 Advances in Technology Innovation , vol. 2, no. 3, 2017, pp. 61 - 67 61 Advanced Manufacture of Spiral Bevel and Hypoid Gears Vilmos Simon Department of Machine and Product Design , Budapest University of Technology and Economics, Hungary. Received 20 January 2016; received in revised form 27 May 2016; accept ed 02 June 2016 Abstract In this study, an advanced method for the manufacture of spiral bevel and hypoid gears on CNC hypoid generators is proposed. The optmal head-cutter geometry and machine tool settings are determined to introduce the optimal tooth surface modifications into the teeth of spiral bevel and hypoid gears. The aim of these tooth surface mo d- ifications is to simultaneously reduce the tooth contact pressure and the transmission errors, to ma ximize the EHD load carrying capacity of the oil film, and to minimize power losses in the oil film. The proposed advanced method for the manufacture of spiral bevel and hypoid gears is based on machine tool setting variation on the cradle-type generator conducted by optimal poly- nomial functions and on the use of a CNC hypoid generator. An algorithm is developed for the exe- cution of motions on the CNC hypoid generator using the optimal relations on the cradle-type machine. Effectiveness of the method was demonstrated by using spiral bevel and hypoid gear examples. Significant improvements in the operating characteristics of the gear pairs are achieved. Ke ywor ds : manufacture, spira l bevel and hy- poid gears, load distribution, EHD lubrication, CNC generator 1. Introduction The new CNC hypoid generators have made it possible to perform nonlinear correct ion mo- tions for the cutting of the face-milled and face-hobbed spiral bevel and hypoid gears. Several studies investigated freeform cutting methods using such machines. Among them, Shih and Fong [1] proposed a flank-correct ion methodology derived direct ly fro m the six-a xis Cartesian-type CNC hypoid generator. A poly- nomia l representation of the universal mot ions of machine tool settings on CNC machines was proposed by Fan in Re f. [2]. Chen and Wasif [3] presented a new mathe matical model to calcu- late the cutter system location and orientation and a generic post-processing method to estab- lish the machine kine matic chain and to co mpute the coordinates of the mach ine a xes for the face-milling process on CNC machines. Zhang et al. [4] derived the re lative mot ion relat ion among the virtual c radle, generating gear, cutter and workpiece on the CNC hypoid generator. To achieve maximu m life in a gear set, appro- priate bearing pattern location with low tooth contact pressure and low loaded transmiss ion error must coexists. The maximum tooth contact pres- sure and t rans mission e rro r depend substan- tially on tooth geometry. In order to reduce the tooth contact pressure and the transmission errors, and to decrease the sensitivity of the gear pair to errors in tooth surfaces and to the relative positions of the mating members, carefully chosen tooth surface modifications are usually applied to the teeth of one or both mating gears. As a result of these modifications, a point contact replaces the theoretical line contact of the fully conjugated tooth surfaces. These modifications are introduced into the gear tooth surfaces by applying the appropriate machine tool setting for the man u- facture of the pinion and the gear and/or by using a head-cutter with optimized geometry. The new CNC hypoid generators have made it possible to perform vary ing correction motions during the cutting of face-milled and face-hobbed spiral bevel and hypoid gears. In this paper, a method is presented to determine optimal head -cutter geometry and optima l polyno mial functions for the conduction of mach ine tool setting variat ion in pinion teeth finishing simu ltaneously redu c- ing ma ximu m tooth contact pressure and trans- mission errors , ma ximizing EHD load carry ing capacity of the oil film, and min imizing the power losses in the oil film. The developed optimization procedure relies heavily on the loaded tooth contact analysis for the predict ion of ma ximu m tooth contact pressure and trans- * Corresponding aut hor, Email: simon.vilmos@gt 3.bme.hu Advances in Technology Innovation , vol. 2, no. 3, 2017, pp. 61 - 67 62 Copyright © TAETI mission errors and on the e lastohydrodynamic lubrication analysis for the calculat ion of EHD load carrying capacity of the oil film, and power losses in the oil film. The load distribution and transmission error ca lculation method e mployed in this study was developed by the author of this paper [5, 6]. The EHD lubrication calculat ions are based on the method presented in Refs. [7, 8] The optimization is based on machine tool setting variation on the cradle-type generator conducted by optima l polynomial functions and on optimal head-cutter geometry. In the second step an algorith m is developed for the e xecution of motions on the CNC hypoid generator using the re lat ions on the c rad le -type mach ine. Effectiveness of the method was demonstrated by using spiral bevel and hypoid gear e xa mp les. Significant reductions in the ma ximu m tooth contact pressure and transmission errors , and improve ments in lubricat ion performances were obtained. 2. Manufacture of Spiral Bevel and Hypoid Gears on Cradle-Type Generator The concept of an imag inary generating crown gear is used in the gen erating cutting process of the face -hobbed spiral bevel and hypoid pinion and gear teeth (Fig. 1). The ma- chine tool settings are: The t ilt angle of the cutter spindle with respect to the cradle rotation axis (κ), the swivel angle of cutter tilt (μ), the radia l mach ine tool setting (e), and the tilt d istance fro m t ilt centre to re ference plane o f head-cutter (hd). To obtain the tooth surface in the generating process, the work gears are rolled with the i m- aginary generating gear. The coordinate sys- tems   eeee z,y,xK and   iiii z,y,xK are attached to the head-cutter and to the pinion/gear, respec- tively. The teeth-surfaces of the pinion and of the gear are defined by the following system of Eqs. (9)-(10):           , 1 , , 0 3 2 1 4 3 2 1 i e ig h rd t i i i i i c c ec c           r M M M M M M M r    , 0 0 0 i c i c c  v e (1) tO cO Generated gear Generating crown gear (c) (t) (w) c0y e 01y c0x Head cutter ty t0y tz ' tz t0x tx ' t0z Fig. 1 Spiral bevel gear hobbing 2.1. Variation of Machine Tool Setting Para m- eters The variations of the tilt and swivel angles, tilt d istance, radial mach ine tool setting, and the ratio of roll a re conducted by polynomia l fun c- tions of fifth-order:       1 10 1 10 1 10 10 11 12 2 5 15 .... c c c c c c c c c c                        1 10 1 10 1 10 20 21 22 2 5 25 .... c c c c c c c c c c                        1 10 1 10 1 10 30 31 32 2 5 35 .... c c c c c c d h c c c c                       1 10 1 10 1 10 40 41 42 2 5 45 .... c c c c c c e c c c c                        1 10 1 10 1 10 1 50 51 52 2 5 55 .... c c c c c c g i c c c c                  (2) where 1c  is the angle of rotation of the imag- inary generating cro wn gear in pin ion tooth surface generation. T h e re fo re , th e ma xi mu m t o o th c on ta ct pressure, ma ximu m transmission error, EHD load carrying capacity, and frict ion factor de- pend on 33 manufacture parameters: Advances in Technology Innovation, vol. 2, no. 3, 2017, pp. 61 - 67 63 Copyright © TAETI   1 2 55 max max 0 1 0 , , , prof prof ji t ij i j mp r r p p r c                   1 2 55 2 max 2 max 0 1 0 , , , prof prof ji t ij i j mp r r r c                       1 2 55 0 1 0 , , , prof prof ji t ij i j mp r r W W r c                   1 2 55 0 1 0 , , , prof prof ji T T t ij i j mp r r f f r c                 (3) 2.2. The Optimization of Machine Tool Settings and Head-Cutter Geometry An optimization method is applied to sy s- tematica lly define optima l head-cutter geometry and machine tool settings to simultaneously minimize ma ximu m tooth contact pressure and angular displacement er ror o f the driven gear, to ma ximize the EHD load carrying capacity of the oil film, and to minimize power losses in the oil film. The proposed optimization procedure relies heavily on the loaded tooth contact analy- sis for the prediction of ma ximu m tooth contact pressure and transmission errors and on the EHD lubrication analysis to calculate the EHD load carrying capacity of the oil film, and the frict ion factor. The employed methods are developed in Refs. [5 - 8] The goal of the optimization is to minimize tooth contact pressure and transmission errors , to ma ximize the EHD load carrying capacity of the oil film, and to minimize power losses in the oil film while keeping the loaded contact pattern inside the physical tooth boundaries of the pin- ion and the gear. The applicab le object ive func- tions can be expressed as       max 2 max max 0 2 max 0 p mp mp mp p f c c p          (4)       0 0 T W f T mpmp mp fW f c c W f     (5) where 0max p and 0max2  are the ma ximu m tooth contact pressure and transmis sion error, 0 W and 0T f are EHD load carry ing capacity of the oil film and the friction factor obtained for the initial va lues of manufacture parameters; p c , c , Wc , and fc are non-negative weight coeff i- cients, expressing their relative importance. The proper constraints are due to the re- quire ments that the contact pattern remains inside the possible contact area defined by load distribution calculation and inside the physical tooth boundaries of the pinion and the gear. It leads to the requirement that the contact load outside the instantly possible contact area should be zero. Therefore, the constraint can simply be denoted by   0mpC  (6) whe re C is the tota l o f tooth su rfa ce po ints with instantaneously not e xist ing contact loads. There fo re , it depends on the tooth surface topography th rough the manu factu re p ara me - ters mp. The optimization p roble m formu lated ac- cording to Eqs. (4), (5), and (6) is a nonlinear constrained optimization proble m. Functions  mpf and  mpC are not available analytically, they exist numerica lly through the load distri- bution calculation and EHD lubrication analysis. Therefore, the computer simulat ion of load distribution and EHD lubrication must be run, repeatedly, in order to compute the various quantities needed by the optimization algorith m. The load distribution calculation and the EHD lubrication analysis are based on highly nonlin- ear systems of equations. An approximate and iterative technique is used to perform the load distribution calculation and the EHD lubricat ion analysis . This causes that the calculation of partial derivatives for gradient-based optimiza- tion algorith ms to be quite impractical. For this reason, a nonderivative method is selected to solve this particular optimizat ion problem. He re, the pattern search method is used. Advances in Technology Innovation , vol. 2, no. 3, 2017, pp. 61 - 67 64 Copyright © TAETI 3. Manufacture of Face-Hobbed Hypoid Gears on CNC Hypoid Generator The CNC machine for generation of spira l bevel and hypoid gears is provided with six degrees -of-freedom for three rotational mot ions ( ,  ,  ), and three translational motions (X, Y, Z, Fig. 5). The six a xes of CNC generator are directly driven by the servo motors and able to imple ment prescribed functions of motions. The face-hobbing method requires simu ltaneous six-a xis control (the face-milling method re- quires only five-a xis control). The following coordinate systems are applied to describe the relations and motions in the CNC generator (Fig. 2) : Co o rd in a t e s yst e ms   tttt z,y,xK a nd   iiii z,y,xK a re rig id ly co nn ect ed to t he head-cutter and the pinion/gear, respectively. The coordinate transformat ion fro m system t K to system i K performs the following equation: i0 iy , y Cy Y t0z , X Cx i0x t0x CO 1O TiO Z ix t0y iz i0 cz ,z tz ty tx Head-Cutter Workpiece Fig. 2 Machine-tool setting for pinion tooth-surface finishing on CNC gnerator       0 , , , i i ti CNC t t ti t X Y Z         r M M M r Μ r (7) The location and the orientation of the tool with respect to the pinion/gear are given in coordinate systems that are represented for a conventional, crad le-type generator (Fig. 1). An algorith m is developed for the execution of motions on the CNC generator using the rela- tions valid for the cradle-type mach ine. This algorith m is based on the conditions that the relative position of the a xes of the head -cutter and the pinion rotations, 0t z and 0i y , and the a xia l re lative position of the head-cutter and the pinion/gear should be the same whether the pinion/gear is cut on a cradle-type or on a CNC hypoid generator. To ensure the same relative position of the two a xes, 0t z and 0i y , on both the cradle -type and CNC hypoid generating machines, the ele ments of the coordinate transformation ma- trices should be equal. On the basis of Eqs. (1) and (7) the following condition should be satis- fied:        0 00 00 0234120 ,,, t tt z tti z tcccii z i ZYX eM eMMMMMe r rr    (8) The same re lative position of the head-cutter and the pinion along their a xes in the case of both machines, is satisfied by applying the following condition       0 2 1 4 3 2 0 0 0 t t t O O i i i c c c t O ti t         r M M M M M r M r (9) 4. Results and Discussion A computer p rogra m was developed to i m- ple ment the formu lation provided above. By applying this program the optimal mach ine tool settings were calcu lated and functions were developed for the execution of motions on the CNC hypoid generator using the relations on the cradle-type machine. Fig. 3 Tooth contact pressure distribution in hypoid gear pair when the pinion and gear tooth surfaces are fully conjugate The load distribution calculat ion was per- formed for 21 instantaneous positions of the pinion and the gear rolling through a mesh cycle. The tooth contact pressure distributions along the potential contact lines for 21 instantaneous positions and for a ll the adjacent tooth pairs engaged for a part icular position of the mat ing Advances in Technology Innovation, vol. 2, no. 3, 2017, pp. 61 - 67 65 Copyright © TAETI me mbe rs, fo r the case when no modificat ions are introduced into the pinion teeth of the hypoid gear pair, na me ly straight-lined head-cutter profile and the basic values of machine tool settings are applied, are shown in Fig. 3. In this case the pinion and gear tooth surfaces are fu lly conjugate. The obtained ma ximu m tooth contact pressure is 369.5 MPa and the ma ximu m angular displacement error of the driven gear is 8.87 arcsec. The tooth contact pressure distribution for the case when the pinion teeth are manufac- tured by the head-cutter of optimized geometry and by optimal variation in machine tool settings governed by Eq. (2) is shown in Fig. 4. It can be observed that the ma ximu m tooth contact pre s- sure is reduced to MPa4.332p max  and the maxi- mum transmission error to secarc79.0 max2  . Similar reductions in the ma ximu m tooth contact pressure and in the ma ximu m displace ment error of the driven gear a re obtained in the case of a spiral bevel gear pair (Figs. 5 and 6). Fig. 4 Tooth contact pressure distribution in hypoid gear pair when the pinion tooth is manufactured by the h ead -cutter of optimized geo metry and by op timal variation in machine tool settings Fig. 5 Tooth contact pressure distribution in the spiral bevel gear pair when the pinion and gear tooth surfaces are fully conjugate Fig. 6 Tooth contact pressure distributions along the potential contact lines when the pinion tooth is manu fa ctu red by opt imi zed head-cutter and machine tool settings Fig. 7 Pressure distribution in the oil film in spiral bevel gear pair for the basic values of machine tool setting parameters Fig. 8 Pressure distribution in the oil film in the spiral bevel gear pair for the optima l values of machine tool setting parameters By applying the optima l co mbination of head-cutter geometry and machine tool settings the lubrication performances of the spiral bevel gear pair are imp roved. In Figs. 7 and 8 it can be considered that there is a considerable increase in the EHD load carrying capacity and reduction in the power losses in the oil film. Advances in Technology Innovation , vol. 2, no. 3, 2017, pp. 61 - 67 66 Copyright © TAETI -50 0 50 100 150 200 250 227 228 229 230 231 232 233 234 235 236 237 238  [deg.]  , [ d e g .] , X ,Y ,Z [ m m ]]   X Y Z Fig. 9 Motion graphs for the CNC hypoid gen- erator for fin ishing the pinion in fun ction of the rotation angle of the head-cutter on the CNC generator -0,04 -0,03 -0,02 -0,01 0 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 -15 -10 -5 0 5 10 15  t [deg.]  x ,  ,  [ d e g .] ,  X ,  Y ,  Z [ m m ]    X Y Z Fig. 10 Diffe rences in motions on the CNC hypoid generator as results of using head-cutte r o f opt imized geo met ry , optima l polynomia l functions for the conduction of variation in machine tool settings and modified roll fo r pin ion tooth flank generation The graph shown in Fig. 9, represent the e xecution of motions on the CNC hypoid ge n- erator for finishing the pinion teeth governed by Eq. (2). The variat ion in mot ion para meters is e xpressed in function of the rotation angle of the head -cutt er on th e CNC ge ne rato r. The dif- ferences in the values of mot ion para meters on the CNC hypoid generator, as results of using optima l polynomia l functions for the conduction of variat ion in machine tool settings and mod i- fied roll for p inion tooth flank generation, are shown in Fig. 10. 5. Conclusions An advanced method for the manufacture of spiral bevel and hypoid gears on CNC hypoid generator is presented. The optimal head -cutter geometry and mach ine tool settings are det er- mined to introduce the optima l tooth modific a- tions into the teeth of spiral bevel and hypoid gears in order to reduce the tooth contact pre s- sure and transmission errors , to ma ximize the EHD load carry ing capacity of the oil film, and to minimize power losses in the oil film. The method is based on machine tool setting varia- tion on the cradle-type generator conducted by polynomia l functions of fifth -order. An algo- rith m is developed for the execution of mot ions on the CNC hypoid generator using the optima l relations on the cradle-type machine. By apply- ing the head-cutter of optima l geometry and the optima l variation in machine tool settings the following operating parameters are improved: (1) In the case of the hypoid gear pair mo de rate reduction in the ma ximu m tooth contact pressure of 10% and a drastic reduction in the transmission errors of 91% were o b- tained. (2) For the spiral bevel gear pair significant reductions in the ma ximu m tooth contact pressure of 62% and in the transmis sion er- rors of 73% were achieved. (3) The EHD load carrying capacity of the oil film is drastically increased for 252% and the power losses in the oil film are reduced for 61% in the case of the spiral bevel gear pair. References [1] Y. P. Shih and Z. H. Fong, “Flank correction for spiral bevel and hypoid gears on a six-a xis CNC hypoid generator,” ASME Journal of Mechanical Design, vol. 130, pp. 062604-1-8, 2008. [2] Q. Fan, “Tooth surface error correction for face-hobbed hypoid gears,” ASME Journal of Mechanical Design, vol. 132, pp. 011 004-1-8, 2010. [3] Z. C. Chen and M . Wasif, “A generic and theoretical approach to progra mming and post-processing for hypoid gear machin ing on mu lti-a xis CNC face-milling machines,” Inte rn at iona l Jou rna l o f Advan ced Man- ufacturing and Technology, vol. 81 , pp. 135-148, 2015. Advances in Technology Innovation, vol. 2, no. 3, 2017, pp. 61 - 67 67 Copyright © TAETI [4] W. Zhang, B. Cheng, X. Guo, M. Zhang, and Y. Xing, “A mot ion control method for face hobbing on CNC hypoid generator,” Mechanism and Machine Theory, vol. 92, pp. 127-143, 2015. [5] V. Simon, “Load distribution in hypoid gears,” ASME Journal of Mechanical Design, vol. 122, no. 4, pp. 529-535, 1998. [6] V. Simon, “ Load distribution in spiral bevel gears,” ASME Journa l of Mechanica l De- sign, vol. 129, pp. 201-209, 2007. [7] V. Simon, “ Elastohydrodynamic lubricat ion of hypoid gears ,” Proc. of Third International Po we r T rans miss ion an d Ge a rin g Con- ference, ASM E Journal of Mechanica l De- sign, vol. 103, pp. 195-203, 1981. [8] V. Simon, “ Influence of machine tool set- ting para meters on EHD lubrication in hy- poid gears,” Mechanism and Machine Theory, vol. 44, pp. 923-937, 2009. [9] V. Simon, “In fluence of tooth modificat ions on tooth contact in face-hobbed s piral bevel gears,” Mechanism and Machine Theory, vol. 46, pp. 1980-1998, 2011. [10] V. Simon, “Opt imization of face-hobbed hypoid gears,” Mechanism and Machine Theory, vol. 77, pp. 164-181, 2014.