Mathematical models for manufacturing a novel gear shaper cutter

Transcription

1 Journal of Mechancal Scence and Technology 24 (2010) 383~390 DOI /s z Mathematcal models for manufacturng a novel gear shaper cutter Shen-Wang Ln 1, Cheng-Shun Han 2,3,, Ju-Bn Tan 4 and Shen Dong 3 1 Department of Industral Engneerng & Management, Far East Unversty of Scence and Technology,Tanan,Tawan 2 Post-doctor Staton of Instrumentaton scence and Technology, Harbn Insttute of Technology, Harbn, Chna 3 School of Mechatroncs Engneerng, Harbn Insttute of Technology, Harbn, Chna 4 Insttute of Ultra-precson Optoelectrc Instrument Engneerng, Harbn Insttute of Technology, Harbn, Chna (Manuscrpt Receved December 23, 2008; Revsed June 23, 2009; Accepted July 13, 2009) Abstract The desgn prncple and models for novel error-free shaper cutters are dscussed to mprove the accuracy and servce lfe of gear cuttng tools. The modfed methods for desgnng the tooth profle of the shaper cutter are developed by ncludng the nverse envelope method, mdpont-mdlne method and mdpont-dsplacement method based on the theory of geometrcal reverse problem. The unversal mathematcal models for manufacturng optmal tooth profle of shaper cutters are presented. The feasblty and relablty of the proposed prncple, methods and models are proved by combnng the numercal examples wth practcal applcaton. The desgn method suggested n ths paper s superor to the tradtonal desgn scheme, and wll be helpful n facltatng the desgn and manufacture of shaper cutters. Keywords: Mathematcal model; Modfed tooth profle; Profle error; Shaper cutter Introducton Gear shapng s wdely used to produce varous gears, such as helcal gears, external gears and nternal gears for dstnct features of the manufacturng process. The tradtonal desgn scheme on ordnary gear shaper cutters s smlar, and a detaled dscusson s presented n [1]. Snce accuracy and servce lfe of the cuttng tool drectly affect the manufacturng cost and productvty n gear producton, wth the ncreasng demands for hgh productvty and low manufacturng cost of gear producton, the elevaton of manufacturng precson and servce lfe of gear shaper cutters have attracted consderable nterest. Jannnck[2, 3] presented the ntegrated desgn method of shaper cutters for varous applcatons. Rogers et al. presented an offset based method to obtan the tooth specfcatons requred for the shapng processes and the strength of the meshng tooth par[4]. Km and Km analyzed the basc theory of pnon cutter shape and developed computer software to desgn a pnon cutter[5]. Other researchers analyzed the tooth profle undercuttng condtons of generatng gears by shaper cutters, tooth-profle shftng and gear clearance[6-9]. Tsay et al proposed a complete geometrcal mathematcal model of a Ths paper was recommended for publcaton n revsed form by Assocate Edtor Dae-Eun Km Correspondng author. Tel.: , Fax.: E-mal address: hancs@ht.edu.cn KSME & Sprnger 2010 spur shaper cutter, ncludng the protuberance, the nvolute regon and sem-toppng n manufacturng standard or nonstandard spur gears[10]. Bar nvestgated not only the profles of the shaper cutter whle consderng protuberance and semtoppng, but also the relatonshp between the shaper cutter parameters and the gear tooth profle for the manufacture of helcal gears wth small numbers of teeth[11]. On the other hand, mprovement of tool materals propertes was used for ncreasng servce lfe of tools[12-14]. Some researchers studed the nfluence of cuttng edge form and cuttng speeds on tool wear resstance[15]. Km developed the on-lne tool-lfe montorng system usng acoustc emsson sgnals n gear shapng[16]. In most of these shaper cutter related studes, tooth profle modfcatons are frequently appled on desgn and manufacture of shaper cutters for gear accuracy and qualty, and materals or cuttng condton are nvestgated on tool lfe. However, lttle research work related to the accuracy desgn wth tool servce lfe smultaneously. It s well known that longer qualfed tooth length can mprove the servce lfe of a shaper cutter and lower the gear producton cost. Tool lfe s not only determned by the materals and cuttng condton, but also has close relatonshp wth the qualfed tooth length. A cuttng tool can carry on cuttng work by repeated sharpenng tll t s abandoned. In each sharpenng, the volume of cutter body s reduced, and a new cuttng edge sharpened may have formatve error. After the last, oversze error or other reasons cause the cutter be aban-

2 384 S.-W. Ln et al. / Journal of Mechancal Scence and Technology 24 (2010) 383~390 doned. The qualfed tooth length s the length from the cuttng edge of a new shaper cutter(called frontal edge for short) to the cuttng edge of the fnal shaper cutter that cannot be used after repeated sharpenng(called rear edge for short). The longer qualfed tooth length that can be resharpened more tmes makes the longer tool lfe. The desgn accuracy, especally the flank surface of shaper cutter, s benefcal to extend qualfed tooth length. However, tradtonal desgn approaches for shaper cutter, for example, the nvolute shaper cutter wth module 8, the number of teeth 13, wthn the tooth length 6mm, the error of cuttng edge projecton on the end surface to the theoretcal nvolute reaches mm[17]. The desgn error s also large for a non-nvolute shaper cutter, such as epcyclodal shaper cutter, lke the numercal example n [18]; even though the allowable error of tooth top s mm, and the allowable error of the flank s mm, the usable tooth length s only 0.52mm. Therefore, n gear shapng related desgn, t s mportant to take both accurate tooth profle and longer qualfed tooth length nto consderaton, whch can reduce producton cost and ncrease productvty. The authors have tred to establsh new desgn methods for novel error-free shaper cutters wth longer qualfed tooth length; the desgn prncple and general mathematcal models are dscussed. In ths paper, desgn prncple of contnuous functons and correspondng mathematcal models for errorfree novel shaper cutter are ntroduced frstly. Then based on the gear generaton mechansm and the theory of gearng, the reference cross secton of the grndng wheel s evaluated by the reversal problem method--reversal envelope method. The absolute value of the tooth profle error of every pont on the lne from the top to the root s less than 10-4 mm, the lne s the mdlne of the qualfed tooth length gven, so that the reference cross secton of the grndng wheel can be modfed by mdpont-mdlne method. Accordng to the error dstrbuton requrement on tooth surface wthn qualfed tooth length, the mdpont of contact lne between the grndng wheel and the qualfed tooth surface of theoretcal shaper cutter should devate a lttle n order to make the absolute value of errors on the frontal edge and on the rear edge approxmately equal; thus the grndng wheel profle s obtaned fnally by mdpont-dsplacement method. Ths paper presents the unversal mathematcal models of a new type of desgn and manufacturng method for dfferent knds of tooth profles of shaper cutter; the feasblty and relablty of the prncple, methods and models are verfed by numercal examples and vrtual realty. 2. Desgn prncple and mathematcal models for errorfree novel shaper cutter Snce the cuttng edge clearance of the shaper cutter, tp dameter and tooth thckness of the shaper cutter decrease gradually from front to back, ths causes dfferent modfcaton coeffcent on dfferent secton. Takng nvolute gears as an example, the profle of the shaper cutter s also nvolute. (a) General shaper cutter structure (b) Rake angle γ and rear angle α e of shaper cutter Fg. 1. Dagram of shaper cutter. Hence rear faces should also be nvolute helcod. When rake angle γ = 0, the tooth profle of the rake face s nvolute, so there s no desgn error. But n general case, γ 0, the rake face becomes a cone, the cuttng edge s a space curve that s ntersected by the rake face and the rear face, and ts projecton on the end secton s no longer nvolute. Ths causes tooth profle error, and the error ncreases wth absolute value of rake angle and the rear angle becomes larger. Many researchers have studed gear desgn and manufacturng wth tooth profle shftng. But there stll exst tooth undercuttng at the generated gear tooth surfaces, under certan condtons, such as small number of teeth, large gear module, bg modfcaton coeffcent and large absolute value of rake angle γ and rear angle α e. Especally for hard and brttle characterstcs of cemented carbde materals, the rake angle γ of carbde shaper cutter must be negatve, of whch the absolute value should be greater to prevent tppng. Such shaper cutter causes generated gear flank-concave tooth surfaces. Fg. 1(a) and (b) show dagrams of structure and tool angles(rake angle γ and rear angle α e ) of shaper cutter, respectvely. Fg. 1(b) specally takes carbde shaper cutter wth negatve rake angle as an example n whch z-axs s the axs of shaper cutter and x-axs perpendcular to z-axs. The new desgn method tres to guarantee projecton of every resharpened cuttng edge on the end surface concde wth theoretcal curve, namely error-free desgn. Consderng the meshng relaton between the blade curve s projecton on the end of the shaper cutter and the gear tooth profle, tradtonal desgn for the shaper cutter takes the ntersectant curve as the blade curve, whch the cylnder generated by the conjugate tooth profle of the gear movng around the axs of the gear ntersects the rake face of the shaper cutter.

3 S.-W. Ln et al. / Journal of Mechancal Scence and Technology 24 (2010) 383~ Ths leads to larger desgn error. To solve ths problem, on the base of gear theory and gear generatng mechansm, the developed desgn prncple s that the conjugate curve can be calculated from the conjugate tooth profle of the gear wth modfcaton coeffcent ξ1, and that s also the blade curve s projecton on end surface of the shaper cutter. For there s rear angle α e, decrease of tp dameter and tooth thckness of the shaper cutter from front to back causes dfferent modfcaton coeffcent ξ on dfferent secton. All resharpened cuttng edges form the flank surface of the sde edge of the shaper cutter farther. So whle ξ vares contnuously, the flank surface of the sde edge wll be more accurate than that ntroduced n [1]. In gear shapng, the velocty of cuttng moton s larger than that of generatng moton; thus the cylnder generated by the blade curve of the shaper cutter s regarded conjugatve wth the gear approxmately. Therefore, the desgn prncple ntroduced here s not no-error but t s more accurate than tradtonal method. The dstance from the no-error desgn s too lttle to nfluence the precson of shaper cutter. Based on knematcs prncple, the moton of shaper cutter can be superposed on the generated gear. On any secton perpendcular to the gear axs, formatve curve of the shaper cutter can be treated as an envelope of gear profle famly by contnuous movement when secton crcle of cylnder formed by recprocatng moton of gear ptch crcle along shaper cutter does pure rollng movement. Fg. 2 depcts the ntal poston relatonshp between the shaper cutter and the gear. Coordnate systems σ =[O; x, y, z] and σ 1 =[O 1 ; x 1, y 1, z 1 ] are rgdly attached to the shaper cutter and the gear, respectvely. z-axs and z 1 -axs are the rotaton axes of the shaper cutter and the gear, respectvely. Orgn ponts O and O 1 are both on the same plane perpendcular to z-axs. Suppose the gear modfcaton coeffcent s ξ 1 n coordnate system σ 1, the transversal curve equaton of gear tooth profle on coordnate plane xy can be expressed as follows: r = { x ( θ ), y ( θ ),0} = { x, y,0} (1) Let the ptch crcle wth center O 1 roll angle α along the ptch crcle of the projectng curve of shaper s blade curve on coordnate plane xy wth center O (shown n Fg. 3), and the moton s pure rollng. Hence coordnate system σ 1 has shfted to σ α =[O 1 ; x 1, y 1, z 1 ]. Eq. (1) can be represented n coordnate system σ as follows: r = { x1sn( α + α1) y1cos( α + α1) + ( R+ R1)cos α, x1cos( α + α1) y1sn( α + α1) + ( R+ R1)sn α, 0} = {, xy,0} where R and R 1 are the rad of ptch crcles wth center O and O 1, respectvely, and there exsts: (2) Fg. 2. Intal poston of two ptch crcles of shaper cutter and generated gear. Fg. 3. Relatve poston of two ptch crcles after rollng angle α. where α and α 1 are angles between lne OOα and x-axs and x α -axs respectvely. Based on envelope theory, the transversal curve on coordnate plane xy of cylnder generated by the blade curve of the shaper recprocatng n lne s the conjugated curve of curve Eq. (1). There exsts a formula of envelope condton: x= x( θα, ) y = y( θα, ) z = 0 xy θ α xy α θ = 0 Then the transversal curve on coordnate plane xy can be deduced: r = { x, y,0} x x1 + y1 y1 R1 ( x1 snα1 + y1 cos 1 α1 ) = Let x / x 1 + y 1 = snϕ, then the second equaton of Eq. (5) can be rewrtten as: (4) (5) α = Rα / R (3) 1 1 α = R arccos(( x snϕ + y cos ϕ) / R ) / R+ Rϕ/ R (6)

4 386 S.-W. Ln et al. / Journal of Mechancal Scence and Technology 24 (2010) 383~390 Fg. 4. Cross secton of shaper cutter. where ϕ = arctan( x 1 / y1 ) Substtutng Eq.(7) nto Eq.(6), then nto Eq.(5), solvng Eq.(5) becomes easy, and the correspondng curve equaton of the soluton can be smplfed as r = { x ( θ ), y ( θ ),0} Smlarly, the correspondng equaton of the generated cylnder can be expressed as ˆ r = { x ( θ), y ( θ), λ} = {ˆ, x yˆ, zˆ} where λ s a parameter. It should be notced that Eq. (9) s ndependent of the coordnate orgn. Gven the rake angleγ? shown n Fg. 1(b), the equaton of the rake face s r2 = { ucos v, usn v, ξ mtan γ ξ mcot αe} = { x, y, z } (7) (8) (9) (10) where α e denotes rear angle of the shaper cutter lke shown n Fg. 1(b), m represents the module of the shaper cutter/gear and coeffcent ξ s modfcaton coeffcent of the blade curve s projecton on end surface of the shaper cutter, that s to say ξ s also the modfcaton coeffcent of ptch crcle wth radus R. u and v are parameters. From Eqs. (9) and (10), the followng can be derved: 2 2 u = x ( θ ) + y ( θ ) y ( θ) v = arctan x ( θ ) (11) Notably f ξ s gven, the blade curve can be solved from Eqs. (9) and (10) or from Eqs. (10) and (11). Whle ξ vares contnuously, the accurate flank surface of the sde edge wll be obtaned. 3. Reference secton of grndng wheel Usually, gear shaper cutter s ground wth a grndng wheel whch recprocates n lne quckly and synchronously to do slower generatng moton. Snce the recprocaton s so quck t can be regarded as the cylnder generated by recprocatng moton of the grndng wheel profle does the generatng moton, and the envelope of the famly of cylnders s the shaper s backlash flank surface. Once the flank surface of the sde edge has been solved, the envelope can be obtaned. Reversal evaluaton of the cylnder equaton n a famly of cylnders s an nverse envelope problem. On the other hand, regardng the shaper cutter dong generatng moton relatve to the grndng wheel, the nverse envelope problem becomes the envelope problem. Furthermore, ths problem can be smplfed to a 2-D problem as shown n Fg. 4. There exst envelope features between the cylnder generated by the grndng wheel and the transversal of the secton perpendcular to z-axs and passng through the mdpont P of the qualfed flank surface of the shaper cutter obtaned n the secton 2. The nstantaneous coordnate systems σ = [ O; x, y, z] and σ 2 = [ O2; x2, y2, z2] are rgdly connected to the shaper cutter and the cylnder generated by the grndng wheel, respectvely. Then the followng wll dscuss models of the aforementoned envelope features. Let coordnates of orgn O n σ be (0, 0, z 0 ) obtaned n secton 3. The equaton of the transversal by the flank surface and plane x y may be expressed as: r ˆ = { x ˆ( ξ ), yˆ ( ξ ), z 0} = { xˆ, yˆ, zˆ} (12) Hence, the transversal of the cylnder generated by the grndng wheel can be regarded as a tooth profle of a gear rack. Accordng to gear generatng mechansm, the equaton of the transversal by tooth profle of the rack and the generated cylnder can be obtaned. r ˆ ˆ ˆ ˆ 3 = { xcosα + ysn α, xsnα + ycos α + Rα, z0} = { x, y, z } (13) n whch R s ptch radus of the shaper cutter, and α s a parameter that satsfes: α = arcsn(( x ˆ sn ϕ + yˆ cos ϕ ) / R ) ϕ1 (14) where ϕ = arctan(( x ˆ / ˆ ) (15) 1 y Let the curve of Eq. (13) move along the straght generatrx {snα e, 0, -cosα e }; t wll generate the cylnder, then the axal cross secton of the grndng wheel generatng the cylnder yelds: 2 r {( ˆcos ˆ s = x α + ysn α R)cos αe, xˆsnα + yˆcos α + Rα, (16) ( xˆ cosα + yˆsn α R) cosαesn αe} = { x, y, z } s s s

5 S.-W. Ln et al. / Journal of Mechancal Scence and Technology 24 (2010) 383~ It should be notced that the cross secton of the grndng wheel expressed by Eq. (16) s not the optmal one, and t wll cause larger profle errors at the top and root of the gear tooth. So that s called the reference secton of the grndng wheel. 4. Actual flank surface and optmzaton methods for grndng wheel desgn The theory of solvng the actual flank surface of the sde edge should be ntroduced frstly, such as mdpont-mdlne pont method and center devaton method. Then the correspondng models wll be presented. Notably, accordng to what was dscussed n the secton 3, suppose the curve of (13) s a drectrx, and {snα e, 0, -cosα e } s a straght generatrx; there s generated a cylnder. The actual flank surface s enveloped by the generatng moton of the cylnder relatve to the shaper cutter. Mdpont-mdlne method: the mdpont P of the qualfed flank surface of the shaper cutter satsfyng the accuracy requrement s taken as the contact pont between grndng wheel profle and the shaper cutter. The absolute value of the tooth profle error of every pont on the mdlne from the top to the root on the flank surface s less than10-4 mm n order to modfy the reference secton of the grndng wheel. The absolute values of profle errors of the ntal and the last resharpened cuttng edge therefore approach to equal and satsfy accuracy demands. Mdpont devaton method s proposed on the numercal calculatng results based on mdpont-mdlne method and manufacturng realty. The absolute value of profle error on the mdlne s small, and the absolute error value of the ntal and resharpened blade s profle approaches to equal as mentoned above; however, the errors are not equal after all. As long as the mdpont of the contact lne between the grndng wheel and the shaper s flank surface shfts moderately towards the absolute value of errors ncrescent drecton, the absolute values of profle error of the whole flank surface wll be reduced further. Based on the aforementoned prncple, methods, and combnng secton 3 of ths paper, the correspondng mathematcal models are yelded. In processng, the functon of the cross secton of the grndng wheel as expresson Eq. (16) and the cylnder generated by the drectrx expresson Eq. (13) movng along straght generatrx {snα e, 0, -cosα e } s obvously equvalent, and the cylnder s equaton can be expressed as r = { xˆcosα yˆsnα R+ µ sn αe, xˆsnα + yˆcos α + Rα, z 0 µ cos αe} = { x, y, z } (17) Let the z-axs coordnate of an arbtrary pont on the mdlne through the mdpont P be ẑ, then the value of parameter µ appearng n Eq. (17) s ( z0 zˆ )/cos e µ = α (18) Hence, the correspondng pont on the generated cylnder by grndng wheel to every pont on the mdlne of the shaper s flank surface can be obtaned by substtutng Eq. (18) nto Eq. (17). Based on the theory of gearng, the coordnates of actual ponts on the mdlne of the shaper s flank surface mentoned above are x1 = x2 + y ( x cos ϕ1 x2)/( Rϕ1) y1 = y2 + y ( y sn ϕ1 y2)/( Rϕ1) z1 = z x y1 x y 0 x ϕ ϕ = 1 1x n whch there exst: /2 x2 = ( xˆ + R ϕ1) cos( ϕ ˆ 1 arctan( Rϕ1 / x)) /2 y ˆ ˆ 2 = ( x + R ϕ1 ) sn( ϕ1 arctan( Rϕ1 / x)) (19) (20) where R and ϕ 1 are same as the aforementoned ones, and the fourth equaton of Eq. (19) s the envelope condton equaton. The mdlne of qualfed flank surface can be derved from Eq. (10) u = ( ξm tanγ ξmcot α zˆ ) tanγ (21) e Hence, coordnates (x 2, y 2, z 2 ) of correspondng ponts on the desgned flank surface can be determned. Let x 1 =x 2 (22) Solvng smultaneous Eqs. (20) and the fourth equaton of (19), x 1 and ϕ 1 wll be obtaned, and then substtute them nto y = y y (23) 2 1 and replace y n Eq.(17) wth y = y + y teratve solvng from (18) to (22) tll 4 { y } (24) max < 10 (25) Suppose ξ 1 and ξ 2 are modfcaton coeffcents of ntal blade and fnal resharpened blade, respectvely, then tooth errors on top and root of the ntal blade and fnal blade may be solved by smulatng Eqs. (21), (22) and (23). Furthermore, modfed grndng wheel can be obtaned by shftng the contact pont P on the mdlne to the drecton of maxmal absolute value of the tooth error. The procedure s smlar to that of the prevous secton, so the detals are omtted here. What should be emphaszed s that the constrant of Eq. (25) s

6 388 S.-W. Ln et al. / Journal of Mechancal Scence and Technology 24 (2010) 383~390 3 { y } max < 10 (26) If the constrant can be satsfed, the cross secton of the grndng wheel wll be the optmal soluton. The precson of the qualfed tooth surface s the best, and the error dstrbuton tends to be consstent from frontal edge to the rear edge on the whole qualfed tooth length. 5. Numercal example and desgn methods compared Involute shaper cutters are taken as computatonal examples here n order to verfy the superorty of the proposed desgn method. Gven parameters of shaper cutter, profle angle α 0, base crcle radus R 0 and base helx angle β 0 after modfcaton, accordng to [1], the correspondng formulae are α 0 = arctan(tan α0/(1 tanαe tan α0) (27) R 0 = mz0cos α 0/ 2 (28) β 0 = arctan(snα0tan α e ) (29) In the expressons above, parameters m, α 0, α e and z0 represent gear module, orgnal profle angle, rear angle, tooth number, respectvely. The equaton of flank surface of the sde edge s (cos( η ψ) ηsn( η ψ)) = ( sn( η ψ) + ηcos( η ψ)) x = R0 + y R 0 z = ψr0cot β 0 (30) where η s nvolute parametrc varable and ψ s parametrc varable of helcal rear face. The ntersecton between two functon surfaces of Eqs. (14) and (9) s the cuttng edge of the shaper cutter, and whle ξ takes dfferent values, dfferent cuttng edges can be obtaned. The dfference between the projecton equaton of cuttng edge on end face and standard nvolute s a desgn error of the shaper cutter. On the other hand, the desgn error of ths method ntroduced by ths paper above s zero apparently. Table 1 lsts errors of calculaton example for tradtonal desgn method when m=2, α e = 6 and α 0 = 20. It s obvous from Table 1 that a lesser tooth number and larger absolute value of negatve rake angle causes larger desgn error. The followng dscusses the machnng scheme and correspondng models of the novel desgn method. The followng are examples of the error-free desgn method gven by ths paper. Example 1 Smulaton for grndng nvolute carbde shaper cutter wth modulus m=2, tooth number 50, rake angle γ = and gven tooth length 6mm. Table 2 lsts calculatng results of coordnates of cross-secton curve of the grndng wheel, and Table 3 lsts the correspondng profle errors of shaper flank surface wthn tooth length 6mm. Example 2 Smulaton for grndng general nvolute shaper cutter wth γ =5 0, and other parameters s the same as that of Example 1. Table 4 lsts the profle errors of shaper flank surface wthn tooth length 6mm. If the profle error s lmted to ± 0.007mm, t s found that the qualfed tooth length can reach 26mm after calculaton. So t can be verfed that the tool lfe has been greatly prolonged. Example 3 The transversal curve equaton of desgned shaper cutter tooth profle for machnng epcyclods gear s: 2.5sn γ x1 ( θ ) = sn(21 θ) snθ 2.5[21snθ sn(21 θ) + ] 2 2cos(20 θ ) cosγ y1( θ ) = cosθ cos(21 θ) 2.5[21cosθ cos(21 θ) ] 2 2cos(20 θ ) γ [0, ), θ [ ,9 ) (31) As was descrbed n the ntroducton to ths paper, the qualfed tooth length s 1.5mm wth the desgn methods and mathematcal models of ths paper, compared to the tradtonal desgn value of the qualfed tooth length whch s only 0.52mm. Fg. 5. Parameter dagram of shaper cutter. Table 1. Tooth profle errors of tradtonal desgn method (mm). Ordnary shaper cutter Carbde shaper cutter γ ξ z 0 Tooth top Tooth root Tooth top Tooth root Tooth top Tooth root Tooth top Tooth root

7 S.-W. Ln et al. / Journal of Mechancal Scence and Technology 24 (2010) 383~ Table 2. Coordnates of cross-secton curve of the grndng wheel (mm). x s y s Table 3. The profle errors of shaper flank surface (mm). Top of the tooth Root of the tooth Intal blade Mdlne of blade Fnal blade Table 4. The profle errors of shaper flank surface (mm). Top of the tooth Root of the tooth Intal blade Fnal blade Ths ndcates that the error tself of the tradtonal desgn method s larger; the tradtonal desgn method s nferor to the error-free desgn method gven by ths paper. The methods have been appled n batch productons. Besdes what has been dscussed above, ths study can also be used n manufacturng other gear shaper cutters, lke nonnvolute shaper and preshavng shaper. The generatng machnng s carred on the SYKE9C machnng tool made n England. Seres products have been developed (some are lsted n Table 5), and the correspondng parameters are shown n Fg Dscusson Frst, ths study presents the general desgn prncple and mathematcal models, whch can be appled to dfferent knds of tooth profle desgn. There only exsts a dfferent rake angle γ between ordnary shapers and the carbde shaper. For ordnary shapers the sgn of γ s postve and for the carbde shaper, t s negatve. As for the nvolute and non-nvolute tooth profle, only the functon form of Eq. (1) s dfferent. Second, n the ntroducton of ths paper t was emphaszed that the qualfed tooth length should be as long as possble. But the models dscussed n ths paper manly concentrate on satsfyng the tooth profle precson. It seems be mutually contradctory. In fact, hgh desgn precson can prolong qualfed tooth length to a certan extent. From the actual producton condtons, the qualfed tooth length should be taken nto account for the ordnary shaper cutter desgn; on the other hand, the tooth profle precson should be consdered to the carbde shaper because the qualfed tooth length has been gven already. For the ordnary shaper cutter desgn, on the bass of precson requrement, namely y of Eq. (26) s less than the acceptable error, a longer qualfed tooth length conformng wth the accuracy requrement wll be obtaned by change modfcaton coeffcents of ntal blade and fnal resharpened blade ξ 1 and ξ 2 to lengthen the effectve tooth profle reasonably. The mathematcal models developed heren can extend the servce lfe of shapers. Moreover, regardng specal requred shaper cutters, such as the aforementoned example 3, the tolerance s y [0,0.05) top of the tooth y = y ( 0.1,0) root of the tooth (32) The tolerance s bgger than that of other knds of tooth profle shaper cutter. 7. Conclusons Because of the lmted length, ths paper only gves the man models and omts the detals. The detals can be easly derved from the methods mentoned above. On the bass of the dea, the prncple, mathematcal models and numercal examples mentoned n ths study, and practcal producton n Harbn No.1 Tools Factory, the conclusons are generalzed as follows: (1) The novel desgn scheme s error-free. It s obvously better than the tradtonal desgn scheme, especally for desgn carbde shaper cutters wth large module and large negatve rake angle or specal profle shaper cutters. (2) The novel error-free desgn scheme and the proposed methods and mathematcal models n ths paper can produce hgh-accuracy carbde shaper cutters and Table 5. Products table of BC 2 seres. Product seral number Tooth number De (mm) B (mm) B1 (mm) Short wdth coeffcent Eccentrcty (mm) BC

8 390 S.-W. Ln et al. / Journal of Mechancal Scence and Technology 24 (2010) 383~390 greatly ncrease qualfed tooth length of general shaper cutters. (3) The desgn scheme, prncple and mathematcal models presented n ths study have generalty, whch can be used to desgn and produce shape cutters of varous shapes. (4) Based on the proposed desgn prncple, the reversal envelope method for solvng the cross secton of grndng wheel, mdpont-mdlne method and mdpont-dsplacement method not only can be used to mprove the tooth profle precson of shaper cutters but also be appled n complex cuttng tools desgn and manufacture, such as broaches and hobs for hgh accuracy and long servce lfe of the tools. Hence, ths paper has reference value not only to shaper cutters, but to other complex cuttng tools as well. Acknowledgments The authors gratefully acknowledge the fnancal supports from Natonal Natural Scence Foundaton of Chna (No ) and Specal Scentfc and Technologcal Foundaton of Helongjang Provnce government, Chna (No. QC06C031). References [1] Z. J. Yuan, H. M. Lu and Y. S. Tang, Desgn of gear cutters, Xn Sh Da Publcaton, Bejng (1983). [2] W. L. Jannnck, Shaper cutters-desgn and applcaton, Part 1. Gear Technol, 7 (2) (1990) [3] W. L. Jannnck, Shaper cutters-desgn and applcaton, Part 2. Gear Technol, 7 (3) (1990) [4] C. A. Rogers, H. H. Mabe and C. F. Renholtz, Desgn of spur gears generated wth pnon cutters, Mechansm & Machne Theory, 25 (1990) [5] J. D. Km and D. S. Km, Development of software for the desgn of a pnon cutter, J. Mater. Process. Technol, 68(1997) [6] H. Yoshno, M. Shao and A. Ishbash, Desgn and manufacture of pnon cutters for fnshng gears wth an arbtrary profle, Int. J. JSME Ser. III. 35(1992) [7] H. Y. La and D. S. Wu, An enhanced DFM model for shaper cutters, Internatonal Journal of Advanced Manufacturng Technology, 19 (2002) [8] S. L. Chang and C. B. Tsay, Computerzed tooth profle generaton and undercut analyss of noncrcular gears manufactured wth shaper cutters, ASME J. Mech. Desgn, 120 (1998) [9] C. F. Chen and C. B. Tsay, Tooth profle desgn for the manufacture of helcal gear sets wth small numbers of teeth, Int. J. Mach. Tools Manufact., 45(2005) [10] C. B. Tsay, W. Y. Lu and Y. C. Chen, Spur gear generaton by shaper cutters, J. Mater. Process. Technol., 104 (2000) [11] B. W. Bar, Computerzed tooth profle generaton on ellptcal gears manufactured by shaper cutters, J. Mater. Process. Technol, 122 (2002) [12] K. D. Bouzaks, W. Köng and K. Vossen, Use of powder metallurgcal hgh speed steel n gear hobbng and gear shapng, CIRP Annals - Manufacturng Technology, 31 (1982) [13] E. W.Huang, Wafer tools-the dsposable shaper cutter. AGMA Paper, (1985) 11. [14] K. Frtz, andk. Claus, Reducng producton costs n cylndrcal gear hobbng and shapng, Gear Technology, 17 (2000) [15] P. Salonen, The tool wear n face mllng and gear shapng of Austempered Ductle Iron (ADI), Trbologa, 24 (2005) [16] J. S. Km, M. C. Kang, B. J. Ryu and Y. K. J, Development of an on-lne tool-lfe montorng system usng acoustc emsson sgnals n gear shapng, Int. J. Mach. Tools Manufact, 39 (1999) [17] G. M. L, J. T. Jn and Y. Y. Tang, A study on CAD/CAM geometrc model of nvolute gear shapng cutter, Journal of Shandong Mnng Insttute, 13 (1994) [18] W. S. Du, Z. Y. He, Manufacturng Technque of Specal Profle Pnon Cutter and Its Pattern Research, Journal of Changchun Unversty of Technology, 11 (1990) Shen-Wang Ln receved hs B.A. n Industral Educaton from Natonal Tawan Educaton College n He then receved hs M.A. n Industral Scence from Mssour State Unversty, U. S. A. and Ph.D. degrees n Precson Machnes from U. of H F. T. n Dr. Ln s currently an Assocate Professor n the Dept. of Industral and Busness Management at Far East Unversty, Tawan. Dr. Ln s a lfe member of the Chnese Metrology Socety and Chnese Mechancal Engr. Socety. Hs research nterests nclude precson nstruments and evaluaton and desgn of the profle of mllng cutters. Cheng-Shun Han receved her B.S. n Optcal Engneerng n 1988, M.S. n Mathematcs n 2001 and Ph.D. n Mechancal Engneerng n 2005 from Harbn Insttute of Technology n Harbn, Chna, respectvely. Dr. Han s currently an assocate Professor at the School of Mechancal Engneerng at Harbn Insttute of Technology. Her research nterests nclude precson and ultra-precson technology, and mathematcal models n mechancal engneerng.