of Fluds Engneerng NOVEMBER 2007, Vol. 129 / 1433

7 Fg. 12 Resultng force over a blade Fg. 11 Spectrum of vbraton measured n the rotatng shaft. Spots ndcate harmoncs of blade passng frequency. the harmoncs of the RSI force and the relatonshp among ther ampltudes. Ths should be corresponded wth the predcted force n Table 1 and/or Table 5. The response analyzed above relates the mpeller-runner to the nearest accessble poston to the mpeller-runner on the shaft surface. Because of constructve characterstcs of large hydraulc turbomachnery, dsplacement probes to measure shaft vbraton cannot be mounted on the foundaton and are usually mounted on the bearng housng whch vbrates sgnfcantly. To avod the effect of the bearng housng, the shaft vbraton s measured usng telemetrc on-board accelerometers that rotate wth the shaft. Ths measurng poston avods the uncertantes ntroduced by the response of the bearng flud flm and the bearng housng. The vbraton measured on the shaft surface s referred to the rotatng system of reference and can be converted to the nertal system of reference combnng both perpendcular drectons Rodrguez et al. 23. The nstrumentaton used ncludes two accelerometers, a multchannel telemetry system, and an acquston system. The spectrum of the vbraton wth respect to the nertal system of reference s shown n Fg. 11. The frequences above 50 tmes the rotatng speed are not shown because ther energy represents less than a 0.1% of the rms global value below 5000 Hz. For the frequency range below 50 cycles per revoluton, the harmoncs of the blade passng frequency represents the 99.3% of the rms global value. Table 6 lsts the ampltude of the harmoncs n ths frequency range and the relatve mportance between them. The hghest ampltude found n the measured vbraton corresponds to the second harmonc 2f B, whch s predcted by both Table 7 Table 6 Harmonc Harmonc Harmoncs of f B expected n the cut-water effect Ampltude mm/s % Harmoncs of f B found n the measured vbraton Ampltude mm/s % Total methods. Kubota et al. ndcate that ths vbraton corresponds to a dameter Mode 2. The analyss carred out n ths artcle ndcates that t could correspond to two possble cases: an eccentrcty or the cut water. The next harmonc presentng the hghest ampltude s f B.To predct ths harmonc, Kubota et al. need the exctaton of the dameter Mode 9. A dameter Mode 9 s a dffcult vbraton pattern to obtan, especally because the mpeller-runner has seven blades. Moreover, t s expected that some other dameter modes should be excted before the nnth because t s ndcated that lower dameter modes are more prone to be excted. For ths partcular case, Kubota et al. propose excted dameter Modes 2 and 9. The analyss carred out n ths artcle ndcates that the f B could correspond to the effect of the cut water, then t proposes a combnaton of an eccentrcty and a cut-water effect. If t s consdered that the ampltude of f B comes from the cut water, then the ampltudes for the rest of the harmoncs should be consstent wth Table 6. In Table 7 are lsted the expected ampltudes for a cutwater effect that produces a component at f B wth ampltude 1.21 mm/s rms. If Table 7 s compared wth Table 6, t can be notced that the ampltudes are smlar except for the second and the thrd harmonc. If we assume that the dfference wth the second harmonc comes from the fact that there s a combnaton of the eccentrcty and the cut water, then the rest of 2f B can be obtaned from the eccentrcty and the only dsagreement wth Table 6 s the thrd harmonc. Usng the computer, the exctaton for each blade can be determned. Ths exctaton s shown n Fg. 12 where the numbers over the peaks ndcate whch gude vane s nteractng wth the blade. Gude Vane 1 corresponds to the cut water. The combnaton of both effects can be observed n Fg. 12, where the sngle rse n the ampltude occurs when the blade passes n front of the cut water. Large hydraulc turbomachnery s especally dffcult to algn and to balance; for that reason, the eccentrcty effect cannot be avoded. Nevertheless, the sngle rse n the magntude of the nteracton n the cut water can be reduced. From ths analyss, t can be proposed to perform a drll n the cut water to decrease the pressure dfference encountered by the blade when t passes from the back to the front of the cut water. Ths correctve proposton was performed by the power plant and the global vbraton levels were reduced. 4 Conclusons Ths artcle presents a theoretcal analyss that allows us to descrbe the characterstcs n the frequency doman of the vbraton orgnated n the RSI. The analyss s based on the sequence of nteracton. Ths approach allows us to nterpret the vbraton as a consequence of a modulaton n the ampltudes of the nteractons and not as a consequence of an excted dameter mode as has been usually consdered. Ths nterpretaton suggests drectons for the soluton of desgn and condton montorng problems. It helps to determne the orgn of the harmoncs that gves gudelnes to get lower ampltude at specfc harmoncs or to avod them. For nstance, a clearly sngle rse n the ampltude of the nteractons wll show that there s a specfc problem n a gude vane, and a drll n ths gude vane wll reduce the vbraton levels / Vol. 129, NOVEMBER 2007 Transactons of the ASME

8 Ths approach looks attractve, but s contrasted wth a sngle practcal case and t s necessary to analyze more cases to corroborate the approach. Nomenclature A ampltude of the modulatng sgnal F ampltude of the exctaton N N referred to the N-dameter mode T perod of one revoluton s T V statonary blade passng perod s number of movng blades Z V number of statonary blades a modulatng sgnal f exctaton as a functon of tme N h postve nteger q excted harmonc of the movng blade passng frequency s postve nteger z postve nteger t RSI tme nterval between two consecutve nteractons s rotatng speed rad/s 0 ntal angular poston rad phase rad Subscrpts referred to the th harmonc n the exctaton of each movng blade j referred to the jth harmonc of the movng blade passng frequency k referred to the kth harmonc n the ampltude modulatng sgnal m referred to the mth statonary blade n referred to the nth movng blade x referred to the x axs of the nertal system of reference References 1 Ohura, Y., Fuj, M., Sugmoto, O., Tanaka, H., and Yamagata, I., 1990, Vbraton of the Powerhouse Structure of a Pumped Storage Power Plant, U2, IAHR Symposum, Belgrade, Yugoslava. 2 Tanaka, H., 1990, Vbraton Behavor and Dynamc Stress of Runners of Very Hgh Head Reversble Pump-Turbnes, A1, IAHR Symposum, Belgrade, Yugoslava, Sec. U2. 3 Egusquza, E., Nascmento, L. P., Valero, C., and Jou, E., 1994, El Dagnóstco De Daños En Grupos Hdroeléctrcos Medante El Análss de Vbracones, Ingenería del Agua, 1 3, pp ; also avalable at: Egusquza, E., Mateos, B., and Escaler, X., 2002, Analyss of Rotor-Stator Interacton n Operatng Pump-Turbnes, Proceedngs of the XXI IAHR Symposum on Hydraulc Machnery and Systems, Lausanne, Swtzerland. 5 Egusquza, E., 2003, Comportament Dnàmc de Màqunes Hdràulques, Edcons UPC, Catalan. 6 Fsher, R. K., Sedel, U., Grosse, G., Gfeller, W., and Klnger, R., 2002, A Case Study n Resonant Hydroelastc Vbraton: The Cause of Runner Cracks and the Solutons Implemented for the Xaolangd Hydroelectrc Project, Proceedngs of the XXI IAHR Symposum on Hydraulc Machnery and Systems, Lausanne, Swtzerland. 7 Mateos, J. B., 2005, Contrbuton to Rotor-Stator Interacton n Pump Turbne Prototypes, Ph.D. thess, Techncal Unversty of Catalona UPC, Barcelona, Span. 8 Tsukamoto, H., Uno, M., Hamafuku, N., and Okamura, T., 1995, Pressure Fluctuaton Downstream of a Dffuser Pump Impeller, The Second Jont ASME/JSME Fluds Engneerng Conference, Forum of Unsteady Flow, FED, Vol. 216, pp Sh, F., and Tsukamoto, H., 2001, Numercal Study of Pressure Fluctuatons Caused by Impeller-Dffuser Interacton n a Dffuser Pump Stage, ASME J. Fluds Eng., 123, pp Wang, H., and Tsukamoto, H., 2001, Fundamental Analyss on Rotor-Stator Interacton n a Dffuser Pump by Vortex Method, ASME J. Fluds Eng., 123, pp Zhang, M., and Tsukamoto, H., 2005, Unsteady Hydrodynamc Forces Due to Rotor-Stator Interacton on a Dffuser Pump Wth Identcal Number of Vanes on the Impeller and Dffuser, ASME J. 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Turbomach., 112, pp Guo, S., and Okamoto, H., 2002, An Expermental Study on the Flud Forces Induced by Rotor-Stator Interacton n a Centrfugal Pump, The Nnth Internatonal Symposum on Transport Phenomena and Dynamcs of Rotatng Machnery, Honolulu, HI, Feb Guo, S., and Maruta, Y., 2005, Expermental Investgatons on Pressure Fluctuatons and Vbraton of the Impeller n a Centrfugal Pump Wth Vaned Dffusers, JSME Int. J., Ser. B, 48 1, pp Kubota, Y., Susuk, T., Tomta, H., Nagafuj, T., and Okamura, T., 1983, Vbraton of Rotatng Bladed Dsc Excted by Statonary Dstrbuted Forces, Bull. JSME, 26, pp Franke, G., Fsher, R., Powell, C., Sedel, U., and Koutnk, J., 2005, On Pressure Mode Shapes Arsng From Rotor Stator Interactons, Sound Vb., 39 3, pp Rodrguez, C. G., 2006, Feasblty of On Board Measurements for Predctve Mantenance n Large Hydraulc Turbomachnery, Ph.D. thess, Techncal Unversty of Catalona UPC, Barcelona, Span. 21 Rodrguez, C. G., Egusquza, E., Escaler, X., Lang, Q. W., and Avellan, F., 2006, Expermental Investgaton of Added Mass Effects on a Francs Turbne Runner n Stll Water, J. Fluds Struct., 22, pp Lang, Q. W., Rodríguez, C. G., Egusquza, E., Escaler, X., Farhat, M., and Avellan, F., 2007, Numercal Smulaton of Flud Added Mass Effect on a Francs Turbne Runner, Comput. Fluds, 36, pp Rodrguez, C. G., Egusquza, E., Escaler, X., and Lang, Q. W., 2006, Feasblty of On-Board Measurements n a Hydraulc Turbnes, 23rd IAHR Symposum on Hydraulc Machnery and Systems, Yokohama, Japan, Paper Code 265. Journal of Fluds Engneerng NOVEMBER 2007, Vol. 129 / 1435