Rotor profile design in a hypogerotor pump

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1 Jounal of Mechancal Scence and Technology ( ~470 Jounal of Mechancal Scence and Technology DOI 0.007/s y oo pofle desgn n a hypogeoo pump Soon-Man Kwon *, Han Sung Kang and Joong-Ho Shn Depamen of Mechancal Desgn and Manufacung Engneeng, hangwon Naonal Unvesy, 9 Sam-dong, hangwon, Kyongnam 64-77, epublc of Koea (Manuscp eceved May, 008; evsed Ocobe 0, 008; Acceped Augus 4, Absac A geomec appoach fo he oue-oo pofle as a conjugae o he nne-oo n a hypoochodal oo pump (hypogeoo pump s poposed by means of he pncple of he nsananeous cene and he homogeneous coodnae ansfomaon. The nne-oo pofle s defned by he combnaon of wo ccula acs. Nex, he adus of cuvaue of he oue-oo s deved wh he elaonshps of he ochod ao and he nne-oo ooh sze ao. Then by examnng he mnmum adus of cuvaue of he exended hypoochodal oue-oo pofle on he convex secon, an explc fomula o avod undecung n he hypogeoo pump s poposed. I s found ha undecu o selfnesecon does no occu so long as he mnmum value of he adus of cuvaue on he convex secon s no less han zeo. Desgn examples ae pesened o smulae he opeaon and o demonsae he feasbly of he appoaches usng a compue-aded desgn pogam developed on ++ language. Keywods: Hypogeoo pump; Inne-oo; Oue-oo; Insananeous cene; Tp cleaance; Tochod ao; Inneoo ooh sze ao; adus of cuvaue; Undecu Inoducon Numeous applcaons n hydaulc and lubcaon sysems jus eque he cculaon of he flud. In such cases low nose emssons and lle pessue pples ae moe mpoan han hghly effcen ansmsson of enegy. The geoo pump s deal pncple fo such applcaons. ompaed o convenonal exenal gea pumps, he sucon and pessue connecon of he geoo pump s axal o he dvng shaf. Ths also suppos he compac consucon. Due o he sold ooh shape, he geoo pump s essan o hydaulc and mechancal mpac loads. The long duably of he geoo pump s based on he elavely low sldng speed beween he nne and he oue oo. Fuhemoe, hs pump s chaacezed by an exemely good smoohness and a low nose Ths pape was ecommended fo publcaon n evsed fom by Assocae Edo Won-Gu Joo * oespondng auho. Tel.: , Fax.: E-mal addess: smkwon@changwon.ac.k KSME & Spnge 009 level. Desgnes of engnes, compessos, machnes ools, acos, and ohe equpmen equng hydaulc sysems can now buld pump componens negally no hese mechansms. Some mpoan leaues on he basc geomey and s elaed opcs of he geoo pump can be found: fo example, olboune [] poposed a geomey mehod o fnd he envelopes of ochods ha pefom a planeay moon. Lvn ad Feng [] used dffeenal geomey o geneae he conjugae sufaces of epochodal geang. Demenego e al. [] developed a ooh conac analyss (TA compue pogam and dscussed avodance of ooh nefeence and apd weang hough modfcaon of he oo pofle geomey of a cyclodal pump whose one pa of eeh s n mesh a evey nsan. Usng he mehod fo deemnng and acng he lm cuve, Mmm and Pennacch [4] obaned anscendenal equaons fo he calculaon of he lm dmensons o avod undecung. On he whle, Ye e al. [5] pesened smple explc fomulae by examnng he

2 460 S.-M. Kwon e al. / Jounal of Mechancal Scence and Technology ( ~470 adus of cuvaue on he convex secon fo calculang he lm dmensons o avod undecung n he nne-oo. Howeve, mos eale sudes focused on he commecally avalable geoo pump usng he equdsan shoened epochod cuve o he auhos bes knowledge. To mpove he cayove phenomenon of he adonal geoo desgn, mos ecenly Hwang and Hseh [6] pesened a geomey desgn pocedue based upon he heoes of envelope and conjugae sufaces fo he hypoochodal gea pump (abbevaed as hypogeoo pump n hs pape usng he equdsan exended hypoochod cuve. They also pesened non-undecung condons of he oue-oo usng he heoy of geang [7]. Howeve, he pocedue fo obanng he nonundecung condons of [6] s somewha complcaed wh he added dsadvanages ha equaons mus be solved numecally. Ths pape pesens he mehod on oo pofle desgn of a hypogeoo pump. The oue-oo pofle as a conjugae o he nne-oo s defned by he pncple of he nsananeous cene and he homogeneous coodnae ansfomaon n Secon, and he nne-oo pofle s defned by he combnaon of wo ccula acs n Secon. Nex, he adus of cuvaue of he oue-oo s deved wh he elaonshps of he ochod ao and he nne-oo ooh sze ao n Secon 4. Then by examnng he mnmum adus of cuvaue of he oue-oo on he convex secon followng he mehodology of [5], an explc fomula fo he lm dmensons o avod self-nesecng o undecung s poposed n Secon 5. I s found ha self-nesecon does no occu so long as he mnmum value of adus of cuvaue on he convex secon s no less han zeo. Wh he esul obaned n hs pape, he calculaon becomes a much smple ask han ha of [6]. Based on developed analycal expessons, some dscussons ae addessed n Secon 6 o demonsae he feasbly of he appoaches.. Oue-oo ooh pofle A hypogeoo pump (see Fg. consss of wo man componens: an nne-oo and an oue-oo ha has one moe ooh han he nne-oo. The nne-oo cenelne s posoned a a fxed eccency fom he cenelne of he oue-oo. As he oos oae n he same decon abou he especve axes, flud s Fg.. Typcal hypogeoo pump. O(I ω Yf ω O(I P(I Lnk (Gound E Lnk (Inne oo Lnk (Oue oo Xf Fg.. Insananeous cenes n a hypogeoo pump. dawn no he enlagng chambe up o a mum volume. As oaon connues, he chambe volume deceases, focng flud ou of he chambe. Ths pocess, used pmaly n lqud anspoaon and many flud powe applcaons, occus consanly fo each chambe, povdng a smooh pumpng acon. We have dsplayed a schemac of he hypogeoo pump n Fg.. The numbe of eeh of he nne-oo s always one less han he oue-oo,.e., hey have N and ( N + eeh, especvely. We can choose any shape fo he nne-oo eeh, and he oue-oo s hen geneaed conjugae o he nne-oo. We descbe hee only he nne-oo havng N acs of ccle n he placemen of fom s cene wh he adus of. The cene dsance beween oos (o eccency s E. I can be egaded knemacally as a mechansm of hee-lnks and hee-jons: he fame coespondng o E= O O as Lnk, he oue-oo as Lnk, and he nne-oo as Lnk, especvely. The oue-oo (Lnk uns abou O, and he nne-oo (Lnk uns abou s cene O, he angula velocy ao beng N :( N +. Two pons

3 S.-M. Kwon e al. / Jounal of Mechancal Scence and Technology ( ~ O and O ae pemanen nsananeous cenes I and I, especvely. We wll denoe he wo pch ad = O I = O P and = OI = OP whch ae unknowns o be deemned below. The magnude of he velocy V a pch pon I ( P can be deemned by V = ω = ω ( The angula velocy ao can be wen as ω N ω = = + N ( Fom Eq. (, we can easly deemne he locaon of he pch pon I wh he ad of elaon = E and Kennedy s heoem [8] as follows: = E( N +, = EN ( Befoe devng he pofle equaon of he oueoo, hee coodnae sysems coespondng o he hypogeoo pump should be defned as shown n Fg. : one saonay efeence sysem S f aached o O, and wo moble efeence sysems S and S aached o O and O, especvely. The angles of φ and φ ae pofle defnon paamees of he efeence sysems S and S, especvely. In Fg., he conac pon s and he common nomal o boh oos passes hough he pch pon I. Theefoe, he mesh pon n S -coodnae sysem and he coespondng leanng angle ψ can be deemned as below: x = + cosψ y = sn ψ (4 snφ ψ = an µ cosφ (5 whee he paamee of µ = / s used conssenly houghou hs pape and s efeed o as he ochod ao. I s ecommended ha he desgne of he hypogeoo pump should adop he cuae hypoochod cuves (.e. µ > o avod selfnesecon phenomenon. The ogn of coodnae sysem does no concde wh ha of he oue-oo n Eq. (4. In such a case he coodnae ansfomaon may be used based on he applcaon of homogeneous coodnaes and 4 4 maces ha descbe sepaaely oaon abou a saonay axs and dsplacemen of one coodnae sysem wh espec o he ohe [7]. Fo he homogeneous coodnae ansfomaon fom he conac pon of n S -efeence sysem o ha of n S -efeence sysem, he followng max equaon s defned: = M = M ( φ M ( φ (6,,f f, whee he max fom S -sysem o j M, j descbes ansfomaon S -sysem, and cosφ snφ 0 0 snφ cosφ 0 0 M ( φ = (7,f cosφ snφ 0 E snφ cosφ 0 0 M ( φ = (8 f, = [ + cosψ sn ψ 0 ] T (9 Y Y Yf O(I O(I X φ φ O(I ψ P(I cosφ conac pon, N φ X Xf ψ N P(I snφ Fg.. Thee coodnae sysems fo oue-oo pofle defnon. X Xf whee he supescp T n Eq. (9 means he anspose of he max. The esulng expesson of Eq. (6 s ( + cosψ cos( φ φ sn ψsn( φ φ Ecosφ + + ( + cosψ sn( φ φ = sn ψcos( φ φ Esnφ + 0 (0

4 46 S.-M. Kwon e al. / Jounal of Mechancal Scence and Technology ( ~470 Fom Eq. (, we have he followng elaon, Y ω dφ/ d φ N = = = ( ω dφ / d φ N + If we defne φ by he geneaed paamee of oupu moon, we can oban φ = Nφ and φ = ( N + φ. Subsung hese elaons no Eq. (0 leads o he followng lobe pofle paamec equaons n S - efeence sysem: +δ Offse oue-oo O X x = cosφ+ cos( φ+ ψ + Ecos( Nφ (a y = snφ+ sn( φ+ ψ Esn( Nφ (b whee sn( N φ ψ an + =, ( 0 φ π ( µ cos( N + φ The eal pofles ae manufacued wh echnologcal gaps due o many paccal consdeaons, such as pecson of machnng ools, pevenon of jammng condons, and applcaon of lubcans. Alhough gea eeh gaps ae nevable, hey may lead o flud losses and occuence of addonal dynamc foces, decease sably and ncease nose and vbaon, especally a hgh speeds. The equemen fo a pope p cleaance s a ade-off poblem. The equdsan cuve pncple s appled o ealze he pope oleances fo he oue-oo pofle, and as a consequence, he offse pofle o he deal one of Eqs. ( s obaned. Ths equdsan offse pofle wll be geneaed as equdsan of Eqs. ( wh equdsan adus lage o smalle han he heoecal one ( by a p cleaance, δ, as follows: x( δ = cosφ+ ( + δ cos( φ+ ψ + Ecos( Nφ y( δ = snφ+ ( + δ sn( φ+ ψ Esn( Nφ (4a (4b These gve a unfomly enlaged equdsan cuve (Fg. 4 when δ > 0. On he ohe hand, hese gve a unfomly conaced equdsan cuve (Fg. 5 fo δ < 0. Eqs. (4 ae he equaons of he pofle on he non-deal oue-oo. We can obseve ha Eqs. (4 can be degeneaed Fg. 4. Equdsan offse oue-oo lobe pofle (dashed lne. δ = 0 (exended hypoochod cuve δ = - (hypoochod cuve = E =.5 N = 8 = 4 Fg. 5. Oue-oo shape n case of δ =. no he well-known sandad hypoochod equaons n he foms fo he case when δ = (see Fg. 5: x = cosφ+ Ecos( Nφ (5a y = snφ Esn( Nφ (5b Nex, we consde he oue-oo oaed by he amoun of θ (see Fg. 6 fo he sake of genealzaon. In ha case, we can descbe he oue-oo pofle n he saonay S f -efeence sysem as follows: M ( θ (6 f = f,

5 S.-M. Kwon e al. / Jounal of Mechancal Scence and Technology ( ~ Y Yf Y B f θ X + τ N B- O Xf f N+ α + δ δ γ α β N- O N X Fg. 6. Oue-oo pofle n case of oaon of θ. Fg. 7. Defnon of nne-oo paamees. whee cosθ snθ 0 0 snθ cosθ 0 0 M ( θ f, = ( cosφ+ ( + δ cos( φ+ ψ + Ecos( Nφ snφ+ ( + δ sn( φ+ ψ Esn( Nφ = 0 (8 O Y N+ (f Ω Ω α+ α mn B ' f f B N N X -efeence sysem be- The oue-oo pofle n comes f f S f x = cos( φ+ θ + ( + δ cos( φ+ ψ+ θ + Ecos( Nφ θ (9a y = sn( φ+ θ + ( + δ sn( φ+ ψ+ θ Esn( Nφ θ (9b. Inne-oo ooh pofle and flow ae The nne-oo (see Fgs. 7 and 8 can be defned by he combnaon of wo ccula acs: nne-oo eeh ccula acs of Secon I ( α Φ β and γ Φ α +, and flle ccula acs of Secon II ( β Φ γ. The poson angles shown n Fg. 7 ae π α = ( X, ON = (,( =,,,, N N (0a Fg. 8. Schemac fo deemnaon of ( X, O ( X, O l f. β = = α + δ (0b γ = = α δ (0c + + Hee he secon dscmnaon angle δ can be deemned fom O + N+ of Fg. 7 as δ cos + = whee f f f f ( = l + l cosτ (a ( lf ( + f l + + cosτ = f f l = O B = cosω f ( cos ( f + Ω + + (b (c

6 464 S.-M. Kwon e al. / Jounal of Mechancal Scence and Technology ( ~470 and Ω = ( α+ α / = π/ N. Eq. (a s obaned by OB, Eq. (b by ONB, and Eq. (c by ON + B o ONB of Fg. 8, especvely. The ccula ac equaons of Secon I (Fg. 9 and Secon II (Fg. 0 n he saonay S f -efeence sysem can be wen, especvely, as follows: Y N+ B Yf f (Φ N D' D f ( xd cos( α + θ f + ( yd sn( α + θ = f ( xd lf cos( α + Ω + θ f + ( yd lf sn( α + Ω + θ = f (a (b whee he angle θ epesens he oaon angle of he nne-oo. The elaon beween he oaon angles of he oos s θ/ θ = N/( N +. The S f -coodnaes of pon D n Fgs. 9 and 0 ae O Fg. 9. Secon I of nne-oo ooh pofle n Y N+ D' D B Yf f α Φ θ β N S f X Xf -sysem. x = ( Φ cos( Φ+ θ (4a f D f y = ( Φ sn( Φ+ θ (4b D (Φ γ Φ β θ X whee f y ( ( X, O an φ Φ = = θ f (5a x ( φ f f x = x E, f y = f y (5b Hee we defned he poson angle Φ as Eq. (5a (see Fg. n ode o calculae he chambe aea easly. I allows he same poson daa beween oos. Subsung Eqs. (4 no Eqs. (, we fnd ( Φ = cos( Φ α + + cos ( Φ α, ( α Φ β ( Φ = l cos( Φ ( α + Ω + θ f f lf + lf cos ( Φ ( α + Ω + θ, ( β Φ γ ( Φ = cos( Φ α ( γ Φ α + + ( α + + cos Φ, + (6a (6b (6c Fg. 0. Secon II of nne-oo ooh pofle n Y O Yf O Fg.. Defnon of geneaed paamee Φ. D Φ θ S f X Xf Xf -sysem. Once he geneaed shape and conjugae shape ae known, he volume dsplaced by he wokng pocke, as hs pocke goes hough a complee cycle fom mum volume ( A H o mnmum volume ( Amn H, can be deemned. Hee A, A mn and H epesen he mum chambe aea, he mnmum chambe aea and he oo hckness, especvely. Fo hs goal, he evaluaon pocedue of he -h chambe aea, A, a any nsan should be peceded.

7 S.-M. Kwon e al. / Jounal of Mechancal Scence and Technology ( ~ I can be caed ou fom Fg. numecally as follows: A D d D (7 whee Φ n n = ( k k Φ ( k k Φk Φ0 k= f f k = x( φk + y( φk (8a f f D = x ( Φ + y ( Φ (8b k D k D k Φ = Φ Φ (8c k k ( k and Φ 0 and Φ n ae he sa and he end poson angles of he -h chambe, especvely. These angles ae calculaed fom he schemac of Fg. fo deemnaon of conac angle θ (, Yc ( 0 θc( an Φ = = θ Xc ( Yc ( n θc ( an + Φ = + = θ X ( + Y O Yf Φ0 D c Φ k Φ(k- Φ n Fg.. Schemac fo chambe aea. Yf Y O E Yf O * α N+ α = EN θc(+ θc( θ P(I N c θ A X Xf ( Xc(, Yc( δx m δy X Xf, Xf Fg.. Schemac fo deemnaon of conac angle, (9a (9b θ c. whee * Xc( = cosα + m m * Yc( = snα + m (0a (0b * m = IN = µ + µ cosα (0c * and α = α + θ. To deemne he wokng aea of A = A Amn wh he ad of Eq. (7, should be noed ha boh A and A mn occu smulaneously a θ = ( j π/ N (whee j =,,, N fo he case when N s even numbe, whle A a θ = ( j π/ N and A mn a θ = ( j π/ N, especvely, fo he case when N s odd numbe. Snce A o A mn occu N mes fo evey oaonal un of he nne-oo, he specfc flow ae V h (o heoecal dsplacemen pe un evoluon can be deemned as Vh = A H N. Theefoe, he ol flow ae of he hypogeoo pump s calculaed as q= η V pm ( V h whee η V s he volumec effcency manly dependng on p cleaance and face cleaance, and ( pm s he oaonal speed of he nne-oo. 4. adus of cuvaue fo oue-oo Dung he desgn sage of he hypogeoo pump, he sze ( and he placemen ( of he cylndcal nne-oo eeh ae hose of mpoan dmensons. If s lage han a mum value o s less han a mnmum value, hen he enveloped ooh pofle of he oue oo wll self-nesec (see Fg. 4. The ooh pofle of he oue-oo wll heefoe be undecu. Ths wll poduce backlash beween oos dung unnng and become a poenal poblem, e.g., a decease n volumec effcency. I s heefoe mpoan o calculae lm dmensons o avod undecung on he oue-oo when desgnng he geoo pump. I s also well known ha he wea ae can be educed by nceasng he adus of cuvaue of he lobes. The adus of cuvaue s a funcon of he sze and he placemen of he nne-oo eeh whch geneae he lobe shape.

8 466 S.-M. Kwon e al. / Jounal of Mechancal Scence and Technology ( ~470 As s well known, he fomula fo he adus of cuvaue of a paamec cuve s ( ( x + y ρ = xy xy ( whee ( x, y ae coodnaes of he paamec cuve, ( x, y and ( x, y ae he fs and he second devaves of ( x, y wh espec o paamee, especvely. If ρ > 0 n Eq. (, hen he locaon of he cene of cuvaue s o he gh of he pah (.e., convex pofle. If he mesh pons of n Eqs. ( ae dffeenaed wh espec o φ, hen he esulng fomula fo he adus of cuvaue of he ooh pofle of he oue-oo wll be vey complcaed because of he em of ψ. I wll be mpossble o oban decly an explc fomula. Howeve, wll be ovecome wh he noducon of he adus of cuvaue fo he sandad hypoochod cuve (see Fg. 5. When δ = n Eqs. (4, we can oban he sandad hypoochod cuve as n Eqs. (5. Wh he adus of cuvaue ρ N of he sandad hypoochod cuve, he adus of cuvaue of he oue-oo pah (exended hypoochod cuve aced by he mesh pon, a a specfed npu poson φ, can be found as ρ= ρ ( N To fnd ρ N, we consde Eqs. (4 and (5,.e., x = x( δ = and y= y( δ =. Subsung Eqs. (5 no Eq. ( yelds a smple fomula wh paamee of φ = ( N + φ as follows: ( µ + µ cosφ / ρ = N µ µ ( N cosφ o n he nomalzed fom of adus of cuvaue (4a o zeo: o N µ µ ( N cos φ = 0 (5a N µ cosφ = µ ( N (5b Snce N and µ ae posve and eal values, hen cosφ > 0,.e., 0 φ π/. Theefoe, an nflecon pon wll occu when N µ 0< µ ( N 5. Non-undecu condon fo oue-oo (6 To demonsae he nefeence (o selfnesecng phenomenon, wo oue-oos ae depced smulaneously n Fg. 4. The same desgn paamees ( =, N =8, E =.5 have been used n Fg. 4, wh he excepon of (equal o 4 n he smalle oo and 0 n he lage oo. Shown n Fg. 5 s he elaonshp beween adus of cuvaue ρ of he ooh pofle and he geneaed paamee φ. Fom Fg. 5, we can obseve ha he adus of he nne-oo ooh nceases, hen ρ deceases. If s lage han a lm value, he mnmum adus of cuvaue on he convex secon wll be negave and he ooh pofle of he oue-oo wll be nesecng. Ths wll poduce backlash beween he oue-oo and nne-oo dung unnng. To avod hs self-nesecng, he pon wh zeo adus of cuvaue mus be avoded,.e., he mnmum value of ρ of he ooh pofle on he convex secon should no be less han zeo. =0 =4 undecu ( µ + µ cosφ / ρ ρ = = λ µ N µ µ ( N cosφ (4b =, N=8, E=.5 whee λ = / s he nne-oo ooh sze ao. The anson beween concave and convex poons esuls n he adus of cuvaue becomng nfne. Ths nflecon pon wll occu n he hypoochodal pah when he denomnao n Eqs. (4 ends Fg. 4. Oue-oo pofle desgn example fo showng undecu.

9 S.-M. Kwon e al. / Jounal of Mechancal Scence and Technology ( ~ =, N=8, E=.5 These fs and second local exema ae he adus of cuvaue of he nal geneaed pon a he boom of he lobe fo nenal conac and he adus of cuvaue occung a he op of he lobe fo nenal conac, especvely. adus of cuvaue, ρ =4 =0 ρ mn <0 convex pofle secon(should be ρ>0 (ase. onsde B = 0 ;.e., Eq. (8b may be wen as + cos = 0 (4 µ µ φ Fo Eq. (4 o be a possble soluon, he elaonshp beween he ochod ao and he geneang angle s Geneaed paamee, φ (deg. Fg. 5. adus of cuvaue fo he desgn example. In ode o calculae ρ mn on he convex secon, Eqs. (4 ae dffeenaed wh espec o φ and seng he esul equal o zeo. Afe eaangng, he equaon s obseved o be of he fom A B = 0 (7 whee A = snφ (8a B µ µ cosφ = + (8b = N + µ ( N + µ ( N cosφ (8c I s clea fom Eq. (7 ha hee ae hee dsnc cases whee saonay cuvaue n he hypoochodal pah could occu: when A = 0, and/o B = 0, and/o = 0. As s mpoan o undesand each case, hey ae pesened now n some deal. µ = cosφ ± cos φ (4 Snce µ > 0, hen he only value of φ whch sasfy hs condon s φ = 0. Subsung hs value no Eq. (4 gves µ =. Then subsung µ = no Eq. (9, we see ha he fs local exemum ρ = ( ρn = 0. Ths defnes a cusp n he pah of pon N and s a specal case whch may no be a paccal soluon. (ase. onsde = 0 ;.e., Eq. (8c may be wen as N + µ ( N + µ ( N cosφ = 0 (4a o N + µ ( N + cosφ = (4b µ ( N As cosφ, he hd local exemum occus when he ochod ao s (ase. onsde A = 0 ;.e., snφ = 0. The values of he geneaed paamee whch sasfy hs condon ae φ = 0 o φ = π. Subsung φ = 0 and φ = π no Eq. (4a and smplfyng, he fs and he second local exema ae ( µ ρ= ρφ = 0 = µ + N ( µ + ρ = ρφ = π = N µ (9 (40 N + < µ < N + (44 Ths equaon s he mos geneal esul fo he ochod ao of a paccal hypoochodal geoo. Subsung Eq. (4b no Eqs. (4 and smplfyng, he hd local exemum s o / ρ = ( µ ( N + N / * ρ = ( µ ( N + λ µ N (45a (45b

10 468 S.-M. Kwon e al. / Jounal of Mechancal Scence and Technology ( ~470 If he ochod ao sasfes Eq. (44 hen all hee local exema wll occu on he pah. Howeve, f he ochod ao µ ( N + /( N +, hen only he fs and second local exema can occu on he pah. As saed befoe, o avod self-nesecng, he value of ρ mn on he convex secon should no be less han zeo. Seng ρ = 0 esuls n an explc fomula fo calculang he mum value of λ of he nne-oo ooh sze ao o avod undecung on he oue-oo Table. Desgn paamees fo compason. Paamees gven n [6] Desgn consan ase N=4, =0, E=6.9, =4. ( =6.9 ase N=6, =0, E=.0, =.0 ( =0.7 / λ ( = = ( ( N µ N µ + (46 If has been deemned befoehand, hen he mnmum dsance mn can be calculaed by he followng explc fomula deved fom Eq. (46: (a ase ( N = + 7( N + mn (47 Usng Eqs. (46 and (47 s vey easy o calculae lm dmensons. Fo an example, f he desgn paamees ae gven by =, N =8, and E =.5 as n Fgs. 4 and 5, he mum nne-oo ooh adus o avod undecung s ( = Dscusson Based on he obaned esuls, a compue-aded package HypoGeoo V.0 has been developed o desgn he hypogeoo pump usng ++ language n connecon wh OpenGL. Ths AD pogam has he chaacescs of he gaphc use neface and he smulaon of he eal opeaon fo he hypogeoo pump. To valdae of he poposed appoach, we evs he exsng esul of Hwang and Hseh [6]. They pesened wo specal cases as n Table. These wo cases have no undecung on he pofles, causng he desgn values of o be lowe han he ( values. As can be shown n Fg. 6, ou esuls ae n exac ageemen wh hose of [6]. Accodng o he esul of Saenko and Gobayuk [9], he heoecal dsplacemen of he epochodal geoo pump s appoxmaely evaluaed as Vh 4πE( H. In ohe wods, V h nceases n ha pump wh he ncease of and E, bu deceases wh he ncease of. Howeve, (b ase Fg. 6. ompason wh he exsng esul [6]. should be noed ha he heoecal dsplacemen V h of he hypogeoo nceases as nceasng of, E and. Ths end s somewha dffeen fom ha of he epochodal geoo pump. The ochod ao µ fo all commecally avalable hypogeoo pump wll have a value ha sasfes Eq. (44;.e. he local exemum gven by Eqs. (45 s he mos common mnmum adus of cuvaue on he convex secon of he ooh pofle. Fo an llusave pupose, he mum nne-oo ooh sze ao of Eq. (46 s gaphcally epesened n Fg. 7 wh he vaaon of he ochod ao unde he lm condon of Eq. (44. Fom Fg. 7, we can obseve ha (a pemssble λ nceases as µ nceases; (b pemssble λ deceases wh he ncease of N ; and (c he ange of µ fo paccal pupose s geng wde as N nceases because of he lm condon of Eq. (44.

11 S.-M. Kwon e al. / Jounal of Mechancal Scence and Technology ( ~ λ =( / N= 4 Lm condons N+ <µ< N The pesen esuls ae easy o undesand and exac. ( Smple explc fomulae fo no nefeence condons ae pesened by examnng he mnmum adus of cuvaue on he convex secon of he oue-oo pofle. ( The developed desgn mehodology has been successfully appled o he hypogeoo pump usng a compue-aded pogam, and some examples have been pesened o vefy he valdy of he developed mehodology µ=/ Fg. 7. Maxmum ooh sze ao wh he vaaon of ochod ao. Besdes, n he hypogeoo pump desgn, o avod conac wh o nefeence beween he wo neghbo nne-oo eeh o exsence of he flle adus of ( f mn, he mum pemssble value of he nneoo ooh sze ao s denoed as λ c and he nneoo ooh adus o he nne-oo ooh sze ao may be consaned by he followng elaon (see Fg. 8: 0< < snω (48a 0< λ< λ c = snω (48b Howeve, because Eqs. (48 only deemne he desgn ange of he nne-oo ooh sze, Eq. (46 (he equaon of undecung mus be employed o he feasble desgn. 7. onclusons The exac oue-oo pofle and some explc fomulae fo he lm dmensons o avod undecung n he hypogeoo pump have been obaned by he pncple of he nsananeous cene, and by examnng he mnmum adus of cuvaue on he convex secon of he lobe pofle, especvely. The followng conclusons can be dawn: ( The paamec lobe pofle equaons of he oueoo n a hypogeoo pump ae analyzed and obaned by he pncple of he nsananeous cene. Acknowledgmen Ths eseach s fnancally suppoed by hangwon Naonal Unvesy n efeences [] J.. olboune, The geomey of ochod envelopes and he applcaon n oay pumps, Mechansm and Machne Theoy 4 ( [] F. L. Lvn and P. H. Feng, ompueze desgn and geneaon of cyclodal geang, Mechansm and Machne Theoy ( [] A. Demenego, D. Vecchao, F. L. Lvn, N. Nevegna and S. Manco, Desgn and smulaon of meshng of a cyclodal pump, Mechansm and Machne Theoy 7 (00 -. [4] G.. Mmm and P. E. Pennacch, Nonundecung condons n nenal geas, Mechansm and Machne Theoy 5 ( [5] Z. Ye, W. Zhang, Q. Huang and. hen, Smple explc fomulae fo calculang lm dmensons o avod undecung n he oo of a cyclod oo pump, Mechansm and Machne Theoy 4 ( [6] Y.-W. Hwang and.-f. Hseh, Geomey desgn usng hypochod and nonundecung condons fo an nenal cyclodal gea, Tansacons of he ASME, Jounal of Mechancal Desgn 9 ( [7] F. L. Lvn, Gea geomey and Appled Theoy. PT Pence Hall, Englewood lffs, (994. [8] J. E. Shgley and J. J. J. Ucke, Theoy and Machnes and Mechansms. McGaw-Hll, (980. [9] V. P. Saenko and. N. Gobayuk, On geoo hydaulc machne desgn, ussan Engneeng eseach 4 (7 (

12 470 S.-M. Kwon e al. / Jounal of Mechancal Scence and Technology ( ~470 Soon-Man Kwon eceved hs B.S., M.S., and Ph.D degees n Mechancal Engneeng fom Yonse Unvesy, n 989, 99, and 00, especvely. D. Kwon s cuenly an Assocae Pofesso a he Depamen of Mechancal Desgn & Mfg. Engneeng a hangwon Naonal Unvesy n Koea. D. Kwon s eseach neess ae n facue and fague behavos of maeals, and ceave mechansm and mechancal desgn. Joong-Ho Shn eceved hs Ph.D. degee fom Oho Sae Unvesy n 986. D. Shn s cuenly a Full Pofesso a he Depamen of Mechancal Desgn & Mfg. Engneeng a hangwon Naonal Unvesy n Koea. D. Shn s eseach neess nclude compue aded mechansm and mechancal desgn.