Analytical model of a particular type of positive displacement blower

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1 Analtical modl of a aticula t of ositiv dislacmnt low GIVANNI MIMMI and PAL PENNACCHI Diatimnto di Mccanica Stuttual, Univsità dgli studi di Pavia, Via Fata, I-700 Pavia, Ital Astact Man as ist in litatu which dal with th twin-scw comsso; this usuall has two diffnt otos, a mal and a fmal, and is commonl usd to oduc comssd gas fo industial uss. Howv, a diffnt t of ositiv dislacmnt ota comsso with two scws is somtims usd; on of its tical alications is in ca ngin suchaging. Psnt a dals with th latt t, which is dfind as a two-scw low. This low has two idntical hlical otos, ach with th los. Th kinmatics and th gomt of th otos a analsd h, and a comlt mathmatical modl fo th oto is dfind. Moov diffnt ossil shas of th otos, dnding on th dsign aamts, a analsd and th limitations in th choic of th dsign aamts a sntd. Finall an analsis of th thotical scific sliing of th otos is sntd, showing which zons of th ofil a th most stssd. This modl will usful fo futh studis on oto ssu loads and low dnamics. Kwods: ota comssos, suchags, gas, 3D analtical modl, aamtic dsign. Notation C - olut cus oint; h - limit atio of and ; M - scw motion mati; P0 - itch oint; - scw angula itch; R - 3D vcto snting sufac Σ; - D vcto snting cuv ; - as adius; - ti adius; i - inn adius; - itch adius; S - fnc sstm; w - lativ vlocit; u - vlocit; α - u limit valu fo aamt ϑ; - cuv; γ - angl sutnding a half of th ti tooth; δ - angl sutnding a comlt olut ac; ε - angl sutnding an olut ac fom as to itch cicl; θ - olut ssu angl; ϑ - aamt; Σ - sufac; σ - scific sliing; τ - tangnt in th olut cus; Ξ - limit quation; υ - aamt; ω - angula vlocit. INTRDUCTIN Fo a tt undstanding, it ma usful to mhasis som considations, which will lain th oigin of this aticula t of machin. Th us of ositiv dislacmnt lows, divn th cankshaft, as suchaging dvics in intnal comustion motos is not cnt. Ths comssos hav not had widsad alication, aat fom th imotant ction of acing cas, sciall if th diffusion of th tuochags, wh th low is divn th haust gas tuin, is considd, sinc this t which hav had oad alication. Analtical modl of low - v. doo f -.doc 3/05/06 :4 - -

2 Th ositiv dislacmnt lows usd in autos w usuall th Roots t. This t was v siml, ut its fficinc was v low, as tt laind lat, thfo its alication was not advisal, sciall if on consids that th ow ndd to div th low was usuall otaind th moto shaft. As fa as gads to th industial alications, som ositiv dislacmnt lows, which hav a high fficinc, a widl usd. Ths machins a th scw comssos, which a chaactisd two otos, a mal and a fmal, whos ofil is ath coml and quis a v accuat machining. Both Roots lows and twin scw comssos hav ositiv and ngativ chaactistics. Th Roots lows hav qual oto sha, whos gomt is wll known and dfind mathmatical cuvs with siml algaic ssion such as ccloids (, ) o oluts (3). Howv thi fficinc is quit low, as otd man authos (4-7) on th asis of imntal and thotical studis. Som authos (7) hav oosd diffnt shas fo th otos too in od to imov th fficinc. Th twin scw comssos hav th otos chaactisd a gomt ath coml that ld to sval studis aout th ofil sha otimisation and th dvlomnt of scial dsign softwa (8-4) in cnt as. Th comlit of th ofil is wll ssd in (0) wh th a dfind as som kind of ci, wh th ingdints hav n oints, staight lins and ats of mathmatical cuvs that a mid in suital ootions. Moov th was a ctain intst in dsigning th cutt sha fo th machining of th mal and fmal otos (5) with a aticula intst in th fomancs otaind on th asis of a aticula cutt sha and of th manufactuing cost. A gat advantag of twin scw comssos is thi fficinc that is ath high as otd in man studis oth thotical and imntal (8, 4-). Du to th diffusion of ths comssos, som authos hav also sntd studis aout th dnamic ffcts on th otos (-4) stating fom an analtical modl of th otos. Th low analsd h mgs diffnt chaactistics of th two ts viousl dscid. Th tagt in this cas consists of th comination of th Roots oto siml gomt on th aial sction with th high fficinc of th scw lows. In fact, this comsso has two idntical otos, such as th Roots t, ut with a hlical sha, which coms fom scw comssos. Sinc this low is mlod fo suchaging intnal comustion motos, th fomanc imovmnts a usuall valuatd in tms of toqu o ow incasing o missions duction ath than fficinc. Sval as stss th fomanc imovmnt achivd this t of low (5-7). Th aim of this a is to giv a 3D analtical modl fo th oto sha. Th knowldg of this analtical modl givs th asis fo th stud of th dnamic ffcts of th ssu loads on th otos. This ma caid out with a mthod simila to that sntd in (, 4, 8), which nds th analtical modl of th otos in od to dfin th contact lins, th scto of sufacs wh th ssu acts, tc. Thfo th main at of snt a will dal with th dfinition of th analtical modl of th otos considing th analtical ssion of all th sufacs that comos th oto and discussing th ffct of th dsign aamt changs. nl fw considations will sntd aout th gnation of th ofils mans a siml ack, sinc this ocdu on on hand can asil lain th gnation of th ofils ut on th oth hand it is not so usful to dtmin th 3D analtical modl of th oto. RTR DESIGN It is not tml difficult to dfin th oto sha if it is considd that th oto ofils hav to conjugatd and consquntl th acs of cuvs, which comos th ofil, a Analtical modl of low - v. doo f -.doc 3/05/06 :4 - -

3 likl to long to a finit st of functions onl. Thfo th analsis gins considing an aial sction of th otos which is shown in Figu. Th fnc sstms a usd: th fid sstm Sf and th moil sstms S and S, igidl connctd with ach oto and otating with it. Th oigins of sstms S and S a in th gomtic cnts of th otos, whil that of Sf is at th midoint of sgmnt. Th asolut otations of th two moil fnc sstms a qual in modulus, ut in oosit dictions. In Figu a ctain liknss twn ths tth and th olut gas is vidnt. Th itch cuvs of th kinmatic ai a th cicumfncs, with adius (s Figu ). Th itch oint P0 coincids with th oigin of fnc sstm Sf. Moov it can considd that a suital altnativ to th ccloidal fom fo th low otos is th olut fom as otd in (3). Howv, in this cas th sac ( in Figu ) is tml ducd twn th ti of a tooth and th inn at of th van of th oth oto as comad to standad modula dsign. ω f ω T P P f 0 f θ Figu. Aial sction of two-scw low otos.. Pofil gnation ack As noticd th oto ofil on th aial sction snts olut acs that can asil otaind a siml ack (), which is smmtic with sct to f and antismmtic with sct to f (Figu ). This nsus idntical ofils on oth otos. If th ack consists of two lins, A-B and B-C, th ack to oto gnation will sult in actl th sam ofil, ut th olm of undcut ma ais. In fact, using a ack onl a fid minimum num of tth can otaind avoiding undcut, s f.i. (9). In th cas otd in Figu, nmin will aoimatl qual to 6. A ossil ocdu to avoid undcut consists in modifing th ack mans of a conv cuv, sa a cicl, which is ut smoothl on th ack twn oints D and E. So not onl will th undcut convnintl aolishd, ut also th adius of th cuvatu can usd as anoth ofil aamt to gt a famil of otos in which th otimal ofil with sct to dislacmnt and manufactuing ma found. Th considation of th ack lains th ofil gnation, ut it is also usful to dtmin th analtic ssions in a aamtic fom of th cuvs which comos th ofil, thfo a diffnt ocdu will adotd considing ach ac that comos th ofil. As a consqunc of th smmt, th analsis is stictd to a half of a tooth. Analtical modl of low - v. doo f -.doc 3/05/06 :4-3 -

4 f f a P f 0 f B C D E A f θ θ Figu. Aial sction of two-scw low otos.. Inn and ti cicls Figu 3, which snts a magnification of a oto half tooth, shows that two acs of cicumfnc, and, dlimit th tooth. Pacticall saking, th cosond to th inn and ti cicl of olut gas. So, if th ti adius and th inn adius i a dfind, th aamtic quation of th acs, and i, in th fnc sstms S and S is v siml, ct fo th otation in th cosonding fnc sstm fo th coct angula ositioning. Th limitation of th aamts ϑi and ϑ is considd lat. i Figu 3. Ti and inn acs. Figu 4. Eitochoid ac. ( ϑ ) ( ϑ ),,,, : cosϑ sinϑ,,, 0 ϑ α (),, Analtical modl of low - v. doo f -.doc 3/05/06 :4-4 -

5 ( ϑ ) ( ϑ ) cosϑ i, i, i i, i, : sinϑ i, i, i i, 0 ϑ α () i, i,.3 Eitochoid ac Duing th otation, th inn and ti acs a not alwas in msh, thfo it is ncssa to dtmin what kind of cuv is nvlod th tm oints of th ti ac of on oto on th oth. Th dtmination of an ac of such a t of cuv, lalld in Figu 4, is a common olm in th stud of ositiv dislacmnt machins and it has n dl analsd th authos in (30-3) wh th analtical ssions a dvlod. Th solution is givn an itochoid ac i,, whos aamtic quation in fnc sstms S and S is: ( ϑi, ) cosϑi, cosϑi, ϑi, α, ( ϑ ) sinϑ + sin ϑ i, i, : 0 i (3) i, i, i, i, Also in this cas, th cuv is dfind ct fo a otation aound th oigin of th lativ fnc sstm. Th low limit fo th aamt ϑi is 0, which cosonds to th tm of th loo oosit th doul oint of th cuv, whil th u limit αi will discussd in th nt aagah. C τ i Figu 5. Involut ac. Figu 6. Involut cus and tangnt in th cus τ..4 Limits of itochoid ac and olut as cicl dtmination Fo th comlt dtmination of th oto ofil, onl th fouth ac, lalld in Figu 5, has to dfind. If it is considd that th ac h amind is alwas and onl in msh with th cosonding ac of th oth oto and that th acs a qual to ach oth, it can infd that thos acs should olut acs (s.g.,, 33). Moov, th olut has a cus on its as cicl, so th tangnt τ in th cuv cus C is adial (s Figu 6). Analtical modl of low - v. doo f -.doc 3/05/06 :4-5 -

6 Th latt considation suggsts th cition fo th continuit condition twn th olut, and itochoid i, acs. Th two cuvs must hav th sam tangnt at th junction oint, thfo th tangnt to th itochoid has to coincid with τ in th fillt twn th two acs; sids it has to ass though th oigin, of th latd fnc sstm (s Figu 6). This allows us to dtmin th valu of th aamt αi and th coodinats of th tm oint C of ac i,. Th tangnt to cuv i, in a gnal oint (0, 0) has th following analtic ssion: i ( ϑi ) 0 i ( ϑi ) 0 (4) i i ϑ ϑ i i Considing now that th tangnt has to ass though th oigin of th fnc sstm, quation (4) coms: ϑ ϑ i i i i ϑ ϑ i i α α i i i i ( α ) i ( α ) i (5) B lacing quation (3) and its divativs in quation (5), th valu αi in closd fom as function of and is otaind: α i accos (6) 3 + It is wothwhil noticing that th tm oint C of th ac i, which has coodinats ( α ), ( α )), longs also to th olut as cicl (s Figu 6). Fo th ( i i i i vious considations, th as adius can thn dtmind as: 4 i( α i) + i( αi) cosαi (7) 3 A wll known lationshi twn th as adius and th itch adius fo th olut ofils ists, so it is ossil to dtmin th ssu angl θ of th olut and to not that it is not an indndnt aamt in this cas, sinc it is a function of th adii as shown in quation (7): θ accos (8).5 Involut ac Now th quation of th ac,, lalld in Figu 5 can found out. Th aamtic ssion of th olut is otd fo instanc in () and, ct fo a otation in th lativ fnc sstm S and S fo th coct angula ositioning, is givn : Analtical modl of low - v. doo f -.doc 3/05/06 :4-6 -

7 ( ϑ, ) cosϑ, + ϑ, sinϑ, ϑ, α, ( ϑ ) sinϑ ϑ cosϑ,, : 0 (9),,,,, Evidntl th ac, gins on th as cicl and nds at th intsction with th ti cicl, so th valu of th aamt α, can otaind th following quation, and also considing quation (9): ( α ) + ( α) α (0) Now it is imotant to valuat th angl γ which sutnds th ti ac, and which vidntl coincids with th angl sutnding th inn ac i,. Not that this valu also coincids with th limit valus of aamts αi and α. Du to th smmt, th ac sutndd a half of a tooth on th is qual to π 6 (s Figu 7), whil th angl δ sutndd th comlt ac is qual to: ( α) δ actan () ( α ) γ π/6 δ ε i i Figu 7. Dtmination of angl γ. γ Figu 8. Comlt oto: and cicumfnc acs, itochoid ac, olut ac. W can now dtmin th valu of th aamt ϑ which cosonds to th intsction twn and th itch cicl, using th following: ϑ () ˆ ˆ ˆ ( ) + ( ϑ) ϑ Analtical modl of low - v. doo f -.doc 3/05/06 :4-7 -

8 and otain th valu of th angl ε, which sutnds th olut ac fom th as cicl to, as follows: ˆ ( ϑ) ε actan ( ˆ (3) ϑ ) Finall angl γ is qual to: π γ δ + ε (4) 6 Th assml of acs dtmind (), (), (3) and (9), with suital otations in sstms S and S, allows us to dfin th ofil of a half tooth and finall, using smmt, that of th oto (s Figu 8)..6 Pofil dtmination in 3D sac nc th functions which snt th acs i,, i and a dtmind, th can suital otatd and th D analtical modl of th oto dfind as alad shown in Figu 8. B stating fom th D modl, it is ath siml to dfin th 3D modl of th oto too, which is mad th sufacs Σi, Σ, Σi and Σ. Ths a dvlod fom th cosonding ac a scw motion. Fo th analtical ssion it is fist ncssa to dfin th vcto, which snts th ac in D. B considing fo aml th itochoid, it is: [ ( ϑ ) ( ϑ ) 0 ] T i( ϑ i) i i i i (5) If is th angula itch of th scw otos, th scw motion mati M(υ) is qual to: cosυ sinυ 0 0 sinυ cosυ 0 0 M ( υ) 0 υ π (6) 0 0 υ In quation (6) it is stssd that th otos of th comsso und considation fom onl a 0 scw otation. Thfo sufac Σi is sntd in 3D Ri Σ R ( ϑ, υ) M( υ) ( ϑ ) (7) i : i i i i B itating th ocdu illustatd viousl fo all th acs which comos th oto sction, an analtical modl of th oto is finall otaind as otd in Figu 9. Analtical modl of low - v. doo f -.doc 3/05/06 :4-8 -

9 Figu 9. Roto analtical modl. 3 DESIGN PARAMETERS AND LIMITS A ctain liknss twn th analsd oto and th olut gas, as shown in Figu and Figu 8, has alad n stssd. Howv, th low num of tth of ths otos quis an tnal snchonis. Moov, in this cas th conct of modula dsign cannot usd, caus nith an actual addndum no a ddndum ists. nl two dsign aamts a indndnt, among all thos which aa in th vious quations: th num of tth which is usuall qual to th and also th inn adius which can ssd as function of th itch and ti adius as follows: i (8) Th ssu angl θ dnds on adii and too, as shown quation (8). In aticula, aanging quation (8) it is otaind: ( 4 3cos θ ) (9) Th vious ssion shows that tth chaactisd th sam ssu angl θ a sntd in th lan lins though th oigin of quation h. Ths lins chaactis simila otos. Thfo it is suital to slct thos two adii as indndnt dsign aamts. Now th limits of th dsign aamts a discussd. Two ounda valus a quit as to otain valuating th limit shas of th otos which cosond to, whn th oto ofil tuns into a cicumfnc, and to whn inn adius has a null valu. Actuall th condition: < < (0) Analtical modl of low - v. doo f -.doc 3/05/06 :4-9 -

10 is too wid, sinc it dos not tak th fact that th tooth is shand into account, i.. th angl γ of quation (4) is zo. B again considing quations (0) and (), and sustituting th valu of givn quation (7), it is otaind: ˆ ϑ () 4 ( ) ( ϑ ) 4 α () ˆ 4 Th angls δ and ε can also hav a diffnt ssion if th gomtic maning of th olut is considd. As fa as gads to Figu 0, th following quations a valid: A, C, AB ϑˆ, CD α, AB AÔB actan actanϑˆ A CD CÔD actan actanα C (3) Equations (), () and (3) allow us to dtmin: δ α actanα ϑ actan ϑ (4) ε ϑ actanϑ (5) C α ϑ^ A ε α ϑ^ D δ B Figu 0. Dtmination of δ and ε. Ξ( h) Figu. Rsntation of limit function Ξ(h). h If th limit condition cosonding to γ 0 is valuatd using quation (4) and lacing quations (4) and (5), on otains: π ˆ ϑ 6 actan ˆ ϑ actan ˆ ϑ (6) and aling th tangnt function to oth ats of quation (6): Analtical modl of low - v. doo f -.doc 3/05/06 :4-0 -

11 π tan tan ˆ ϑ 6 π + tan tan ˆ ϑ 6 ˆ ϑ ˆ ˆ 3 tan ˆ 3 3 ϑ ϑ + ϑ + ϑ 3 ( + ϑ ) tan ˆ ϑ 0 (7) Taking into account that a limit lation such as h is lookd fo, quation () and (7) consquntl com: ϑ h 4 h (8) h h h Ξ( h) + tan tan h h h 3 h h 0 (9) 4 Limit quation Ξ(h) 0 is shown in Figu. Among th infinit solutions () that quation (9) has in th h on intval (;), which snts inqualit (0), th low on has to chosn that is qual to h Thfo th sac of th admissil dsign aamt valus coms (s Figu ): < <.5074 (30) Figu. Sac of dsign aamt and. CNCLUSINS Th snt a dals with th dsign of a aticula t of scw low otos. Th taditional mthod fo gnating th oto ofils ack is initiall takn into account; it is usful to gnat th ofils ut it dos not giv an licit 3D analtical modl of th oto () Point h is a scond t discontinuit (34) fo function Ξ(h), sinc / lim Ξ ( h). Analtical modl of low - v. doo f -.doc 3/05/06 :4 - - h

12 as givn th mthod sntd h. This analtical modl is th asis fo futh studis on th dnamic of th low otos. Th analtical modl is otaind considing all th sufacs, which comos th oto, and giving thi aamtic ssions. n th asis of th oosd modl of th otos, a stud aout th dsign aamts and limits is oosd and aamt sac dfind. ACKNWLEDGEMENTS Th authos wish to thank M.U.R.S.T. (Ministo dll Univsità dlla Ricca Scintifica Tcnologica) and th Univsità di Pavia fo funding th sach. REFERENCES Buckingham, E. Analitical Mchanics of Gas. Dov, Nw Yok, 963. Litvin, F.L. Ga Gomt and Alid Tho. Pntic-Hall, Englwood Cliffs, NJ, Mitt, H.E. Gas. Pitman, London, Ston, R.C. Th Efficinc of Roots Comssos and Comssos with Fid Intnal Comssion. Poc Instn Mch Engs vol 0 no A3, 988, Kstin, J. and wczak, J.A. Th ssion of Wok in a Roots Blow. Poc Instn Mch Engs, 5-3, B, 95, Glaist, E. Communication. Poc Instn Mch Engs, 5-3, B, 95, Shu, P.C., Tang, Y. and Png, X.Y. A sustitut of Roots Blow monotooth ota comsso. In Pocdings of 994 Intnational Comsso Engining Confnc at Pudu (Ed W. Sodl), Pudu Univsit, Wst Lafatt, Indiana, Vol., 994, Singh, P.J. and Bowman, J.L. Effct of dsign aamts on oil-floodd scw comsso fomanc. In Pocdings of 986 Intnational Comsso Engining Confnc at Pudu (Eds J.F. Hamilton and R. Cohn), Pudu Univsit, Wst Lafatt, Indiana, Vol., 986, Znan, X. and Dagang, X. Stud on actual ofil sufac and ngaging claanc of scw comsso otos. In Pocdings of 986 Intnational Comsso Engining Confnc at Pudu (Eds J.F. Hamilton and R. Cohn), Pudu Univsit, Wst Lafatt, Indiana, Vol., 986, Edstöm, S.E. A modn wa to good scw otos. In Pocdings of 99 Intnational Comsso Engining Confnc at Pudu (Ed J.F. Hamilton), Pudu Univsit, Wst Lafatt, Indiana, Vol., 99, Bnnwitz, C. Softwa suot fo scw oto dsign, manufactu and qualit contol. In Pocdings of 99 Intnational Comsso Engining Confnc at Pudu (Ed J.F. Hamilton), Pudu Univsit, Wst Lafatt, Indiana, Vol., 99, Zhang, L. and Hamilton, J.F. Main gomtic chaactistics of th twin scw comsso. In Pocdings of 99 Intnational Comsso Engining Confnc at Pudu (Ed J.F. Hamilton), Pudu Univsit, Wst Lafatt, Indiana, Vol., 99, Zichng, Z. Comut aidd dsign of a twin-scw figant comsso. In Pocdings of 99 Intnational Comsso Engining Confnc at Pudu (Ed J.F. Hamilton), Pudu Univsit, Wst Lafatt, Indiana, Vol., 99, Stosic, N. and Hanjalic, K. Dvlomnt and otimization of scw ngin oto ais on th asis of comut modlling. In Pocdings of 994 Intnational Comsso Engining Confnc at Pudu (Ed W. Sodl), Pudu Univsit, Wst Lafatt, Indiana, Vol., 994, Flming, J.S., Tang, Y. and Andson, H. Twin scw comsso fomanc and its lationshi with oto cutt lad sha and manufactuing cost. In Pocdings of 994 Intnational Comsso Engining Confnc at Pudu (Ed W. Sodl), Pudu Univsit, Wst Lafatt, Indiana, Vol., 994, Sjöholm, L. Diffnt oational mods fo figation twin-scw comssos. In Pocdings of 986 Intnational Comsso Engining Confnc at Pudu (Eds J.F. Hamilton and R. Cohn), Pudu Univsit, Wst Lafatt, Indiana, Vol., 986, Moi, H., Kasua, K., Fujiwaa, M., Matsuaa, K., Suzuki, A. and Aoki, M. Singl stag, oil-f scw comsso with a comssion atio of ight. In Pocdings of 986 Intnational Comsso Engining Confnc at Pudu (Eds J.F. Hamilton and R. Cohn), Pudu Univsit, Wst Lafatt, Indiana, Vol., 986, Tang, Y. and Flming J.S. Simulation of woking ocss of an oil floodd hlical scw comsso with liquid figant injction. In Pocdings of 99 Intnational Comsso Engining Confnc at Pudu (Ed J.F. Hamilton), Pudu Univsit, Wst Lafatt, Indiana, Vol., 99, 3-0. Analtical modl of low - v. doo f -.doc 3/05/06 :4 - -

13 9 Tang, Y. and Flming J.S. taining th otimum gomtical aamts of a figation hlical scw comsso. In Pocdings of 99 Intnational Comsso Engining Confnc at Pudu (Ed J.F. Hamilton), Pudu Univsit, Wst Lafatt, Indiana, Vol., 99, Mioshi, K. Analsis of scw comsso fomanc asd on indicato diagams. In Pocdings of 99 Intnational Comsso Engining Confnc at Pudu (Ed J.F. Hamilton), Pudu Univsit, Wst Lafatt, Indiana, Vol., 99, Kanging, C., Zhnwu, T. and Hngi, C. Th dvlomnt and alication of a wat-injctd twin scw comsso. In Pocdings of 99 Intnational Comsso Engining Confnc at Pudu (Ed J.F. Hamilton), Pudu Univsit, Wst Lafatt, Indiana, Vol., 99, Adams, G.P. and Sodl, W. A mthod fo comuting th comssion loads in twin scw comssos. In Pocdings of 994 Intnational Comsso Engining Confnc at Pudu (Ed W. Sodl), Pudu Univsit, Wst Lafatt, Indiana, Vol., 994, Adams, G.P. and Sodl, W. Dnamic simulation of oto contact focs in twin scw comssos. In Pocdings of 994 Intnational Comsso Engining Confnc at Pudu (Ed W. Sodl), Pudu Univsit, Wst Lafatt, Indiana, Vol., 994, Adams, G.P. and Sodl, W. Comutation of Comssion Loads in Twin Scw Comssos. ASME Jounal of Mchanical Dsign, Vol. 7, (995), Huttauck, D., Puchas, C., Fick, W. and Joos, K. Entwicklungskonzt ds Mcds-Bnz- Vizlind-ttomotos mit mchanisch Aufladung fü di C-Klass. MTZ, No 56 Dc 95, Huttauck, D., Puchas, C., Fick, W. and Joos, K. Dvlomnt of th Mcds-Bnz 4-Clind Gasolin Engin with Comsso. In Pocdings of VDI-Intnationals Win Motonsmosium, 5-6 Ail 996, Schmitz, T., Holloh, K.D. and Jugns, R. Potntial in mchanischn Zusatzaufladung fü Nutzfahzugmoton. MTZ, No 55 Ma 94, Mimmi, G. and Pnnacchi, P. Comutation of oto loads in th scw ums. To intd on ASME Jounal on Mchanical Dsign. 9 Ssini,. Mccanica Alicata all Macchin Cinmatica. Vol., Casa Editic Amosiana, Milan, Ital. 30 Mimmi G. and Pnnacchi P. Flow Rat Rgulait in Rota Tochoidal-Lo Pums. Machin Elmnts and Machin Dnamics, G.R. Pnnock d., DE-Vol.7, 994, Th Amican Socit of Mchanical Engining, Nw Yok, Mimmi, G. and Pnnacchi, P. Dsign of th-scw ositiv dislacmnt ota ums. Contact Mchanics II - Comutational Tchniqus, M.H. Aliaadi and C. Alssandi ds., 995, Comutational Mchanics Pulications, Southamton Boston. 3 Mimmi, G. and Pnnacchi, P. Dtmination of Tool Pofil fo th Milling of Th Scw Pum Roto. Mccanica - Intnational Jounal of th Italian Association of Thotical and Alid Mchanics, Vol 3, no 4, 997, Hniot, J. Inganaggi: tattato toico atico. Tcnich Nuov, Milano, Amio, L. Analisi Matmatica. vol., UTET, Toino, 986. Analtical modl of low - v. doo f -.doc 3/05/06 :4-3 -