ROTORDYNAMICS/DISCHARGE-LINE WATER-HAMMER COUPLING VIA SEALS IN PUMP ROTORDYNAMICS. A Thesis KAIKAI ZHANG

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1 ROTORDYNAMICS/DISCHARGE-LINE WATER-HAMMER COULING VIA SEALS IN UM ROTORDYNAMICS A The by KAIKAI ZHANG Submtted to the Offe of Gaduate Stude of Texa A&M Uvety patal fulfllmet of the equemet fo the degee of MASTER OF SCIENCE May 00 Majo Subjet: Mehaal Egeeg

2 ROTORDYNAMICS/DISCHARGE-LINE WATER HAMMER COULING VIA SEALS IN UM ROTORDYNAMICS A The by KAIKAI ZHANG Submtted to Texa A&M Uvety patal fulfllmet of the equemet fo the degee of MASTER OF SCIENCE Appoved a to tyle ad otet by: Daa W. Chld Cha of Commttee Lu Sa Ade Membe Joe M. Roeet Membe Joh A. Weee Head of Depatmet May 00 Majo Subjet: Mehaal Egeeg

3 ABSTRACT Rotodyam/Dhage-le Wate Hamme Couplg Va Seal ump Rotodyam. May 00 Kaka Zhag, B.S., ekg Uvety Cha of Advoy Commttee: D. Daa W. Chld A ew loed-loop fequey-doma model developed to opoate the wate hamme effet wth pump otodyam, ode to vetgate the ub-yhoou tablty poblem obeved a feld pump. Seal flow-ate petubato due to eetty ae alulated fom Soula ad Sa Ade eal ode. A omplete tafe futo matx betwee oto moto ad eato foe due to peue petubato developed detal. Stablty aaly wth tafe-futo add- module oduted XLTRC. Seal leaae ad the eato foe agle ae foud to be mpotat hftg atual fequee ad dampg. The ub-yhoou tablty obeved the feld duplated uefully wth double-leaae eal.

4 v ACKNOWLEDGMENTS I am deeply gateful to my upevo D. Daa Chld. H faal uppot made th wok poble. He guded me fom the tat ad gave me lot of dea, uggeto ad eouagemet. H h kowledge the feld of tubomahey otodyam ad ght to the poblem made th job a lot eae fo me. I beeftted a geat deal fom my duo wth D. Chld. The yea I pet at Texa A&M Uvety may beome the teaue of my whole lfe. I patulaly appeate D. Sa Ade fo h pemo to ue the eal ode, ad alo fo h aeful evew of the daft, ad all h helpful adve ad oeto. I am vey pleaed to thak Avjt ad Lu fo the kd help ug XLTRC. It my pleaue to akowledge the atae I eeved fom Athu pepag the fal exam ad oetg the the. My thak ae alo exteded to Jo ad Buga the Tubomahey Laboatoy, who haed good tme wth me the lab. I am alway gateful to my paet. Wthout the old uppot, I would ot have gotte to th pot at all. I would dedate th the to my glfed, Wag Tgtg, fo he love.

5 v TABLE OF CONTENTS age ABSTRACT... ACKNOWLEDGMENTS...v TABLE OF CONTENTS...v LIST OF FIGURES...v NOMENCLATURE...x I. INTRODUCTION... II. RESEARCH OBJECTIVES...4 III. ROTOR MODEL IN XLTRC...5. ump deg Roto model... 6 IV. CLOSED LOO MODEL.... Stadad wate-hamme aaly fo ppele.... Fte elemet developmet fo ppele.... Seal flow-ate petubato alulato Couplg of oto moto to wate-hamme dyam Couplg of peue ollato to oto moto Complete loed-loop model... V. MODEL RESULTS AND DISCUSSION...5. Effet of the eato foe agle β Effet of eal leaae.... Effet of dhage ppe ofguato Effet of ug peed Compao wth feld eult... 6 VI. CONCLUSIONS...9 REFERENCES...4 AENDIX A BEARING, SEAL, IMELLER COEFFICIENTS IN XLTRC...4. eue dam beag otodyam oeffet XLTRC... 4

6 v age. Cetfugal pump mpelle otodyam oeffet XLTRC Aula eal otodyam oeffet XLTRC AENDIX B FINITE ELEMENT MODEL DEVELOMENT Wave equato ad hape futo Tafe futo of dhage le Tafe futo of let le Complete tafe futo... 6 AENDIX C INUT DATA FOR SEAL FLOW-RATE CALCULATION...65 AENDIX D EIGENVALUE CALCULATION WITH GENERAL FORM TRANSFER FUNCTIONS...67 VITA...70

7 v LIST OF FIGURES age Fgue Outle Dawg of the 6*6*0 DVMF Shppg ump... 5 Fgue Roto Model XLTRC... 6 Fgue Ft Damped Mode Shape wth Ogal-leaae Seal... 7 Fgue 4 Seod Damped Mode Shape wth Ogal-leaae Seal... 7 Fgue 5 Thd Damped Mode Shape wth Ogal-leaae Seal... 8 Fgue 6 Fouth Damped Mode Shape wth Ogal-leaae Seal... 8 Fgue 7 Fgue 8 Fgue 9 Rotodyam Damped Natual Fequey Map wth Ogal-leaae Seal 0.65mm Rotodyam Root Lou lot wth Ogal-leaae Seal 0.65mm Rotodyam Damped Natual Fequey Map wth Double-leaae Seal.7mm Fgue 0 Rotodyam Root Lou lot wth Double-leaae Seal.7mm Fgue Wate-hamme Example of ump... Fgue etubed Wate-hamme Flow-ate ad eue Model... Fgue Coodate Sytem... 5 Fgue 4 etubed Seal Flow-ate pe Ut Moto Veu Extato Fequey wth Ogal-leaae Seal at Rug Speed of 4700 RM... 8 Fgue 5 etubed Seal Flow-ate pe Ut Moto Veu Extato Fequey wth Double-leaae Seal at the Rug Speed of 4700 RM... 8 Fgue 6 etubed Seal Flow-ate pe Ut Moto Veu Extato Fequey wth Ogal-leaae Seal at the Rug Speed of 750 RM... 9

8 v age Fgue 7 etubed Seal Flow-ate pe Ut Moto Veu Extato Fequey wth Double-leaae Seal at the Rug Speed of 750 RM... 9 Fgue 8 etubed Wate-hamme Flow-ate ad eue Model wth etubed Flow-ate Ijeto δq& eal fom a Seal... Fgue 9 ump Volute Dhage... Fgue 0 Complete Cloed-loop Sytem... Fgue Cloed-loop Model Showg the Tafe Futo betwee the Roto Dplaemet Veto ad the Reato Foe... 4 Fgue Dffeet Dhage pe Cofguato... 6 Fgue Rotodyam Root Lou lot wth Ogal-leaae Seal ad Abupt Expao at the Rug Speed of 4700 RM... 6 Fgue 4 Rotodyam Root Lou lot wth Double-leaae Seal ad Abupt Expao at the Rug Speed of 4700 RM... 7 Fgue 5 Rotodyam Root Lou lot wth Ogal-leaae Seal ad Gadual Expao at the Rug Speed of 4700 RM... 7 Fgue 6 Rotodyam Root Lou lot wth Double-leaae Seal ad Gadual Expao at the Rug Speed of 4700 RM... 8 Fgue 7 Rotodyam Root Lou lot wth Ogal-leaae Seal ad Abupt Expao at the Rug Speed of 750 RM... 8 Fgue 8 t Damped Mode Shape wth Wate-hamme Couplg, Doubleleaae Seal, β 5... o 9 Fgue 9 d Damped Mode Shape wth Wate-hamme Couplg, Doubleleaae Seal, β 5... o 9 Fgue 0 d Damped Mode Shape wth Wate-hamme Couplg, Doubleleaae Seal, β 5... o 0 Fgue 4 th Damped Mode Shape wth Wate-hamme Couplg, Doubleleaae Seal, β 5... o 0

9 x age Fgue Rotodyam Damped Natual Fequey Map wth Ogal-leaae o Seal, Ogal Dhage Tato, β 5... Fgue Rotodyam Root Lou lot wth Ogal-leaae Seal, Ogal o Dhage Tato, β 5... Fgue 4 Rotodyam Damped Natual Fequey Map wth Double-leaae o Seal, Ogal Dhage Tato, β Fgue 5 Rotodyam Root Lou lot wth Double-leaae Seal, Ogal o Dhage Tato, β Fgue 6 Rotodyam Root Lou lot wth a Ifte Temato ad Double-leaae Seal... 5 Fgue 7 Hootal Shaft Vbato at the Iboad Beag, Coley[]... 7 Fgue 8 Hootal Shaft Vbato at the Iboad Beag Afte Itallato of Log Tape Tato, Coley []... 8

10 x NOMENCLATURE δx & eal plaa oto moto at eal [ L / T ] δ eal flow-ate petubato at mpelle eal let [ L / T ] q& eal δ Q & pump petubato of the ext flow-ate of the pump [ L / T ] δ pump dhage peue petubato [ F / L ] pump δ f mpelle the eato foe [ F ] R atteuato fato Coley model τ d tme delay Coley model [T ] β eato foe agle 5 R etae [ FT / L ] d veloty of popagato of the peue wave [ L / T ] u veloty of the flud [ L / T ] peue [ F / L ] X, Y, Z oto moto oodate ytem x, y, eal-eetty-oeted oodate ytem R, ompoet of oto poto veto x, y, ytem[ L ] x R y R, ompoet of oto poto veto X, Y, Z ytem [ L ] X R Y Z qx, Z qy tafe futo betwee oto moto ad eal flow-ate petubato petubed eal flow-ate pe ut moto [ L / T ]

11 x agle betwee X ax ad x ax e 0 the tat eetty e X e0y 0, ompoet of tat eetty δ q &, δ & the global eal flow-ate petubato x ad y deto [ L / T ] x q y & & δ q x, δq x eal pat ad magay pat of x δ q& [ L / T ] & & δ q y, δq y eal pat ad magay pat of y δ q& [ L / T ] Ω extato fequey [ / T ] ω ug peed [ / T ] Laplae tafom opeato [ / T ] j A the pojeted aea of the dhage mpelle [ L ] mp T tafe futo betwee eal flow-ate petubato ad dhage G,, 5 peue petubato [ FT / L ] G, G G ompoet of the omplete tafe futo matx [ F / L ]

12 I. INTRODUCTION Coley [] te a ae tudy volvg lage wate jeto pump. Afte about oe yea of opeato, ome of the pump beame utable at a fequey ea the pump ft atual fequey. Someoe oted that oly pump wth a abupt deeae damete at the pump let ad eae at pump dhage had the poblem, whle pump wth a gadual deeae o eae damete dd ot. The tablty wa elmated oe of the pump by eplag the abupt ppg expao tato wth a gadual ppg expao tato at pump dhage ad a gadual otato tato at pump let. Coley [] ubequetly developed a model to expla the tal tablty ad t eoluto. He mplemeted a feed-bak loop model betwee plaa oto moto at a eal δ X & eal ad a eato foe developed by a flow petubato at the dhage. H feed-bak model volve the followg poedue:. etubed moto of the oto δ X & eal at a eal loato ext to the ext mpelle aue a leakage petubato at the mpelle eal letδ of the ext flow of the pump;.e., δ Q & δq&. pump eal q& eal, yeldg the ame petubato. δq & pump aue a petubato the pump dhage peue δpump petubato of the pump H-Q uve. defed by a Th the follow the fomat of ASME Joual of Tbology.

13 . Ietg δ t to the dhage le of the pump aue a egatve, delayed, pump ad atteuated peue wave δ t τ R δ t, wth the atteuato ppe d fator <. Th model mple that a potve tep-futo peue petubato to the dhage le would podue afte a tme delayτ d a egatve tep-futo efleted wave at edued ampltude. Coley obtaed the atteuato fato R fom a aout aaly. pump 4. The efleted wave δppe at o the pojeted aea of the mpelle to podue the eato foe δ f mpelle δppe Amp, that loe the loop betwee oto moto δx & eal ad the eato foe δ f mpelle, whh may ethe be tablg o leadg to moe dplaemet the eal, depedg o the tme delay ad oto peed. Coley ued a dgtal ompute ode that mulate a aalog ompute to pefom epeated taet aalye o h model. He howed that a tablty ould be podued by vayg the tme delayτ d. Alo, the oto atual fequey ould be vaed fom aoud 65HZ to 50HZ by vayg the tme delay. Hee Coley multaeouly detfe a tablty meham ad how how the oto atual fequey a be hfted due to a teato wth the dhage le. Coley taet modelg appoah aot be ued fo ovetoal tablty aaly e t eque multple tme mulato to map out the tablty haatet. Moeove, h tme-delay aout model pefe two dffeet peue at oe loato a above poedue how, whh a evdet defey. Coley plaa model

14 ompatble wth mot uet otodyam model, whh debe oto moto two plae. I th the, a det fequey-doma model developed to eplae Coley taet modelg appoah fo tablty aaly. The ppele dyam modeled va fte elemet method ug a tapeed flow elemet. Soula ad Sa Ade [] eeteal flow omputatoal oluto ued fo the eal flow-petubato veu oto-moto tafe futo. The omplete oto-moto/eato-foe tafe futo developed a a plug tafe futo module to XLTRC a otodyam ode developed by Tubomahey Laboatoy. The eultat loed-loop tafe-futo model ug wth dffeet eal leaae ad dhage ppe ofguato, to vetgate the tablty haatet. Compao wth tet eult wee oduted to evaluate the ew model.

15 4 II. RESEARCH OBJECTIVES The eeah aed out hee awe the followg queto:. How doe eal eetty fluee otodyam haatet?. How doe eal leaae fluee the otodyam haatet?. How do the otodyam haatet hage wth ppe ofguato? 4. How doe the eato-foe agle β affet the otodyam haatet? 5. How doe the ug peed affet the otodyam haatet? eult? 6. Ca the Wate-Hamme/Rotodyam Couplg model mulate oetly the feld

16 5 III. ROTOR MODEL IN XLTRC. ump deg The wate jeto hppe pump Fgue a lage, two-tage, double-uto volute pump. The hppe ad boote pump ae detly dve by a two-haft ombuto ga tube. The 6 6 0DVMF pump ha a ovetoal peue dam ad babbtted leeve beag to uppe ol whl at the hghe peed. Wea g fo the pump ae of ovetoal deg wth a tellte ovelay ad leaae of 0.5 to 0.65 mm 0.0 to Gea-type ouplg wee ued o the ta. The ogal pump had a dhage tato fom 0.406m 6 to m 4 ove a pa of 0.5 m 0. Fgue Outle Dawg of the 6*6*0 DVMF Shppg ump

17 6. Roto model Fgue how the oto model XLTRC, whh baed o the fte elemet method. Half the ma of the ouplg added at eah ed of the pump. Seal ad peue dam beag oeffet wee alulated fom the bult- pogam of XLTRC ad added to the model. The mpelle fluee wa alo take to aout by addg uppot fle, whoe oeffet wee alulated a debed by Chld [4]. 0 Shaft Radu, he 0 0 Shaft Couplg Beag mpelle eal 55 Shaft 6 60 Beag Couplg Axal Loato, he Fgue Roto Model XLTRC The oto ha a total legth of mete 8 ad ma of 840 kg 85 lbm, ludg the ouplg. Fgue to Fgue 6 how the ft fou damped mode hape wth the ogal-leaae 0.65mm 0.05 eal. Note that the ft two mode ae boue

18 7 ad pth mode, whh ae gd-body mode. The thd oe bakwad whlg. The fouth oe uodal, wth low ampltude at the beag loato, ad maxmum ampltude at the eal loato. Damped Egevalue Mode Shape lot fowad bakwad f0.6 pm d.574 eta N750 pm Fgue Ft Damped Mode Shape wth Ogal-leaae Seal Damped Egevalue Mode Shape lot fowad bakwad f499.6 pm d.606 eta N750 pm Fgue 4 Seod Damped Mode Shape wth Ogal-leaae Seal

19 8 Damped Egevalue Mode Shape lot fo w a d bakwad f.7 pm d.570 eta N750 pm Fgue 5 Thd Damped Mode Shape wth Ogal-leaae Seal Damped Egevalue Mode Shape lot fow ad bakwad f498. pm d.899 eta N750 pm Fgue 6 Fouth Damped Mode Shape wth Ogal-leaae Seal Wth the ogal-leaae eal 0.65 mm 0.05, the oto ha a damped tal peed at 50 RM, whle wth double-leaae eal.7 mm 0.05, t deeae to 900 RM. Alo, the dampg ato deeae. No tablty poblem wa pedted ethe ae fom the otodyam oot lou plot. Fgue 7 to Fgue 0 how the damped tal peed map ad otodyam oot lou plot wth ogal-leaae ad double-

20 9 leaae eal epetvely. All thee fequey ompoet ae well damped exept the oe jut above 6000 CM. Nomally, oto ytem wth double-leaae eal wll gve lowe atual fequee ad dampg ato. Howeve, fo thoe fequey ompoet aoated wth a mode that ha a vbato ode the mddle of the oto, doubleleaae eal how o obevable fluee. Rotodyam Damped Natual Fequey Map 8000 Natual Fequey pm Roto Speed, pm Fgue 7 Rotodyam Damped Natual Fequey Map wth Ogal-leaae Seal 0.65mm 0.05

21 0 Rotodyam Root Lou lot Dampg Rato Coal Cyldal Bakwad mode Natual Fequey, pm Fgue 8 Rotodyam Root Lou lot wth Ogal-leaae Seal 0.65mm 0.05 Rotodyam Damped Natual Fequey Map 8000 Natual Fequey pm Roto Speed, pm Fgue 9 Rotodyam Damped Natual Fequey Map wth Double-leaae Seal.7mm 0.05

22 Rotodyam Root Lou lot.00 Dampg Rato Coal Cyldal Bakwad mode Natual Fequey, pm Fgue 0 Rotodyam Root Lou lot wth Double-leaae Seal.7mm 0.05

23 IV. CLOSED LOO MODEL. Stadad wate-hamme aaly fo ppele The mple wate-hamme example of Fgue ot of a let le oetg a tak to a pump that povde flow to a dhage le, whh ed at a valve wth etae R. d Fgue Wate-hamme Example of ump Fgue how a typal petubed flow-ate ad peue model, whee the ext petubed flow fom the let le deotedδ let ad ext ae deotedδ,δ let dh age q& pump, the peue petubato at the pump, A a dampg oue that eulted fom the pump H-Q uve, whh model the peue dffeee ao the pump. δ ex deote the peue petubato jut upteam of the valve ad δq& ex deote the petubed flow-ate at the valve.

24 Fgue etubed Wate-hamme Flow-ate ad eue Model Th model a be ued to ode flow ad peue petubato the ytem due to dtubae the ytem, e.g., a petubato the dhage valve poto. The waveequato model fo the let ad dhage le omally oveted to a ytem of oday dffeetal equato that a be developed va fte-elemet aaly.. Fte elemet developmet fo ppele Y. Road [5] gve the wave equato goveg the popagato of plae wave ho. Ethe equato o equato a be ued to aout fo the expao o otato the dhage le o let le. u u u u t t Wheeu veloty of the flud alog the ppe, the peue, ad the veloty oud. Equato ued fo let le whee veloty the deed vaable; Equato

25 4 ued fo dhage le whee peue the deed vaable. Laplae tafomato [6] ae take to obta the tafe futo. Appedx A how the full developmet of the tafe futo betwee eal flow-ate petubato ad petubed peue.. Seal flow-ate petubato alulato Soula ad Sa Ade [] gve the defto of the petubed axal flow-ate ad the global petubed axal flow-ate, whh gve a a tegal of the petubed axal flow-ate ove the umfeee. They alo peet the eult obtaed fo the detemato of the global petubed axal flow-ate at the yhouou fequey a a futo of the eal tat dmeole eetty at the let ad ext plae of the eal teted by Maquette[7]. Smla alulato wee oduted fo the eye eal of the dhage mpelle. Eah eal ha damete 4. mm.75, axal legth 6.5 mm.5. Appedx C gve the put data fo eal flow-ate petubato alulato. Fo both the ogal-leaae eal ad double-leaae eal, 0. tat eetty the tat dmeole eetty wa take fom the tat load defleto alulato XLTRC. Th aaly alulate the oto defleto at eta ug peed wth beag oeffet oly. The eult at eal loato ae take fo eal flow-ate petubato alulato. wa ued. The global flow-ate petubato due to geeal moto : δ q& Z R Z R eal qx x qy y

26 5 Whee R, R x y ae ompoet of the oto poto veto that ae paallel ad pepedula epetvely, to the tat eetty veto e 0 ; Z, Z ae tafe futo betwee oto moto ad eal flow-ate petubato paallel ad pepedula to the tat eetty veto epetvely. Se the flow-ate petubato alulato ae pefomed by ug aothe oodate ytem X, Y, Z, whee Z alog the oto ax qx qy ad Y vetal, a oodate tafomato betwee X, Y, Z ad x, y, eeay hee. Fgue how the tatoay ytem X, Y, Z ad eal eetty-oeted ytem x, y,. The x ax of the x, y, ytem deted alog e 0 ; hee,, the agle betwee the x ad X : e0y ta 4 e 0 X y Y x X Fgue Coodate Sytem

27 6 I th model, alo take fom the tat load defleto eult XLTRC. The ompoet of the eal dplaemet veto the x, y eal-eetty-oeted oodate ytem ae: R x R o R, R R R o 5 X Y y X Y Fom equato, the petubed flow-ate due to moto wth a eal at Z : δq& Z R X qx [ Z [ R qx X o R o Z qy Y ] Z ] R Y qy [ Z [ R qx X Z R qy Y o ] o ] 6 Fgue 4 ad Fgue 5 how the alulated eult of Z, Z veu extato fequey at the ug peed 4700 RM, wth ogal-leaae 0.65 mm 0.05 ad double-leaae.7 mm 0.05 eal epetvely. The alulated eult of Z qx, Z veu extato fequey at the ug peed 750 RM ae alo how qy a Fgue 6 ad Fgue 7, wth ogal-leaae 0.65 mm 0.05 ad doubleleaae.7 mm 0.05 eal epetvely. Fou dffeet ee how thee fgue epeet eal pat ad mgay pat of Z, Z, a label how. Note that the eal qx qy qx qy pat of Z qx pat of Z qx δq& o R x x δq& o R x x lghtly eag wth hagg fequey, whle the magay eae lealy wth eag fequey. A tafe futo betwee moto the x deto ad the flow petubato model a be modeled a:

28 7 δ q& R x x a0 a jω a0 a Z qx 7 The eult foδ q& y, δ q& y eal pat ad mgay pat of δ q& y ae moe omplated. The eal pat of Z qy o δq& R y x eae a moe o le quadat mae wth eag fequey. It eahe a mmum value at 400 RM ad the beg to eae. The magay pat of Z qy o δq& R y x eae lealy wth extato fequey. Note that the eal pat of Z qy muh malle tha the othe flow petubato tem ad loe to eo. Fo mplty, a eaoable tafe futo betwee moto the y deto ad the flow petubato a be modeled a: δ q& R y y b Ω j b Zqy 8 Alo, ote that the petubed flow-ate wth double-leaae eal ha muh hghe δq& R x x about thee tme tha that of ogal-leaae eal, whh foud late to play a mpotat ole makg the pump otodyam ytem utable.

29 8 etubed Flow Rate veu Extato Fequey Rug Speed 4700 RM, Ogal Seal Cleaae 0.65 mm 600 etubed Flow Rate pe Ut Moto Lte/m/m Re Zqx Im Zqx Re Zqy Im Zqy Extato Fequey RM Fgue 4 etubed Seal Flow-ate pe Ut Moto Veu Extato Fequey wth Ogal-leaae Seal at Rug Speed of 4700 RM etubed Flow Rate veu Extato Fequey Rug Speed 4700 RM, Double-eal Cleaae.7 mm 600 etubed Flow Rate pe Ut Moto Lte/m/m Re Zqx Im Zqx Re Zqy Im Zqy Extato Fequey RM Fgue 5 etubed Seal Flow-ate pe Ut Moto Veu Extato Fequey wth Double-leaae Seal at the Rug Speed of 4700 RM

30 9 etubed Flow veu Exato Fequey Rug Speed 750 RM, Ogal Seal Cleaae 450 etubed Flow Rate pe Ut Moto Lte/m/m Re Zqx Im Zqx Re Zqy Im Zqy Extato Fequey RM Fgue 6 etubed Seal Flow-ate pe Ut Moto Veu Extato Fequey wth Ogal-leaae Seal at the Rug Speed of 750 RM etubed Flow Rate veu Exato Fequey Rug Speed 750 RM, Double-eal Cleaae etubed Flow Rate pe Ut Moto Lte/m/m Re Zqx Im Zqx Re Zqy Im Zqy Exato Fequey RM Fgue 7 etubed Seal Flow-ate pe Ut Moto Veu Extato Fequey wth Double-leaae Seal at the Rug Speed of 750 RM

31 0 4. Couplg of oto moto to wate-hamme dyam Equato 6 defe the petubed flow-ate that podued at a eet eal due to oto moto. The queto : How doe the petubed flow the eal fluee the uteady moto the let o dhage le? etubed moto at the weag-g eal fo the ft tage of a pump would podue petubed flow at t ext that would the detly petub the let flow of the pump. Th ot of peue ollato at the pump let would pobably eate a axal ollato of the pump oto veu a adal oto petubato. etubed moto of a weag-g o te-tage eal at a teo mpelle would ot a detly fluee the flow at ethe the let o dhage flow of the pump. Fo both taght-though ad bak-to-bak pump, petubed flow fom the let of the balae-pto eal wll detly petub the ext flow of the pump; hee, fo th duo, oly the mple queto wll be odeed: How doe petubed moto at the ext balae-pto eal fluee uteady flow the dhage le? Fgue 8 llutate the petubed flow-ate fom the eal δq& eal beg added to the petubed pump flowate δ. The et petubed flow δ q& dh e jeted to the dhage-le ad q& pump podue evetually a petubed peue dtubae at the pump dhage. ag

32 Fgue 8 etubed Wate-hamme Flow-ate ad eue Model wth etubed Flow-ate δq& Ijeto eal fom a Seal 5. Couplg of peue ollato to oto moto The petubed peue ollato at the pump dhage at o pojeted aea of the dhage mpelle ad loe the loop o the model. Fgue 9 how the dhage ppe of a volute paallel to the X ax. The foe atg o the oto due to a peue petubato o the oto oeted wth epet to the Y ax by the eato foe agle β. Th agle hould eaoably age fom -90 to 90 degee. Moe peely, t pobably loe to 0 degee th ae. The eultat foe o the dhage mpelle δ A, whee A mp the pojeted aea of the dhage mpelle. It veto defto : dh ag e m δf δf X Y δ dh ag e A m β o β 9

33 Y X Fgue 9 ump Volute Dhage 6. Complete loed-loop model Fgue 0 how the omplete loed-loop model. Statg wth the otodyam model, a petubato the oto poto at the balae-pto eal loato R, R aue a petubato the flow-ate at the let to a eal δ &. Th flow X Y petubato at the let to the pump dhage le aue a hage the pump dhage peueδ dh ag e. Th petubato, δ dh ag e, at o the pojeted aea of the mpelle ad develop the eato foe ompoet δ f, δf. Fo th double-uto dhage X Y mpelle, δ & doubled a a gle petubato fo oveee, whh eaoable q eal e the two eal ae petty loe to eah othe. q eal

34 Stat hee Rotodyam Model Eet-eal oluto Y R X, R δf X, δf Y δ F δ dh A ag e m Seal Model ojeted Aea of Impelle δq& eal δdh ag e Wate-hamme Model wave equato Fgue 0 Complete Cloed-loop Sytem Steete ad Wyle [8] how how to alulate the fte temato mpedae Z, whh ued to mulate the effet of the log hppg ppe. Fgue how the omplete tafe futo betwee oto dplaemet R, R ad the eato foe ompoet δ f, δf, a a tafe-futo matx. X Y X Y C δf X δ δf Y G G G G dh ag β e Am δq& T A o β RX RY mpelle β o β 0 Whee:

35 4 G G G G [ Z [ Z qx qx [ Z [ Z o Z Z qx qx o Z Z qy qy ] T A o ] T A qy qy ] T A o ] T A mpelle mpelle mpelle mpelle β β o β o β T the tafe futo alulated fo the ppele a dued Appedx B. The pojeted aea of the mpelle etmated to be m 0. The omplete tafe futo mght have the ame ode 4 N the umeato ad the deomato, whh deped o the umbe of mode N take to aout the watehamme model. Kleyha ad Chld [9] ad Appedx D how how to develop a tatevaable model ad alulate the omplex egevalue wth geeal umeato-deomato fom tafe futo. δf X, δf Y Rotodyam Model R X, RY G j Fgue Cloed-loop Model Showg the Tafe Futo betwee the Roto Dplaemet Veto ad the Reato Foe

36 5 V. MODEL RESULTS AND DISCUSSION The loed-loop model eult ae peeted th eto. Oly the dhage le odeed hee, e the model ludg the let-le had mmal fluee o the eult. The ogal-leaae eal of 0.65mm 0.05h ad double-leaae eal of.7mm 0.05h ae ued to vetgate the effet of eal leaae. The eato-foe agle β alo vaed to tudy t effet o otodyam tablty. Dffeet dhage ppe ofguato ae examed to vetgate the fluee o otodyam haatet. Fgue how two dhage ppe ued the model. The ogal dhage tato ha left damete of 0.406m 6 ad ght damete of , ove a pa of 0.5 m 0. The modfed gadual expao dhage tato ha the ame left damete ad ght damete, ove a loge pa of.8 m 7. Dffeet ug peed ae examed to how how the otodyam haatet hage wth ug peed. The feed-bak eato foe agle β wa foud to have gfat fluee o otodyam haatet. Fgue to Fgue 7 how the otodyam oot lou plot wth dffeet eal leaae, dhage ppe ofguato ad ug peed, whee the eato foe agle β vae fom -90 to 90 degee. Fgue 5 how the ft fou o damped mode wth wate-hamme ouplg, double-leaae eal ad β 5.

37 6 Ogal dhage ppe: LD0.406m, RD0.609m, Legth0.5m Gadual Expao ppe: LD0.406m, RD0.609m, Legth.8m Fgue Dffeet Dhage pe Cofguato.00 ROOT LOCUS LOT Ogal eal Cleaae, Ogal Dhage pe, Rug Speed 4700 RM, Chagg Dampg Rato o β Damped Natual Fequey CM Fgue Rotodyam Root Lou lot wth Ogal-leaae Seal ad Abupt Expao at the Rug Speed of 4700 RM

38 7 Root Lou lot Double-eal Cleaae, Ogal Dhage pe, Rug Speed 4700 RM, Chagg Dampg Rato Damped Natual FequeyCM Fgue 4 Rotodyam Root Lou lot wth Double-leaae Seal ad Abupt Expao at the Rug Speed of 4700 RM.00 Root Lou lot Ogal eal Cleaae, Gadual Expao pe, Rug Speed 4700 RM, Chagg Dampg Rato o β Damped Natual Fequey CM Fgue 5 Rotodyam Root Lou lot wth Ogal-leaae Seal ad Gadual Expao at the Rug Speed of 4700 RM

39 8 Root Lou lot Double-eal Cleaae, Ogal Dhage pe, Rug Speed 4700 RM, Chagg Dampg Rato Damped Natual FequeyCM Fgue 6 Rotodyam Root Lou lot wth Double-leaae Seal ad Gadual Expao at the Rug Speed of 4700 RM Root Lou lot Double-eal Cleaae, Ogal Dhage pe, Rug Speed 750 RM, Chagg Dampg Rato Damped Natual Fequey CM Fgue 7 Rotodyam Root Lou lot wth Ogal-leaae Seal ad Abupt Expao at the Rug Speed of 750 RM

40 9 Damped Egevalue Mode Shape lot fow ad bakwad f54. pm d.488 eta N997 pm Fgue 8 t Damped Mode Shape wth Wate-hamme Couplg, Double-leaae Seal, β 5 o Damped Egevalue Mode Shape lot fowad bakwad f090.7 pm d.56 eta N997 pm Fgue 9 d Damped Mode Shape wth Wate-hamme Couplg, Double-leaae Seal, β 5 o

41 0 Damped Egevalue Mode Shape lot fow ad bakwad f85.9 pm d -.74 eta N997 pm Fgue 0 d Damped Mode Shape wth Wate-hamme Couplg, Double-leaae eal, β 5 o Damped Egevalue Mode Shape lot fow ad bakwad f604.4 pm d.086 eta N997 pm Fgue 4 th Damped Mode Shape wth Wate-hamme Couplg, Double-leaae Seal, β 5 o. Effet of the eato foe agle β Fgue to Fgue 7 how the otodyam oot lou plot whee the eatofoe agle hagg the data pot wth the ame olo have dffeet eato-foe

42 agle. Fgue 8 to Fgue how the ft fou damped mode wth wate-hamme o ouplg, double-eal leaae ad β 5. The followg pheomea a be ee fom thee fgue: Two damped atual fequee ema almot uhaged wth hage to the eato-foe agle. Reallg the mode hape gve Fgue to Fgue 6, th eult eaoable e the eal petubato loato ode wth the vbato ode. The othe atual fequee ae gfatly hfted by the eato foe agle, ethe tabled o detabled. Itablte ae pedted eta age of agle β betwee 0 ad 90 degee wth a uodal mode hape. Th age hage wth dffeet eal leaae, ug peed ad dhage tato ofguato. Thee uetaty the eato foe agle β, at whh tablty ould be pedted peely. Wth ogalleaae eal ad abupt dhage tato at the ug peed 4700 RM, th age about 8 to 85 degee. Wth double-eal leaae ad abupt dhage tato at the ug peed 4700 RM, th age about 0 to 90 degee. Wth ogal-leaae eal ad gadual dhage tato at the ug peed 4700 RM, th age about 8 to 90 degee. Wth double-eal leaae ad gadual dhage tato at the ug peed 4700 RM, th age about 5 to 90 degee. Alo, the tablty fequey hage damatally wth the eato foe agle β, fom 000 CM to CM. 4 No tablty epoted by ompute mulato f the eato-foe agle loated the age fom -90 degee to 0 degee.

43 . Effet of eal leaae Compao of Fgue ad Fgue 4 how that eal leaae ha a gfat effet o the otodyam haatet. Bede what metoed befoe eto III., a double-leaae eal a elage the flow-ate petubato a how Fgue 4 to Fgue 7; hee, makg the poblem woe. The tablte obeved Fgue 4 wth double-leaae eal have muh lage egatve dampg ato tha thoe fgue 0 wth the ogal-leaae eal. Alo, tablty appea oly fo the eato foe agle β agg fom 8 to 90 degee Fgue, whle Fgue 4, tablty appea a age of β fom 5 degee to 90 degee. Smla eult hold fo Fgue 5 ad Fgue 6. Fgue ad Fgue how the otodyam atual fequey map ad oot lou o plot wth ogal-leaae eal, ogal dhage tato ad β 5. Fgue 4 ad Fgue 5 how the otodyam atual fequey map ad oot lou plot wth the o double-leaae eal, ogal dhage tato ad β 5. Wth the ogal-leaae eal, the ut magally table though out the peed age. Wth double-leaae eal, a ub-yhoou tablty ompoet how up wth a peeo fequey of 00 RM. The utable fequey eae wth ug peed ad appoahe to aothe atual fequey, whh loe to what wa obeved the feld.

44 Rotodyam Damped Natual Fequey Map Natual Fequey pm Thd mode Roto Speed, pm Fgue Rotodyam Damped Natual Fequey Map wth Ogal-leaae Seal, Ogal Dhage Tato, β 5 o Rotodyam Root Lou lot.00 Dampg Rato Thd mode Natual Fequey, pm Fgue Rotodyam Root Lou lot wth Ogal-leaae Seal, Ogal Dhage o Tato, β 5

45 4 Rotodyam Damped Natual Fequey Map Natual Fequey pm Utable mode Roto Speed, pm Fgue 4 Rotodyam Damped Natual Fequey Map wth Double-leaae Seal, o Ogal Dhage Tato, β 5 Rotodyam Root Lou lot.500 Dampg Rato Utable mode Natual Fequey, pm Fgue 5 Rotodyam Root Lou lot wth Double-leaae Seal, Ogal Dhage o Tato, β 5

46 5. Effet of dhage ppe ofguato A how Fgue, the modfed tapeed pee ha a tato fom m 6 to m 4 ove a pa of.8 m 7 tead of ogal 0.5 m 0. Fgue ad Fgue 5 how the otodyam oot lou plot wth the ogal abupt expao tato ad the modfed gadual expao tato epetvely. Smlaty foud betwee thee two fgue. A fte temato wthout a tapeed tato pee alo examed. Fgue 6 how the otodyam oot lou plot wth a fte temato ad double-leaae eal. Compao wth fgue 6 otodyam oot lou plot wth doubled leaae eal ad wthout wate hamme ouplg how that the fte temato ha o vble fluee o the otodyam haatet. Rotodyam Root Lou lot.00 Dampg Rato Natual Fequey, pm Fgue 6 Rotodyam Root Lou lot wth a Ifte Temato ad Double-leaae Seal

47 6 4. Effet of ug peed Fgue 4 ad Fgue 7 how the otodyam oot lou plot wth double-leaae eal, the ogal dhage tato fo peed equal to 4700 RM ad 750 RM, epetvely. Compao how that otodyam haatet ae qute mla. Itablte ae pedted a aowe age of agle β at 750 RM. Theefoe, a lowe ug peed ha a lttle bette pefomae. 5. Compao wth feld eult The feld tet Coley[] how that the ut wth ogal-leaae eal ha a welldamped tal at 800 RM. I the utable ut, at a peed of appoxmately 00 RM, a beatg pheomeo appea the vbato patte, whh mea two dete fequee ae loe togethe. A the peed eae futhe, thee two fequee epaate fom eah othe. The ub-yhoou vbato ompoet ha a lght eae fequey wth eae ug peed, whle the ampltude eae damatally, whh a be ee Fgue 7. Afte eplag the abupt expao dhage tato ppe wth a gadual expao tato ppe, the ub-yhoou vbato ompoet dappeaed. Fgue 8 how the hootal haft vbato at boad beag afte tallato of the log tape tato. Compao of the peet model eult ad feld eult how that th model a pedt a ub-yhoou tablty wth a double- o leaae eal ad the ogal abupt expao dhage ppe at β 5, wth a peeo fequey at aoud 00 RM, a Fgue 5 how. Afte eplag of abupt

48 7 dhage tato wth gadual dhage tato, the ew model doe t yeld the ame fluee o tablty a feld tet how, whh mple that thee mght be ome othe meham dve th poblem. Fgue 7 Hootal Shaft Vbato at the Iboad Beag, Coley[]

49 Fgue 8 Hootal Shaft Vbato at the Iboad Beag Afte Itallato of Log Tape Tato, Coley [] 8

50 9 VI. CONCLUSIONS A feedbak fequey-doma otodyam/wate-hamme ouplg model ha bee developed to mulate a feld pump vbato poblem. The model uefully duplate the feld eult of the ub-yhoou tablty. Couplg of moto at eal ad the eato foe a makedly hft atual fequee ad dampg. Seal leaae, eato foe agle, ad dhage ppe tape-ofguato a fluee the otodyam haatet gfatly. The followg oluo ae daw fom the above developmet, alulato ad ompao:. The mpelle eato foe ha a gfat effet o otodyam haatet.. Th wate-hamme ouplg model a pedt ub-yhoou tablty obeved the feld uefully, wth double-leaae eal.. The eae of eal leaae a edue dampg ad tffe. Wth watehamme ouplg, the eae of eal leaae eult geate petubato flow-ate, thu dug tablty. 4. The eato-foe agle gfatly fluee the otodyam haatet. Itablty ompoet ethe ub-yhoou o upe-yhoou may be dued fo eta age of β. Uetaty foud egadg the fluee of β o the otodyam haatet. It yet to udetad how to alulate the eato foe agle feld.

51 40 5. By vayg the eato-foe agle β, mla otodyam haatet ae obeved wth abupt ad gadual dhage tato. A ftely log temato doe ot fluee the otodyam haatet. 6. The pedted otodyam haatet ae mla at ug peed 750 RM ad 4700 RM.

52 4 REFERENCES [] Coley, J.E., 978, Subyhoou Vbato a lage Wate Flood ump, oeedg, 7 th Tubomahey Sympoum, pp.0-0. [] Coley, J.E., 994, A ump Itablty Theoy Ug a Aout Feedbak Meham, IGTI pape 94-GT-0, peeted at Iteatoal Ga Tube ad Aeoege Coge ad Expoto, De Hague, The Nethelad. [] Soula, T. ad Sa Ade, L., Jue 00, Ivetgato o the Relatohp betwee the Fequey-Depedet etubed Ft-Ode Flow-ate ad the Seal Eetty, pvate ommuato. [4] Daa W. Chld, 99, Tubomahey Rotodyam, heomea, Modelg, ad Aaly, Joh Wley & So, I.. [5] Y. Road, 960, Geeal Dyam of Vbato, Fedek UNGAR ublhg Co., New Yok, pp [6] Wllam T. Thomo, 960, Laplae Tafomato, ete-hall, INC., pp. -5. [7] Maquette, O., 995, Expemetal Veu Theoetal Compao of the Stat ad Dyam Chaatet of Oe Smooth ad Two Gooved Lqud Aula Seal wth L/D of 0.457, Tehal Repot No. TL-SEAL-5-95, Tubomahey Laboatoy, Texa A&M Uvety Sytem, College Stato, TX. [8] Steete, Vto L. ad Wyle, E. B., 967, Hydaul Taet, MGaw-Hll book Co., New Yok, pp [9] Geoge F. Kleyha, Daa W. Chld, 997, The Aout Ifluee of Cell Depth o the Rotodyam Chaatet of Smooth-Roto/Hoeyomb-Stato Aula Ga

53 4 Seal, ASME Ta., Joual of Egeeg fo Ga Tube ad owe, Vol. 9, No. 4, pp

54 4 AENDIX A BEARING, SEAL, IMELLER COEFFICIENTS IN XLTRC. eue dam beag otodyam oeffet XLTRC XLeDm Speadheet fo eue Dam Beag Dyam Coeffet Veo.0, Copyght by Texa A&M Uvety. All ght eeved. Ttle: edam Beag_kk Beag Damete 6.75 he Seleted Lubat Beag Axal Legth.88 he ISO 00 Supply eue 0 p Lube Flm Tempeatue 58 Deg F Dhage eue 0 p Voty at Tflm 8. p Cavtato eue 0 p Dety at Tflm 5.9 lb/ft kt o Gv Axal Node 0 -- Lad Axal Node 0 -- ad Numbe ad Lead Edge ad A Le eload Agle Mahed Cleaae eload Dam/Gv A Legth Dam/Gv Axal Legth Dam/Gv Depth kt/gv C Node -- degee degee degee he -- degee he he Lad C Node X Load Y Load Speed Kxx Kxy Kyx Kyy Cxx Cxy Cyx Cyy lb lb pm lb/ lb/ lb/ lb/ lb-/ lb-/ lb-/ lb-/ E05.04E E E05.66E0-9.97E0 -.07E0.7E E05.9E E E05.64E0-9.4E0-9.89E0.59E E05.6E05-6.4E05 6.E05.64E0-9.E0-9.4E0.49E E05.5E05-7.0E05 6.7E05.6E0-8.80E0-8.87E0.4E E05.60E E05 7.E05.59E0-8.45E0-8.47E0.5E E05.69E05-8.4E E05.56E0-8.5E0-8.6E0.0E E05.79E05-9.0E05 8.5E05.5E0-7.9E0-7.9E0.5E E05.90E E05 9.E05.5E0-7.7E0-7.7E0.E0 Speed E X E Y Max owe Lo Q out Keq WFR Meq pm he he p hp ft /e lb/ -- lb E E E-0.6E E E E E-0.55E E E-04-9.E E-0.85E E E-04-8.E E-0.4E E E E E-0.4E E E E E-0.69E E E E E-0.96E E E-04-5.E E-0 4.E E0

55 44. Cetfugal pump mpelle otodyam oeffet XLTRC XLImpl Speadheet fo Cetfugal ump Impelle Rotodyam Coeffet Veo.0, Copyght by Texa A&M Uvety. All ght eeved. Ttle: mpelle Impelle Type Choe: A : at 000 pm ad 0 deg C B : at 000 pm ad 0 deg C C : at 000 pm ad 0 deg. C Da : at 000 pm ad 60 deg. C Db : at 4000 pm ad 60 deg. C D : at 4000 pm ad 60 deg. C Dd : at 000 pm ad 0 deg. C Note: leae efe to Selet Impelle Type, fom Table o the ght ad above Fgue, hapte 6 of the text by, that bet debe the Geomety ad Opeatg Codto Chld, D., 99, "Tubomahey Rotodyam: Impelle Type C heomea, Modelg, Lqud Dety.94 lug/ft ad Aaly," Impelle Radu 9 Joh Wley & So I., Thoat Wdth b fo futhe fomato Rotodyam Coeffet Speed Kxx Kxy Kyx Kyy Cxx Cxy Cyx Cyy Mxx Mxy Myx Myy pm lb/ lb/ lb/ lb/ lb-/ lb-/ lb-/ lb-/ lb-**/ lb-**/ lb-**/ lb-**/ E E

56 45. Aula eal otodyam oeffet XLTRC XLASeal Speadheet fo Aula Seal Dyam Coeffet Veo.0, Copyght by Texa A&M Uvety. All ght eeved. Ttle: Aula Tet Seal_kk Seal Damete.75 he Max Iteato Seal Axal Legth.5 he Mometum Relaxato Fato Ilet Radal Cleaae 0.05 he eue Relaxato Fato Ext Radal Cleaae 0.05 he Tempeatue Relaxato Fato eload 0 he Roto Relatve Roughe Numbe of Lobe -- Stato Relatve Roughe Lead Edge Loato 0 degee Moody' Coef Amod Moody' Coef Bmod Moody' Coef Expo Etae Lo Coef No. C. Gd ot -- Ext Seal Coef 0 -- No. Axal Gd ot 6 -- Supply Tempeatue 9 0 F Ilet Swl Rato Seleted Lubat X Stat Eetty 0 -- Wate teal value Y Stat Eetty 0 -- Voty at Tupply p Momet Coef Opto Do Not Compute Dety at Tupply 5.00 lb/ft Fequey Aaly Opto Syhoou Aaly Compeblty.5E-06 /lb Cotat Shaft Rpm 000 pm Spef Heat BTU/lb 0 F Themal Codutvty BTU/ft-h 0 F Stat Rotato About X 0 ada Coef Them Exp / 0 F Stat Rotato About Y 0 ada Temp V Coef / 0 F Seal vot Loato 0 he XLASeal Speadheet fo Aula Seal Dyam Coeffet Veo.0, Copyght by Texa A&M Uvety. All ght eeved. Ttle: Aula Tet Seal_kk Seal Damete 9.5 he Max Iteato Seal Axal Legth.75 he Mometum Relaxato Fato Ilet Radal Cleaae 0.05 he eue Relaxato Fato Ext Radal Cleaae 0.05 he Tempeatue Relaxato Fato eload 0 he Roto Relatve Roughe Numbe of Lobe -- Stato Relatve Roughe Lead Edge Loato 0 degee Moody' Coef Amod Moody' Coef Bmod Moody' Coef Expo Etae Lo Coef No. C. Gd ot -- Ext Seal Coef 0 -- No. Axal Gd ot 6 -- Supply Tempeatue 9 0 F Ilet Swl Rato Seleted Lubat X Stat Eetty 0 -- Wate teal value Y Stat Eetty 0 -- Voty at Tupply p Momet Coef Opto Do Not Compute Dety at Tupply 5.00 lb/ft Fequey Aaly Opto Syhoou Aaly Compeblty.5E-06 /lb Cotat Shaft Rpm 000 pm Spef Heat BTU/lb 0 F Themal Codutvty BTU/ft-h 0 F Stat Rotato About X 0 ada Coef Them Exp / 0 F Stat Rotato About Y 0 ada Temp V Coef / 0 F Seal vot Loato 0 he

57 46 AENDIX B FINITE ELEMENT MODEL DEVELOMENT. Wave equato ad hape futo Y. Road [5] gve the wave equato goveg the popagato of plae wave ho, whh a be ued hee to aout fo the expao o otato the dhage le o let le. t u u u u B t B wheeu veloty of the flud alog the ppe, the peue, ad the aout veloty of the peue wave. Fo dhage le, peue hould be take a depedet vaable, hee, equato B odeed. The weak fom of equato B : tg tg d tg t tg δ δ δ δ B Takg oe-dmeoal two-ode tapeed elemet ad ug the tat oluto of the wave equato a the hape futo:

58 47, t t t φ φ B4 φ φ B5 The ma matx ad tffe matx fo eah elemet a be deved fom equato B a: d d d d d tg K d tg M j j j j φ φ φ φ B6 Subttuto of equato B5 to B6 gve: M tg B7 M M 6 tg B8 M tg B9 tg K B0

59 48 tg K K B tg K B Ad the ght had of the equato would be: { } T T tg tg Q Q Q B Codeg that: t q A t u B4 Equato B a be ewtte a: { } T T t q t q Q Q Q B5 Note that: tg tg, B6 The ma matx, tffe matx ad Q veto a be ewtte a:

60 49 l M l M M l M B7 l K l K K l K B8 { } T T t q t q Q Q Q B9 Theefoe, wth peue a the depedet vaable, the ytem equato beome: [ ]{} [ ]{ } { } Q K M & & B0 Epeally, whee { } Q ha the fom: { } T t q t q Q... 0 B Fo let le, the flow-ate hould be take a depedet vaable, hee, equato B wth flow veloty a depedet vaable odeed. Smlaly, the weak fom of equato B a be wtte a:

61 50 u tg u tg d u u tg u u tg t u u tg δ δ δ B Codeg that: u t u B The ght had de of equato B a be wtte a: u tg t tg u tg t tg RHS B4 Takg a two-oded elemet ad ug the tat oluto of equato B a hape futo:, t u t u t u φ φ B5 φ B6 φ B7

62 5 A matx fom equato a be deved fom B by ubttutg B5, B6, B7 to t: [ ]{ } [ ]{ } { } Q U K U M & & B8 Whee: { } T T j j j j j u tg t tg u tg t tg Q u u U d d d d d tg K d tg M,, φ φ φ φ φ φ B tg M 0 tg M M tg M tg K tg K K

63 5 tg K Subttutg equato B6 to B9, ma matx, tffe matx ad RHS a be wtte a: l M 0 l M M B l M l K l K K B l K T u tg t u tg t Q

64 5 Note the tem u tg ad u tg ght had de, whh a be moved to left, o that the modfed tffe ad RHS beome: l l K l l K B T t t Q Ue flow-ate a the depedet vaable tead of the flow veloty, RHS of equato B8 beome: T t t Q 4 4 B Theefoe, wth flow-ate q& a the depedet vaable, the ytem equato beome: [ ]{ } [ ]{ } { } Q U K U M & & B4 Epeally, whee { } Q ha the fom: { } T t t Q B5

65 54 The aout veloty a homogeeou, lttle ompeble lqud ad a gd odut detal wth that of oud the lqud. Due to the elatty of the ppe-le, the wave veloty a odut le tha that of oud ad atually beome a futo of ppe damete. Fo th-walled ppe-le of ula o-eto, the atual veloty may be foud fom the elato: K D ee B6. Tafe futo of dhage le The equato fo peue ued fo dhage le. The ytem equato have bee gve a B0: [ M ]{} & [ K]{ } { Q} The ma matx ad tffe matx ae quae mate, povded that elemet wee take fo the dhage ppe. The atual fequee ad mode a be obtaed by olvg the geeal egevalue poblem: ω M λ Kλ B7 Deote the egevalue ad aoated egeveto a: λ, Φ,,...

66 55 Thu the atual fequee ad mode hape ae:,...,, 0.5 Φ λ ω Let the tafomato matx Φ be: [ ]... Φ Φ Φ Φ Φ B8 Ad defe the modal oodate: { } [ ]{ } ˆ Φ B9 It eaoable hee that oly the ft eveal mode tead of all mode ae luded the tafomato matx. All the poedue wll hold exept that the umbe of modal oodate wll be dffeet. The ytem equato B0 a be dagoaled modal pae a: m m m Q K M ˆ ˆ& & B40 O matx fom: Φ ˆ ˆ... ˆ ˆ... q q && && && && && && ω ω ω ω

67 56 Whee: I M M m Φ Φ Φ Φ... K m K ω ω ω ω Q m Q Φ By takg a Laplae tafomato, t a be wthed to -doma: Φ ˆ ˆ... ˆ ˆ... q q & & ω ω ω ω B4 Equato B4 a be olved ealy, e t uoupled:,..., ˆ Q m ω B4 Whee:,..., Φ Φ q q Q m & & B4

68 57 Theefoe, the phyal oodate a be olved ug equato B9: Q Φ ˆ m j j Φ j j j ω j B44 Epeally, ad ae: q& β q& B45 q& β q& B46 Whee: j β Φ j j Φ Φ j, ω j Φ j j ω j, β j Φ j j Φ Φ j j ω Φ j j ω j B47 If the dhage le loe the ght ed, the q& 0, theefoe the tafe futo betwee ad q& a be foud to be a defed equato B47. If the dhage le ope the ght ed, the 0, the lat equato of the ytem equato B0 a be emoved. Gog though the ame poedue debed above, the tafe futo betwee ad q& a be foud to be a defed equato B47. Note that thee two ae have dffeet atual fequee ad mode, thu the eult totally dffeet, although the expeo ae the ame a:

69 58 q G & B48 The mot ofte ae of bouday odto ae wth etae d R ove the ght ed. q R d & B49 Togethe wth equato B44 ad B45, the tafe futo wth etae a be olved a: β β β d d R R q G & B50 I th pojet, the dhage ha mle log otat ppe. Ifte temato mpedae aodg to Steete ad Wyle [8] a be ued to model th ofguato. 0.5 C L R Z C B5 Whee: 5,, 8 A C A L q D f R m & B5 µ VD N N b D e b a f R R m, 4 / ae Moody fto fato ad Reyold umbe epetvely. The put date fo fte temato mpedae ae:

70 59 Mea flow-ate: q &.68m / e Flud dety: 986kg / m Flud Voty: µ 0.00a e pe Damete: D m Flow veloty: V 4.974m / e Aout veloty: 80m / e e / D 0.00, a.8e, b.0 4, b.0e 6 E Reyold Numbe: N R Moody fto fato: f m Retae: R 47.6kg / m 5 / e Capatae: C.80E 0m e / kg Ieta: L kg / m 5

71 60 The alulato how that the eal pat of the mpedae almot otat, whle the magay pat of the mpedae look a the followg Fgue B. Theefoe, the mpedae a be expeed a: Z CW KW B5 C I th ae, K , C W W Imagay at of Impedae Immpedae Fequey ad/e Fgue B Imagay at of Ifte Temato Impedae

72 6. Tafe futo of let le The wave equato wth flow-ate a vaable ued fo the let le. The ytem equato ha bee gve above a equato B4. [ M ]{ U& } [ K]{ U} { Q} Smla poedue a be oduted to obta the tafe futo. The atual fequee ad mode a be obtaed by olvg the geeal egevalue poblem: ω M λ Kλ Deote the egevalue ad aoated egeveto a: λ, Φ,,... Thu the atual fequee ad mode hape ae ω 0.5 λ, Φ,,... Let the tafomato matx Φ be: Φ [ Φ Φ Φ Φ ]... B54 Ad defe the modal oodate:

73 6 { } [ ]{ } U U ˆ Φ B55 Moe ofte, oly the ft eveal mode ae take to edue the omputato wthout loog auay, o that omally the modal oodate have dffeet dmeo fom phyal oodate. The uoupled ytem equato modal pae ae: m m m Q U K U M ˆ ˆ& & O matx fom: Φ ˆ ˆ... ˆ ˆ... ˆ ˆ... ˆ ˆ... A A U U U U U U U U & & && && && && ω ω ω ω B56 Afte takg a Laplae tafomato, equato B56 beome: Φ ˆ ˆ... ˆ ˆ... A A U U U U ω ω ω ω B57 Equato B57 a be olved qukly e t uoupled.,..., ˆ Q U m ω B58

74 6 Whee:,..., Φ Φ A A Q m B59 Gve the modal pae oluto, the phyal pae oluto a be obtaed va equato B55. ˆ Φ Φ Φ Φ j j j j j j j j A A U U ω B60 Thu, povded that 0, the tafe-futo of let le : Φ j j j A U G ω B6 4. Complete tafe futo Gve equato B50 ad B6, togethe wth: eal pump e dh q q q & & & δ δ δ ag B6 e dh e dh let q A ag ag & δ δ δ B6 The omplete tafe futo fo the ppele a be obtaed a: ag G A G G G q T eal e dh δ & δ B64

75 64 Whee A the haatet tematg mpedae of the hppe pump alulated fom the H-Q uve.

76 65 AENDIX C INUT DATA FOR SEAL FLOW-RATE CALCULATION Iput data fo Heal.exe: Roto damete: 0.4 m Flm lad legth: m Flm lad leaae: 0.65 mm ad.7 mm Dmeole tat eetty: 0. the X deto Supply peue: 5.58 Ma Dhage peue: 0 a Flud: wate Tempeatue: 57 degee Etae lo oeffet: 0. Dhage lo oeffet: 0 Dety: 000 kg/m^

77 66 Voty: 0.00 a e Cumfeetal gd pot umbe: 4 Axal gd pot umbe: Roto ad tato elatve oughe: E-

78 67 AENDIX D EIGENVALUE CALCULATION WITH GENERAL FORM TRANSFER FUNCTIONS A gle degee feedom ytem take a a example to llutate the poedue detal. Th gle degee feedom ytem ha ma, dampg, pg haatet m,, k epetvely. A geeal tafe futo wth umeato ad deomato polyomal wa opoated. Fgue D how the feedbak loe loop. F X N D Fgue D Feedbak loe loop wth geeal tafe futo opoated N A geeal fom of tafe futo H a be expaded to a fom lke: D N A B C H D a ω D Theefoe, equato of moto ould be wtte a:

79 68 F X C B X a A X k m ω D Two dffeet type of addtoal tem ae omg fom tafe futo, a we a ee fom equato D. Fo mplty, oly ode oe tem fo eah ae, Laplae doma, by defg the ew vaable: X a A X, X e C B X ω D The followg model obtaed: 0 0 X e X C B X a AX F X X X k m ω D4 I tme doma, the tate-vaable fomat of equato D4 : X ev CX BV V V X ax AX X F X m X m X m k V m V V X ω & & & & & D5 Note that the tate vaable fom ha 5 depedet tate, whh ae moe tha that wthout tafe futo, oe of them omg fom ft type, whle two of them omg fom eod type tem. Equato D5 a be tated matx fom a:

80 69 [ ] [ ][ ] [ ] F X K X & D6 The loe-loop egevalue a be alulated detly fom[ ] K. Fo a gle degee feedom ma, dampg ad pg ytem, the ogal tate matx look lke: m m k 0 D7 Afte addg ow ad olum, the ew tate matx look lke: e B C a A m m m m k ω D8

81 70 VITA KAIKAI ZHANG A. eoal Htoy Adde: 685 Rokhold Ave, Sa Gabel, CA, 9775, hoe: Emal Ade: Ctehp: eople Republ of Cha B. Eduatoal Htoy. Texa A&M Uvety, College Stato, Texa, 7784 Majo: Mehaal Egeeg Degee: Mate of See, May 00. ekg Bejg Uvety, Bejg,.R. Cha, 0087 Majo: Meha Degee: Bahelo of See, Jue 00 C. ofeoal oto Gaduate Reeah Atat, Tubomahey Laboatoy Texa A&M Uvety, at-tme poto, 08/00-/00 D. Membehp ofeoal Aoato Amea Soety of Mehaal Egee Studet Membe