Manufacturing Method of Large-Sized Spiral B e v e l G e a r s in Cyclo-Palloid S y s t e m U s i n g M u l t i - A x i s C o n t r o l a n d M u l t i - T a s k i n g Machine Tool

K. Kawasaki, I. Tsuji, Y. Abe and H. Gunbara

Management Summary The large-sized spiral bevel in a Klingelnberg cyclo-palloid system are manufactured using multi-axis control and a multi-tasking machine tool. This manufacturing method has its advantages, such as arbitrary modification of the tooth surface and machining of the part minus the tooth surface. The pitch circular diameter of the treated in this study is more than 1,000 mm (approx. 40"). For this study, we first calculated the numerical coordinates on the tooth sur- faces of the spiral bevel gears and then modeled the tooth profiles using a 3-D CAD system. We then manufactured the large-sized spiral bevel gears based on a CAM process using multi-axis control and multi-tasking machine tooling. After rough cutting, the workpiece was heat treated and finished by swarf cutting (Ed.’s note: The removal and cutting of metal in which the axis of the cutting tool is varied with respect to the part being machined) using a radius end mill. The real tooth surfaces were measured using a coordinate measuring machine and the tooth flank form errors were detected using the measured coordinates. Moreover, the gears were meshed with each other and the tooth contact patterns were investigated. As a result, the validity of this manufacturing method was confirmed.

Introduction duces equi-depth teeth, but can also be ing of large-sized spiral bevel gears has Large-sized spiral bevel gears are produced using a face system, grown. often used for power / which produces tapered teeth (Refs. This article discusses the manufac- thermal power generation applications 1–3). The spiral bevel gears in this sys- ture of large-sized spiral bevel gears (pulverizing, etc.). Due to the increase tem are usually generated by a contin- in the Klingelnberg cyclo-palloid sys- of energy demand in the world, the uous-cutting procedure using special tem using multi-axis control and multi- demand for large-sized spiral generating machines. However, tasking machine tooling. The mate- gears has increased accordingly and the availability of those generators for rial of the workpiece was 17CrNiMo06 may continue so for some time. These this use has declined recently, while and was machined using a coated car- gears are usually manufactured based production costs have not. Therefore, bide end mill. As a result, the detect- on a cyclo-palloid system, which pro- the demand for high-precsion machin- ed tooth flank form errors were small.

56 GEARTECHNOLOGY August 2011 www.geartechnology.com

 0  X (θ) =  y + r c o s θ  (1) c  ci c   zci - rc sinθ 

X(v,θ ) = C(θ1)Xc (θ) + D(v) (2)

cosθ1 – sinθ1 0  C(θ ) sin θ cosθ 0  1 =  1 1   0 0 1  M v θ (v) = d + Θ 1 r 0 2 2 2 Rm + rc – M d cos Θ0 = (3) 2Rm rc   – M d sin(v – θ0 )   – θ D(v) =  M d cos(v 0 )   0 

Moreover, the tooth contact patterns which represents inner curved line. The W(Xψ) between and gener- 2 2 – 2 of the manufactured large-sized spiral inner cutting edge X is expressed on ated gearM atd the+ R momentm rc when generat- c cosθ0 = bevel gears were observed and those plane y z in Oc-x y z by the following ing angle is 2ψM, thed R mequation of meshing c c c c c0  positions were good. equation:  0  between the two gears is as follows   Tooth Surfaces X c (θ) = yci + rc0c o s θ   (Refs. 8–9): (1) X c (θ) =  yci + rc c o s θ  (1) of Spiral Bevel Gears    (1) X c (θ) = zcyic-i +rcrcscin o θ s θ (1) zci - rc sinθ   The generator and cutter heads that Nψ (v, θ;ψ ) • W ( v, θ; ψ ) = 0 (4) (4) z - r sinθ  Klingelnberg manufactures are typi-  ci c cally utilized in cut- From Equation 4 we have θ = θ (v, ting in the cyclo-palloid system. The The surface of the locus described ψ). Substituting θ (v, ψ) into Xψ and equi-depth teeth of the complemen- by X in O-xyz is expressed as: Nψ, any point on the tool surface of c tary crown gear are produced one after the crown gear and its unit normal are X(v,θ ) = C(θ1)Xc (θ) + D(v) (2) another by the rotating and turning X(v,θ ) = C(θ1)Xc (θ) + D(v) (2) continued (2) motions of the cutter in this method— X(v,θ ) = C(θ1)Xc (θ) + D(v) (2) i.e., the tooth trace of the complemen- where C is the coordinate transfor- tary crown gear is an extended epicy- mation matrix for the rotation about z z cloid. Therefore, the spiral bevel gears axis: z in this system are generated by a con- Extended epicycloid cosθ1 – sinθ1 0  tinuous cutting procedure. cosθ1 – sinθ1 0  Extended epicycloid C(θ )= sicnoθsθ c–osiθnθ 0 0   Figure 1 shows the basic concept C(θ1 )= sin θ1 1 cosθ1 1 0   1  1 1   that produces an extended epicycloid. C(θ1 )= s0in θ1 co0sθ1 1 0  y  0 0 1   c O-xyz is the coordinate system fixed O M d v0 0 1  to the crown gear and the z axis is the θ1 (v) = M d v + Θ0 q yc θ1 (v) = r + Θ0 crown gear axis. O is the center of Mr d v c θ (v) + Θ O 1 = 2 2 0 2 Q R 2r r 2 – M 2 both the rolling circle R and the cutter. m + c d (3) q R cos Θ Rm + rc – M d x (3) 0 = 2 2 2 The cutter fixed to the rolling circle cos Θ0 = R 2Rr r– M (3) m2+Rmcrc d Fig. 1: Extended epicycloid cos Θ0 = m c (3) R rotates under the situation. OOc is Q  – M 2sRin(rv – θ )  zc  d m c – θ0  Figurex 1—Extended epicycloid.R the machine distance and is denoted  – M d sin(v 0 )    (y , z ) by M . When the rolling circle R of D(v) = M– Md cdos(inv(–v θ– 0 θ ) 0 ) ci ci d D(v) =  M cos(v – θ )  Fig. 1: Extended epicycloid  d 0   radius r (Md–q) rolls on the base circle D(v)  M co0s(v – θ )  r = d 0   zc c Q of radius q, the locus on the pitch 0  2 2 0 2  M d2 + Rm2 – rc2 (yci, zci) surface described by the point P which θ M + R – r O cos 0 = d 2 m 2 c 2 c P cosθ0 = x is a point fixed to the circle R is an M2dM+dRRmm– rc c yc cosθ = 2M d Rm r extended epicycloid. When the spiral 0 c 2M d Rm bevel gear is generated for hard cutting In Equations 2 and 3, v is a param- Cutter on the special generator after heat treat- eter which represents the rotation angle Oc Fig. 2: Cutting edges oPf cutter x y ment, a cutter with circular-arc cut- of the cutter aboutθ ψ the• z axis, θ andψ R c c Nψ (v, θ;ψ ) • W ( v, θ; ψ ) = m0 (4) ting edges is used. These circular-arc is the meanNψ (v cone, ; distance) W ( v(Fig., ; 3).) = X0 (4) Nψ (v, θ;ψ ) • W ( v, θ; ψ ) = 0 (4) cutting edges provide a profile modi- expresses the equation of the tooth Cutter fication to the tooth surfaces of the (tool) surface of the complementary Fig. 2: Cutting edges of cutter generated gear. Therefore, a cutter with crown gear. The unit normal of X is Figure 2—Cutting edges of cutter. circular-arc cutting edges is considered expressed by N. in this article. The complementary crown gear is Figure 2 shows the cutter with cir- rotated about the z axis by angle ψ y y Extended c cular cutting edges. O -x y z is the and generates the tooth surface of the 0 c c c c epicycloid coordinate system fixed to the cutter. spiral bevel gear. We call this rotation P

Oc is the cutter center; zc is the cut- angle ψ of the crown gear the generat- xc ter axis; rc is the cutter radius; γ is the ing angle. When the generating angle is pressure angle of the inner cutting edge ψ, X and N are rewritten as Xψ and Nψ R Oc of the cutter; ρ is the radius of the cur- in O-XYZ assuming that the coordinate vature of circular arc cutting edge; y , system O-xyz is rotated about the z axis ci 0 x O,z zci are the coordinates of the center of by ψ in the coordinate system O-XYZ Q curvature of circular arc in plane x = fixed in space. When ψ is zero, O-XYZ c Fig. 3: Locus of cutting edge 0, and are expressed as a function of coincides with O-xyz. γ and ρ (Ref. 7); θ is the parameter Assuming the relative velocity Figure 3—Locus of cutting edge.

www.geartechnology.com August 2011 GEARTECHNOLOGY 57 defined by a combination of (v, ψ), spiral bevel gears. Table 2 shows the respectively (Ed.’s note: Or normal cutter specifications and machine set- vector—the normal to a surface is a tings in the calculation of the design; vector perpendicular to it. The normal the PCD (pitch circle diameter) of the unit vector is often desired, sometimes gear is 1,350 mm (approx. 53"). known as the “unit normal.”). When The determined coordinates are Figure 4—Tooth profile of gear mod- the tool surface of the complementary changed by the phase of one pitch after eled using 3-D CAD system. crown gear in O-XYZ is transformed the tooth surfaces Xg Xg', Xp and Xp' are into the coordinate system fixed to the calculated. This process is repeated and generated gear, the convex tooth sur- produces the numerical coordinates on face is expressed. A similar expression other convex and concave tooth sur- is applied to the concave tooth surface. faces. When the range-of-existence of In this case, the difference of the turn- the workpiece that is composed of the ing radius between inner and outer cut- root cone, face cone, heel and toe, etc.,

ting edges Exb that provides a crowning is indicated, the spiral bevel gear is to the tooth surface of the generated modeled. gear should be considered. The convex Figures 4 and 5 show the tooth and concave tooth surfaces of the gear profiles of the gear and mod- Figure 5—Tooth profile of pinion mod- are expressed as Xg and Xg', respec- eled using a 3-D CAD system based on eled using 3-D CAD system. tively. The concave and convex tooth the calculated numerical coordinates. surfaces of the pinion are expressed The tool pass is calculated automati-

as Xp and Xp' respectively. Moreover, cally after checking tool interferences,

the unit nomals of Xg Xg' Xp and Xp' are choosing a tool and indicating cutting

expressed as Ng Ng' Np and Np'. conditions. In this way the CAM pro- CAD/CAM System cess is realized; when the numerical The numerical coordinates on the coordinates of the tooth surfaces are

tooth surfaces Xg Xg' Xp and Xp' of the calculated, the tooth surfaces are esti- spiral bevel gears were calculated mated by the smoothing of a sequence based on the concept in the previous of points, removal of the profile of section. Moreover, those unit normals undercutting, offset of tool radius and Figure 6—Gear workpiece on multi- Ng Ng' Np' and Np were also calculated. generation of NURBS (non-uniform tasking machine. Table 1 shows the dimensions of the rational basis-spline) surface (generat- ed from a series of curves). Moreover, by virtue of calculations of intersecting Table 1— Dimensions of spiral bevel gear. curved lines of convex and concave Pinion G ear tooth surfaces—and sectional curved Number of teeth 16 40 line—an approximation of straight line Pitch circle diameter 540.0 mm 1,350.0 m m is conducted. This approach “escape” Pitch cone angle 21.801 deg. 68.199 de g. is added in order to avoid the interfer- Hand of spiral Right Left ence. When the attitude of the tool and Normal module 24.9799 coordinate transformation is conduct- Shaft angle 90 deg . ed, NC data and IGES (initial graphics Spiral angle 32 deg . Pressure angle 20 deg . exchange specification) data for the machining and display are obtained. Mean cone distance Rm 727.0 mm Face width 185.0 mm Manufacturing of Whole depth 56.21 mm Large-Sized Spiral Bevel Gears The gears were manufactured based on CAD/CAM system mentioned T able 2 — Cutter specifications and machine settings. above. The manufacturing processes Cutter radius r c 450.0 m m were divided into three parts—rough- Radius difference Exb 4 .5 m m ing, semi-finishing and finishing ρ ρ Radius of curvature of circular arc , ( ') 3,500 m m machining. Cutter blade module 23.0 m m Manufacturing of gear. The gear Pressure angle 20 de g. was machined by a ball end mill uti- Base circle radius q 546 .9441 m m lizing a vertical, three-axis machin- Machine distance Md 610 .4189 m m ing center. However, the gear could

58 GEARTECHNOLOGY August 2011 www.geartechnology.com not be machined efficiently due to axis control machine (Mori Seiki Co., responds to large-sized spiral bevel the machining using only one point Ltd. NT6600) was utilizied for pinion gears. Figure 9 shows the measured on the end mill. This manufacturing machining. The radius end mills made result of the gear and Figure 10 shows method was not suitable for the large- of cemented carbide for a hard cutting that of the pinion. The tooth flank sized gear with a PCD of more than tool were used in machining the tooth form errors are no more than about ± 1,000 mm. Moreover, the accuracy of surface. The number of edges was 12, 0.06 mm and pitch accuracy is Class-1 machining was lacking. Therefore, a and the diameters of end mills were JIS (Japanese Industrial Standards) for five-axis control machine (DMG Co., 20 mm and 16 mm, respectively. Ball both the cases of the gear and pinion. Ltd. DMU210P) was utilized. In this end mills were used in the machining We de not believe that these errors case, the plural surfaces—but not the of the tooth bottom. The number of will have an influence on the tooth installation surface—can be machined edges was 12 and the diameters of end contact patterns for large-sized spiral and a tool approach from an optimal mills were 10 mm and 5 mm, respec- bevel gears. direction can be realized using multi- tively. The material of the pinion was The gears were set on a gear mesh- axis control, as the structure of the two the same as that of the gear. The pinion ing tester and the experimental tooth axes of the inclination and rotation, in blank was rough-cut and heat treated. continued addition to 3 axes of straight line, are The pinion was then semi-finished with added. It is therefore possible to use a the machining allowance of 0.2 mm thicker tool. This is expected to reduce after heat treatment. Moreover, the pin- the machining time and to obtain better ion was finished with the machining roughness values. Cemented carbide allowance of 0.05 mm by swarf cut- radius end mills for hard cutting were ting. Table 4 shows the conditions for used in the machining of the tooth sur- semi-finishing and finishing in pinion face. The number of edges was 12, machining. Figure 8 shows the pinion and the diameters of end mills were 20 on the multi-axis control and multi- mm and 10 mm, respectively. Ball end tasking machine. The machining time mills were used in the machining of of one side in rough cutting was about the tooth bottom. The number of edges eight hours, and with semi-finishing Figure 7—Swarf cutting of gear. was again 12, and the diameters of and finishing, about 32 hours. The the end mills were 10 mm and 5 mm, machining was again finished without respectively. The gear blank made out trouble. of 17CrNiMo06 was prepared. The tool Tooth flank form error and tooth pass was 1 mm for the large-sized gear. contact pattern. The real gear and First, the gear blank was rough-cut and pinion tooth surfaces were measured heat treated. The gear was then semi- using a coordinate measuring machine finished with the machining allowance and compared with nominal data using of 0.2 mm after heat treatment. Finally, the coordinates and the unit surface the gear was finished with the machin- normals (Refs. 10–13). A Sigma M&M ing allowance of 0.05 mm by swarf 3000 developed by Gleason Works was Figure 8—Pinion workpiece on multi- cutting that is machined using the side utilized. This measuring machine cor- tasking machine. of the end mill. Machining utilizing the advantages of multi-axis control and Table 3 — Conditions of gear machining. multi-tasking machine tooling in swarf Diameter Rev olution of Feed Depth of cut Time/one cutting should deliver high accuracy Processes of end mill main spindle (mm/min.) (mm) side (min.) and high efficiency. (mm) (rpm) Table 3 shows the conditions for Semi- semi-finishing and finishing in gear 20.0 2,000 1,150 0.3 110 finishing machining. Figure 6 shows the gear Finishing 20.0 2,200 1,100 0.05 310 workpiece on the multi-axis control and multi-tasking machine; Figure Table 4 — Conditions of pinion machining. 7 shows the cutting of the gear. The Diameter Rev olution of Feed Depth of cut Time/one machining time of one side in rough- Processes of end mill main spindle (mm/min.) (mm) side (min.) cutting is about six hours; and with (mm) (rpm) semi-finishing and finishing, about Semi- seven hours. The machining was com- finishing 16.0 2,800 1,100 0.2 480 pleted with no complications. Finishing 16.0 3,300 1,100 0.05 1,440 Manufacturing of pinion. A five- www.geartechnology.com August 2011 GEARTECHNOLOGY 59 contact patterns were investigated. sides, respectively, although the length However, production of the machine Figures 11 and 12 show the results of of the tooth contact pattern on the drive tools corresponding to the large- the tooth contact patterns on the gear side is somewhat smaller. From these sized spiral bevel gears has recently tooth surfaces of the drive and coast results the validity of the manufactur- decreased and the machines themselves sides, respectively. The tooth contact ing method using multi-axis control are expensive. pattern is positioned at the center of and multi-tasking machine tooling was In this paper, a manufacturing the tooth surface and its length is about confirmed. method of large-sized spiral bevel 50% of the tooth length, based on the Summary/Conclusions gears in the Klingelnberg cyclo-palloid analysis of the tooth contact pattern. Large-sized spiral bevel gears are system using multi-axis control and The experimental tooth contact patterns usually manufactured based on a cyclo- multi-tasking machine tooling was pro- are positioned around the center of the palloid system by a continuous cutting posed. For this study, first the numeri- tooth surfaces of both drive and coast procedure using a special generator. cal coordinates on the tooth surfaces of the spiral bevel gears were calculated and the tooth profiles were modeled using a 3-D CAD system. The large gears were manufactured based on a CAM process using multi-axis con- trol and multi-tasking machine tooling. After rough cutting, the workpiece was heat treated and finished by swarf cut- ting using radius end mills. The real tooth surfaces were measured using a coordinate measuring machine and the tooth flank form errors were detect- ed using the measured coordinates. Moreover, the gears meshed well and the tooth contact patterns were inves- Figure 9—Measured result of gear (µm). tigated. As a result, the validity of this manufacturing method was confirmed. (This work was supported in part by Advanced Technology Infrastructure Support Services pro- moted by Ministry of Economy, Trade of Industry in Japan.)

References 1. “Design of a Bevel Gear Drive According to Klingelnberg Cyclo- Palloid System,” KN 3028 Issue No. 3, (1994), p. 20. 2. Townsend, D. P. Dudley’s Gear Handbook—the Design, Manufacture, and Application of Gears, 2nd Ed., Figure 10—Measured result of pinion (µm). McGraw-Hill, New York, 1991, pp. 20.42–20.45. 3. Stadtfeld, H. J. Handbook of Bevel and Hypoid Gears—Calculation, Manufacturing and Optimization, Rochester Institute of Technology, RIT., 1993, pp. 9–12. 4. Nakaminami M., T. Tokuma, T. Moriwaki and K. Nakamoto. “Optimal Structure Design Methodology for Compound Multiaxis Machine Figure 11—Tooth contact pattern of Figure 12—Tooth contact pattern of Tools—I (Analysis of Requirements drive side. coast side. and Specifications),” International 60 GEARTECHNOLOGY August 2011 www.geartechnology.com Journal of Automation Technology, Vol. 1, No. 2, 2007, pp 78–86. Yoshikazu Abe is president of Iwasa Tech Co., Ltd. Based 5. Moriwaki T. “Multi-functional in Tokyo. Japan. He is also currently the chairman of Machine Tool,” Annals of CIRP, Vol. Japan Gear Manufacturers Association (JGMA). Abe is a graduate of both Gakushuin University’s Department of 57, No. 2, 2008, pp. 736–749. Economics and Washington State University’s Department 6. Kawasaki K., H. Tamura and Y. of Government. Iwamoto. “Klingelnberg Spiral Bevel Gears with Small Spiral Angle,” Proc. 4th World Congress on Gearing and Power Transmissions, Paris, 1999, pp. 697–703. 7. Kawasaki, K. and H. Tamura. “Duplex Spread Blade Method for Cutting Hypoid Gears With Modified Tooth Surface,” ASME Journal of Hiroshi Gumbara received his M-Eng. in 1976 and D-Eng. in Mechanical Design, Vol. 120, No. 3, 1994 from Tohoku University. He has been a professor at the 1998, pp. 441–447. Department of Mechanical Engineering at Matsue National College of Technology, Matsue, Japan from 2000 to pres- 8. Sakai, T. “A Study on the Tooth ent. His main research areas are the design and manufacture of gears. Profile of Hypoid Gears,” Trans. JSME, Vol. 21, No. 102, 1955, pp. 164–170 (in Japanese). 9. Litvin, F. L. and A. Fuentes. Gear Geometry and Applied Theory, 2nd Kazumasa Kawasaki received a B-Eng. in 1987 and M-Eng. in 1989 from Niigata University. He previously served as a research asso- Ed., Cambridge University Press, UK, ciate at the faculty of engineering at Niigata University, from 1989 to 2004, pp. 98–101. 2001. He received his D-Eng. in 1998 from Kyoto University. He was 10. Kato, S. and T. Akamatsu. a Visiting Scholar at the Mechanical Engineering school of the University of Illinois at Chicago from 2000 to 2001 and has been “Measuring Method of Hypoid Gear an associate professor at the Center for Cooperative Research Tooth Profiles,” SAE Technical Paper, and the Institute for Research Collaboration and Promotion of Niigata 1982, No. 810105. University, Niigata, Japan from 2001 to present. Kawasaki’s main research areas are the design, manufacture and measurement of gears. 11. Gosselin, D., T. Nonaka, Y. Shiono, A. Kubo and T. Tatsuno. “Identification of the Machine Settings of Real Hypoid Gear Tooth Surfaces,” Isamu Tsuji is a gear technology engineer at Iwasa Tech. Co., Ltd., Tokyo, Japan. He specializes in the research ASME Journal of Mechanical Design, and development of new gear manufacturing methods for Vol. 120, 1998, pp. 429–440. straight, skewed and spiral bevel gears used in power 12. Fan, Q., R.S. DaFoe and J.W. plants and internal gears used in wind turbines. He holds BS and MS degrees in mechanical engineering from Keio Swangaer. “Higher-Order Tooth Flank University, Tokyo, Japan. Form Error Correction for Face-Milled Spiral Bevel and Hypoid Gears,” ASME Journal of Mechanical Design, Vol. 130, 2008, 072601–1–7. 13. Kawasaki, K. and I. Tsuji. “Analytical and Experimental Tooth Contact Pattern of Large-Sized Spiral Bevel Gears in Cyclo-Palloid System,” ASME Journal of Mechanical Design, Vol. 132, 2010, 041004–1–8.

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