PERIPHERAL SURFACE PROFILING OF THE HOB CUTTER FOR SCREW COMPRESSOR ROTORS COMPONENTS II. MALE ROTOR

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1 U.P.B. Sci. Bull., Series D, Vol. 75, Iss., ISS PERIPHERAL SURFACE PROFILIG OF THE HOB CUTTER FOR SCREW COMPRESSOR ROTORS COMPOETS II. MALE ROTOR Camelia POPA, icolae OACEA Profilarea suprafeţei periferice a sculei melc reciproc înfǎşurǎoare cu suprafeţele elicoidale ale flancurilor rooarelor compresorului elicoidal, consiuie o problemǎ de generare de speţa a II-a (referioare la suprafeţele reciproc înfǎşurǎoare cu conac punciform). În lucrare, pe baa eoremelor referioare la descompunerea unei mişcǎri elicoidale, se propune o meodǎ de profilare a suprafeţei periferice primare a sculei melc generaoare a roorului conducǎor din componenţa compresorului elicoidal. Sun preenae formele consrucive ale lobilor roorului conducǎor, forma cremalierei generaoare şi forma profilului aial al sculei melc, reciproc înfǎşurǎoare aceseia. De asemenea, pe baa unui produs sof dedica, se preinǎ eemple numerice ale formelor aiale ale sculelor melc generaoare penru roorul conducǎor penru un compresor cu rapor de ransmiere 4/6, respeciv /5. Peripheral surface profiling of he hob cuer represens a second degree generaion problem referring o poin conac surfaces. In his paper, according o heorems referring o he decomposiion of a helical moion, we propose a surface profiling mehod of he peripheral surface of he hob cuer for he male roor, screw compressor componen. The consrucive form of he male roor lobes, he shape of he generaing rack and he shape of he aial profile of he hob cuer are presened. Also, we presen fronal applicaions of he aial shapes of he hob cuer generaing for he male roor, screw compressor componen, gear raio 4/6, and /5 respecively. Keywords: helical surfaces, hob cuer, meshing surfaces. Inroducion Mehods of profiling ool drives, envelope a vore of surfaces associaed wih a couple in rolling aodes, are: Oliver II heorem, for enveloping surfaces wih conac poin, and Gohman heorem, hrough he inermediae surface; helical movemen decomposiion mehod, icolaev [],[],[4],[6],[8]. We also proposed D and D graphic soluions in order o solve hese problems [7], [9]. Assis, Deparmen of Machine Elemens and Graphics, Dunărea de Jos Universiy of Galai, Galai, Romania, lpopa@ugal.ro Prof., Deparmen of Manufacuring, Roboics and Welding Engineering,, Dunărea de Jos Universiy of Galai, Galai, Romania, noancea@ugal.ro

2 Camelia Popa, icolae Oancea Using Béier polynomials, surface represenaion is possible o be done, precisely enough. This is he soluion for he helical surfaces of screw compressor roors, male and female.. Coordinaes sysems; generaion moion In Fig., i is represened he aes sysem, and he absolue movemens of he reference sysems associaed o hese aes. The reference sysems are he following: - y is he global sysem wih ais, ais of roaion of he ais associaed o he generaing surfaces roller; - y he global sysem, wih ais y superposed o he primary peripheral surface ais of he hob cuer; - X Y Z he relaive sysem aached o he ais of he aode, A ; - ξζ he relaive sysem aached o he ais of he rack generaing (fla surface superposed o he plane ζ), A ; - X S Y S Z S he relaive sysem associaed o he peripheral surface of he hob cuer. Fig.. Coordinaes sysems; generaion moions The kinemaics of he generaion process is known as follow: roaion of he aode A (radius R rp ), ϕ angular moion parameer,

3 Peripheral surface profiling [...] for screw compressor roors componens; II.Male roor ( φ ) T = ω X. () displacemen of he aode A - l moion parameer, = ξ - a, R r p a = ; () roaion of he sysem X S Y S Z S around he ais Y, wih φ angular moion parameer, T ( φ ) =ω X S. () The following condiions are known: condiion of rolling of he aodes A and A, = Rr φ p ; (4) dependence generaed by he shape of he peripheral surface of he hob cuer (consan seep parallel-ype heli, wih p helical parameer), = pφcosω ; (5) ransformaion beween fied coordinaes sysems, A y = cosω sinω y, (6) sinω cosω where A is he disance beween he aes, respecively, of aode A and of he helical surface V. The relaive moion of he sysem aached o he aode A, of he generaing surface XYZ, compared wih he reference sysem associaed o he rack generaing space ξζ, is given by he ransformaion: ξ = ω T ( φ ) X - a. (7) The curren poin on he generaing surface, as a cylindrical helical surface is: X ( ) = X u ; Σ Y = Y ( u) ; (8) Z = p υ, for u discree variable known hrough a reduced number of values ( or 4 poins); υ angular roaion parameer around he ais Z ; p helical parameer of he conducive roor.

4 Camelia Popa, icolae Oancea. Mehod of definiion for he shape of he generaing rack surface According o (7) and (8) we define: ξ cosφ ( ) R sinφ X u rp = sinφ cosφ Y ( u) Rr φ p. (9) ζ pυ The enveloping condiion is ' ' [ X X ( u) ] X [ Y Y ( u) ] - u - Yu =, () where: X = Rr cos φ ; p Y = Rr sin φ. p () Equaion () represens condiion of normal [] (R rp is he radius of he cylindrical aode A, (Fig. )). Fig.. Surface Σ, generaing surface known hrough 4 poins : A[X A,X A ], B[X B,X B ],C[X C,X C ], D[X D,X D ] We can accep ha, he direcri of he surface represening he cylindrical flank of he generaing rack, (equaions () (8)), can be subsiued by a Béier polynomial, given by he following equaions : ξ = Aξ ( ) Bξ ( ) Cξ ( ) Dξ ; I : () = A ( ) B ( ) C ( ) D. All he poins which deermine he direcri of he cylindrical surface (he flank of he blank) is given by: A[ A, y A ], B[ B, y B ], C[ C, y C ], D[ D,y D ], ( Fig. ).

5 Peripheral surface profiling [...] for screw compressor roors componens; II.Male roor 4. Peripheral surface profiling mehod for he hob cuer Knowing he surface of he generaing rack flank (in approimae shape, as Béier polynomial), we propose o deermine he characerisic (conac curve) using he mehod of decomposiion of he helical moion [5], (Fig. ). The helical movemen of peripheral surface of he hob cuer, (V, p) decomposes ino a sum of equivalen movemens: a ranslaion, upon he uni vecor of he generari of cylindrical surface of he flank rack and he roaion of he ais A. The disance: π a = p an ω, () represens he disance o he helical surface ais, V, ( Fig. ). Fig.. Helical movemen decomposiion mehod; coordinaes sysems The condiion o deermine he characerisic curve is provided as follows:. (4) The normal vecor o he cylindrical surface I, is always normal o is own generari; characerisic curve depends only on he roaion around he ais A. Therefore, he characerisic curve, during he helical movemen of ais V and he parameer p, is defined as he projecion ais A on he surface I.

6 4 Camelia Popa, icolae Oancea We define, see Fig. : - he ais A, in he y sysem, k sin j cos A ω ω =, (5) - he normal on he surface I, k j i j I =, (6) or : = k j y i k j y i I, (7) (see formula () flank surface of he ool-rack); - he vecor r i O O r i =, (8) where, r is he vecor of he cylindrical surface I. According (), he coordinae ransformaion y, d d r R y p β β ζ ξ cos sin =, (9) leads o he epression of surface shape of he rack I in he sysem y : ( ) ( ) ( ) ( ) ( ) ( ) = = =. cos ; sin ; d d r D C B A y R D C B A p β β ξ ξ ξ ξ () The sie of he parameer ω is, see Fig. 4: β d ω s ω =, () where: -ω s is he angle of hob cuer ais, in relaion o he fron plane of he male roor (righ screw); -β d - angle of he male roor heli, cylinder radius R rp ; -ω - il angle of he heli of hob cuer, wih he radius R rs, = lobs r r s cos R R arcsin ω s p β. ()

7 Peripheral surface profiling [...] for screw compressor roors componens; II.Male roor 5 Fig.4. Til ais of hob cuer The enveloping condiion is: ( A, I, r i ) =. () or, y ( ) ( ), a y, cos β d =. (4) cosω sinω The condiion (4) is a funcion of parameers and : q (, ) =, (5) where,. The equaions () and (5) are a locus surface I. The values of he parameers l and fulfilling (5), replaced in Béier form for rack flanks, deermine he mari: T C I = y, (6) i i i represening he coordinaes of he characerisic curve C I. Helical movemen (V,p) of he curve C I, generaes peripheral surface of he hob cuer. The coordinae ransformaion, see Fig. 4,

8 6 Camelia Popa, icolae Oancea X Y Z S S S = [ R R ] i r p i rs cosω sinω y, (7) i sinω cosω i=,...,n so ha, he helical movemen of (V,p) of he characerisic C I, becomes: X S = X [ ] S, y, φ ; i i i [, y, φ ]; Π YS = YS i i i= n. (8) i Z = Z [, y, φ ], S S i i i and represens he primary peripheral surface of he hob cuer - surface Π. Giving he surface Π he condiion Z S =, (9) he aial secion Π A is obained in he form: X [ ] S = X S, y,,φ ; i i i Π i= n, () Y = Y [, y,,φ ], S S i i i he variable [ φ ] represening he sie of he parameer corresponding o he aial secion, see (9). 5. umerical applicaions The profile of generaing rack of male roor (see also Fig. 5) has he following profile porions. ٠Firs applicaion (screw compressor, raio 4/6) Table The consrucive daes of he reference rack, version I ( Fig. 5) R r u ma ψ ma [ ] υ ma [ ] υ ma [ ] u ma L p Rr c Fig.5.Rack generaing profile

9 Peripheral surface profiling [...] for screw compressor roors componens; II.Male roor 7 Table and Fig. 6 describe numerically he aial secion coordinaes of he primary peripheral surface of he hob cuer. Table Aial secion of he primary peripheral surface of he hob cuer X S Y S X S Y S Fig. 6. The aial secion profile of he hob cuer Fig. 7. The relaive posiion beween he hob cuer and he male roor

10 8 Camelia Popa, icolae Oancea The Fig.7, represens he relaive posiion beween he hob cuer and he male roor. ٠Second applicaion (screw compressor, raio /5) Table The consrucive daes of he generaing rack ( Fig. ) R r u ma ψ ma [ ] υ ma [ ] υ ma [ ] u ma L p c Rr In Table 4, he aial secion coordinaes of he hob cuer peripheral surface are described. Table 4 The aial secion of he hob cuer s primary peripheral surface X Y X Y Fig. 9. The shape of he aial secion for he male hob cuer

11 Peripheral surface profiling [...] for screw compressor roors componens; II.Male roor 9 In Fig. 9 is represened he aial secion shape of he hob cuer for he main roor. In Fig. is represened, in D modeling, he relaive posiion beween he hob cuer and he male roor. Fig.. The relaive posiion beween he hob cuer and he male roor 6. Conclusions. The helical surfaces ha edges he screw compressors componen roors, are comple surfaces (cylindrical helical surfaces and consan sep). Generaing his ype of surfaces is done wih a specific rack, whose ransverse profile is deermined from echnological and funcional consideraions in order o obain an opimum yield.. The male roor generaors can be epressed by simple analyical forms, or approimaion of Béier polynomials.. The hob cuer profiles are generaed according o he composiion and decomposiion of he helical moion, using he inermediae surfaces mehod. The mehods D of he ools primary peripheral surfaces were also represened. B I B L I O G R A P H Y [] Livin, F. L., 989, Theory of Gearing, ASA RP- (AVSCOM 88-C-5), Washingon, D.C.; [] Houson, R., Mavripilis, D., Oswald, F., Liu, Y.S., 994, A basis for solid modeling of gear eeh whih applicaion in design manufacure, Mech. Mach. Theory 9 (5):7-7; [] Oancea,., 4, Mehode numérique pour l éude des surfaces enveloppes, Mech. Mach. Theory (7): ; [4] Radevich, S.P., 8, «Kinemaic geomery of surface machining, CRC, Boca Raon, ISB ;

12 4 Camelia Popa, icolae Oancea [5] Teodor, V.,, Conribuion o he elaboraion of a mehod for profiling ools-ools which generae by enwrapping, Lamber Academic Publishing, ISB ; [6] Rodin, R., P., 99, Osnovy proekirovania rehushich insrumenov (Basics of desing of cuing ools) Vishcha Shkola, Kiev;(in Russian) [7] Veliko, I.,Gencho,., 998, Profiling of roaion ools for forming of helical surfaces, In.J. Mach Tools; manuf.8:5-48; [8] Liukshin, V., S.,968, Teoria vinovih poverhnosei, Mashinosroenie, Moskva; [9] Baicu, I., Oancea,., Profilarea sculelor prin modelare solida ( Profiling of ools using solid modelling) Ediura TEHICA-IFO,Chişinǎu,, ISB X. (in Romanian)