ANALYSIS OF FORCED SPATIAL VIBRATIONS OF A CENTRIFUGAL PUMP IMPELLER WITH AXIAL FORCES BALANCING DEVICE

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1 Int. J. of Applied Mechanic and Engineeing 6 vol. No. pp.77-7 DOI:./ijame-6- ANALYSIS OF FORCED SPATIAL VIBRATIONS OF A CENTRIFUGAL PUMP IMPELLER WITH AXIAL FORCES BALANCING DEVICE A. ZHULYOV and V. MARTSINKOVSKY Depatment of Geneal Mechanic and Machine Dynamic Sumy State Univeity Rimky-Koakov St. Sumy 7 UKRAINE alexey.zhulyov@gmail.com mat@omdm.umdu.edu.ua C. KUNDERA * Faculty of Mechatonic and Machine Deign Kielce Univeity of Technology Al. -lecia PP 7 - Kielce POLAND kundea@tu.kielce.pl In thi pape a model of a pump impelle with annula eal and a balancing device ued a a combined uppot-eal aembly i conideed. The foced coupled adial angula and axial vibation of the oto ae detemined with conideation of lineaized inetial damping gyocopic poitional and ciculating foce and moment acting on the impelle fom the ide of the fluid flow in annula eal. The theoetical analyi i upplemented with a numeical example the amplitude fequency chaacteitic ae hown. Key wod: vibation adial angula axial oto impelle patial ytem elf-centeing haftle pump.. Intoduction In the otay machine the oto i the main aembly which detemine the eliability featue. The common featue of centifugal pump impelle i that they otate in annula eal which caue lage peue dop to occu. The eulting adial hydodynamic foce and moment acting on the impelle have a deciive influence on it dynamic and accodingly on the dynamic of the whole machine. The hydotatic foce a well a inetial damping gyocopic and ciculation foce and moment geneated in the cleaance of the balancing device ae known to contibute to change in the natual and citical fequencie of the oto and a lo in it dynamic tability a decibed fo example by Child [] San Ande [] and Goiewki []. In ecent theoetical appoache (Cheng et al. []; Li et al. []; Faia and Mianda [6]) the tanvee vibation of the oto ae analyzed by applying a non-linea model of dynamic fluid foce geneated in the annula eal popoed by Muzynka and Bently [7]. The eult of the numeical calculation baed on thi complex non-linea model wee epeented in the fom of dynamic tajectoie of the impelle cente Poincae map and bifucation diagam. A dynamic analyi of the impelle-cleaance eal ytem baed on lineaized model i alo eential a it povide u with analytical elationhip to be ued in engineeing pactice. In thei ealie wok (Kundea and Matynkovky [8]; Matynkovky et al. [9]) the autho decibe a ingle-tage pump with an impelle diectly connected to a balancing device in which the annula * To whom coepondence hould be addeed Download Date /9/8 : AM

2 78 A.Zhulyov V.Matinkovky and C.Kundea and face cleaance eal act a a elf-egulating coupled jounal-and-thut beaing. The pape peent a deign of the balancing device and analyze the effect of coupled adial and axial vibation of the impelle. Thi pape i a follow-up theoetical dynamic analyi of an impelle-balancing device ytem whee the impelle i ubjected to coupled foced adial angula and axial vibation. The dynamic analyi i baed on lineaized hydodynamic foce and moment geneated in the cleaance of the eal of the impelle decibed in the wok by Matynkovky [] Macinkowki and Kundea [] Koczak [] and Jędal []. The eult ae analytical elationhip that can be eaily ued a the fit appoximation at the deign tage of new impelle-baed ytem.. Poblem tatement The fit impelle integated with a balancing device wee patented in the 99 fo example by Matinkovky et al. [] Kubota [] and Chiba et al. [6]. In the patent of Matinkovky et al. [] the deign of the ingle-tage pump impelle i chaacteized by the lack of a claic dive haft connected to the olling element beaing. The beaing node i eplaced by longitudinal and lateal eal cleaance of the impelle and the balancing device. The hydodynamic foce and moment geneated in thee cleaance poition the impelle elative to the pump caing. The diagam in Fig. how a ingle-tage centifugal pump with an impelle diven by a flexible haft via a ball joint. The longitudinal cleaance and 6 the ealing wea-ing cleaance and the lateal cleaance with the annula chambe 9 act a the lateal and longitudinal elf-adjuting hydotatic beaing of the impelle. In the pump caing behind the ea houd of the impelle thee ae adial vane 7 which uppe the otation of the fluid (geneated by the impelle) and pevent a lo of peue in the centipetal diection. Fig.. Geomety of the impelle with a balancing device: - pump caing - impelle - haft; ball joint; 6 longitudinal eal of the impelle; 7 adial vane; 8- line dik of balance device; 9- annula chambe; lateal (face) cleaance. The annula cleaance (thottle) 6 the chambe 9 with vaiable peue p and the lateal cleaance (thottle) coupled with automatic axial foce balancing ytem contol the axial poition of the impelle. A mall mialignment of the haft elative to the axi of the annula eal have little effect on the flow ate coelation between angula and axial vibation can be neglected. Howeve angula and axial vibation ae dependent on adial vibation. The impelle i connected to a flexible dive haft (toion ba). The haft diamete i choen accoding to the tength unde the influence of the engine toional moment. Baed on thi the deign wa Download Date /9/8 : AM

3 Analyi of foced patial vibation of a centifugal 79 called a haftle pump. Since the haft i flexible the impelle i fee to adial angula and axial movement. Unde the action of hydodynamic foce and moment that occu in annula eal and the gyocopic moment of the impelle duing the opeation poce of the pump the impelle elf-cente and take the mot table poition againt extenal ditubance.. Deivation of equation of ytem dynamic The dynamic of the tudied ingle-dic oto model i decibed by five genealized coodinate xyz x y (Fig.). The balancing device add a half of degee of feedom due to the condition of balance flow though it thottling channel. Since the hydodynamic foce in annula eal depend on all genealized coodinate the foced patial vibation of a ingle dic oto (impelle) with a balancing device ae decibed by the coupled ytem of nonhomogeneou diffeential equation of th ode. The deivation of equation of coupled adial angula and axial vibation of the impelle with a balancing device i baed on the expeion of the peue p in the annula chambe (9) (Fig.) that depend both on the face eal cleaance H and on the adial oto offet eh. The peue in the annula chambe affect the value of the axial foce which i defined a a balancing device contol action. Thi peue i detemined fom the equation of continuity of flow though the ytem of eal cleaance: longitudinal (6) and lateal () (Fig.). The dimenionle peue can be witten in the following opeational fom by ubtitution p ddt (Matynkovky et al. [9] whee M p u k k k p (.) z e pn D p D p D p T p M p p (.) T Vpn c Az EQ Q c u (.) c c k k k c (.) p p p p p p p p p (.) e e e e c pn pn pn pn pn pn o o o o o o o o o e c e o o o o o o o Q Q Q g g. (.6) The value that coepond to the teady tate of the oto ae maked by the zeo upecipt. The equation of foced axial vibation i lineaized with epect to to the peue in opeato fom and i expeed a? Download Date /9/8 : AM

4 7 A.Zhulyov V.Matinkovky and C.Kundea D p u d u N p N p. (.7) zz z z e e e In thi equation the pope opeato and deviation action opeato ae expeed by the equalitie Dzz p с p с p с p с dz A kh (.8) T T с с T TT с T A с A (.9) N p T p k A A N p T p k A A (.) e e mh ch T A p A p (.) A n A n T e c 6 A R R A R R A R R A R A A A A A A. A A A A. A B e e c c (.) A A Ae A A Ae A A A A A z H uz u uh H. (.) H H H H The equation of foced adial and angula vibation of the impelle in the annula eal wee olved uing expeion fo the foce and moment ae decibed in the above mentioned wok. Uing the lineaized hydodynamic foce and moment geneated in the cleaance of the impelle eal we can wite the equation of coupled foced adial and angula vibation of the impelle a Matynkovky [] i t e au au au i au au i b b b i b b u u i t. i u u u e (.) The aggegate coefficient fo the font and ea annula eal ae a follow u a a a а а а а а а а а а а а а а а а а (.) Download Date /9/8 : AM

5 Analyi of foced patial vibation of a centifugal 7 b b b b b b b b b b b b j b b b b b b b (.6) (.7) (.8) whee u u iu u x y xy x y x i y H x y l xy (.9) H k u m k k i I ml k l a l. (.) i i i I j H H The coefficient of tiffne fo the haftle pump (Fig.) model ae k k k. The poduct j I I decibe the gyocopic moment of the dic. The coefficient fo the font (i=) and ea (i=) eal ae ai kgi аi kdi kgikii (.) аi k pi mi аi k. gii аi k. di i (.) b a j j b а j k j b а j j i i i c i i c di i i i i c mi bi. аi ji jc bi. аi jc ji bi* bi* mi (.) i kgii i kdii i k di i k рi (.) 7. j 7. j i i i i i i. i i 7. i ji i i ji (.) i q pih i Kii. qi l i. mi (.6). l / H i i i ml ji 6I j ci mlci I R b e R j I m I m I I. (.7) Download Date /9/8 : AM

6 7 A.Zhulyov V.Matinkovky and C.Kundea The fomula fo the hydotatic tiffne coefficient damping and inetia coefficient in the annula eal have the following fom (Macinkowki and Kundea []). Rl kp p Hm Rl ph d k H m l k g Rl (.8) H m Rl kp p H m k. Rl ph H m l d k g Rl.(.9) H m The connection of axial adial and angula vibation i caued by the majoity of the coefficient of hydodynamic foce and the moment that depend on the thottling peue fall on the ea annula eal; р p p. Coefficient of the taight line potential а b and co ciculation foce and moment ae popotional to the fit extent of diffeence р and coefficient of taight line diipation a b and ciculation a b a well a the co potential and gyocope ae popotional р. The thottling peue fall on the font annula eal i p p pe. If the peue pump and the inlet peue ae conideed a the extenal influence the lineaized foce expeion fo the font eal ae obtained fom the expeion fo the ea eal though eplacing p by p e.. Equation of coupled adial angula and axial vibation We obtain the equation of coupled adial and angula vibation having the expeion fo the peue in the annula chambe (.) and the lineaized coefficient of hydodynamic foce and the moment in equation (.). Aanging the tem accoding to the ode of the diffeential opeato we obtain the equation of the ytem in the following fom kн M p D p Ca u d p Cauz Dp D p it k k e Ca Ca e Ca Ca Dp Dp kн M p dp Cb u Dp Cbuz Dp D p it k k e Cb Cb e Cb C b Dp Dp (.) whee D p a p a pa i a pa (.) п n D p b p b pb i b pb (.) Download Date /9/8 : AM

7 Analyi of foced patial vibation of a centifugal 7 d p pn i p (.) d p pn i p (.) C C C C u a n i nu a a u a n i nu a a b nu i n b u b b nu i n b u b (.6) (.7) (.8). (.9) The ytem of foced adial angula and axial vibation conit of Eq (.7) and (.) R p u R p R p u M p am p M p (.) z z a e e pu p pu M p M p M p (.) z z e e whee Zu Z pu M p M p (.) z z z ze e R p d p D p R p D p D p k НC a R p M p C z a (.) p d pd p k НC p D p D p b p M p C z b (.) Z z p p D Z dz (.) zz M p D p a H M p C D p D p k C M p C D p k C e a a a a n (.6) j l M p D p (.7) H Download Date /9/8 : AM

8 7 A.Zhulyov V.Matinkovky and C.Kundea b b n M p C D p D p k C (.8) e b b M z p N p M z ep enep M p C D p k C. (.9) The diffeential patial equation ytem that pefom independent adial angula and axial vibation can be obtained fom the Eq (.)-(.) R p u Ф p Ф Z p u Ф. (.) z z z The ytem of Eq (.)-(.) lead to an equation of joint adial and angula o adial and axial vibation if we take uz o Ru RФ u Ф Ru Ru Ф z z Zu Zu Ф. z z z (.) Thee component of the ocillation can be detemined fom Eq (.)-(.) by Came` ule. We conideed at ome length the adial vibation it a e e e z M a M M R R u M e M M it e e z D M z Mz e e Zz (.) whee the pope ytem opeato i D R R R z Z Z z. (.) z Expanding the deteminant (.) gouping the tem with the ame extenal influence and making the ubtitution p i we obtain u ( i) u ( i) u ( i) u ( i) u ( i ) (.) a e Maθ Zz it Pa it MRZz it Pγ it ua ae ae u e e D D D D P u M M R Z M R R e e it it z z θz zθ па па D D (.) Download Date /9/8 : AM

9 Analyi of foced patial vibation of a centifugal 7 P u M M R Z M R R e e it e e e z ze θz z e ea D D In the expeion decibed above fo example P i the opeato of foced adial vibation actuating peue pump pulation. Component of the angula and axial vibation ae calculated imilaly. Extenal influence vay hamonically with the otation fequency of the oto theefoe the eaction of the linea ytem unde conideation ae hamonic function with the ame fequency. и i t и е a it a е и z i t z и е. (.6) zа The fequency tanfe function ae equal to the atio of eaction to hamonic influence. Subtitute in diffeential opeato p i. Then in view of Eq (.)-(.) the fequency tanfe function of the adial vibation unde the influence of tatic unbalance t a Pa i iе D i i иaaе a a a a ae W U iv A е A a A a aa aa Wi a ia (.7) u a ah =actg V a a. (.8) U ae the amplitude and phae fequency popetie. Similaly the tanfe function ae calculated by othe extenal influence. The amplitude and phae fequency popetie can be obtained a the modulu and agument of the tanfe function. Beaing in mind that и a a H a al H иza za H i accepted a a dimenionle diplacement and fo the abolute value of the amplitude we obtain the fomula a H aa a H A aa a a пa п H A H A aе ea е aa HaA a l a H A l a H пaa l п H A l aе ea е (.9) z H aa za H Az aa za a пa z п z H A z H A. aе ea zе. Numeical example To analyze the coupled adial angula and axial vibation of the oto it i neceay to ue numeical method. Baic data and nominal paamete of the pump ae a follow: р MPa. р MPa п m= kg kg m N m Е Pa R 6 R =7 R =6 R =8 R = R 6 = l =l = l =7 H H H b 9 l c = (all linea ize ae indicated in 9 п e Download Date /9/8 : AM

10 76 A.Zhulyov V.Matinkovky and C.Kundea millimete);. c.. The paamete of the annula channel tapeing i. The teady value of the face cleaance and the flow ate ae: u u Qo. m. The abolute value of the amplitude of coupled adial angula and axial vibation of the oto ae defined by fomula (.9) in the cae when the amplitude of the coeponding petubation ae et: а а ea. Thei appoximate etimation can be obtained fom the following: the pump i high-peed that i why accoding to the global tandad 9-7 fo the balancing of the -th cla of accuacy а 6mm. whee the eccenticity of the dic ma cente i а. m. To calculate the value of the petubation of dynamic unbalance take up a.. Fo coupled adial angula and axial vibation the amplitude fequency chaacteitic of vibation which ae given ie by tatic and dynamic unbalance and pump peue pulation ae made. The oto eaction to the inlet peue pulation diffe little fom eaction to ψ theefoe not hown. In Fig - the dimenional amplitude fo thee paamete of tapeing of annula eal ae peented (.. ). The eonance in the chaacteitic ae abent due to lage damping. The exception ae adial and angula vibation in diffue eal at a low otation fequency (about - ). The amplitude of the adial and axial vibation in the fequency ange ae below micon that i not dangeou. aaμm aa ad 6. 7 z μm aa.. Fig.. Amplitude-fequency epone of adial vibation caued by the tatic unbalance а. Download Date /9/8 : AM

11 Analyi of foced patial vibation of a centifugal 77 a μm. a ad z a μm Fig.. Amplitude-fequency epone of adial vibation caued by the dynamic unbalance. aμm a ad 6. 7 Download Date /9/8 : AM

12 78 A.Zhulyov V.Matinkovky and C.Kundea z a μm.. Fig.. Amplitude-fequency epone of vibation caued by pump peue a. The light inceae of the amplitude obeved on mot figue at indicate that adial angula and axial vibation ae connected. The contitutive impact i done by the annula eal tapeing. The negative tapeing (diffuion aifoil). inceae the amplitude of vibation on the eonant mode thei convegence. educe ignificantly. Concluion The popoed method of dynamic calculation baed on a imple model of the oto balancing ytem allow evaluation of the citical oto peed and amplitude of it foced vibation. Howeve fo moe gounded concluion a moe compehenive and deepe numeical analyi of the implified model and the development and eeach of impoved model i equied in ode to decibe the dynamic popetie of the oto-balancing ytem. Ueful analytical infomation can be obtained by conideing the patial ytem. Equation deived of the foced coupled adial angula and axial vibation include inetial poitional damping gyocopic and ciculation foce. The coefficient in the equation conide the foce and moment that occu in two annula eal. By matching the geometic paamete of annula and face eal the equied dynamic chaacteitic of the ytem can be povided. The numeical evaluation of the influence of annula eal conicity howed that the diffue deteioate vibation tate of the oto unlike the confuo. An analyi of amplitude popetie how the inteelation of adial angula and axial ocillation. The tape of lot channel ignificantly affect the amplitude of vibation. Thi impoe additional equiement to the tiffne of tuctual element becaue poible foce defomation can occu in diffue thottle channel. The numeical analyi of dynamic popetie of the haftle pump how that the annula eal have good damping popetie. Theefoe the amplitude chaacteitic do not have clealy defined eonance and amplitude in the woking ange ae not moe than micon. Thu the calculation confim the woking ability of the haftle pump. Pototype of centifugal pump modenized in thi way have been teted both unde laboatoy and factoy condition. The finding confim thei numeou advantage ove conventional pump and thee include: - bette viboacoutic chaacteitic; - highe eliability and a longe evice life between ovehaul; - eaie opeation aembly and tanpot. It i poible to educe the weight and ize of thi type of centifugal pump by eliminating the extenal beaing of the dive haft. Download Date /9/8 : AM

13 Analyi of foced patial vibation of a centifugal 79 Nomenclatue A dimenionle tatic impelle unbalance Eq.(.) A A A B oute uface aea of the impelle houd [m ] Ae Ac A uface aea of the impelle [m ] A A A e dimenionle uface aea Eq.(.) a eccenticity of the impelle ma cente [m] a i a i adial foce coefficient i=() Eq (.) b i b i gyocopic moment coefficient i=() Eq (.6) C a C b contant (Eq (.6) - (.9)) obtained by ubtituting equation of lineaized foce c o c c c paamete in Eq.(.9) D p pope opeato of independent adial vibation Eq.(.) D zz (p) pope opeato of independent axial vibation Eq.(.8) D p pope opeato of independent angula vibation Eq.(.) d p d p opeato chaacteizing the combination of adial and angula vibation Eq (.) and (.) E compeion modulu [N/m ] e adial diplacement of the impelle axi H H width of the longitudinal cleaance [m] H width of the lateal cleaance [m] I I ma moment of inetia [kgm ] K i coefficient chaacteizing the local acceleation impact on the damping adial foce [ - ] k k coefficient of adial and angula tiffne of the haft kp kd k g coefficient of hydodynamic foce in annula eal hydotatic tiffne damping and inetial foce epectively Eq (.8) and (.9) k k k T paamete of Eq (.) (.) l c ditance fom the impelle cente to the cental eal l l length of longitudinal cleaance [m] m educed ma of the impelle [kg] N (p) N e (p) petubation opeato Eq (.) p e p inlet peue and outlet peue of the impelle [N/m ] p peue in the annula chambe of the balancing device [N/m ] p n nominal pumping peue [N/m ] p p peue at the inlet to longitudinal cleaance * * A B Q fluid flow ate at the tatic equilibium [m /] q flow though the channel unit width [m /] R R adii of the impelle [m] R R R adii of the balancing device [m] R opeato Eq.(.) T paamete Eq.(.) u initial dimenionle adial and angula diplacement u z dimenionle axial diplacement of the impelle V volume of the annula chambe [m ] z z n axial diplacement of the impelle and it nominal value [m] i i co coefficient of adial and angula vibation i=() Eq (.8) elative adial diplacement of the impelle axi coefficient of hydaulic loe θ elative angula coodinate paamete of conicity of the annula cleaance angula coodinate [ad] x y κ coefficient of wil flow in the annula cleaance Download Date /9/8 : AM

14 7 A.Zhulyov V.Matinkovky and C.Kundea coefficient of the geomety of the impelle in Eq.(.) modified fictional pecific eitance λ fictional pecific eitance μ coefficient of dynamic vicoity [Pa ] ρ fluid denity [kg/m ] m elative coefficient of local eitance e dimenionle peue of Eq.(.) ω angula velocity of the impelle [/] nominal angula velocity[/] Refeence n [] Child D.W. (99): Tubomachiney Rotodynamic Phenomena Modeling and Analyi. John Wiley and Son Inc. [] San Ande L. (99): Dynamic foce and moment coefficient fo hot length annula eal. ASME Jounal of Tibology vol. No. pp.6-7. [] Goiewki Z. (8): Analyi of coupling mechanim in lateal/toional oto vibation. Jounal of Theoetical and Applied Mechanic vol.6 No. pp [] Cheng M. Meng G. and Jing J. (6): Non-linea dynamic of oto-beaing-eal ytem. Achive of Applied Mechanic vol.76 pp. -7. [] Li W. Yang Y. Sheng D. and Chen J. (): A novel nonlinea model of oto/beaing/eal ytem and numeical analyi. Mechanim and Machine Theoy No.6 pp [6] Faia M.T.C. and Mianda W.M. (): Peue dam influence on the pefomance of ga face eal. Tibology Intenational No.7 pp.. [7] Muzynka A. and Bently D.E. (99): Fequency-wept otating input petubation technique and identification of the fluid foce model in oto/beaing/eal ytem and fluid handling machine. Jounal of Sound and Vibation vol. No. pp.. [8] Kundea C. and Matinkovky V.A. (): Static and dynamic analyi of a pump impelle with a balancing device. Pat I: Static analyi. Int. J. of Applied Mechanic and Engineeing vol.9 No. pp [9] Matinkovky V.A. Zhulyov A. and Kundea C. (): Static and dynamic analyi of a pump impelle with a balancing device. Pat II: Dynamic analyi. Int. J. of Applied Mechanic and Engineeing vol.9 No. pp.6-6. [] Matinkovky V.A. (): Rotodynamic of centifugal machine (in Ruian). Editoial of the Sumy State Univeity Sumy. [] Macinkowki W.A. and Kundea C. (8): Deign theoy of noncontacting eal (in Polih). Editoial of the Kielce Univeity of Technology Kielce. [] Koczak A. (): Invetigation of ytem countebalancing the axial thut in multi-tage centifugal pump (in Polih). Scientific Book of the Sileian Univeity of Technology No.679 Gliwice. [] Jędal W. (): Centifugal Pump (in Polih). PWN: Waaw. [] Matinkovky V.A. Demin S.I. Gulyj A.N. and Bachkina A.A. (99): Centifugal pump. Patent SU778. [] Kubota Y. (999): Thut-balancing device. Patent EP9A. [6] Chiba Y. Ihikawa Ch. Katua H. and Abe D. (7): Single-haft multitage pump. Patent US7987B. Received: May 6 Revied: July 6 Download Date /9/8 : AM