5-axis double-flank CNC machining of spiral bevel gears via custom-shaped tools—Part II: physical validations and experiments

h ps://doi.o g/10.1007/s00170-021-08166-0 ORIGINAL ARTICLE 5-axis double-flank CNC machining o spi al be el gea s ia cus om-shaped ools—Pa II: physical alida ions and expe imen s Gaizka G´ omez Escude o1·Pengbo Bo2·Haizea Gonz´ alez-Ba io3·Amaia Calleja-Ochoa3·Michael Ba oˇ n4,5 · Luis No be o L´ opez de Lacalle1 Recei ed: 20 Ap il 2021 / Accep ed: 30 Sep embe 2021 ©The Au ho (s) 2021 Abs ac Recen ly, a new me hodology o 5-axis lank compu e nume ically con olled (CNC) machining, called double- lank machining, has been in oduced (see “5-axis double- lank CNC machining o spi al be el gea s ia cus om-shaped milling ools—Pa I: Modeling and simula ion”). Ce ain geome ies, such as cu ed ee h o spi al be el gea , admi his app oach whe e he machining ool has angen ial con ac wi h he ma e ial block on wo sides, yielding a mo e e icien a ian o lank machining. To achie e high machining accu acy, he pa h-planning algo i hm, howe e , does no look only o he pa h o he ool, bu also o he shape o he ool i sel . The p oposed app oach is alida ed by se ies o physical expe imen s using an ab asi e cus om-shaped ool speci ically designed o a pa icula ype o a spi al be el gea . The po en ial o his new me hodology is shown in he semi inishing s age o gea manu ac u ing, whe e i ou pe o ms adi ional ball end milling by an o de o magni ude in e ms o machining ime, while keeping, o e en imp o ing, he machining e o . Keywo ds 5-axis CNC machining ·Double- lank machining ·Cus om-shaped ools ·Semi inishing ope a ions ·Tangen ial mo abili y ·F ee- o m shape manu ac u ing 1 In oduc ion E icien and highly accu a e manu ac u ing o cu ed geome ies such as ca ansmissions, gea boxes, o o he doubly cu ed engine pa s is a se ious challenge in many indus ies like au omo i e o ae onau ic, o name a ew. Spi al be el gea s, when compa ed o s aigh - oo hed be el gea s, a e able o un a highe speed [1]anda e he e o e indispensable elemen s among gea mechanisms. To achie e smoo h and silen high-speed ansmission, manu ac u ing wi h a e y high p ecision is essen ial, e.g., using di ec ace nanog inding [2]. Mo eo e , high p ecision inc eases du abili y o he manu ac u ed gea s ha is ano he main objec i e o mode n, sus ainable manu ac u ing echnologies [3]. T adi ionally, manu ac u ing o spi al be el gea s equi es specially de iced machines. The e a e se e al mains eam app oaches o manu ac u e spi al be el gea s: Gaizka G´ omez Escude o [email p o ec ed] Ex ended au ho in o ma ion a ailable on he las page o he a icle. gea hobbing wi h pe ime e cu (Gleason) [4], cyclo- palloidal con inuous gene a ion by spi al hobbing (Klin- gelnbe g and Oe likon) [5], and con inuous gene a ion by spi al hobbing wi h conical- ype cu (Klingelnbe g) [1]. Howe e , all hese app oaches a e app op ia e o la ge manu ac u ing ba ches. In con as , he p oposed app oach aims a low-cos manu ac u ing o a single wo kpiece and/o eplacemen o a b oken pa using 5-axis compu e nume ically con olled (CNC) machining. The ecen ends in gea manu ac u ing al eady head his di ec ion [6–8], which is well-sui ed o small ba ches and an economical al e na i e o he adi ional app oach using an expensi e hobbing machine. A he same ime, CNC machining is able o keep high accu acy ha is essen ial o gea manu ac u ing. Ano he signi ican ad an age o 5-axis CNC machining comes om he ac ha i can be combined wi h addi i e manu ac u ing [9]. This ype o hyb id manu ac u ing enables, o example, gea epai , ha is some hing highly desi able, bu no possible wi h he adi ional gea -cu ing me hods. The pu pose o his s udy is o u he ad ance he ecen geome ic modeling simula ions on 5-axis CNC machining wi h cus om-shaped ools [10]. Tha is, he pa h-planning algo i hm does no only look o op imal machining pa hs, / Published online: 25 No embe 2021 The In e na ional Jou nal o Ad anced Manu ac u ing Technology (2022) 119:1647–1658 bu also o he shape o he ool i sel [11–13]. While lank milling wi h cu ed (ba el) ools is known and possible o inpu ee- o m su aces [10,14] simul aneous angen ial con ac on wo sides equi es a speci ic inpu geome y. The ecen nume ical simula ion esul s showed ha o spi al be el gea s, a cus om-shaped ool admi s enough eedom o lank-machine a cu ed alley be ween wo spi al gea ee h wi h a single sweep, ha ing a bi- angen ial con ac h oughou he mo ion. This newly in oduced me hodology, called double- lank, he e o e o e s e en mo e e icien manu ac u ing han adi ional lank machining. On a concep ual le el, he ools used in his wo k a e e y simila o small g inding wheels [15–17], i.e., me al co es coa ed wi h ab asi e pa icles [18,19]. The e o e, he e ec s o he ab asi e g ains on he su ace ha e o be aken in o conside a ion. In his pape , eal machining expe imen s a e conduc ed o physically alida e he ecen ly p oposed double- lank machining me hodology. The esul s show ha his app oach ou pe o ms classical ball end milling by o de o magni ude in e ms o machining ime and, o he pa icula spi al be el gea conside ed in his pape , his app oach is well-sui ed o he semi- inishing s age. The esul s a e also i ually compa ed agains (single) lank machining wi h on-ma ke ba el ools wi h a o able esul s o he p oposed double- lank machining wi h cus om-shaped ools. 2 P e ious wo k Manu ac u ing o spi al be el gea s has been s udied o e se e al pas decades, see, e.g., [1,4,5,20] and o he ele an e e ences in [20]. The whole loop: design, p oduc ion, inspec ion and ins alla ion o spi al- oo hed be el gea s is a complex p ocess ha equi es a special ea men . The e a e se e al geome ic cons ain s ha need o be sa is ied o gua an ee smoo h ansmission be ween he cu ing ool and he ma e ial block: (i) he lank con ac be ween he conjuga e gea pai ( he ool and oo h) has o be along a whole line (cu e), (ii) he line con ac is equally dis ibu ed in he en i e engagemen a ea, and (iii) he line con ac needs o p ese ed a e e y ins an o he mo ion [4]. T adi ional pipeline o manu ac u ing o gea s is aimed o la ge ba ches and g oo e-making machines ha ely on slo ing on cu ed ool. The wo main app oaches: he con inuous indexing me hod, e e ed o as “ ace hobbing”, and he single indexing me hod e e ed o as “ ace milling” p esen some di e ences. The ace hobbing me hod p oduces an epicycloidal shape in oo h leng hwise di ec ion. The ace milling me hod is p ocessed in such a way ha bo h lanks a e manu ac u ed in a single cu , i.e., a cons an slo wid h esul s in he oo h oo due o he ci cula cu e head [21]. While palloid gea s a e p oduced by a conical hob, he cyclo-palloid gea s a e manu ac u ed using a ace hob cu e . This pape belongs o a amily o mode n app oaches ha ocus on gea s manu ac u ing using uni e sal mul i asking machines o i e-axis milling cen e s [22,23]. The main ad an age o his new end s ems in i s e sa ili y as he ool in gene al does no depend on he gea geome y. This ac makes he echnology e y lexible as i can be pe o med on a ious milling cen e s, and no , in con as o he adi ional me hods (Gleason and Klingelnbe g), on one speci ic la ge-scale machine. Suh e al. [8] use a 3-axis milling machine wi h a o a y able; howe e , ball end milling is applied in semi- inishing and inishing s ages. A nume ical app oach o de e mina ion o machine- ool se ings o oughing o a pinion by using a sp ead-blade ace milling cu e is p oposed in [3]. Fi e-cu me hod is applied and he manu ac u ing ime is minimized by maximizing he ma e ial cu du ing he ough-cu ing s age. T adi ionally, s anda d ools a e equen ly used o gea s machining in uni e sal machining cen e s [7]. Howe e , his pape ollows he ecen end whe e ee- o m ools a e used o nea - o-ne -shape machining, especially o complex slo s, such as he oo h space o he gea . Fo ce ain ype o gea s, such as non-ci cula spu be el gea , he i egula shape makes he design and manu ac u ing p ocess e en mo e di icul and o ging s age is in ol ed o dis ibu e he ma e ial acco ding o he shape ea u es o he gea [24]. Ano he issue ha needs o be ca e ully conside ed is su ace oughness. A model o p edic and con ol he ee h su ace oughness o 3+2 axis milling using ball end milling has been p oposed ecen ly [25]. O he ecen wo ks ha e ocused on gea s machining wi h uni e sal machines using milling echnology [26, 27]. Howe e , inno a i e p ocesses wi h highe ma e ial emo al a es a e one o he main objec i es in mode n manu ac u ing. In his line, Supe Ab asi e Machining (SAM) [28] is one e sa ile and easible solu ion ha inc eases gea s’ machining e iciency. Speci ically, SAM p o ides g inding echnology p ecision [29] wi h simila machining eeds and cos s, bu wi h sho e machining imes. This esea ch goes in he di ec ion o e icien spi al be el manu ac u ing using uni e sal mul i asking machines o i e-axis milling cen e s, whe e he main objec- i e is o u he educe machining ime. To his end, he p oposed esea ch aims a he semi- inishing s age using highly e icien double- lank machining whe e no only he pa h o he ool, bu also he shape o he ool i sel a e he unknowns in an op imiza ion-based amewo k. 1648 In J Ad Manu Technol (2022) 119:1647–1658 The es o he pape is o ganized as ollows. Sec ion 3 gi es a b ie summa y o he ma hema ical de i a ion o he cus om-shaped ool design and i s 3D mo ion. Sec ion 4 desc ibes he case s udy and Sec ion 5shows he esul s ob ained. Finally, Sec ion 6discusses he u u e esea ch di ec ions and concludes he pape . 3 Double-flank machining and cus om-shaped ool design The basic building blocks o he double- lank machining algo i hm will be b ie ly ecalled, mo e de ails can be ound in [11]. The p oposed app oach i s ini ializes he mo ion o he ool using a bisec o su ace o he oo h space and es ima es he ini ial shape o he ool (Sec ion 3.1), ollowed by global ool and mo ion op imiza ion (Sec ion 3.2). 3.1 Ini ializa ion o double-flank machining The goal is o app oxima e he space be ween wo ee h o a gea , ecall Fig. 1,byanen elope o a gene al ool  such ha he e is a bi- angen ial con ac be ween and  on bo h sides o he oo h space (double- lank machining). The unknowns in an op imiza ion-based algo i hm a e bo h he machining ool and a uled su ace R( he mo ion o he ool’s axis). The uled su ace is ep esen ed as a (3×1)- enso p oduc B-spline pa ch R( ,s) =(1−s)p( ) +sq( ), [ ,s]∈[0,1]×[0,1],(1) whe e sis he pa ame e in he di ec ion o ulings and is he ime pa ame e o he wo bounda y cubic B-spline cu es p( ) and q( ), o mo e de ails see [11]. Gi en he alley be ween wo ee h o a gea , i s he bo om pa o he alley is immed o (as his pa canno be lank-machined anyway). This imming ope a ion de ines wo side su aces 1and 2.As he machining ool is aimed o ha e angen ial con ac wi h bo h 1and 2, he bisec o su ace Bis compu ed. The bisec o , howe e , is a gene al su ace, and he e o e (a) (b) (c) Fig. 1 (a) Spi al be el gea (5-axis CNC machined using he p oposed me hodology). (b) A zoom-in o one oo h space (aka “ alley”) be ween wo ee h and i s CAD model ha is o med by a doubly cu ed ee- o m su ace (c) spline i ing me hod is used o compu e i s uled su ace app oxima ion, see Fig. 2(b). The ini ial uled su ace de ines he ini ial shape o he machining ool and i s en elopes de ine he ini ial app oxima ion o he wo su aces 1and 2.The wo ( igh and le ) en elopes 1and 2a e equi ed o app oxima e 1and 2, espec i ely, as close as possible, (a) (b) (c) (d) Fig. 2 Design o he cus om-shaped ool. (a) A ca i y bew een wo ee h ep esen ed as a spline su ace. (b) The sel -bisec o B(blue) is i ed by a uled su ace Rini ( ed) and an ini ial shape, ini,o he ool is compu ed. The ini ial ool and i s ini ial posi ion may pene a e he e e ence su ace ( amed in ed). (c) The ool and i s ajec o y Rbo h unde go global op imiza ion o minimize he e o o he le (1) and igh (2) en elopes om . The en elopes a e colo -coded by he dis ance e o dis (i−i),i=1,2, ha mee s ine machining ole ance o 50 μm. (d) The inal double- lank mo ion o he cus om-shaped ool  h ough he gea alley 1649In J Ad Manu Technol (2022) 119:1647–1658 and wi hin he gi en machining accu acy ε=50 μm, see Fig. 2(c). To compu e he bes en elopes 1and 2,i is o mula ed as an op imiza ion p oblem. The unknowns a e he wo cu es p( ) and q( ) (bounda ies o he ule su ace R) and a scala unc ion d(s) ha de e mines he sphe e adius in he uling di ec ion s. To compu e he sel -bisec o Bo , he mo ion o he ool is concep ualized as a wo pa ame e amily o sphe es (one in ime, second in he uling di ec ion) ha should ideally ouch on wo sides, see Fig. 2(b). Bis hen a locus o all such cen e s o sphe es. The igh and le sides (de ined by imming o he bo om pa o he alley) su aces 1and 2a e used, see Fig. 2(b), o compu e B,and hen F(z)=dis (z, 1)−dis (z, 2), (2) whe e z∈R3is he desi ed cen e o he sphe e and dis is he poin -su ace minimal dis ance. The iso-su ace F(z)=0 ha de ines he bisec o Bis compu ed using a a ian o he ma ching cubes algo i hm, see [11] o mo e de ails. 3.2 Tool and mo ion op imiza ion In he p oposed op imiza ion-based amewo k, bo h he ool and i s mo ion, ep esen ed by a uled su ace R,a e op imized. The op imiza ion has wo main objec i es: (i) o app oxima e he su ace wi hin a ine machining e o , i.e., emo e as much ma e ial as possible and (ii) o gua an ee ha he en elope o he ool lies inside he alley , i.e., he e is no o e cu . As discussed in Sec ion 3.1, an ini ial uled su ace Rini is compu ed om he sel -bisec o B. This gi es also, o each alue o s,s∈[0,1], a se o scala alues ha co espond o he dis ance dis (R( , s), ) andbya e aging hese alues o a ious one ob ains an ini ial adial unc ion d(s). This gi es an ini ial pai o en elopes ini 1 and ini 2. Howe e , hese en elopes, in gene al, in e sec  which co esponds o o e cu ing, see Fig. 2(b). To elimina e his phenomenon, he ool ini and i s mo ion Rini bo h unde go global op imiza ion. The goal is o op imize hem such ha ini 1and ini 2become as close as possible o ( emo e as much ma e ial as possible) and hey bo h lie inside he alley (no o e cu ). To achie e his goal, we p oceed as ollows. The uled su ace Ris uni o mly sampled bo h in and spa ame ic di ec ions o ob ain ij := R( i,s j),i=1,...,m,j=1,...,n. In ou disc e e app oach, o each s-pa ame e alue ( ixed j), one ob ains a se o disc e e alues dij which a e he dis ances om  o a ixed poin o he axis as i mo es in ime. Toob ainamo iono  ha is pene a ion- ee wi h (no o e cu ), we de ine dj=min i=1,...,m dij ,(3) whe e dja e he pene a ion- ee adii. In his disc e e se up, he pene a ion- ee adius depends on he sampling densi y. m=100 was se in all he expe imen s; his alue u ned ou o be su icien ly la ge o e u n s able alues o he alley shown in Fig. 1. These adii a e he lowe bounds o he poin -su ace dis ance o each j, i.e., dis ances ha de ine (disc e e) adial unc ion ha co esponds o a pene a ion- ee ool, see Fig. 2(d). Consequen ly, he pene a ion- ee e o is de ined as εj=d j−dj(4) whe e d ja e he samples o he ini ial adial unc ion d. Deno e by da ec o o unknown dis ances d:= (d1,...,d n)and op imize bo h, he uled su ace Rand d. Finally, he objec i e ha a e e y ime ins an he ool is equi ed o be as close as possible o ,bu also pene a ion- ee, is o mula ed. This leads o a minimiza ion p oblem Fp ox(p,q,d)=1 mn n  j=1 m  i=1 dis ( ij ,)−dj−εj2→min (5) subjec o he axis- igidi y cons ain s F igid(p,q)=p( i)−q( i), p( i)−q( i)−L2=0,(6) whe e dis (, ) is a poin -su ace dis ance and Lis he leng h o l. The unknowns in he minimiza ion a e he con ol poin s o he wo B-spline cu es p( ) and q( ),and he ec o o sphe e adii d.m=100 and n=30 was se in he nume ical simula ions. Mo e de ails on he whole ool-op imiza ion p ocedu e can be ound in [11]. 4Cases udy The esul s o he pa h-planning algo i hm desc ibed in [11] we e con e ed in o a CL- ile, and consequen ly con e ed in o a G-code, and es ed in a con en ional machining cen e , Kondia HS1000, see Fig. 3. The semi inishing ope a ions we e ca ied ou wi h bo h a ball end ool and a cus om-shaped ool. The cus om-shaped ool was capable o bi- angen ial machining, which esul ed in a simul aneous semi inishing o bo h walls o he oo h space. One o he objec i es was o educe machining imes in he semi inishing s age as he cus om-shaped ool admi s wide s ips o high accu acy and he e o e only a single pa h is needed, in con as o ball end milling which equi es many milling pa hs. Spi al be el gea was selec ed as a case s udy as i is one o he mos widesp ead componen s in he indus ial sec o . I is also he elemen pa excellence used o ans e powe om one elemen o ano he , by ansmi ing ci cula 1650 In J Ad Manu Technol (2022) 119:1647–1658 Fig. 3 5-axis milling cen e mo ion in e ms o he gea wheel con ac . One o he mos impo an applica ions o gea s is he ansmission o mo emen om he sha o a powe sou ce, such as an in e nal combus ion engine o an elec ic mo o , o ano he , end-e ec o , sha . In ei he case, a high accu acy is highly demanded as he machining e o s in luence signi ican ly he pe o mance. The gea wheels can be manu ac u ed om a wide a ie y o ma e ials o ob ain he igh mechanical p ope ies. F om he poin o iew o mechanical design, s eng h and du abili y, i.e., wea esis ance, a e he mos impo an a ibu es. In gene al, he gea designe should conside he abili y o manu ac u e he gea , om he o ma ion o he gea ee h up o he inal assembly o he gea in a machine. O he conside a ions include weigh , co osion esis ance, noise, and cos . F-1550 s eel (18C Mo4) was selec ed as he es ing ma e ial o manu ac u ing o he spi al be el gea , since i eaches a ai ly good ag eemen wi h all he cha ac e is ics ha s eel needs o possess. Mechanical and physical p ope ies and chemical composi ion o he used ma e ials a e shown in Table 1. This pa icula spi al be el gea was chosen, because i s con ac su ace is la ge compa ed o hose o s aigh - oo hed be el gea s, and his ac poses a g ea challenge when compu ing bo h he ool geome y and he machining pa h. The speci ic cha ac e is ics o he wheel and he heo e ical ma ing pinion o a 5/3 gea a io a e shown in Table 2. The es s we e ca ied ou on a con en ional machining cen e , Kondia HS1000. This machine is a 5-axis milling machine, wi h 3 linea and 2 a e o a y axes. The linea axes a e 2 in he head (X, Z) and one in he able (Y), while bo h o a y axes a e in he indexing able (A, C). The spindle speed capaci y is 24,000 pm. Ini ially a 210-mm-diame e and 120-mm- hick s eel bille was used as a s a ing poin . A se ies o p e ious ope a ions we e ca ied ou o achie e a geome y close o he inal. These ope a ions a e shown in Fig. 4 oge he wi h machining imes needed o each pa icula machining s age. 4.1 Cus om-shaped ool o SAM Rega ding he SAM ool, a cus om-shaped ool was used o pe o m he semi inishing ope a ion o he oo h space. I s coa was a monolaye elec opla ed CBN g inding wi h a g ain size o 300 μm, see Fig. 5. A ool o his ype was chosen due o i s excellen ool wea cha ac e is ics, because he ab asi e g ains a e esha pened as hey b eak up du ing he machining p ocess and also because he SAM p ocess is well adap ed o he calcula ion and manu ac u ing o a cus om-shaped ool. When manu ac u ing he ool co e, he hickness o he binding ma e ial and he ab asi e g ains we e aken in o accoun in o de o ob ain a ool wi h he exac geome y calcula ed in he modeling s age. The adius o he ool a ies om 4.8 o 13.2 mm, see Fig. 6,and he Table 1 Top: F-1550 s eel (18C Mo4) chemical composi ion (%). Bo om: Mechanical and physical p ope ies. The da a a e cou esy o [30] CSiMnP max Smax C Mo Cumax 0.15–0.21 0.15–0.40 0.60–0.90 0.025 0.035 0.90–1.20 0.15–0.25 0.40 Ha dness Yield poin Tensile S eng h Densi y 34 HRC 0.88 GPa 1.08 GPa 7850 kg/m3 1651In J Ad Manu Technol (2022) 119:1647–1658 Table 2 Spi al be el gea dimensions hickness o he ab asi e coa is 0.5 mm. No e ha he ool has nega i e Gaussian cu a u e (i.e., i is no a conical ool), see [11] o mo e de ailed analysis on he ool design. Conce ning cu ing condi ions, hese we e adap ed owa ds he SAM echnology in his so o machining cen e s, in pa icula hey we e adjus ed o he spindle capaci ies, wi h a spindle speed o 24,000 pm, he limi o he machine, and a eed a e o 250 mm/min. I is impo an o no e he e ec o cu ing speed on machining, especially in he con ex o supe ab asi e machining. The op imal SAM condi ions o he speed o o a ion should be a ound 60,000–90,000 e olu ions [31]. Howe e , o achie e hese speeds, high pe o mance heads a e equi ed which con en ional machines do no ha e. 5 Resul s and alida ion To physically alida e he esul s o he modeling algo i hm, he machining pa h o he cus om-shaped ool in he manu ac u ing o a spi al be el gea was applied. Mo e speci ically, he cus om-shaped ool was used in he semi inishing ope a ion, which is he place whe e Supe Ab asi e Machining echnology has i s po en ial niche o wo k. In Fig. 7, he alley be ween spi al be el gea ee h is shown du ing oughing and semi inishing ope a ions wi h milling and SAM ope a ions. Obse e a clea ly isible di e ence in he quali y o su ace smoo hness in Fig. 7(a) and (c). A quali a i e compa ison be ween double- lank machin- ing using a cus om-shaped ool and ball end milling, du ing he semi inishing s age, is p esen ed. Mo eo e , a i - ual compa ison agains single- lank milling using a ba el ool is also made. In pa icula , i is shown ha su ace oughness and manu ac u ing ime a e signi ican ly educed when using double- lank machining wi h cus om-shaped ool. 5.1 Su ace oughness Su ace oughness is one o he key pa ame e s ha in luence a smoo h mo emen be ween gea s, hei ace- ace con ac , and consequen ly he li e o he whole gea . Typically, he su ace oughness is measu ed using a con ocal mic oscope, howe e , due o he di icul Fig. 4 P og ess and machining ime summa y o he whole gea manu ac u ing p ocess. The machining ime is o ma ed as hou s:minu es:seconds 1652 In J Ad Manu Technol (2022) 119:1647–1658 Fig. 5 G ain dis ibu ion and p o ile accessibili y o he aces o he gea , esin was applied in o de o measu e a nega i e o he oo h space. The p ocess o ob aining he nega i e o he ace p oceeds as ollows: i s , he a ea o be measu ed is deg eased wi h he DN1 deg ease cleane p o ided by PLASTIFORM’s own company (PLASTIFORM, Mad id, Spain). Once his is done, a closed a ea mus be o med such ha he luid (liquid esin) co e s bo h sides o he ca i y, and hen he luid is applied o he measu ing a ea using a dispensing gun. Finally, cca 6 min is needed o he solu ion o d y ou , and hen one can emo e he nega i e o he ca i y, see Fig. 8. I was selec ed a luid ype esin, so ha i could low h ough he en i e gea ca i y and hus adap well o he su ace o he aces. Speci ically, he F65 p oduc was used, which allows a semi- lexible geome y o be ob ained, sui able o measu emen by bo h con ac measu ing sys ems and op ical measu emen sys ems. The p ecision ob ained wi h his esin is ±1μm. A Leica DCM 3D con ocal mic oscope was used o analyze he su ace oughness o he esin. Bo h sides o he ca i y we e analyzed, as he amoun o excess o ma e ial was sligh ly di e en on each side a e oughing. The adjus men o he oughness measu emen in his case was a cu ing leng h o 0.8 mm and an e alua ion leng h o 4 mm, acco ding o ISO 4288 [32]. Figu e 9shows he opog aphy and ela ed da a o bo h sides o he ca i y. Topog aphy on bo h sides o he oo h ca i y shows a pe ec ly ecognizable g inding pa e n, in which he e a e ema ked peaks and alleys along he scanned su ace caused by he andom dis ibu ion o he ab asi e g ains. I is no iceable ha sligh ly be e esul s in e m o oughness we e ob ained on he igh ace, as he oughing ope a ion lea es ha ace smoo he and s epless be ween passes, jus he opposi e o he le ace, as i can be seen in Fig. 7(d). Howe e , his ac is no a limi a ion o he p oposed double- lank app oach, bu i is due o he ac ha he oughing s age le he igh ace smoo he . The esul s a e in acco dance wi h “indica i e su ace oughness compa ison” ha many companies handle [33]. The oughness alues ob ained we e accep able o a semi inishing ope a ion because hey a e close o hose ob ained wi h simila s a egies conside ing hem as inishing ope a ions. 5.2 Machining ime Ano he aspec ha was conside ed in his wo k was he analysis o machining ime du ing semi inishing ope a ions on gea ee h. To his end, machining ime o ball end milling ope a ion and double- lank SAM s a egy wi h a cus om-shaped ool was measu ed, and ba el lank milling i ual machining ime was calcula ed. The con en ional semi inishing ope a ion using a ball end mill wi h 4 mm diame e was used wi h a s epo e o 0.33 mm o dep h o cu in 3 la e al s eps in each ace and a eed o 2800 mm/min. On he le ace, 20 axial passes we e epea ed 3 imes axially (60 passes o al) while on he igh ace 20 passes we e su icien . In con as , semi inishing using a cus om-shaped ool was accomplished in a single sweep wi h he ollowing pa ame e s: a eed o 500 mm/min and a spindle speed o 16000 pm. In he case o single- lank milling using a ba el-shaped ool wi h 12 mm o ba el diame e , in o al 20 passes we e done o co e he whole su ace wi h a eed o 848 mm/min. Wi h he abo e-men ioned alues o cu ing pa ame e s o he h ee manu ac u ing semi inishing ope a ions, he ollowing machining ime esul s we e ob ained: (i) con en ional ball end milling: 2 min and 7 s, (ii) double- lank SAM semi inishing wi h he cus om-shaped ool: 24 s, Fig. 6 Cus om-shaped g inding ool and i s geome y 1653In J Ad Manu Technol (2022) 119:1647–1658 Fig. 7 Spi al be el gea semi inishing ope a ion. (a) Manu ac u ing wi h SAM. (b) Vi ual e i ica ion wi h SAM. (c) Manu ac u ing wi h ball end mill. (d) Vi ual e i ica ion wi h ball end mill and (iii) ba el lank milling: 1 min and 8 s. Double- lank SAM domina ed in e ms o machining ime. When compa ing wi h he o he wo, in he case o ball end milling, he semi inishing machining ime was educed by 81.1%, sa ing in o al 43 min pe gea . In he case o ba el lank milling, SAM double- lank machining was 2.83 imes as e han ba el lank milling. See Fig. 4 o he summa y o he machining imes o each pa icula s age. 5.3 Dimensional de ia ion Dimensional de ia ion o he h ee es ed semi- inishing s a egies: ball end mill ope a ion, double- lank SAM wi h a cus om-shaped ool, and ba el lank milling we e quali a i ely compa ed. Figu e 10 shows simula ion esul s o he h ee semi inishing ope a ions using a comme cial so wa e. In he case o he oo su ace o he oo h, he esul s ob ained by all me hods we e e y simila , eaching a oo h su ace excess o ma e ial o up o 0.8 mm. The esul s ob ained on he oo h ace su ace show clea di e ences. In he case o ball inishing ope a ion, a uni o m inish was ob ained along he en i e su ace wi h a s ock a ound 0.23 mm. In he case o he SAM, he e a e wo clea ly di e en ia ed zones. In he i s zone, he igh ace, he alues ob ained in he semi- inishing a e p ac ically close o he inal geome y o he piece. On he le side, an unde cu o 0.1 mm was ob ained along he oo h ace. Compa ing he abo e-men ioned esul s wi h ba el lank milling, i can be seen ha he p oblem on he oo su ace is almos sol ed, excep o he ille adius, whe e is a s ock o mo e han 0.3 mm. Howe e , in he es o he su ace he e is an Fig. 8 Cu ing p ocess o he esin and i s zoomed-in pa a e ha dening 1654 In J Ad Manu Technol (2022) 119:1647–1658 Fig. 9 Su ace a ea and p o ile oughness pa ame e s alues o he le ( op) and igh (bo om) side o he gea oo h ca i y undula ion p o ile ha goes om 0.03 mm in he bo om o 0.21 mm in he peak. 5.4 Discussion and limi a ions The p oposed app oach signi ican ly educes he semi in- ishing ime by using a p ope ly designed cus om-shaped ool. The ool has o be manu ac u ed in ad ance, how- e e , he cus om-shaped ool cos s a e low, in pa icula : cylind ical s eel ba F115 (85e) o c ea e 4 SAM ools, i.e., 21.25epe he s eel co e o he ool, and 45e o add he ab asi e g ains. In o al, he cos o he cus om-shaped ool is 66.25e. In con as , he on-ma ke ool o ball end milling, VF4SVBR0200, cos s 120e. The su ace oughness alues ange Sa = 2.59–3.87 mm using he SAM app oach which is a sligh ly mo e han he numbe s ha can be ob ained by means o con en ional milling [33]. Ne e heless, hese alues a e accep able in he case o a wo kpiece which la e unde goes inishing ope a ions. Ano he sligh limi a ion is ha he e y bo om o he ca i y is no accessible wi h he ool whose shape is designed o double- lank he wo aces o he ca i y. The bo om o he ca i y has o be machined using ball end milling app oach. Compa ing SAM double- lank machining wi h lank milling wi h ba el ools, bo h app oaches o e simila accu acy, bu double- lank machining is as e ( ac o o 2.83) due o he ac ha only a single sweep o he ool is needed. The p esen ed esul s a e p omising, howe e , he double- lank me hodology has been es ed on one speci ic ype o a spi al be el gea , wi h pa ame e s shown in Table 2. Fo gea s wi h la (plana ) ee h, he double- lank app oach is no challenging as he ideal mo ion boils down o a plane-plane bisec o compu a ion. The o he ex eme o small and e en mo e cu ed e e ence geome y such as, e.g., pinions has no been expe imen ed wi h, bu can be a p omising enue o u u e esea ch. 6 Conclusion and u u e wo k We ha e p esen ed a new a ian o 5-axis lank machining, called double- lank. In his machining me hodology, no only he machining pa h, bu also he shape o he ool i sel 1655In J Ad Manu Technol (2022) 119:1647–1658