TOOL WEAR AND ITS COMPENSATION IN SCREW ROTOR MANUFACTURING Professor N. Sosic Cair i Posiive Displaceme Compressor Tecology Ciy Uiversiy, Lodo, EC1V 0HB, U.K. E-mail:.sosic@ciy.ac.uk ABSTRACT Sice screw compressor roors are of a elical sape ad ool pois egaged i eir maufacurig pass differe cuig leg, e ools do o wear uiformly alog e roor profiles. Te ouiformiy depeds o a differece bewee e roor poi agle ad pressure agle ad o e roor elix agle. A differece i relaive moveme bewee e ool ad e roors wi ad wiou e cuig sock is used o quaify a relaive wear of e ool. Tis is ow i couerpar used as a basis for calculaio of a variable roor sock disribuio, wic resuls i a uiform wear of e fiisig ool we i is applied. Fially, e rougig or semi-fiisig ool is geeraed wic cus e roors wi suc a variable disribued sock. Applicaio of is procedure ca subsaially improve e overall ool ecoomy. 1 INTRODUCTION Screw compressors are posiive displaceme roary macies comprisig a mesig pair of elical roors coaied i a casig formig a camber wic volume depeds o eir roaio. Compressor efficiecy depeds o e roor profile ad clearaces bewee e roors ad bewee e roors ad compressor ousig. For suc macies o perform effecively e roors mus mee e mesig requiremes ad sould maiai a seal alog e eire roor coac. Fig 1. Roor ad ool coordiae sysems Te roor ad ool coordiae sysems are preseed i Fig 1. Roors are oday maily maufacured by e use of formed, millig or gridig ools. Tis meas a ools suiable for screw roor producio mus be calculaed by a appropriae mesig procedure. Te evelope meod, wic saes a wo surfaces are i mes if eac evelops e oer uder a specified relaive moio, is suiable for suc a purpose. Te procedure sars wi a give surface (1) wi is derivaives () ad (3), were x() ad y() are e roor (Fig ) coordiaes i e rasverse plae i e X,Y coordiae sysem for wic a
mesig ool surface is o be deermied. A family of suc surfaces is give i parameric form (4) ad eir derivaives i X Y Z roor ad X Y Z ool coordiae sysem respecively are give i (5) as preseed i Fig. 1, were C represes e axis disace, Σ saf agle ad idices ad are roor ad ool. Te evelope equaio of suc surfaces i respec of e roaio agle q is (6). Sice a geeraig surface is defied by e parameer, e evelope ca be used o calculae aoer parameer q, wic is e roor roaio agle, as a codiio o geerae a mesig surface. Te cross produc i e evelope equaio represes a surface ormal wile jr/j is a relaive velociy of wo surfaces i e commo poi, is a ool roaio agle ad p is a roor lead defied for a ui roaio agle. ρ = τ (, ϑ) = [ x, y, z ] = [ x cosϑ y siϑ, x siϑ + y cosϑ pϑ] r (1) r x =, y x y x y,,0 = cosϑ siϑ, siϑ + cosϑ,0 r = [ y, x, p] (3) ϑ ρ, ϑτ, = x, y, z = R cos τ, Rsi τ, ysiσ+ zcosσ = x C, y cosσ z si Σ, y si Σ+ z cosσ ( ) [ ] [ ] [ ] ρ τ [ y, x,0], = [ pϑ si Σ y cos Σ, ( x C) cos Σ, ( x C) si Σ] r r ρ = 0 ϑ τ y y ϑ (7) x x ( C x + p co Σ) x + y + p p C co Σ = 0 Evaluaed from (6) a codiio of mesig is preseed i (7), wic ca be solved by applicaio of a umerical meod. For a give parameer, e poi coordiaes ad eir derivaives are kow. A guessed value of parameer q is e used o calculae x, y ad jy /jx from (1) ad () for give x, y, jx/j ad jy/j. A revised value of q is e calculaed from equaio (7) ad e procedure repeaed uil e differece bewee wo cosecuive values becomes sufficiely small. For give rasverse coordiaes of e roor profile ad eir derivaives, coac agle q, deermied from (7) may be used o calculae e x, y ad z coordiaes of e roor elical surface from equaio (1). Te ool surface ad fially e ool rasverse poi coordiaes R ad z may e be calculaed from (4) as: ( x C) + ( y cosσ z si Σ), z = y si Σ + z Σ R = x + y = cos (8) Aloug derived ere for a screw roor ool geeraio, e mesig codiio, as sow, may be coveiely employed i geeral gearig pracice. Referece exbooks o gears cosider is applicaio oo complex ad oly a case of sraig-sided ools for ivolue geeraio is repored ere. However, Adreev 1961, [1], ad recely Xig 000, [5] i eir books o screw compressors give is full derivaio specifically for screw compressor ools. A similar approac as bee publised i Tag, (6) () (4) (5)
1995, [4], Zag, 1997, [6] ad i Sosic, 1998, [] were addiioally a reverse ool o roor rasformaio (10) was give, wic was laer used i Sosic, 1998, [3] o calculae iflueces of seig errors upo e screw roor maufacure. From e kow ool coordiaes, R ad z ad eir derivaive jz /jr, e roor profile pois x ad y ca be used o calculae e roor rasverse coordiaes x ad y of e roor wi give sock from iverse of e equaio (1) as: [ x, y ] [ x cosϑ + y siϑ, x siϑ + y cosϑ ] for e agle parameer q wic is obaied from: = (9) z z R + z cos ϑ + ( p + C co Σ) siϑ + p co Σ C = 0 (10) R R Oce calculaed, e disribuio of q alog e profile may be used o calculae e coordiaes of mesig ool ad roor pois, as well as o deermie e coac lies ad pas of coac bewee e roor ad ool. Te sealig lie of screw compressor roors is somewa similar o eir coac lie. Sice ere exiss a clearace gap bewee roors, e sealig lie is a lie cosisig of pois of e mos proximae roor posiio. Te mos coveie pracice o obai a clearace gap bewee e roors is o cosider e gap as e sores disace bewee e roors i a cross secio ormal o e roor elicoid. A rasverse clearace gap ca e be obaied from e ormal oe by a appropriae rasformaio. I is ieresig a a maufacurig sock, a surplus of maerial wic is o be removed by a ool acio, may be cosidered as a egaive clearace. I suc a case all rasformaios derived for clearaces ca be applied ad are valid for e sock calculaio. CALCULATION OF THE ROTOR AND TOOL COORDINATES.1 Roor Coordiaes for a Give Sock If a sock, wic is o be cu of e roors, d is give i e plae ormal o e roor elicoids surface, e firs par of e relaio (6), e cross produc of e r derivaives, wic defie e ormal direcio, ca be used o calculae e coordiaes of e roor elicoids wi e sock from e rasversal roor coordiaes wiou e sock ad e sock iself as: δ dy x = x + p, D d were e deomiaor D is give as: y y δ dx p, D d z δ dx = ( x D d = + dx dy dx dy D = p + p + x + y (1) d d d d dy y ) d (11) Tese serve o calculae e ew rasverse coordiaes x ad y of e roor wi sock from equaio (11) for e agle q =z /p. Te x ad y ad origial roor coordiaes x ad y serve ow o calculae e ool coordiaes, wic will cu e roors wi e sock, or rouged or semi-fiised roors ad
roors wiou e sock, or fial roors respecively. A differece bewee ese wo will somewa give a relaive measure of e ool wear.. Tool Wear A relaive moveme bewee e roor ad ool coais a simulaeous slidig ad rollig acio, because bo compoes ad o oly a age oe cou for e ool wear. Cuig process is deermied by e ool speed ad advace. However, if oly a relaive measure of e ool wear is cosidered, ese parameers will simulaeously be coaied i bo, e roug ad fie cuig. Terefore, e ool wear will be depede oly o e differece bewee e coordiaes eier of wo ools, wo roor elical coordiaes or wo roor rasverse coordiaes. Eier e roor or ool moio equally well deermie cuig roug e sock. As e simples, a differece of roor rasverse coordiaes x, y ad x, y is, ereby, preseed ere as relaive measure of e ool wear. s ( x x) + ( y y) = (13) A roor pair i cofiguraio of 5-6 lobes o e mai ad gae roor is used as a example ere. Te mai roor diameer is 18 mm. Te roor elix agle is 40 o. Roor cere disace is 90 mm. Fig. N Roors 5/6-18 mm Fig 3. Tool wear o cu uiform sock Te mesig roors i rasverse cross secio are preseed i Fig. A screw compressor roor ad is formed ool are mesed o o parallel ad o iersecig axes. Due o eir axes o beig parallel, ere is oly poi coac bewee em. Tis poi is a cuig poi. Te eed o saisfy equaio (7) leads o e roor - ool mesig requireme for e give fiised roor rasverse coordiae pois x ad y wi jy/jx. Te roor elical coordiaes x, y ad z are obaied from (1). Tese are sufficie o ge e ool rasverse poi coordiaes R ad z from (8). Te ideical procedure is applied for e x, y roor coordiaes wi 100 mm sock o obai eir cuig ools. I bo cases e roor-ool cere disaces were 105 ad 15 mm for e mai ad gae roor respecively. A saf agle was 45 o. Bo e ool coordiaes R ad z are ploed i Fig 3 as R ad z o visualize e ool wear wic is scaled 50 imes. I is obvious a e ool wear is o uiform alog e ool profile. I is miimal ad equal o e sock applied i all pois were e roor profile ad pressure agle are e same, ad i is larger i all oer cases. A disribuio of a relaive ool wear resulig from suc a cuig is preseed i deail i Figs 4 ad 5 for e mai ad gae roor respecively. Te ool poi umber i Figures 4 ad 5 represes e coordiae poi posiio alog e profile. For e roors preseed i e example e maximal ool wear was abou wice of e miimal oe for e mai roor ad abou 50 % i excess for e gae roor.
Fig 4. Tool wear, mai roor Fig 5. Tool wear, gae roor 3 APPLICATION OF VARIABLE CUTTING STOCK TO MINIMISE WEAR I is quie logical o expec a a variable sock applied aroud e roor profile will cause a differe ool wear alog e ool profile compared wi e uiform oe. If a disribuio of a sock is cose o be reciprocal o e wear caused by e uiform sock, a uiform ool wear ca be obaied. Tis is prove ere by applicaio of suc a sock o e roors, wic leads o e ew roor coordiaes x, y. Te ew coordiaes are compared wi e old oes, x, y ad preseed i Fig 6. Te scale of e roor coordiae differeces is agai 50 imes. Te ew coordiaes are ow used o calculae e ool coordiaes R ad z, preseed i Fig 7 as R ad z wic are compared wi e fiisig ool ad eir differece scaled 50 imes. Tis wear is preseed separaely for e mai ad gae roors i Fig 8 ad 9. I is visible a e ool wear is uiform ow. Te roor poi umber i Figures 8 ad 9 represes e coordiae poi posiio alog e profile. Fig 6. A roor sock for uiform ool wear Fig 7. Uiform ool wear
Fig 8. Sock for uiform wear, mai roor Fig 9. Sock for uiform wear, gae roor Te uiform ool wear appears o be e opimal oe for e ool ecoomy allowig e ool o be used for e loges period bewee wo ool codiioigs eier by dressigs or sarpeig. 4 CONCLUSION A ool wear is always prese i maufacurig of screw compressor roors. I is logical o expec a a variable macie sock wic is removed from e roor will cause a differe ool wear alog e ool profile compared wi e uiform oe. If a sock iversely proporioal o e wear caused by e uiform sock is o be cu off, a uiform ool wear could be obaied. Te evelope eory of gearig as bee applied ere as a mesig requireme for crossed elical gears o calculae e sock disribuio wic would resul i e uiform wear of a fiisig ool. REFERENCES [1] Adreev P.A, 1961: Viovie kompressorie masiii (Screw Compressio Macies), SUDPROM Leigrad, Russia [] Sosic N, 1998: O Gearig of Helical Screw Compressor Roors, Proc IMecE, Joural of Mecaical Egieerig Sciece, Vol 1, 587 [3] Sosic N, Smi I. K, ad Kovacevic A, 1998: Some Aspecs of Tool Desig for Screw Compressor Roor Maufacurig, I Desig Coferece -Desig 98, Dubrovik, Croaia [4] Tag Y, 1995: Compuer Aided Desig of Twi Screw Compressors, PD Tesis, Sraclyde Uiversiy, Glasgow, U.K. [5] Xig Z. W, 000: Screw Compressors: Teory, Desig ad Applicaio, (i Ciese), Cia Macie Press, Beijig, Cia [6] Zag Z, Li J, Si G ad Fu W, 1997: Geeral Program for Millig Cuer Profile Desig of Screw Compressor Roor, d I Compressor Tecique Coferece, Cegdu, Cia