A novel hob cutter design for the manufacture of spur-typed cutters

Transcription

1 journal of materials processing tecnology 29 (29) journal omepage: A novel ob cutter design for te manufacture of spur-typed cutters Jen-Kuei Hsie a, Huang-Ci Tseng b, Sinn-Liang Cang b, a Mecanical Engineering Department, National Ciao Tung University, EE57, Ta Hsue oad, Hsincu 3, Taiwan, OC b Institute of Mecanical and Electro-Mecanical Engineering, National Formosa University, Huwei Townsip, Yunlin County, Taiwan, OC article info abstract Article istory: eceived 23 November 27 eceived in revised form 2 February 28 Accepted 24 February 28 Keywords: Hob cutter Spur-typed cutter Undercutting Spur-typed cutters wit multiple cutting angles are important tools frequently used in te manufacture of many macine elements. Due to teir complex geometry, te production of tese cutters involves milling and several grinding processes. Suc a complex manufacturing process, coupled wit te need for expensive manufacturing tools, makes te cutters costly. Undercutting is a penomenon tat causes weakness at te root of a gear. Engineers use sifted gears, modified toot profiles, or canges in te pressure angle to overcome undercutting in gears. However, in tis paper, by utilizing te undercutting penomenon, a novel design of straigt-sided ob cutter wit multiple pressure angles is proposed for te manufacture of spur-typed cutters. Wit te simultaneous application of multiple pressure angles, tis tool design concept significantly simplifies te manufacturing process. Te effects of cutting angles, te degree of undercutting, and te widt of te top land of te cutter are studied. Te concepts and results proposed in tis paper are beneficial as design guidance for tool designers and manufacturers. 28 Elsevier B.V. All rigts reserved.. Introduction Spur-typed cutter, wic is formed wit multiple cutting angles, is one of te most frequently used tools in te manufacture of macine components. Te production of tese cutters involves milling, roug grinding, and finis grinding. Due to te complex geometry and manufacturing process, various expensive manufacturing tools must be employed, and ence te manufacture of te cutters can be very costly. Te obbing of gears is te most effective manufacturing process found in te gear industry. New suggestions and metods to improve te precision and efficiency of obbing ave been introduced by researcers. Cluff (987) investi- gated ow te generating accuracy of ob cutter was affected by cutter geometric peculiarities and resarpening errors. adakrisnan et al. (982) proposed a metod to obtain te grinding weel profile of te twist drill flute in resarpening. Ainoura and Nagano (987) investigated te conventional obbing using a ob wit its elix running in te direction opposite te gear, and tey found it more effective for te ig-speed manufacture of comparatively small module gears for automobiles. In Koelsc s researc (994), obbing cutters wit different coatings are tested in ig-speed cutting, and cermets were found to posses te best performance in igspeed dry cutting. More specifically, Pillips (994) indicated tat ob cutter coated by titanium nitride made productivity Corresponding autor at: oom 529, Building 2, Te New Campus, National Formosa University, No. 64, Wunua oad, Huwei Townsip, Yunlin County 632, Taiwan, OC. Tel.: ; fax: address: cangsl@nfu.edu.tw (S.-L. Cang) /$ see front matter 28 Elsevier B.V. All rigts reserved. doi:.6/j.jmatprotec

2 848 journal of materials processing tecnology 29 (29) realized. Bouzakis and Antonidais (995) proposed a computational procedure, wic enables te determination of optimum values for te sift displacement and for te corresponding sift amount. In teory, te ob cutter may be considered to be a worm tat is slotted in te axial direction to form a series of cutting blades. In Litvin s publication (989), te axial section of te worm was considered as te rack. Most of te literature uses rack cutters to simulate te generating process of a ob cutter. Tsay (988) investigated elical gears wit involute saped teet, wose matematical description was derived by straigt-sided rack cutter. Cang et al. (997a) proposed a general matematical model of gear generated by CNC obbing macine. ecent researc, relevant to te obbing process, includes te paper of Cang et al. (996), in wic te manufacture of elliptical gears was studied. Cang (996) simulated te obbing process troug a computer numerically controlled (CNC) obbing macine. Cang et al. (997b) acieved design optimization by tuning parameters of modified elical gear train. More recently, Kapelevic (2) investigated te obbing of an unsymmetrical involute toot profile. Cang et al. (22) patented a new ob cutter profile tat generated a elical cutting tool. Te matematical model of tis elical cutting tool is presented in te paper written by Liu and Cang (23). However, te above-publised work presented little systematic and in-dept discussion on te design and generation of te cutting tool toot profile. In particular, tere were no discussions of te design of a ob cutter capable of generating a spur-typed cutter wit multi-cutting angles. Terefore, te possibilities for modifying existing designs are limited. Undercutting is a penomenon tat causes weakness at te root of a gear. Various metods ave been used to overcome tis problem. In tis paper, a novel design for a straigt-sided ob cutter wit multiple pressure angles is proposed, and te undercutting penomenon is used to facilitate te manufacture spur-typed cutters. Te proposed ob cutter can generate te multiple cutting angles simultaneously, wic significantly reduces time and cost required in te traditional process. Te multiple pressure angles of ob cutter include a large pressure angle tat generates te involute surface of te spur-typed cutter, wic acts as te main body. A smaller pressure angle of, possibly, less tan 5, undercuts te spur-typed cutter and forms te radial rake angle, wic is a special application of undercutting in gear geometry. A tird pressure angle, wic is te largest one in te ob cutter, generates te relief and clearance angles. Caracteristics of te spur-typed cutter, including cutting angles, top land widt, and full undercutting, are all studied. Te main teme of tis paper is te concept of using multiple pressure angles to generate te complex multiple cutting angles. Te developed matematical model and te conducted analyses contribute a lot bot in designing and manufacturing te spur-typed cutters. 2. A novel design for a ob cutter A ob cutter wit straigt-sided cutting face is commonly used in te manufacture of involute gear. An improper param- Fig. Normal section of ob cutter. eter design of te cutter will cause te undercutting at te roots of te gear. However, by appropriately designing te parameters of ob cutter, te undercutting can be controlled and become beneficial. Fig. sows te normal section of te novel type ob cutter, wic is also considered as te profile of a rack cutter. Te cutting face of Fig. can be divided into six regions, i.e. left cutting face I, rigt cutting face II, fillet cutting faces III and IV, top land cutting face V, and camfering cutting face VI. Te profile is almost similar to an ordinary ob cutter for involute gears, except for te two large pressure angles, i.e. regions I and VI, and a small pressure angle, i.e., region II. Te cutting face II, wit its small pressure angle, generates te cutting face and radial rake angle of te cutter by undercutting. Te cutting face I, wit its large pressure angle, generates te main body of te cutter. Te cutting face VI, wit te largest pressure angle, generates te clearance and relief angles of te cutter. Te origin of te coordinate system S a (X a, Y a, Z a )is located at te middle of te rack cutter body. Te positive X a axis is set upwards; positive Y a is directed to te rigt, wile te Z a axis can be determined by te rigt-and rule. In Fig., represents te pitc line of te rack cutter, L is te pressure angle of te left straigt-sided cutting face, is te pressure angle of te rigt straigt-sided cutting face, and 3 is te largest pressure angle cutting face VI. HKW is te addendum of te rack cutter and HFW is te dedendum of te rack cutter, wile 2b and P represent te toot tickness and pitc of te rack cutter, respectively. In tis paper, te teory of gearing proposed by Litvin (989), wic considers te locus equation and mesing equation simultaneously, is used to generate te toot profile of te cutter. 2.. Equations for te rack cutter Te equations for te six regions of te cutting face sown in te S a coordinate system can be represented as follows: () Left cutting face I: Te coordinates of te origin point e I of te cutting face I in Fig. can be sown as x (e I) HKW sin L ()

3 journal of materials processing tecnology 29 (29) y (e I) a = b tan(45 + L /2) HKW tan L + cos L (2) Wen parameter l (I) indicates te position on te cutting face I, te equation of tis cutting face can tus be represented in te S a coordinate system as te following equation: r (I) HKW sin L + l (I) cos L b tan(45 + L /2) HKW tan L + cos L + l (I) sin L Te unit normal to tis cutting face is obtained as n (I) sin L i a cos L j a (4) (2) igt cutting face II: Te coordinates of te origin point e II of te cutting face II in Fig. can be sown as x (e II) HKW sin (5) y (e II) b + tan(45 + /2) + HKW tan cos (6) Wen parameter l (II) indicates te position on te cutting face II, te equation of tis cutting face can tus be represented in te S a coordinate system as HKW sin + l (II) cos b + tan(45 + r (II) /2) +HKW tan cos l (II) sin (7) Te unit normal to tis cutting face is obtained as n (II) sin i a cos j a (8) (3) Fillet cutting face III: In Fig., parameter is te radius of fillet. Te geometry sows tat C I = HWK tan L and b I = /tan(45 + L /2). Coordinates of te origin O I of te fillet cutting face III can be sown as x (O I) HKW (9) y (O I) a = b tan(45 + L /2) HKW tan L () Parameter I indicates te position on te fillet cutting face III, wic defines te cutting face position wit te X a axis for I (9 L ). Te equation of tis cutting face can tus be represented in te S a coordinate system as te (3) following equation: HKW cos I ) r (III) b tan(45 + L /2) HKW tan L + sin I () Te unit normal to tis cutting face is obtained as n (III) cos I i a sin I j a (2) (4) Fillet cutting face IV: If parameter defines te radius of te fillet, Fig. sows tat C II = HKW tan and b II = /tan(45 + /2). Coordinates of te origin O II of te fillet cutting face IV can be sown as x (O II) HKW (3) y (O II) b + tan(45 + /2) + HKW tan (4) Parameter II indicates te position on te fillet cutting face III, wic defines te cutting face position wit te X a axis for II (9 ). Te equation of tis cutting face can tus be represented in te S a coordinate system as HKW cos II r (IV) b + tan(45 + /2) + HKW tan sin II Te unit normal of tis cutting face is obtained as (5) n (IV) cos II i a sin II j a (6) (5) Top land cutting face V: Te top land cutting face V is formed by two tangential points d II and d I. Te coordinates of point d II in te S a coordinate system are x (d II) HKW (7) y (d II) b + tan(45 + /2) + HKW tan (8) And te coordinates of point d I in te S a coordinate system are x (d I) HKW (9) y (d I) a = b tan(45 + L /2) HKW tan L (2) Parameter indicates te position of point d II along te Y a direction and (Y d I a Yd II a ). Te equation of tis cutting face can tus be represented in te S a coordinate

4 85 journal of materials processing tecnology 29 (29) Fig. 2 Coordinate system relationsip of rack cutter and generated gear. system as te following equation: HKW r (V) b + tan(45 + /2) + HKW tan + Te unit normal of tis cutting face is obtained as (2) n (V) i a (22) (6) Camfering cutting face VI: Te coordinates of te starting point q on tis cutting face in Fig. can be sown as x (q) HFW e (23) y (q) b + (HFW e) tan L (24) Parameter l (VI) indicates te position on te cutting face VI. Te equation of tis cutting face can tus be represented in te S a coordinate system as te following equation: HFW e + l (VI) cos 3 r (VI) b + (HFW e) tan L + l (VI) sin 3 (25) Te unit normal to tis cutting face is obtained as n (VI) sin 3 i a cos 3 j a (26) 2.2. Locus equations of rack cutter Wen a rack cutter is used to cut gears, te coordinate systems relationsip can be represented as in Fig. 2. In tis figure, te teoretical pitc surface of te rack cutter is tangent to te pitc surface of te spur-typed cutter to be cut wen te sifted distance C =.. As te teoretical pitc surface of te rack cutter, i.e., te Y a Z a plane, moves toward te spurtyped cutter, C becomes negative. Te matematical model proposed ere can ten be used to simulate different spurtyped cutters wit different sifted distances manufactured by te same rack cutter or ob cutter. In te manufacturing of a spur-typed cutter, te active pitc surface Y d Z d translates towards te left wile te generated spur-typed cutter rotates in a counter-clockwise direction. Te locus equation of te rack cutter represented in te S coordinate system and attaced to te spur-typed cutter represents te cutting process. Te generated profile can tus be obtained by solving te locus equation and te mesing equation [(6) and (6)] simultaneously. Te mesing equation correlates te surface parameter of te cutting face to te motion parameter of te generation process. () Locus equation of left cutting face I: Te locus equation of te left cutting face I sown in te generated cutter coordinate system S can be obtained by transforming te cutting face equation from S a to S. Te transformation matrix and locus equation are as follow: cos sin r sin + (r c) cos sin cos r cos + (r c) sin [M a ]= (27) r (I) = [M a] r (I) a (28) (2) Locus equation of rigt cutting face II: r (II) = [M a ] r (II) a (29) (3) Locus equation of fillet cutting face III: r (III) = [M a ] r (III) a (3) (4) Locus equation of fillet cutting face IV: r (IV) = [M a ] r (IV) a (3) (5) Locus equation of top land cutting face V: r (V) = [M a ] r (V) a (32) (6) Locus equation of camfering cutting face VI: r (VI) = [M a ] r (VI) a (33) 3. Profile of spur-typed cutter Te matematical model of te generated toot profile can be obtained by considering te locus equation and te mesing equation [(6) and (6)] simultaneously. Te loci equations are sown in te previous section wile te mesing equation is represented by te following equation: n (n) a V (2) = (34) were n (n) a is common unit vector normal to te two contact surfaces; V (2) is te relative velocity between tese two contact surfaces.

5 journal of materials processing tecnology 29 (29) Te mesing equation means tat te common normal is perpendicular to te relative velocity at te contact point. It is independent of te selected coordinate system. For convenience, te common unit normal and relative velocity will be represented in te S coordinate system in tis paper. From Fig. 2, te common normals of te cutting faces represented in te S a coordinate system are te identical to tose represented in te S coordinate system, i.e., n n. Te velocity of te rack cutter is terefore V (F) = r ω j (35) Te velocity of te contact point of te generated cutter is V () = O ao + r a (36) were O a O = ( r + c)i + r j (37) Te relative velocity is obtained and expressed as = V (F) V() (38) () Te relative velocity between te cutting face I and te generated cutter is as follows: = ω [b tan(45 + L /2) HKW tan L ] + cos L + l (I) sin L r i + ω ( HKW sin L + l (I) cos L c)j (39) Substituting te common unit normal equation (4) and te relative velocity equation (39) into Eq. (34), te mesing l (I) = ( HKW c) cos L [b cot(45 + L /2) HKW tan L r ] sin L (4) Solving tis mesing equation (4) and te locus equation (28) simultaneously, te generated toot profile by cutting face I can tus be obtained. (2) Te relative velocity between te cutting face II and generated cutter is as follows: = ω [( b + tan(45 + /2) + HKW tan cos l (II) sin ) r ] i + ω ( HKW sin + l (II) cos c)j (4) Substituting te common unit normal equation (8) and relative velocity equation (4) into Eq. (34), te mesing l (II) = ( + HKW + c) cos + [ b + cot(45 + /2) +HKW tan r ] sin (42) Solving tis mesing equation (42) and te locus equation (29) simultaneously, te generated toot profile by cutting face II can tus be obtained. (3) Te relative velocity between te cutting face III and te generated cutter is as follows: = ω [b tan(45 + L /2) HKW tan L + sin I r ]i + ω ( HKW cos I c)j (43) Substituting te common unit normal equation (2) and te relative velocity equation (43) into Eq. (34), te mesing { } I = tan b + cot(45 + L /2) + HKW tan L + r HKW c (44) Solving tis mesing equation (44) and te locus equation (3) simultaneously, te generated toot profile by cutting face III can tus be obtained. (4) Te relative velocity between te cutting face IV and generated cutter is as follows: = ω [ b + tan(45 + /2) + HKW tan ] sin II r i + ω ( HKW cos II c)j (45) Substituting te common unit normal equation (6) and relative velocity equation (45) into Eq. (34), te mesing { } II = tan b + cot(45 + /2) + HKW tan r ] HKW c (46) Solving tis mesing equation (46) and te locus equation (3) simultaneously, te generated toot profile by te cutting face IV can tus be obtained. (5) Te relative velocity between te cutting face V and te generated cutter is as follows: ] = ω [ b + tan(45+ /2) + HKW tan + r i + ω ( HKW c)j (47) Substituting te common unit normal equation (22) and relative velocity equation (47) into Eq. (34), te mesing [ ( ) ] = b cot 45 + HKW tan + r 2 (48) Solving tis mesing equation (48) and te locus equation (32) simultaneously, te generated toot profile by cutting face V can tus be obtained.

6 852 journal of materials processing tecnology 29 (29) Table Parameters of rack cutter and spur-typed cutter Parameters of sifted spur-typed cutter Circular pit (cp, mm) 2.8 Module (m, mm).8926 Number of teet (T) 2 Heigt of camfering (e, mm).3 Wole dept (HKW + HFW,, mm).77 Lengt of cutter (mm) 3 Pitc diameter (mm) Sifted distance (c, mm).2..2 Outside diameter (D, mm) oot diameter (d, mm) Parameters of rack cutter Addendum (HKW, mm).422 Dedendum (HFW, mm).348 Toot tickness (2b, mm).9 Tip radius (r, mm).5 Pressure angle of face I ( L, ) 48 Pressure angle of face II (, ) 3 Pressure angle of face VI ( 3, ) 57 Fig. 4 Toot profile and generation simulation of spur-typed cutter (c =.2). (6) Te relative velocity between te cutting face VI and generated cutter is as follows: = ω [b + (HFW e) tan L + l (VI) sin 3 r ]i +ω (HFW e + l (VI) cos 3 c)j (49) Te same procedure of substituting te common unit normal equation (26) and relative velocity equation (49) into Eq. (34) gives a mesing equation of l (VI) = (HFW e c) cos 3 [b + (HFW e) tan L r ] sin 3 (5) Solving tis mesing equation (5) and te locus equation (32) simultaneously, te generated toot profile by cutting face VI can tus be obtained. Example. A spur-typed cutter as a circular pitc cp = 2.8 mm, wit 2 flutes, an outside diameter of.39 mm, and a root diameter of 7.85 mm. Te relevant parameters are sown in Table. Te profile of te rack cutter is first designed as sown in Fig. 3. Using te matematical model developed, te generated spur-typed cutter is sown in Fig. 4. Fig. 4 also sows tat te developed matematical model matces te simulation of te generated rack cutter. Te result proves tat Fig. 5 Generated toot profile wit different sift amount. te novel design of using te proposed rack cutter (ob cutter) is an effective and efficient way of manufacturing a spur-typed cutter. Fig. 5 sows te transverse section of te spur-typed cutter. Tis figure reveals tat te same rack cutter wit a different sift can produce different spur-typed cutters. Fig. 6 sows te transverse section of a spur-typed cutter wit te same outside diameters. 4. Caracteristics of te spur-typed cutter In te preceding section, te matematical model of te generated spur-typed cutter as been derived. However, te cutting angles and te widt of te top land of te cutter will also significantly affect te cutting performance. In tis section, tese important factors are studied. 4.. Full undercutting on te spur-typed cutter Fig. 3 Profile of te novel-design rack cutter. As previously mentioned, te rigt cutting face of te proposed spur-typed cutter is generated by undercutting te circular edge of te rack cutter. A poorly designed rack cutter will generate part of an involute curve on a typical cutting edge. Wen te cutting edge is fully undercut, te starting point of undercutting lies beyond te circle of outside

7 journal of materials processing tecnology 29 (29) Fig. 6 Generated toot profile wit different sift amount (outside diameter is constant). diameter. Hence, te x- and y-components of te equation depicting te rigt side of te cutting face in Eqs. (29) and (42), respectively, are te same as te x- and y-components of te equation depicting te undercut portion sown in Eqs. (3) and (46). Example 2. In Example, wen te sifted distance c =., te radius of te intersection point is r P = 5.725, wic is larger tan te radius of te outside diameter, r = Tis confirms tat te design of te rack cutter as satisfied te requirement of a spur-typed cutter Determination of te cutting angles Te cutting angles of te end section profile sown in Fig. 7 affect te cutting performance significantly. In tis section, te cutting angles of a spur-typed cutter manufactured by te novel ob cutter are investigated. Point A is te intersection point of te undercut portion (region IV) and te curve on te top land of te spur-typed cutter, point B is te intersection point of te camfered angle edge and te top land curve, and point E is te intersection point of te left cutting edge and te camfered angle cutting edge. T A and T a are te tangential and positional vectors of point A in te S coordinate system, respectively. T B is te tangential vector of point B in Fig. 8 Te relief angle at point B. te S coordinate system, and T B is te tangential vector of te camfered angle cutting edge at point B in te S coordinate system. Te radial rake angle A is te angle between T A and T A, te relief angle B is te angle between T B and T B, and te clearance angle E is te angle between T B and T E Analysis of te radial rake angle A Since point A is te intersection point of te undercut portion (region IV) and te curve on te top land of te spur-typed cutter, te following equation is establised: xa 2 + y2 A = r + HFW (5) were (x A, y A ) are te coordinates of point A and r is te pitc radius of te spur-typed cutter. Substituting te point A Eqs. (3) and (46) into te above equation, parameter is obtained. Te positional vector T A and tangential vector T A of point A can tus be obtained. Vector dot production is ten performed to attain A between tese two vectors Analysis of te relief angle B From te involute curve property (7), te involute curve is extended from te base circle. In Fig. 8, te normal vector of te involute curve at point B is tangent to te base circle. Te directional vector OG is tus te same as te tangential vector T B at point B. Te pressure angle at point B, i.e., B is tus obtained as follows: r B cos B = r b (52) were r B is te position vector of point B and r b is te radius of te base circle. As vectors T B and T B are perpendicular to BO and BG, respectively, te relief angle B is obtained as B = 2 B (53) Fig. 7 Definitions of te cutting angles Analysis of te clearance angle E In Fig. 9, T E is te tangential vector at point E wile T E is te vector perpendicular to te position vector r E. Applying te teory of involutometry, te clearance angle E is obtained as

8 854 journal of materials processing tecnology 29 (29) Fig. 9 elationsip between clearance angles E and at point E. Fig. Te cutting angles wen te outside diameters are kept constant Widt of top land of spur-typed cutter Te widt of te top land of te spur-typed cutter can be obtained by solving for te coordinates of points A and B sown in Fig. 2. Te coordinates of point A were solved in te previous section. If te polar angle of point B is solved, te x-component of point B is B x = r B cos and te y-component is B y = r B sin. From te geometric relationsip sown in Fig. 2, can be expressed as = KOE + EOB (56) were Fig. Te cutting angles wen outside diameters are canged according to te sifted amount. EOB = inv B inv E (57) Due to pure rolling between te pitc line of te rack cutter and te pitc circles of te generated gears, te arc lengt of NJ sown in Fig. 2 is equal to te corresponding distance l follows: E = 2 + E (54) were = inv B inv E (55) Effect of sifted distance on te cutting angles By substituting te values from Table, te various cutting angles are obtained and sown in Fig. were te outside diameters are canged according to te sifted distance. If te outside diameters are kept constant, te cutting angles are as sown in Fig.. Fig. 2 Pressure angles at different points.

9 journal of materials processing tecnology 29 (29) Fig. 3 olling position of te largest pressure angle for te rack cutter. teory of differential geometry, and te teory of gearing, te matematical models of te spur-typed cutter profile, including te radial cutting angle, relief angle, and clearance angle ave been derived. A computer simulation involving a parametric study of te tree angles was also carried out. Te major caracteristics of te generated spur-typed cutter were also studied in tis paper. Te proposed metod can be used as design guidance in designing novel ob-type cutters to generate spur-typed cutters. Moreover, it is expected tat te results from tis paper will contribute to te improvement of te manufacture of plain mill cutters and provide te tool industry wit a reference for designing and macining similar tools. Te results can also act as a basis for researcers to optimise and improve teir tool designs. Acknowledgments Te work outlined in tis paper was supported by te national science council under grants NSC9-222-E-5-22 and NSC E references Fig. 4 Widt of top land of spur-typed cutter wit constant outside diameter, and variable outside diameter according to te sifted amount, respectively. sown in Fig. 3. Te angle ı = NOJ is terefore ı = l r (58) were r is te radius of te pitc circle of te generated spurtyped cutter. Terefore, angle is obtained as = inv B inv J + ı (59) Fig. 4 sows te widt of te top land of te spur-typed cutter wen te outside diameters are kept constant, and wen te outside diameters are canged according to te sifted distance, respectively. 5. Conclusion A novel design for a ob cutter capable of generating a spurtyped cutter in one obbing process as been proposed. Te cutting edge in te normal direction as been designed as tree straigt lines wit different pressure angles and two arcs. By applying te equations of te rack profiles of te cutting edges, te principle of coordinate transformation, te Ainoura, M., Nagano, K., 987. Te effect of reverse obbing at a ig speed. Gear Tecnol. 4 (Marc/April (2)), 8 5. Bouzakis, K.D., Antonidais, A., 995. Optimizing of tangential tool sift in gear obbing. Ann. CIP 44 (), Cang, S.L., 996. Gear obbing simulation of CNC gear obbing macines. Dissertation for Doctoral Degree. National Caio Tung University, Hsincu, Taiwan, OC. Cang, S.L., Liu, J.Y., Hsie, L.C., 22. Design of ob cutters for generating elical cutting tools wit multi-cutting angles. Cinese Patent Cang, S.L., Tsay, C.B., Nagata, S., 997a. A general matematical model for gear generated by CNC obbing macine. Trans. ASME J. Mec. Design 9, 8 3. Cang, S.L., Tsay, C.B., Tseng, C.H., 997b. Kinematic optimization of a modified elical simple gear train. Trans. ASME J. Mec. Design 9, Cang, S.L., Tsay, C.B., Wu, L.I., 996. Matematical model and undercutting analysis of elliptical gears generated by rack cutters. Mec. Mac. Teory 3 (7), Cluff, B.W., 987. Effects of ob quality and resarpening errors on generating accuracy. Gear Tecnol. 4 (September/October (5)), Kapelevic, A., 2. Geometry and design of involute spur gears wit asymmetric teet. Mec. Mac. Teory 35, 7 3. Koelsc, J.., 994. Hobs in ig gear. Manuf. Eng. (July), Liu, J.Y., Cang, S.L., 23. Design of ob cutters for generating elical cutting tools wit multi-cutting angles. Int. J. Mac. Tools Manuf. 43 (2), Litvin, F.L., 989. Teory of Gearing. NASA Publication, Wasington, DC. Pillips,., 994. New innovations in obbing. Part I. Gear Tecnol. (September/October (5)), 6 2. adakrisnan, T., Kawlra,.K., Wu, S.M., 982. A matematical model of te grinding weel profile required for a specific twist drill flute. Int. J. Mac. Tool Design es. 22, Tsay, C.B., 988. Helical gears wit involute saped teet: geometry, computer simulation, toot contact analysis, and stress analysis. Trans. ASME J. Mec. Transm. Autom. Design,