THE MODELLING OF INVOLUTE TEETH GENERATION, WITH THE RELATIVE GENERATING TRAJECTORIES METHOD

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1 THE MODELLING OF INVOLUTE TEETH GENERATION, WITH THE RELATIVE GENERATING TRAJECTORIES METHOD Vgl Gabel TEODOR 1, Ncuşo BAROIU 1, Flon SUSAC 1 and Ncolae OANCEA 1 ABSTRACT: The modellng of a cul of sufaces assocated wth a pa of ollng centodes, when t s known the pofle of the ack-gea s teeth pofle, by dect measung, as a coodnate matx, has as goal the detemnng of the geneatng qualty fo an mposed knematcs of the elatve moton of tool egadng the blank. In ths way, t s possble to detemne the geneatng geometcal eo, as a base of the total eo. The geneaton modellng allows hghlghtng the potental eos of the geneatng tool, n ode to coect ts pofle, po to usng the tool n machnng pocess. A method developed n CATIA s poposed, based on a new method, namely the method of elatve geneatng tajectoes. The analytcal foundaton s pesented, as so as an applcaton fo knows models of ack-gea type tools used on Maag teethng machnes. KEY WORDS: poflng tools, geneaton by ollng, CATIA, tajectoes method 1 INTRODUCTION The method of elatve geneatng tajectoes, as complementay method fo study of enwappng sufaces, by ollng method, was pesented (Teodo et al., 2015). A vaant of ths method was developed n the pape enttled The Rack-Gea Tool Geneaton Modellng. Non-Analytcal Method Developed n CATIA, Usng the Relatve Geneatng Tajectoes Method, pesented at IManEE Confeence that was held at Septembe 2016 n Chalkdk, Geece. Ths pape epesent the full veson of the pape and ncludes a CATIA developed applcaton, egadng the modellng of geneaton wth ack-gea tool fo an odeed cul of sufaces (pofles) assocated wth a pa of ccula centodes and also ncludes explanaton egadng the gaphcal method and algothm developed n gaphcal desgn envonment. The ssue pesented n ths pape was appoached n othes foms by Bacu (Bacu, 2002), usng the 3D modellng, as so as by Bebnsch and all. 2001, based on the mnmum dstance complementay theoem (Oancea, 2004). Also, the geneaton modellng s used n ode to dentfy the optmal cuttng scheme at geneaton wth hob mll of the nvolutes teeth by Dmtou and Antonads, Dunăea de Jos Unvesty of Galaţ, Faculty of Engneeng, Depatment of Manufactung Engneeng, Domnească no. 111 st., Galaţ, Romana E-mal: vgl.teodo@ugal.o; ncuso.baou@ugal.o; flon.susac@ugal.o; ncolae.oancea@ugal.o At the same tme, Teodo, 2010 developed an analytcal fom of the tajectoes method, appled also fo geneaton modellng. A vaant of the elatve geneatng tajectoes method s pesented n ths pape, as complementay method fo study of the enwappng sufaces assocated wth a pa of ollng centodes. Obvously, the complementay theoems ae altenatve ways fo expess the fundamental theoem of Wlls (Ltvn, 1984). The gaphcal method poposed n ths pape, appled fo modellng of sufaces geneated wth the ack-gea tool, s based on the elatve geneatng tajectoes method. Ths method detemnes the tajectoes of ponts belong to the tool s cuttng edge n the elatve moton egadng the blank. The gaphcal expesson s based on the specfc algothm, developed fo a tool wth pofle known n dscete fom, as an odeed cloud of ponts. Ths cloud of ponts s obtaned by measung of the geneatng tool. The gaphcal method s chaactesed by smplcty of applcaton because of the CATIA system capabltes. The method s vey goous and ntutve at the same tme. The obtaned esults of the pesented applcatons ae compaed wth esults obtaned by an analytcal method, n ode to pove the gaphcal method s qualty. 2 THE GENERATION MODELLING USING RACK-GEAR TOOL The ollng centodes, efeence systems and pofle of the ack-gea tool ae pesented n Fgue 1. They defne the efeence systems: xy s the global efeence system; 126 ACADEMIC JOURNAL OF MANUFACTURING ENGINEERING, VOL. 14, ISSUE 4 /2016

2 XY the efeence system joned wth the blank and the C 1 centode; elatve efeence system joned wth the C 2 centode. The geneatng ack-gea s pofle s joned wth the C 2 centode. Fgue 1. Centodes and efeence systems; the pofle of ack-gea The tool s pofle s defned n the efeence system, n a dscete fom, 1 1 S, 1 n. (1) n n Fo M M +1 elementay segment whch eplace the ack-gea tool, wth equatons: u cos ; u sn, (2) wth, tan ; 1 ; = +1. (3) Fo = 1 n, wth n sgnfcantly hgh, the pofle of the ack-gea tool can be descbed vey goous. The N nomal veso to the elementay pofle M M +1, N sn cos j (4) s defned. In ths way, the paametcal equatons of the nomal to the elementay pofle, n the pont M (), ae gven by N u cos sn ; u sn cos, (5) N wth as a scala paamete. The decton of veso s gven by (5). In the elatve moton between the ack-gea tool and the blank, R X 3 a; a, (6) R the nomal (6) descbes the famly: X u cos N sn R cos u sn cos R sn ; Y u cos sn R sn u sn cos R cos. (7) In equatons (7), u and ae vaable paametes. Fo = 0, the equatons (7) epesents the tajectoy famly of a pont fom the ack-gea s pofle, n the elatve moton egadng the blank. If the goal s to detemne the S pofle s enwappng, t s necessay that the nomals famly N to pass though the geang pole, P, fo vaous ollng postons (Ltvn, 1984). The dffeent ollng postons ae obtaned fo vaous values of the paamete. If the geang pole, P, s defned n the XY efeence system, XP R cos ; P Y R sn, P (8) then, fom (7) and (8) esults the equatons assembly: u cos R cos usn R sn sn R cos ; u cos R sn usn R cos cos R sn. (9) The vaable scala s elmnated between the two condtons (9) and the enwappng condton specfcally fo the elatve geneatng tajectoes s ACADEMIC JOURNAL OF MANUFACTURING ENGINEERING, VOL. 14, ISSUE 4 /

detemned. Ths condton establsh the blank s flank as envelopng fo the tajectoes famly fo ponts belong to the ack gea tool s flank, egadng the XY efeence system of the blank, u cos cos R sn.

3 detemned. Ths condton establsh the blank s flank as envelopng fo the tajectoes famly fo ponts belong to the ack gea tool s flank, egadng the XY efeence system of the blank, u cos cos R sn. (10) u sn sn R sn The equatons assembly: X u cos R cos usn R sn ; Y u cos R sn usn R cos, (11) and the condton (10), detemne the enwapped pofle the analytcal model of the blank s geneated suface. 3 APPLICATION The nvolute flank geneated wth a ack-gea tool s pesented n Fgue 2. In the same Fgue ae pesented the pa of ollng centodes and the efeence systems ae pesented. Fgue 2. Rack-gea; efeence systems The S pofle of the geneatng ack-gea was obtaned n dscete fom by measung wth a pofle pojecto, see Fgue 3. In Table 1, ae pesented the pont s coodnates measued on the ack-gea s pofle. The equatons assembly (11) and (10) detemnes the geneatng pofle fo the nput data: R = 50 mm, ollng ccle adus; m = 5 mm, modulus and α = 20. Fgue 3. The eal ack-gea flank s measung Table 1. Model N. ct. X [mm] Y [mm] In Fgue 4 depcts the geneated pofle of an nvolute tooth, the analytcal modelled pofle. A gaphcal soluton was developed n CATIA desgn envonment. Accodng to ths method, n DMU Knematcs module, the geneatng pocess was smulatng by ollng fo the tooth s flank, usng the ack-gea tool. The ack-gea-measued flank was modelled usng a splne cuve that has as contol ponts the ponts wth coodnates measued usng the pofle pojecto. In the same fle a nomal lne to the tool s pofle (to the splne cuve) was dawn whch was constaned to pass though one of the contol ponts. 128 ACADEMIC JOURNAL OF MANUFACTURING ENGINEERING, VOL. 14, ISSUE 4 /2016

4 pont between ths nomal and the modelled pofle wll be a pont fom the geneated nvolute pofle. Ths last step s epeated fo each of the measued ponts, obtanng ponts onto the geneated pofle. Because the devaton of the ack-gea tool s pofle exceeds the allowed lmts, we can expect that the devatons of the tooth s pofle exceed the allowed lmts too. In Fgue 5 ae pesented the lmts obtaned fo the gven nput data. Fgue 4. The modelled pofle Fgue 5. The geneated pofle By smulatng the mechansm functon, keepng as fxed element the ack-gea, the poston s found when the pevously dawn nomal lne pass though the geang pole. The ntesecton 4 GRAPHICAL METHOD DEVELOPED IN CATIA In ode to detemne the geneatng eos nduced by devatons fom the tool s theoetcal pofle, the method of nvese geneaton s appled. Accodng to ths method, t stats fom a known pofle of tool (detemned by measung) and detemnes the pofle of pece geneated by ths tool. Fo ths, the oles of pece and tool ae changed, analysng the specfc knematcs fo geneaton wth otay cutte. Fo gaphcal pupuses, n CATIA, a poduct fle whch wll contan an assembly composed s ceated thee elements and named RackGea. The fst element s tself a subassembly that contans two components on type pat. The fst pat s the pece s centode that, n ths case, s a ccle wth adus R, and XYZ efeence system s axs and ogn. The second pat wll contan the ponts detemned onto the pece s pofle and consequently, the pofle of pece. The pece s pofle wll be a splne that has as contol ponts the above-mentoned ponts. The second element s also a subassembly composed fom a pat wth s the tool s centode, n ths case a lne, as so as, the efeence system s axs and ogn. The second pat of ths element s epesented by the tool s pofle, dscetely knows, by measung. The pofle of tools s dawn as a splne that admts as contol ponts those measued on pofle pojecto. A cuent pont s consdeed as pont on cuve coespondng to each of the measued contol ponts. On ths pont a lne s dawn pependcula to the cuve that epesent the tool s pofle. The thd element, named GlobalRS, contans the elements of the global efeence system. Hee s defned the geang pole. The ole of ths element s ACADEMIC JOURNAL OF MANUFACTURING ENGINEERING, VOL. 14, ISSUE 4 /

to assue the elatve poston between pece and tool.

The psmatc jont s defned between the axs of the Toolbassembly poduct and a lne paallel to y axs of the GlobalRS poduct.

The evoluton jont s defned between the Z axs of the Pecebassembly poduct and the z axs of the GlobalRS poduct.

5 to assue the elatve poston between pece and tool. In ths assembly, usng the DMU Knematcs module of the CATIA envonment, a ack jont mechansm s defned (Fgue 6) The ack jont s composed by a psmatc jont and a evolute jont. The psmatc jont s defned between the axs of the Toolbassembly poduct and a lne paallel to y axs of the GlobalRS poduct. In addton, fo ths jont s defne the concdence between xy planes of the Toolbassembly and GlobalRS poducts. The evoluton jont s defned between the Z axs of the Pecebassembly poduct and the z axs of the GlobalRS poduct. The concdence between the xy plane of the Pecebassembly poduct and the smla plane of the GlobalRS poduct s mposed. Ths mechansm has as fxed element the Toolbassembly poduct and as dve element, the otaton angle of pece s efeence system aound the Z axs. Stll hee s detemned and kept the dstance measued fom the geang pole to the nomal that passes though the cuent pont, see Fgue 7.a. Modfyng the otaton angle and the step of ths angle, afte a few attempts, the concdence between the nomal and the geang pole s obtaned, see Fgue 7.b. When the dstance fom the geang pole to the nomal become lowe than a cetan value (establshed by use, usually mm) the smulaton s stopped and the mechansm poston s keeped. a). b). Fgue 7. Dstance between geang pole and nomal to the pofle: a). ntal poston; b). fnal poston 2). Wth the mechansm stopped n ths poston, the pece s pofle s actvated and the ntesecton pont s detemned between the nomal to the pofle and the tool s pofle (Fgue 8) Fg. 6. RackGea assembly In ode to ndentfy the contact ponts between pece and tool we wll apply the followng algothm: 1). The mechansm smulaton s made, usng the SIMULATION command and the poston whee the nomal to the pofle pass though the geang pole s dentfed. The dentfcaton s made by montong the dstance between the geang pole and the lne wth the decton pependcula to the pofle. Fgue 8. Intesecton pont between nomal and pofle; tempoay constants Fo a coect postonng, two tempoay constants ae appled, so the pont belongs at the 130 ACADEMIC JOURNAL OF MANUFACTURING ENGINEERING, VOL. 14, ISSUE 4 /2016

6 same tme to the nomal and to the pofle (pont-nomal constant: constant defnton concdence, pont-pofle constant: constant defnton concdence). Consequently, n ode to keep the pont s poston when the mechansm poston s changed, these two constants ae deleted and the pont s fxed usng the solate command. 3). The tool s pofle s actvated and the nomal s defned as passng though the next contol pont of the tool s pofle (pont detemned by measung wth the pofle pojecto). In ths way, the nomal s poston s changed accodng to a new poston onto the tool s pofle. The fst step s esumed fo the new poston of nomal and the algothm contnues wth steps 2 and 3, untl the numbe of ponts s detemned (enough to dawn the pece s pofle). The pece s pofle s dawn as a splne cuve that admts as contol ponts all the ponts pevously detemned. 5 CONCLUSIONS The elatve geneatng tajectoes method s a complementay method fo study of the enwappng sufaces. The pesented applcatons efes to conjugated pofles assocated wth a pa of ollng centodes the patcula case of the geneaton wth ack-gea egadng the geneaton statng fom the dscete fom of the tool s pofle. The gaphcal soluton developed n CATIA use the capabltes of ths desgn envonment. The analytcal esults and those obtaned wth the gaphcal algothm ae dentcal fom the techncal pont of vew. The gaphcal method s smple, ntutve and easy to apply and, n addton aloe to avod ough eos whch may be dffcult to detect usng only analytcal methods. The geneatng tajectoes may hghlght the poblem lnked wth the pofle s ntefeence at machnng. 6 ACKNOWLEDGMENTS Ths wok was suppoted by a gant of the Romanan Natonal Authoty fo Scentfc Reseach and Innovaton, CNCS-UEFISCDI, poject numbe PN-II-RU-TE REFERENCES Baou N., V. Teodo, Oancea N. (2015). A New Fom of In Plane Tajectoes Theoem. Geneaton wth Rotay Cuttes, Bulletn of the Polytechnc Insttute of Ias, Publshed by the Gheoghe Asach Techncal Insttute of Iaş, Tome LXI (LXV), Fasc. 3, pp , ICMS. Bacu I. (2002). Compaatve applcatons of the poflng tools by the method of the sold model, Romanan Jounal of Techncal Scences, Appled Mechancs, Tome 47, Romanan Academy Publshng House, pp , ISBN , ICMAS, Buchaest. Bebnsch S., Teodo V., Oancea N. (2001) A Gaphcal Method Developed n CATIA Desng Envoment fo the Modelng of Geneaton by Envelopng, The Annals of Dunăea de Jos, Unvesty of Galaţ, Fasc. V, vol. I, pp , ISSM Dmtou, V., Antonads, A. (2008) CAD-based smulaton of the hobbng pocess fo the manufactung of spu and helcal geas, Spnge-Velag, London Lmted, DOI: /s x, pp Oancea N. (2004) faces geneaton by enwappng, vol. I, II, Edtua Fundaţe Unvestae Dunaea de Jos dn Galaţ, ISBN Teodo V. (2010) Contbuton to the Elaboaton of a Method fo Poflng Tools. Tools whch Geneate by Enwappng, Lambet Academc Publshng, ISBN Ltvn F. L. (1984) Theoy of Geang. Refeence Publcaton, 1212, NASA, Scentfc and Techncal Infomaton Dvson, Washngton D.C. ACADEMIC JOURNAL OF MANUFACTURING ENGINEERING, VOL. 14, ISSUE 4 /

Journal of Machine Engineering, Vol. 15, No. 4, 2015

Journal of Machine Engineering, Vol. 15, No. 4, 2015 Jounal of Machine Engineeing, Vol. 15, No. 4, 2015 Received: 09 July 2015 / Accepted: 15 Octobe 2015 / Published online: 10 Novembe 2015 Vigil TEODOR 1* Vioel PAUNOIU 1 Silviu BERBINSCHI 2 Nicuşo BAROIU