Gerotor Lubricating Oil Pump for IC Engines

Transcription

1 98689 Goto Lubiating Oil Pump fo IC Engins S. Manò, N. Nvgna, M. Rundo Politnio di Toino G. mnio, C. Pahtti, R. Tihilo Magnti Malli Copyight 1998 Soity of utomotiv Engins, In. BSTRCT This pap doumnts an xtnsiv study aimd at a btt undstanding of th puliaitis and pfoman of ankshaft mountd goto pumps fo IC ngins lubiation. t diffnt xtnts, th modlling, simulation and tsting of a spifi unit a all onsidd. Mo mphasis, at th modlling phas, is ddiatd to th physial and mathmatial dsiption of th flow losss mhanisms; th oftn intiat aspts of kinmatis bing dlibatly lft asid. Th pssu lif valv is analysd at a onsidabl xtnt as is th modlling of th woking fluid, a typially aatd subsystm in suh appliations. Simulation is goundd on MESim, a lativly novl tool in th fluid pow domain, that povs fftiv and ompliant with us dds and objtivs. Tsting, at stady-stat onditions, foms th basis fo th pogssiv tuning of th simulation modl and povids signifiant insight into this typ of volumti pump. INTRODUCTION Th dution of dvlopmnt tim of nw IC ngins pototyps is a ky issu in poviding automotiv industis with nhand omptitiv stngth. To ah this goal a nd xists fo th assssmnt of auat dsign appoahs that will signifiantly and bnfiially impat subsqunt tsting. Simulation is tainly a majo bnfit in ahiving optimal pfoman in systms and omponnts bing dsignd. It is th pupos of this pap, in viw of th simulation of th omplt IC ngin lubiation systm, to povid an fftiv mathmatial pdition of goto lubiating oil pumps. Basially, th unit onsists of a pai of ga shapd lmnts matd so that ah tooth of th inn ga is always in sliding ontat with th out ga to fom sald pokts of fluid. Both gas otat in th sam dition at low lativ spds with th inn ga bing slightly fast. Fluid Figu 1: Pump o shmati nts th hamb with inasing volum, is tappd in th spas btwn th tth and is tanspotd to th outlt. shmati of th pump o is shown in Fig. 1. Th quivalnt hydauli iuit is psntd in Fig.. Oil is takn in fom th tank (oil sump), passs though th inlt dut and is filtd (stain F 1 ). Dlivd flow is futh filtd in F, and, pnding on pssu lvl, xss flow is iulatd to inlt though a pssu limiting valv VL. Downstam of juntion G4, pot P onnts pump dlivy to lubiating oil onsums (U). Figu : Equivalnt hydauli iuit 1

2 MTHEMTICL MODEL Th mathmatial modl onsids th pump as a stadyflow opn thmodynami systm. Th onsvation of ngy lads to th dtmination of pssu vaiations in ah volum though q. (1) wh Q and Q u psnt th inlt and dlivy flow ats, V th ontol volum and β th isntopi fluid bulk modulus. dp dα = β Vω Q dv 1 Q ω u 1 dα (1) Fo as of analysis th angl of otation α of th xtnal ga is hosn as th indpndnt vaiabl; by so doing funtions that xpss volums hav, gadlss of th numb of hambs, th sam piod qual to π. Th ontol volums a mad oinidnt with N vaiabl volums of individual hambs plus two onstant volums bound to inlt and dlivy duts; thus, a systm of N+ diffntial quations simila to q. (1) is obtaind. SINGLE CHMBER MODEL Figu 4: Rfn position and numbing of hambs To dtmin dlivy flow at th following quations valid fo stitos with stady stat flow hav bn usd: Q m = sgn ( p m p amb ) C m -- p ρ m p amb Q m = k m ( p m p amb ) p p p < p () Th fist is valid fo tubulnt gim whil th sond fo lamina flow. Th paamt k is so dtmind that in oinidn with th pssu dop that disiminats th two flow gims q. () yild th sam flow at: j-th hamb C m -- p = k ρ m p (4) Figu : Equivalnt hydauli modl fo a singl hamb Th quivalnt hydauli modl fo a singl hamb is shown in Fig.. Volum V j ( α) psnts th vaiabl volum of th j-th hamb, V m and V a a volums assoiatd with th dlivy and inlt duts sptivly whil vaiabl stitos aaa(α) and aam(α) psnt flow passag aas. Load is modlld with a stito of stion m. Modl quations a as follows: dp j β dv j = Q dα V j ω, j Q uj, ω 1 j = 1,,, N 1 dα N dp m β = dα V m ω 1 Q uj, Q m () 1 N dp a β dα V a ω 1 Q j, + Q = a 1 Th fn ondition α = 0 fo th ngaging gas is shown in Fig. 4; hambs a onvntionally numbd with ising indis in ountlokwis dition; th minimum volum hamb bing idntifid by th indx (N+1)/. FLOW RTES and thfo: Th disiminant pssu dop p* is assumd qual to 0.1 ba, and th flow offiint C = 0.6. FLOW PSSGE RES C -- p ρ k = p Lt us follow (s Fig. 5) a hamb in its pogss duing a omplt volution of th xtnal ga. W obsv that, if that hamb is initially isolatd, th xists a pis instant in whih it osss th im that limits th dlivy stion (b.d. in Fig. 6a). flow passag opns up towads dlivy onfind by ontat point P and b.d. (s Fig. 6b). Whn, in tun, th oth ontat point P 1 osss th im, th flow passag will no mo b stitd and will bom oinidnt with th hamb plan sufa (Fig. 6). This situation will stay unhangd until th lading point P osss th opposit im of th dlivy stion.d. (Fig. 6d). In this instan on obsvs a pogssiv das of th flow passag that vanishs whn that hamb (5)

3 impat th flow apaity and should, thfo, b pditd as auatly as fasibl. Lakag builds up on fou spaat paths (s Fig. 8): lakag flow aoss th spa at tth tips (Q f1 ); it is mad up of flow du to th pssu diffn btwn adjant tooth spas and ntaind flow du to th fluid whih adhs to moving tth tips. lakag flow aoss th sids of th gas and, namly, () btwn th ga fas and th sid plat Q f and () btwn th ga fas and th poting plat (intgal with th pump asing) Q f lakag flow to th oil sump. Figu 5: Inlt and dlivy stions boms again isolatd (Fig. 6). Obviously, th sam onsidations hold valid fo th inlt stion. Figu 8: Lakag flows Figu 6: Signifiant gas positions Fig. 7 shows, ov a omplt xtnal ga volution, th dlivy (aam) and inlt (aaa) flow passags. aa [mm ] α 1im α fm α 1fm α ia α 1ia α fa α 1fa α im aam α [gadi] [dg] Figu 7: Inlt pot aaa and outlt pot aam stions LEKGE FLOWS ND THE COMPLETE MODEL Owing to manufatuing tolans, pump gomty and opating onditions, lakag phnomna ou that aaa Taking into aount flow losss, th modl quations must b modifid as follows: dp j β dv j = ---- Q dt V, j Q uj, + Q f,, j Q f, u, j + Q f, a, j + Q fmj,, ω j 1dα N N dp m β = dt V m Q u, j Q f, m, j Q f, m, a Q fm,, Q m (6) 1 1 N N dp a β dt V a Q j, Q f, a, j + Q fma,, + Q = a 1 1 with j = 1,,..., N. Fig. 9 psnts th omplt hydauli modl. Considing hamb j w assum positiv (i.. ingoing) th flow Q f,,j stmming fom hamb j-1, whil w assum ngativ (i.. outgoing) th flow Q f,u,j fom hamb j+1. Flows Q f,m,j and Q f,a,j btwn hamb j and ith dlivy o inlt, a onsidd positiv if nting th hambs. LEKGE CROSS TEETH TIPS Spa btwn gas givs is to two distint lakag flows: fom hamb j to hamb j+1 (Q f1,u,j ) and fom hamb j-1 to hamb j (Q f1,,j ). Th laan btwn gas is obtaind by inasing th intnal diamt of th xtnal ga (DiR, s Fig. 10); by so doing, gas bing st in th fn position α = 0, th two pofils sha no ontat points (s Fig. 10a). Supposing that nts of gas stay fixd, th intnal on an tun an angl of ±dϕ

4 pio to stablishing points of ontat P 1 o P with th xtnal ga (s Fig. 10b). Figu 9: Complt hydauli modl Figu 11: nalysis of baklash ϕ Figu 10: Claan btwn gas ga. Th angl dϕ (baklash) is thn: Th baklash dϕ an b valuatd as follows (s Fig. 11): onsid a gni point P i of th intnal ga having oodinats x i ( ϑ) and y i ( ϑ) and th il ntd in O with adius R( ϑ) = O P i, as th inn ga tuns, point P i will mov along th il. Th intstion P btwn th il with adius R( ϑ) and th iula a of adius S (s Fig. 1) that foms th tooth of th xtnal ga an b idntifid solving th following systm of quations: ( x K os( j α) ) + ( y K' sin( j α) ) = S j = 0,, N 1 ( x + ) + y = R ( ϑ) thaft it is possibl to xpss th distan P P i. Point P i, among all thos foming th intnal pofil, suh that P P i = min(p P i ), will b th ontat point in that it is th on that must tavl th minimum path to touh th xtnal (7) P P i dϕ = asin (8) R( ϑ) It is an asy matt to vify that if th xtnal ga oupis a diffnt position than th fn on, th alulatd laan will b diffnt and this indiats that dϕ = dϕ( α). Lt us now onsid two gas in psn of laan and in a gni position (α fo th xtnal, ϕ + dϕ( α) fo th intnal). Points P 1 P blonging to th two pofils will b at a minimum distan if tangnts though thm a paalll on anoth (s Fig. 1); in this as th minimum laan will math th distan P 1 P. Howv, sin th two points a not known a pioi, on should valuat tangnts though all points that build up th two pofils by ontinuous and indpndnt vaiation of th position of P 1 P. It an nvthlss b notid that in as tangnts happn to b paalll thn an othogonal lin though both points also intpts th nts of u- 4

5 Figu 1: Evaluation of h min vatu of th pofils. In patiula, fo th xtnal ga this implis a oinidn with th nt of th il of adius S. odingly, to obtain th minimum laan, it is possibl to valuat distan P C whil only point P is vaiabl and thn idntify th minimum as: h min = min(p C ) S Rpating this podu, as α vais, it is possibl to dtmin th minimum laan histoy btwn th two gas. Howv, to astain dϕ( α) in a suffiintly pis mann, a ath lag amount of points of th inn ga must b onsidd and this havily bas on omput tim; fo this ason and th fat that os a suffiintly small, th baklash was hld onstant and lativ to th fn position wh α = 0, that is dϕ( α) = dϕ( 0). Fig. hmin [mm] α [gadi] [dg] Figu 1: Minimum laan btwn gas 1 shows h min ( α) holding dϕ onstant. limitation of this appoah sts on th assumption that th gas nts main fixd. To alulat th laan it is nssay to loally appoximat th inn ga pofil with its osulatoy il (s Fig. 14). Th adius of uvatu, basd on pofil paamti quations x i ( ϑ), y i ( ϑ), an b alulatd to b: -- ( x i + y i ) = x i y i y i x i (9) (10) In a Catsian fn x-y with oigin in point O and th y- axis passing though nts of iumfns C and C i, th laan hight an b valuatd basd on quations of th two ils: (11) wh th adius of uvatu must b aountd with its pop sign (positiv if nt of C i lays intnal to th pofil, ngativ if xtnal). Lakag flow though a vaiabl hight tangula passag is givn by: dp x hmin = d By intgation on obtains: (1) (14) wh v s is th lativ pofil vloity and onstants I 1 I a: (15) Intgation limits x=a and x=-a a adquatly fa fom point O and namly wh th hight h(x) is to som xtnt lag whn ompad with h min. Lakag flow Q f1,u,j fom hamb j to j+1 an b obtaind fom q. (1) and q. (14) wh p = ( p j p j + 1 ); instad fo flow fom hamb j-1 to j it will b pop to us: p = ( p j 1 p j ). LEKGE CROSS GER FCES ND SIDE PLTE Lakag aoss th xtnal ga fa and th sid saling plat an b modlld as a tangula passag of width b f, and lngth l f, (s Fig. 15): wh: Figu 14: Fam of fn fo th hight h(x) hx ( ) = sgn( ) x S x + S + + h min bh min dp Q f = (1) 1µ dx To alulat pssu vaiations along th x-axis, th two dimnsional Rynolds quation an b wittn: dp dx dh 6µv dx 6µv s 6µv s I 1 p = h min I h min a a 1 I = dx I = dx h a min h a min b f, = S K + G l f, = l ϑ f (16) (17) 5

6 ϑ f = aos x --- l (18) DRi DiRi b f, i = () wh: DiRi = DRi 4 psnts th insid diamt of th inn ga. To alulat th lngth l f,i it is nssay to find oodinats of an intstion point P (x p,y p ) btwn th tooth pofil and th il that uts th tooth at its mid hight and thn valuat th a lngth w obtain: P P. By so doing ) y p l f, i = RmRi atan x p () wh: RmRi = 1 -- DRi DiRi man adius of intnal ga (4) Figu 15: Lakag path on th xtnal ga tooth fa Th inoming lakag flow in hamb j, bing h f th laan, boms: Th lngth x is obtaind by solving th systm of quations of ils of adius S and sptivly l : h f b f, b f, i Q f, j, = ( p 1µ l f, l fi, j 1 p j ) (5) and fom this: wh: x + y = l ( x K) + y = S K + l S x = K b f, l = K S (19) (0) (1) Th lakag passagway btwn th inn ga and th saling plat may b thought of as bing tangula of width b f,i and lngth l f,i (s Fig. 16). whil th outgoing flow is: h f b f, b fi, Q f, uj, = µ l f, l fi, ( p p j j + ) 1 LEKGE CROSS GER FCES ND PORTING PLTE (6) s to lakag aoss ga fas and poting plat th flow ontibutions a distinguishd: (i) Q f,m,a fom dlivy to inlt, Q f,a,j fom vaiabl volum hambs and inlt, Q f,m,j fom vaiabl volum hambs and dlivy. Futhmo, two situations nd b analysd: if a tappd hamb xists th will b lakag btwn suh hamb and both inlt and dlivy though passagways shown in Fig. 17, othwis th will only b osspot lakag as indiatd in Fig. 18. mo dtaild haatisation of gomty is shown in Fig. 15. Passagways may again b thought of as bing tangula of width b f, and b f,i fo th xtnal and intnal gas sptivly, whil fo th lngth two ass must b analysd. dlivy inlt Figu 16: Lakag path on th intnal ga tooth fa Th width b f,i osponds to th tooth hight and thfo: Figu 17: Lakag aoss ga fas (tappd hamb) In th situation shown in Fig. 18 w an, to a fist appoximation, assum as lngths l f, and l f,i whil in th as of Fig. 17 w may apt l f, / l f,i / onsiding that th path 6

7 dlivy assumd (.g ). Flows a wittn as follows: inlt Figu 18: Lakag aoss ga fas (without tappd hamb) involvs only half of th tooth xtnsion. No tappd hambs xist fo th following angula positions of th xtnal ga (s Fig. 7): α < α < α 1fm + j α ia + j α < α < α 1fa + j α j = 01,,, N im + j α < α < π + j α (7) in suh a as ossflow xists at two loations and, pisly, btwn nd inlt-bgin dlivy as wll as btwn nd dlivy-bgin inlt. odingly lakag flow ov th xtnal and intnal ga tooth sids should b doubld. Fo ah sid fa w wit: Q f, m, a = b f, h f ( 1µl p p ) m a f, xtnal ga tooth b f, i h f ( 1µl p p ) + m a f, i intnal ga tooth (8) Q f, a, j Q fmj,, b f, b f, i b f, b l f, l fi, l h f = + + f, b xt. ga int. ga tooth bas ( p a p j ) µk () (4) whil on th oth nd (s th ight potion of Fig. 8) pu ossflow xists (dlivy to inlt): (5) Evntually, a tappd hamb xists at nd inlt-bgin dlivy (s Fig. 17) fo th following angula positions: (6) in this instan lakag flows fom th j-th hamb, dlivy as wll inlt ambints ou that a psntd by q. () and q. (4) sptivly, whil at nd-dlivy bgin inlt on th oth sid w fa lakag flows dtaild by q. (5). b f, b f, i b f, b l f, l f, i l h f = + + f, b ( p m p j ) µk xt. ga int. ga tooth bas Q f, m, a = Q f, m, a + Q f, m, a + Q f, m, a fa + j α < α < α im + j α j = 01,,, N Flow at th xtnal ga tooth bas is: b f, b h f Q fma,, = ( 1µl p p ) m a f, b (9) dlivy Th width b f,b is dtmind by th diffn of th xtnal ga adius R st and th adius of th il that omplts th xtnal ga G. s to th lngth l f,b w assum it appoximatly qual to th diamt of th iula a, i.. S. ount must also b givn of lakag though th intndd laan that xists btwn th xtnal ga and th asing with a hight h f, and a width that quals th ga thiknss: H h f Q f, m, a = ( 1µl p p ) m a fb, Consquntly, th total lakag flow amounts to: Q fma,, = Q fma,, + Q f, m, a + Q fma,,, (0) (1) On tappd hamb xists at nd dlivy-bgin inlt (s th lft potion of Fig. 8) fo th following angula positions of th xtnal ga: fm + j α < α < α ia + j α j = 0, 1,, N () in suh a as lakag flows xist btwn th j-th tappd hamb, inlt and dlivy. Whil widths of lativ passags may b assumd qual to b f, and b f,i, th lngth is psntd by valus low than l f, l f,i sin only a potion of tth flanks is involvd; moov, sing that this lngth vais with angula position, a duing fato k is Figu 19: passagway fom dlivy to oil sump LEKGE FLOW TO THE OIL SUMP Flow Q f,m, fom dlivy volum to th oil sump, Fig. 19,is xpssd as: b m Q f, m, = ( 1µl p p ) m m, wh p is taind qual to ambint pssu., h f LEKGE DUE TO ENTRINED FLOW (7) Bsids lakag at tth tips du to pssu diffn btwn adjant spas an ntaind flow must b a ountd du to visosity of th fluid whih adhs to moving tth tips. Fo a gaing pai of axial thiknss H, th ntaind flow is wittn as: v s h min H Q t = (8) bing v s th gas lativ (sliding) vloity at h min. Looking 7

8 at Fig. 0 and alling that th nomal to onjugat pofils osss th instantanous nt of otation C, th sliding vloity at ontat point P is: v s = v i osδ i v osδ (9) l f, = l f0 l f, = x 0 x l f, = ε x < x 0 l f0 x 0 l f0 < x < x 0 ε x > x 0 ε (46) Figu 1: Lakag passag lngth in pssu lif valv Sin q. (46) is valid only if l f, is not too small (in fat, if l f, tnds to zo lakag tnds to infinity) a limit to th valu ε was nfod. PRESSURE RELIEF VLVE Tangntial vloitis a: (40) pplying th sin thom to tiangls CO P and CO 1 P w hav: wh: Figu 0: Gas sliding vloity v i = ρ i ω v = ρ ω 1 R 1 sinδ = sinψ ρ osδ = 1 ( sinδ ) C R 1 sinδ i = sinψ ρ i osδ i = 1 ( sinδ i ) C ρ sinψ i = ---- sinψ ρ ; ρ C ( R 1 + x p ) = + y p i (41) (4) (4) lso in this as th will xist ntaind flows Q t,u,j btwn hambs j (outgoing fom j) and Q t,,j btwn hambs j and j-1. Flows Q f,u,j and Q f,,j to b usd in q. (6) a so wittn: Th modlling of th pssu lif valv involvs th stps: (i) th witing of th dynami quilibium of th spool (ii) th dtmination of its instantanous position as a pquisit fo th quantitativ knowldg of th flow passag aa (iii) th dtmination of its dishag flow at. s to th fist stp, by nglting Coulomb fition and making fn to Fig., w may wit th dynami quilibium of th spool as follows: wh: m v R x k F 0 p m p S mx + v x R x + kx + F 0 ( p m p )S = 0 (47) = spool mass plus on thid of th sping mass = visous fition offiint = flow fos = sping stiffnss = sping pload = dlivy pssu = pssu in th sping sid of th spool = spool aa Q f,, j = Q f1, j, + Q f,, j + Q t,, j Q fuj,, = Q f1, uj, + Q f, uj, + Q t, u, j (44) LEKGE T RELIEF VLVE SPOOL Lakag btwn dlivy and inlt ambints though th lif valv spool (s ahad) taks pla in an annula passag with a hight h f,. Th flow an b xpssd as: D πh f, Q f, = ( 1µl p p ) m v f, (45) wh th lngth l f,, bing ε in th od of a fw tnth of a millimt, an b xpssd as follows, (s Fig. 1): dain hannl Figu : Rlif valv and fos ating on spool PRESSURE IN THE SPRING SIDE OF THE SPOOL Pssu p an b valuatd ith onsiding o nglt- 8

9 ing th apaitan fft of th sping hamb volum V m. If ngltd, th flow at though dain hannls (s Fig. ) an b wittn fo a iula gomty and lamina gim as: d Q d n h d π ( p 8µl p a ) π D = = x 4 (48) wh: d h l n d D and on obtains: = dain hannl hydauli diamt = l 0 + x = dain hannl lngth = numb of dain hannls = spool diamt µl D x p = p a n d d h (49) R x Figu : Contol volum fo flow fos = C q x ( ) p m p v os( ϑ) 4C q x ( ) d π (55) Th fflux angl θ is alulatd as a funtion of th dishag flow aa (x) by intpolation of availabl xpimntal data that also aount fo laans btwn spool and slv. By so doing th bak pssu p is only dpndnt on spool vloity. Instad, if th apaitan fft is onsidd, mass onsvation must b applid to volum V m : (50) (51) bing V m0 th nt hamb volum at x = 0. In this as th bak pssu also dpnds on th hamb dimnsions. FLOW FORCES dp dt V m V m0 x d = ----π 4 β dv mx = Q V d m dx dv m d = ----π dx 4 oding to Nwton s sond law and onsiding th ontol volum in Fig., th flow fo, nglting its unstady omponnts, is wittn as: R x = ρq v os( ϑ) + ρq 1 v 1 (5) Flow though th vaiabl stion (omposd of fou iula hols) is: Q = Q 1 = C q x ( ) -- p ρ m p v (5) DISCHRGE FLOW Solving th sond od diffntial quation (q. (47)) th instantanous spool position an b dtmind. Howv, th solution is only admissibl if th position x falls btwn 0 and x max (osponding to maximum spool displamnt, and gnally gat than x 1 ). Basd on spool position, th dishag flow aa an b alulatd: if 0 < x < x 0 th valv is still losd whas if x 1 < x < x max th flow aa is at its maximum and is xpssd as: d f = n f ---- π (56) 4 wh: n f = numb of hols in th spool d f = hol diamt Fo situations wh x 0 < x < x 1 and looking at Fig. 4 on obtains: ϑ f 1 x x 0 = aos d f d f = n f f = n f ---- ( ϑ 8 f sinϑ f ) (57) (58) wh p v is th valv downstam pssu that diffs fom inlt pssu owing to distibutd losss in th iulatd path. s to vloity v 1, said d th inn spool diamt, w obtain: d 4C q x ( ) Q 1 = v π v 4 1 = d π -- p ρ m p v (54) By substitution into q. (5): Figu 4: Dishag flow aa (on hol) fo x 0 < x < x 1 Th dishag flow is wittn as: 9

10 Th flow offiint C q aoding to [14] is: with: (59) (60) (61) wh: C = maximum flow offiint λ = valu of λ at tansition fom lamina to tubulnt flow D h ν = hydauli diamt = kinmati visosity Th hydauli diamt basd on th hol gomty (s Fig. 4) is: MODELLING OF THE WORKING FLUID (6) Th woking fluid is onsidd as an aatd subsystm. Th volum fation of ai in a liquid an b xpssd as [14]: (6) with V a and V l fing to th volums of ai and, sptivly, oil within a apaity Vtot. Th amount of dissolvd ai in a satuatd liquid is wittn, aoding to th Hny-Dalton, law as: V da = V tot0 χ ---- p (64) p o bing χ 0.09 th Bunsn (solubility onstant). Th volum fation of spaatd ai is dfind as: Thn: (66) and if p > p sat w hav θ = 0, whas if p < p sat w hav: p bing: y = Q lim = C q ρ -- p m p v λ C q = C tanh λ λ = D h ν -- p ρ m p v d f ( ϑ f sinϑ f ) D h = ϑ f sin ϑ f x = θ V a V a + V l V as = (65) V a V as = x θ ( V l + V a ) = x θ V tot θ = ( 1 y) 5 ( 1 + 4y + 10y + 0y + 5y 4 ) p sat (67) s to dnsity of th oil-ai mixtu th following modls a onsidd: p p atm x ρ mix ρ ρ β = l p > p 1 x ai sat xθ xθρ ai + ρ ρ l ( 1 xθ) 1 xθ ai ρ mix = p < p xθ p atm γ p p atm sat β + ( 1 xθ) p (68) Finally, th bulk modulus of th mixtu is aountd aoding to th following dfinition: SYSTEM SIMULTION dp β mix = ρ mix dρ mix (69) Having psntd all th modlling aspts of th pump, Fig. 5 shows th omplt systm that foms th objt of th simulation studis that a addssd haft. Th inlt dut (immsd in th oil sump) is assoiatd with a onstant apaity V t that oil nts aft ossing a stain. On in th pump asing, oil gos in th vaiabl volum hambs V j though passag aas aaa and xits though passag aas aam both bing vaiabl stitos dpndnt on th gaing angula position. Fom th dlivy ambint V m, th oil flow an, at vaious dgs, b iulatd to inlt (though th pssu lif valv and iulation dut) and dlivd to th IC ngin lubiation systm. THE MESIM ENVIRONMENT nalyss a pfomd by us of th dvand Modlling Envionmnt fo Simulation (MESim 1 ). Th al and omplt systm dpitd in Fig. 5 is potayd in th MESim window by a dtaild assmbly of submodls. Eah submodl is synthsisd with an ion and a tag that allows its uniqu idntifiation vn in as of multipl us of th sam ion. n instan numb appndd to th idntifiation tag movs all possibl ambiguitis. Ions a intonntd with lins (Vsion 1.5a of MESim maks us of on lintyp only to idntify pow, pilot and dain lins) and nods (filld dots) gnally indiat th psn of lins banhing. dashdot lintyp is usd to onvy signal infomations. Th ativity in th MESim nvionmnt is shduld and assssd into fou diffnt domains. Th skth phas wh th systm is assmbld though us of ions and lins; th submodls phas wh ah ion is assignd a spifi submodl (in fat, divs mathmatial modlling dsiptions flxiv of physial ality and phnomna an b assoiatd to th sam ion); th paamts phas wh pulia numial valus a assignd, in an odly fashion, to all submodls involvd (.g. displamnt of a pump, angula vloity of a pim mov, t.); and last th simulation phas wh stating valus a allottd to all stat vai- 1.a tadmak of IMGINE Invstissmnt, Roann, Fan 10

11 Figu 5: Th omplt systm abls and intgation of th ODE (odinay diffntial quations) systm bgins in a tim fam xpssd by th us. Dspit th fat that an xtnsiv walth of infomations in tms of ions and submodls is povidd as a standad fatu of MESim, a majo advantag lis in its opn ahittu. This patially allows that ativ dvlopmntal wok b xploitd in ah and all phass itd abov. pquisit to stp into this powful ustomization fatu is MESt, a ompanion utility nvionmnt to MESim. In this ontxt, xpind uss dvlop thi own ions and, mo impotant, thi own submodls. In pfoming this task, an fftiv aid is povidd by an automati gnation of od skltons in ith Fotan o C. THE MESIM SYSTEM Th omplt skth of th systm upon whih simulation studis a pfomd is shown in Fig. 6. Clos to ah omponnt a tag indiats th submodl nam and its instan numb (pliations of th sam submodl). Th ovall systm an, fo as of dsiption, b dividd into 5 potions: a - Th pim mov, signals and ovall gomty n idal pim mov PM001 gnats a vloity signal supplid to a mhanial pot () at RCON0-1 (a otay nod tansfing a vloity to two pots and adding two toqus). Th infomation is splittd and lavs unaltd though two mhanial pots B and C. Th latt is linkd to an angula position (DT50-1) and an angula vloity (WT00-1) tansdu. Submodl VC6-1 pfoms alulations of volums vaiations of hambs (vto ays mthod) and of volums (by intgation); th idal toqu and slip vloitis btwn gas a valuatd as funtions of angula position and vloity signals; in addition, th lin of ontat points is dtmind. Submodl C5-1 alulats by intpolation of stod data fils, th instantanous flow passag aas btwn hambs, inlt and dlivy ambints; ths valus a thn tansfd to submodls VOR50 (s ahad). Submodl P55-1 maks it possibl that signifiant gomty fatus of th unit (numb of hambs, ntiity, gas gomty, poting angls, laans) bom availabl to all submodls quiing suh infomations. Submodl ET50-1 allows, und stablishd stady stat onditions, th alulation of th unit volumti ffiiny: th atual flow at bing supplid by submodl QT01-1 (an hydauli flow at snso with offst and gain) and th angula vloity by WT00-1 (a otay spd snso with offst and gain). b - Th hydauli subsystm Stating fom th oil sump (TK00-) and pogssing futh th following submodls a idntifid;: a stito (OR000-): fatuing th stain; 11

12 Figu 6: Th MESim systm a fixd apaity (V0L50-): assoiatd with th inlt dut; a fixd apaity (V0L50-4): assoiatd with th inlt volum; vaiabl multipl stito (V0R50-): valuats ingoing flow ats to ah hamb though passag aas (C5) on upstam and downstam pssus a availabl; vaiabl volum hambs (V057-1): a numb of N vaiabl volum hambs in paalll an b analysd. Instantanous pssus a dtmind via ODE intgation on divativs and volums thmslvs a inputd (VC6) along with nt flow ats ossing th th hydauli pots; vaiabl multipl stito (V0R50-1): valuats outgoing flow ats fom ah hamb though passag aas (C5) on upstam and downstam pssus a availabl; a fixd apaity (VOL50-): dlivy volum ; stito (OR000-1): suddn hang in flow oss stion at dlivy pip ntan; fixd apaity (VOL50-1): assoiatd with th filt; vaiabl stito (VOR00-1): load; piwis lina signal sou (UD00-1): load tim histoy (oss stion of load stito); oil sump (TK00-1). - Th iulation path This stats at th banh (nod) ompisd btwn VOL50- and OR000-1 and inluds: th pump lif valv (RV91-1); Bsid th modlling dtails psntd in Pssu lif valv on pag 8, this submodl quis additional a at th simulation stag. This stms fom disontinuitis onsqunt to th psn of nd stops lativ to th allowd displamnt of th spool. To dal with thm fiv swith stats and an intg vaiabl hav bn intodud: i(1) = - valv shut, spool at stand still, sping fo gat than pssu indud fo; i(1) = -1 valv shut, spool at stand still, sping fo qual to pssu indud fo; i(1) = 0 spool in motion; i(1) = 1 valv ompltly opn, spool at stand still, i(1) = sping fo qual to pssu indud fo; valv ompltly opn, spool at stand still, sping fo low than pssu indud fo; In aod with th patiula stat of th systm, th govning diffntial quation hangs as follows: mx = 0 if i(1) = - mx + F 0 ( p m p )S = 0 if i(1) = -1 mx + v x R x + kx + F 0 ( p m p )S = 0 if i(1) = 0 mx R x + kx max + F 0 ( p m p )S = 0 if i(1) = 1 mx = 0 if i(1) = (70) 1

13 a stito (OR000-4) to aount fo pssu losss along th iulation path. d - Lakags lakag flows btwn hambs (ME50-1); this submodl ivs an input vto fom VO57-1 with pssu infomations of all vaiabl volum hambs. Slip vloitis and th xtnal ga angula position a also inputs stmming fom VC6-1. ME50-1 yilds as output a vto aying lakag flow ats among hambs; ME60-1: oss pot lakag; ME65-1: hambs - inlt lakag; ME70-1: hambs - dlivy lakag; ME80-1: xtnal lakag (dlivy to oil sump); - Lins HL01-1, HL01-, HL0-1, HL0-, HL0-: ompssibility plus fition submodl of a hydauli hos o pip; SYSTEM FLOW-PRESSURE SIMULTION (STEDY- STTE) Taking advantag of th possibility gantd by th MESim nvionmnt of pfoming stady stat simulations, th systm modl, with all paamts st at thi dsign valus, has fomd th objt of vaious invstigations. Of patiula intst is that aimd at th attainmnt of flowpssu haatistis of th lubiating oil pump fo a numb of angula vloitis (idling to full spd). Th load flow-pssu haatistis of an idal pump, und ontol of an idal pssu lif valv, a psntd by a nt of hoizontal (ah at a givn angula spd) and vtial lins (ah at a givn st pssu of th lif valv). t fixd angula spd and st pssu (aking pssu) th hoizontal sgmnt at full load flow (th lif valv bing shut), though lina, boms slantd of a tain angl if fluid losss a aountd fo th pump. Similaly, th vtial sgmnt has an inlination if lif valv atual bhavio is onsidd. In this instan, linaity is only appoximatly psvd owing to sping stiffnss. Though this maks pssu linaly dpndnt on spool tavl, still a non-linaity xists in th flow atpssu tubulnt modl. t diffnt pump spds, th aking pssu stays th sam and th lif valv (slantd) lins, volv paalll on anoth; pssus at zo load flow (intpts with th x-axis) inas with pump spd. Th full load flow lins a paalll on anoth if flow losss a dmd indpndnt of spd bing only dtmind by pssu (a lina dpndn in lamina flow gim). Instad, if pssu and spd a both influntial on th loss mhanisms, th inlination lssns th high th pump spd and, onsquntly, paalllism is lost. s to th odinat axis intpts (zo gaug pssu) a mak is appopiat: load flow is gnally low than idal flow owing to lakags divn by intnal pssu gadints. In simulation analyss, fo ah vloity (submodl PM00-1), th stito oss stion (submodl VOR00-1) is vaid. This mans that th load (i.. ngin lubiation onsums) is lumpd into a singl, yt vaiabl, flow sistan. In od that on haatisti lin b obtaind, at last 15 diffnt opating points must b invstigatd. On an avag, ah point quis about 60 sonds of simulation tim on a SunSpa 10 wokstation. Th stadystat simulatd flow-pssu haatistis a shown in Fig. 7. ] ni m/ L[ t a w olf li o s m u s n o pm 5000 pm 4000 pm 000 pm 000 pm Figu 7: Flow-pssu haatistis (simulatd) Futh, anoth intsting haatisti has bn simulatd and is shown in Fig. 8, that psnts th oil flow at dlivd to a spifi load (1.5 ba at 1000 v/min) as a funtion of th IC ngin spd. In this as kinmati visosity is st at 5 St. SYSTEM TESTING dlivy pssu [ba] Figu 8: Flow-spd haatisti Th ig on whih xpimnts hav bn pfomd (s Fig. 9) is a spially dsignd tst bnh and onsists of two distint oil hydauli subsystms: th bnt-axis piston moto (Volvo F11-5) supplid by an indpndnt flow gn- 1

14 Figu 9: Th tst ig Figu 0: Font viw of th tst ig with pump unit, oil sump and dlivy hos ating unit and th pump und tst fittd on a ddiatd and spially manufatud suppoting modul (s Fig. 0). Th layout of th omplt hydauli iuit is shown in Fig. 1. On th moto sid, ontol of th shaft angula vloity was fist attmptd though us of a two-way flow gulating valv. Th hoi, basially owing to a non mathing siz of th valv, did not pov satisfatoy and us of a Danfoss PVG lto-hydauli popotional ontol valv was pfd. On th pump sid, th filt FL was isolatd (high pssu dop pvntd appopiat studis of th hoizontal potion of th flow-pssu haatistis) and th load simulatd with a manually vaiabl stito (m). Th absobd toqu and th angula spd w masud with a Vibomt TG10/BP toqu mt onnting, though tosionally stiff ouplings, th moto and th pump. Th instumnt is quippd with indutiv tansdus fd with a ai fquny of 8 khz at 10 V RMS. Toqu signals a amplifid, dmodulatd and filtd and thn ad ditly in Nm on a digital display. Rotational fquny is snsd though a toothd whl, a magnti Figu 1: Layout of th hydauli iuit pikup and a fquny-voltag onvt. Th fftiv flow going though th load (stito m) is masud with a tubin flow mt Km Küpps HM 11E installd in th dlivy lin (F1). Calibation uvs at 10, 0 and 50 St, as povidd by th manufatu, hav bn appoximatd by sixth-od polynomials. Th appaisal of fluid visosity quid knowldg of its tmpatu. So, fo tmpatu masumnts, a platinum sistan thmomt PT 100 was fittd in th oil sump at diffnt loations. Sin th ading was lagly position dpndnt, that adout (T1) was omplmntd by th tmpatu infomation stmming fom a Choml-luml K-thmooupl (T). Th old juntion tmpatu was masud with a thmomt also mbddd in a thmally insulatd polystyn nlosu. Th diffn in voltag btwn th juntions was monitod with a Kithly multimt and onvtd to a tmpatu diffn. Mo igoous podus w not xisd fo two asons. In th fist pla th visosity-tmpatu lationship was attaind with in hous masumnts (Engl visosimt) and in th sond pla intpolation of manufatu s data was in all ass nssay. This is du to th patial diffiulty of attaining and maintaining stabl isothmal onditions in th woking fluid (a SE 15W40 automotiv oil). Th situation boms itial whn iulation ous in onsqun of pssu lif valv intvntion. mo auat ontol of th figant fluid flow (tap wat) to th hat xhang (SC) was unavailabl at th tim of tsting. sw plug tansdu adapt Figu : Tansdu at sping sid loation of lif spool Fou pssu adout loations w stablishd. In th 14

15 inlt dut, aiming at th pssu ontinuous omponnt, a muy pizomt was usd (G8 and TP1); on th dlivy lin, pssu was masud by an Entan stain gaug tansdu (EPI-M4-15) with full sal ading of 15 ba (G4 and TP); a sond Entan stain gaug tansdu (EPI-M4-15) was fittd in th sping sid of th pssu lif valv spool (G9 and TP, s also Fig. ). Finally, a pizolti tansdu (Kistl 701) to povid instantanous dlivy pssu infomations was fittd in th poximity of filt FL (G5 and TP4) pm 5 ] ni m/ L[ t a w olf li o s m u s n o pm 000 pm 000 pm dlivy pssu [ba] Figu : Flow-pssu haatistis (xpimntal vs simulatd) STEDY STTE EXPERIMENTL RESULTS Masud pump flow at-pssu haatistis a potd in Fig. and ontastd with simulatd sults; fluid tmpatu, fo all tsts, bing kpt within th ang dgs C. If a ompaison is now mad with outoms of th simulation (Fig. 7) it is fai to stat that a ot pditiv apability has bn ahivd. Moov, whil a haatisti lin is simulatd in oughly 15 minuts, th task is tainly muh mo laboious and tim onsuming at th tst ig. fw maks may b addd: - Ca should b paid to th iulation path of th lif valv: as losss inas with dishag flow so is th upstam pssu in th sping sid of th spool. This is futh aggavatd by a onomitant pssu inas in th inlt ambint: high flow iulation allows that small quantitis of oil b laimd fom th sump; this lows losss in th inlt dut and, onsquntly, pssu iss in th inlt ambint. Und ths iumstans an inas in spool bakpssu (sping sid) lows th slop of th (idally) vtial potion of th haatisti and, at zo flow at (totally iulatd flow), lads to high pssu valus than thos attainabl by vnting th sping hamb. This is, in fat, povd by xpimntal sults (s Fig. 4) attaind at two angula vloitis wh ontasts btwn vntd and non vntd solutions a adily appant. ] ni m/ L[ t a w olf s m u s n o li o pm 000 pm 4000 pm vntd 000 pm vntd dlivy pssu [ba] Figu 4: Contasts btwn vntd and non-vntd solution - at high angula vloitis (ov 4000 pm) th is vidn of phnomna that a not aptud by th simulation modl. Th qustion that involvs flow fos, aation and, likly, avitation is yt to b fully invstigatd. 15

16 f f i f f Finally, Fig. 5 and Fig. 6 show simulatd and xpimntal volumti ffiiny of th unit at two angula vloitis. i t m ul o v Th poss of xpimntal validation of th mathmatial simulation has bn ntd on pump stady stat pfoman. Th good agmnt with xpimntal sults qualifis th modl as a valuabl tool in th dsign of lubiating oil pumps. y n i i i t m ul o v y n i simulatd xpimntal dlivy pssu [ba] Figu 5: Volumti ffiiny (000 pm) simulatd xpimntal dlivy pssu [ba] Figu 6: Volumti ffiiny (4000 pm) dditional ffots a quid to aiv at a ot simulation of omplx wavfoms that typially ais in th inlt and dlivy lins and snsibly ontibut to fluid bon nois. odingly Fig. 7 shows a dit ompaison btwn xpimntal and simulatd pump pssu ippl (000 pm, oil tmpatu 50 dg C). Simulatd sults hav bn obtaind by m substitution of a singl submodl i.. th lin onnting th pump to th load stito that, instad of bing basd on lumpd paamts, now aounts fo unstady fluid flow and fquny dpndnt fition. Th basi similaity that an b notid lis in th ippl bing gnatd p ah shaft volution by th tn pumping lmnts. Diffns, on th oth hand, a tainly mo appant. Th simulatd ippl posssss a onstant amplitud whas th xpimntal xhibits a modulatd amplitud afftd by a low fquny supimposd signal (doubl of otational fquny and on fifth of th pumping lmnts fundamntal fquny). Rasons fo this a not fully laifid; on possibl xplanation sts on th hypothsis that nts of th two gas do not stay fixd as is in fat th as in th mathmatial modl. CONCLUSIONS Th ontinuous stiv fo passng a ful onomy also lis on possibl impovmnts in lubiating oil pump pfoman. s spifiations bom mo dmanding and vhil dsign yls shot, th us of modlling and simulation is in apid gowth in all ngining filds as a ost fftiv and valuabl plamnt of th build and tst itations podu. It is in this fam of asoning that th psnt pap has bn ointd. With fn to a untly manufatud lubiation pump fo a passng a a vy dtaild simulation pogam has bn dvlopd in th MESim nvionmnt. Th xpimntal validation, mainly at stady-stat, allows a onfidnt us of this spifi pogam fo th dsign of simila units fo diffnt ngins. In fat, th simulation modl of ankshaft mountd goto lubiation pumps has so bn dsignd to allow that gomti paamts, load haatistis, angula vloity and fluid poptis an all b hangd at will. t high angula vloitis a modst dispany btwn simulatd and xpimntal sults may b th onsqun of th simplifid modl of th aatd oil. Howv, oil aation psnts a still unsolvd issu and nds futh invstigations. Us of MESim has povd patiulaly fftiv in shotning th tim ndd fo modls dvlopmnt CKNOWLEDGMENT This sah wok was pfomd at th Politnio of Tuin, Italy und Contat with Magnti Malli S.p.., Division Componnti Manii. uthos aknowldg pmission of publishing th psnt matial gantd by Magnti Malli S.p.. CONTCT N. Nvgna: -mail nin@polito.it S. Manò: -mail saman@polito.it Wb addss BIBLIOGRPHY 1 1. bdallah, Y.. G., Rad,. G., Tunbull, U. P. M.: Conditions 1.To th authos psnt knowldg no vidn was found in th opn thnial litatu of sah wok addssing in dpth studis of goto pumps. It is fo this ason that no fn to pviously publishd matial is povidd in this pap and a bibliogaphy is instad poposd. 16

17 ] a b [ u s s p y v i l d xpimntal simulatd angula position [dgs] Figu 7: Pssu ippl fo minimum tooth tip losss in xtnal ga pumps, Fluid Pow Symposium, nsdal, R. F.: Mathmatial analysis of th NSU Wankl RC Engin, Iliff, London, Botoluzzi, E.: Sviluppo di un modllo matmatio p la simulazion lo studio dll pstazioni di pomp olodinamih ad inganaggi stni, Tsi di Laua, Politnio di Toino, Conad, F., Tostmann, E., Zhang, M.: Expimntal idntifiation and modlling of flow and toqu losss in goto hydauli motos, Fluid Pow, Editd by Mada, E & FN SPON, n Impint of Chapman & Hall, 199, pp D gostino, V., Russo, M.: Studio dll possibili pstazioni di pomp a lobi, Univsità di Napoli. 6. Eisnmann, S., Häl, C., Shib, B.: Compaison of diffnt lubiating oil pump systms fo ombustion ngins 7. Fahl, E., Haas,., Kut P.: Konstuktion und Optimiung von Ölpumpn fü Vbnnungsmoton, FEV Motonthnik GmbH & Co. KG, ahn. 8. Hawoth, D. C.: Dynami fluid flow analysis of oil pumps, SE Tansations 1996, No Jiang, Y., Pzkwas,.: Computational analysis of oil pumps with an impliit pssu basd mthod using unstutud mixd lmnts gids, SE Tansations 1996, No Klung, M.., Fussn, D. R.: pfoman ompaison of vaious automati tansmission pumping systms, SE tansations, 1996, No Kondoh, K., Kosug, T., Takda, Y.: Lubiation pump mad of apidly solidifid aluminum alloy fo high pfoman ngin, SE Tansations, 1996, No Pipping, J. J.: Th goto stoy: Compaison - Displamnt lmnts. 1. Rii, G.: Wight and atd haatistis of mahins: positiv displamnt pumps, motos and ga sts, Mhanism and mahin Thoy, Vol. 18, 199, pp MESim and MESt Manual - IMGINE - Roann, Catto, R: Modllo matmatio simulazion di una pompa a pistoni adiali ilindata vaiabil p impighi violistii, Tsi di laua, Politnio di Toino, Fabiani, M.: Studi toii spimntali su pomp di lubifiazion goto, Tsi di Laua, Politnio di Toino, Lamb, W.S.: Cavitation and ation in hydauli systms, BHR, Canfild, England, Koh, F., Maassn, F.: Dvlopmnt of modn ngin lubiation systms, SE pap Fabiani, M., Manò, S., Nvgna, N., Rundo, M.: Modlling of goto gaings in viw of lubiating oil pumps simulation. (to b publishd). 17