 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; accepted 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 mod- ifications is to simultaneously reduce the tooth contact pressure and the transmission errors, to maximize 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. Keywords : manufacture, spiral 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 from the six-axis Cartesian-type CNC hypoid generator. A poly- nomial representation of the universal mot ions of machine tool settings on CNC machines was proposed by Fan in Ref. [2]. Chen and Wasif [3] presented a new mathematical model to calcu- late the cutter system location and orientation and a generic post-processing method to estab- lish the machine kinematic chain and to compute the coordinates of the machine axes for the face-milling process on CNC machines. Zhang et al. [4] derived the relative mot ion relat ion among the virtual cradle, generating gear, cutter and workpiece on the CNC hypoid generator. To achieve maximum 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 erro 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 manu- 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 optimal polynomial functions for the conduction of machine tool setting variat ion in pinion teeth finishing simultaneously reduc- ing maximum tooth contact pressure and trans- mission errors, maximizing 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 maximum tooth contact pressure and trans- * Corresponding author, Email: simon.vilmos@gt3.bme.hu Advances in Technology Innovation, vol. 2, no. 3, 2017, pp. 61 - 67 62 Copyright © TAETI mission errors and on the elastohydrodynamic 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 calculation 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 optimal polynomial functions and on optimal head-cutter geometry. In the second step an algorithm is developed for the execution of motions on the CNC hypoid generator using the relat ions on the crad le -type machine. Effectiveness of the method was demonstrated by using spiral bevel and hypoid gear examples. Significant reductions in the maximum tooth contact pressure and transmission errors, and improvements in lubricat ion performances were obtained. 2. Manufacture of Spiral Bevel and Hypoid Gears on Cradle-Type Generator The concept of an imaginary generating crown gear is used in the generating 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 radial machine tool setting (e), and the tilt d istance from 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 im- 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 Param- eters The variations of the tilt and swivel angles, tilt d istance, radial machine tool setting, and the ratio of roll are conducted by polynomial func- 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 dh 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 gi c c c c                  (2) where 1c is the angle of rotation of the imag- inary generating crown gear in pin ion tooth surface generation. Therefore, the maximum too th con tact pressure, maximum 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 2max 2max 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 sys- tematically define optimal head-cutter geometry and machine tool settings to simultaneously minimize maximum tooth contact pressure and angular displacement error o f the driven gear, to maximize 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 maximum 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 maximize 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 2max max 0 2max 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 0maxp and 0max2 are the maximum tooth contact pressure and transmission error, 0W and 0Tf are EHD load carry ing capacity of the oil film and the friction factor obtained for the initial values of manufacture parameters; pc , c , Wc , and fc are non-negative weight coeffi- cients, expressing their relative importance. The proper constraints are due to the re- quirements 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) where C is the total o f tooth surface po ints with instantaneously not exist ing contact loads. Therefore, it depends on the tooth surface topography th rough the manufacture parame- ters mp. The optimization p roblem formulated ac- cording to Eqs. (4), (5), and (6) is a nonlinear constrained optimization problem. Functions  mpf and  mpC are not available analytically, they exist numerically 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 algorithm. 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 algorithms to be quite impractical. For this reason, a nonderivative method is selected to solve this particular optimizat ion problem. Here, 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 spiral 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 axes of CNC generator are directly driven by the servo motors and able to implement prescribed functions of motions. The face-hobbing method requires simultaneous six-axis control (the face-milling method re- quires only five-axis control). The following coordinate systems are applied to describe the relations and motions in the CNC generator (Fig. 2) : Coord inate s ystems  tttt z,y,xK and  iiii z,y,xK are rig id ly connected to the head-cutter and the pinion/gear, respectively. The coordinate transformat ion from system tK to system iK 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 algorithm is developed for the execution of motions on the CNC generator using the rela- tions valid for the cradle-type machine. This algorithm is based on the conditions that the relative position of the axes of the head-cutter and the pinion rotations, 0tz and 0iy , and the axial relative 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 axes, 0tz and 0iy , on both the cradle-type and CNC hypoid generating machines, the elements 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 relative position of the head-cutter and the pinion along their axes 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 rogram was developed to im- plement the formulation provided above. By applying this program the optimal machine 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 all 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 members, fo r the case when no modificat ions are introduced into the pinion teeth of the hypoid gear pair, namely 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 maximum tooth contact pressure is 369.5 MPa and the maximum 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 maximum tooth contact pres- sure is reduced to MPa4.332pmax  and the maxi- mum transmission error to secarc79.0max2  . Similar reductions in the maximum tooth contact pressure and in the maximum displacement error of the driven gear are 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 head-cutter of optimized geometry and by optimal 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 manufactured by opt imized 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 optimal values of machine tool setting parameters By applying the optimal combination of head-cutter geometry and machine tool settings the lubrication performances of the spiral bevel gear pair are improved. 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 eg .] , X ,Y ,Z [ m m ]]   X Y Z Fig. 9 Motion graphs for the CNC hypoid gen- erator for fin ishing the pinion in function 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 eg .] ,  X ,  Y ,  Z [ m m ]    X Y Z Fig. 10 Differences in motions on the CNC hypoid generator as results of using head-cutter o f opt imized geometry, optimal polynomial 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 execution of motions on the CNC hypoid gen- erator for finishing the pinion teeth governed by Eq. (2). The variat ion in mot ion parameters is expressed in function of the rotation angle of the head -cutter on the CNC generato r. The dif- ferences in the values of mot ion parameters on the CNC hypoid generator, as results of using optimal polynomial functions for the conduction of variat ion in machine tool settings and modi- 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 machine tool settings are deter- mined to introduce the optimal tooth modifica- tions into the teeth of spiral bevel and hypoid gears in order to reduce the tooth contact pres- sure and transmission errors , to maximize 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 polynomial functions of fifth-order. An algo- rithm is developed for the execution of mot ions on the CNC hypoid generator using the optimal relations on the cradle-type machine. By apply- ing the head-cutter of optimal geometry and the optimal variation in machine tool settings the following operating parameters are improved: (1) In the case of the hypoid gear pair moderate reduction in the maximum tooth contact pressure of 10% and a drastic reduction in the transmission errors of 91% were ob- tained. (2) For the spiral bevel gear pair significant reductions in the maximum tooth contact pressure of 62% and in the transmission 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-axis 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 programming and post-processing for hypoid gear machin ing on multi-axis CNC face-milling machines,” Internat ional Journal o f Advanced 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 motion 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 Journal of Mechanical De- sign, vol. 129, pp. 201-209, 2007. [7] V. Simon, “Elastohydrodynamic lubricat ion of hypoid gears,” Proc. of Third International Power Trans miss ion and Gearing Con- ference, ASME Journal of Mechanical De- sign, vol. 103, pp. 195-203, 1981. [8] V. Simon, “Influence of machine tool set- ting parameters on EHD lubrication in hy- poid gears,” Mechanism and Machine Theory, vol. 44, pp. 923-937, 2009. [9] V. Simon, “Influence of tooth modificat ions on tooth contact in face-hobbed spiral 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.