Determination of a Major Design Parameter for Forward Extrusion of Spur Gears

Jong-Ho Song Yong-Taek Im Professor, Fellow ASME e-mail: ytim@webmail.kaist.ac.kr Computer Aided Materials Processing Laboratory, Department of Mechanical Engineering, ME3227, Korea Advanced Institute of Science and Technology, 373-1 Guseong-dong, Yuseong-gu, Daejeon 305-701, South Korea Determination of a Major Design Parameter for Forward Extrusion of Spur Gears In this study, a major design parameter was identified for cold forward extrusion of solid or hollow spur gears by investigating the effect of gear geometries and forming variables on formation of gear teeth by finite element simulations. A limiting extrusion ratio was determined for reducing the likeliness of underfilling in the die cavity. An equivalent radius of the cross-sectional geometry of a gear was also determined to predict the forming load requirement from an axi-symmetric approximation. Based on this approximation, a modified empirical equation was determined for simple determination of forming loads required. DOI: 10.1115/1.1688379 1 Introduction In recent years, the forging process has been applied in gear manufacturing in place of conventional machining owing to the reduction of production cost. It has economic advantages such as high productivity, material saving, and improved mechanical properties compared with machining. However, process design with high precision is not easy to achieve in order to acquire the competitiveness and productivity of forged gears since close control of material flow is not simple in gear forming for near netshape manufacturing. Therefore, many efforts have been made to improve design skill in gear forging in the last few decades. The most important aspect of process design in gear forging might be forming a gear tooth with a reasonable forming load in a close tolerance. It is well known that the level of forming load heavily affects the life of forming dies and tools and the quality of a manufactured gear is directly related to the complete formation of gear teeth. So far, the upper bound method 1 has been widely applied to predict the forming load at the final stage in closed die forging of a spur gear because of the simplicity of the analysis. It was also extended to preform design of bevel gears based on a reversing algorithm introduced by Osman et al. 2. While this algorithm is easy to apply for a rather simple geometry, the approach has its limitation in understanding the effect of various process variables during the forming procedure for a better process design. Therefore, the application of the finite element FE method has been increased in order to obtain various knowledge such as material flow, distribution of stress and strain, and required forming load with better solution accuracy 3,4. In the area of die design and tooth formation with precision, various studies have been carried out. Rahman et al. 5 have proposed the secondary grinding process once a spur gear was hot forged in order to obtain higher dimensional accuracy of forged gears. Sadeghi et al. 6 have proposed some layouts for designing a preform for precision forging of straight and helical spur gears and remarked upon the effect of gear geometry and process variables on forging pressure and ejection load. Doege et al. 7 designed the tool system consisting of spring assemblies in closed die forging for improving the forming accuracy. Kondo et al. 8 used the bulged specimen in order to prevent underfilling near the central portion of the product in ring gear forging with a divided flow method. In the design of the cold extrusion process without or with a mandrel, one of the most important issues is to determine the Contributed by the Manufacturing Engineering Division for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received Sept. 2002; Revised Oct. 2003. Associate Editor: R. Smelser. major design parameter to guarantee complete filling of the die cavity for each tooth and to predict the required forming load with a reasonable accuracy in a simplified way. Thus, the major design parameter in manufacturing the solid or hollow spur gears will be identified in the present investigation by three-dimensional finite element analyses. For a convenient prediction of forming load requirement, the three-dimensional complex gear geometry will be transformed into an equivalent axi-symmetric one. For these investigations, an in-house program CAMPform3D 9 was used. In order to determine the validity of the FE program used, the simulation results were compared with experimental results available in references 10,11. The number of teeth, module, profile shift coefficient, pressure angle, and clearance ratio were considered in the FE simulations to investigate the effect of product geometry on the formability of the teeth. Additionally, the effect of forming variables such as entrance angles, friction conditions, radii of the mandrel, and initial cylindrical billet dimensions was investigated for the extrusion of gears. Finally, the forming load estimation based on the axi-symmetric approximation with an equivalent radius geometry was compared to the actual three-dimensional simulation result. Based on this comparison a modified empirical equation similar to the one available in the literature 12 was proposed for solid and hollow spur gears, respectively. 2 Background As mentioned previously, in gear extrusion, predictions of complete formation of a gear tooth and required forming load are essential in achieving a successful design. In general, the gear geometry greatly influences forming characteristics of material deformation behavior and forming load. Normally, the geometry of a gear can be described by basic and detailed specifications as shown in Fig. 1. The basic specification is composed of the description of the number of teeth n, module m, pressure angle, profile shift coefficient p, and clearance ratio c. Once the values of the basic specification are given, detailed specification such as the radii of addendum (R o ), pitch (R p ), base (R b ), and root (R r ) can be calculated accordingly based on standard equations as given in Fig. 1. In addition to such specifications, area ratios were introduced in the present investigation to represent variations in the gear geometry as follows: AR T/G A T /A G, AR G/R A G /A R (1) where, A T, A G, and A R represent the cross-sectional areas of total gear teeth, a gear, and root of the gear in that order. Journal of Manufacturing Science and Engineering MAY 2004, Vol. 126 Õ 255 Copyright 2004 by ASME

Fig. 1 Definition of the geometry of a spur gear and its graphical representation In Fig. 2, the standard geometry of a die layout with the mandrel used in the present investigation is described for a given basic specification. The radius of the billet container R was determined to preserve an area reduction ratio of 50% on the basis of the radius of the addendum R o. Based on this value, detailed dimensions of the die geometry were to be determined from the formulas given in Fig. 2. The fillet radius at the die entrance was set to be 1.5 mm. As a representative design parameter, an extrusion ratio defined as the ratio between cross-sectional areas of the workpiece before and after extrusion has been widely used in the axi-symmetric forward extrusion. In general, a higher value of extrusion ratio is not preferable in order to prevent the formation of forming defects caused by severe deformation of the workpiece. However, in gear extrusion, an extrusion ratio is more difficult to define due to its complex geometry. Therefore, in the present study, the following extrusion ratios were introduced to determine the major design parameter that achieved complete filling in gear extrusion. At first, the extrusion ratios based on the radius of the addendum (ER o ) and radius of the root (ER r ) of the gear were defined in consideration of the different material flow at the tooth and other regions of the gear as follows: ER o R 2 R m 2 / R o 2 R m 2, ER r R 2 R 2 m / R 2 r R 2 m (2) Here R and R m represent the radii of the billet container and the mandrel used in the process, respectively, as shown in Fig. 2. If mandrel is not necessary in the process like a solid gear extrusion case, R m 0 will be applied in Eq. 2 and hereafter. Since the material flow varies at the root and tooth tip of a gear, Fig. 2 The standard design geometry of the die for cold forward extrusion of a spur gear with a mandrel 256 Õ Vol. 126, MAY 2004 Transactions of the ASME

Table 1 The experimental conditions of cold forward extrusion Extrusion without mandrel 10 Extrusion with mandrel 11 Material property MPa 357.7 0.156 441.5 0.275 Shear friction factor 0.12 0.12 Workpiece dimension 29.5 75 29.5 80 Diameter mm Height mm Radius of a mandrel mm 0 5 the ratio between these extrusion ratios was also defined as Eq. 3 and considered as a possible design parameter that estimated the level of underfilling in gear extrusion. ER r/o R 2 o R 2 m / R 2 r R 2 m (3) For a simple calculation of forming load, an axi-symmetric approximation with a shape factor has been widely utilized for a product with complex geometry 12. Similarly, the radius of an equivalent axi-symmetric geometry of the gear R e was determined by introducing the following analogy in the cross-sectional area: A G R 2 e R 2 m (4) Then, the extrusion ratio based on such an equivalent radius with a mandrel used could be defined as ER e R 2 R 2 m / R 2 e R 2 m (5) For a reasonable prediction of forming load requirement, an existing empirical rule available in the literature 12 was determined in the present investigation as follows: P CA o o 3.45 ln ER e 1.15 (6) where P is the forming load required, C the shape factor 1.1 and 1.2 for solid and hollow types, respectively, A o the area of the initial billet, and o the yield stress of the material used. 3 FE Simulations In this study, an in-house FE program CAMPform3D 9 was used for investigating the influence of process variables on gear extrusion. It is composed of a solver based on the thermo-rigid viscoplastic approach and remeshing module adopting a master grid approach and an octree based refinement algorithm 13 16. For initial mesh generation, the commercialized program I-DEAS 17 was used for convenience. The accuracy of the FE program was examined through comparison with the experimental results reported by Han et al. and Yang et al. 10,11 with and without using a mandrel, respectively. Table 1 summarizes the experimental conditions, in which Al2024-O was used as the workpiece material and a friction factor was determined as 0.12 by the ring compression test. In this study, the average forming pressures according to various shear friction conditions were calculated and compared with those in the references. Also, distortion patterns were compared with those of experiments at the cross-sections of the root and tip of the tooth. In order to find the major design parameter controlling filling status of the tooth, FE simulations were carried out. The workpiece used in analyses was AISI1010 whose material property can be represented here as given in the document of ICFG 18. 650 0.25 MPa (7) where, is the effective stress and the effective strain. The punch velocity was assumed to be 1.0 mm/sec. Due to geometrically rotational symmetry, only half of a tooth was used in simulations. Process variables investigated in this study were the geometry of the product and forming variables as shown in Fig. 3. To consider the geometry effect, the number of teeth was first varied from 7 to 21, while the other basic specifications were kept constant. Then, the other basic specifications such as module, pressure angle, profile shift coefficient, and clearance ratio were varied for the extrusion process of a gear with 14 teeth. For Fig. 3 The investigated process variables in cold forward extrusion of a spur gear Journal of Manufacturing Science and Engineering MAY 2004, Vol. 126 Õ 257

Table 2 Comparison of average forming pressures between FE simulation and experimental results Extrusion without mandrel FE Simulation Experiment 10 Friction condition 0.06 0.08 0.12 Average pressure MPa 290.06 313.2 365.7 317.5 Extrusion with mandrel FESimulation Experiment 11 Friction condition 0.12 0.17 0.20 Average pressure MPa 522.9 608.6 670.9 617.9 investigation of the effect of forming variables, the die entrance angle, constant shear friction factor, radii of the billet container and mandrel were varied as shown in Fig. 3. 4 Results and Discussion 4.1 Verification of Simulation Program. As mentioned earlier, the accuracy of the currently used FE simulator was examined through comparisons with experimental results reported in references 10,11. Table 2 compares the experimentally measured average pressures from the references and the currently predicted values according to various friction conditions. As shown in this table for the no-mandrel cases, the estimated average pressure was somewhat higher than the experimental data. On the other hand, when a mandrel was used, the average pressure was under predicted in comparison to the experimental result. In this table, it was observed that the friction conditions that gave the smallest difference of average pressures between FE results and experiments were 0.08 and 0.17 in the extrusion cases without and with a mandrel, respectively. The accuracy of workpiece deformation obtained from the FE simulations using the friction factors of 0.08 and 0.17 was examined through cross-sectional grid distortion patterns. Figure 4 compares the cross-sectional grid distortions at two locations of the root O-B and the tip of the tooth O-A between the FE simulations and experiments for the processes without and with a mandrel. As can be seen in this figure, the overall predicted distortion patterns were in good agreement with the experimental findings reported in references. However, a small discrepancy of the distortion patterns between the simulation and experiment exists at the tip of the tooth O-A in Fig. 4 a. This was partly due to the occurrence of surface cracking or so-called fish skin that was formed on the workpiece in the experiment 10. From these observations it was found that the current simulations with shear friction factors of 0.08 and 0.17 for the extrusion processes without and with a mandrel, respectively, might be acceptable. These findings show that the FE simulator used in this study can be used for the design of cold forward extrusion of a spur gear with reliability. 4.2 Effect of Gear Geometry. The effect of gear geometry on formability of gear teeth was investigated by varying basic specifications. At first, the total teeth number was varied from 7 to 21, while the other basic specifications were fixed as module 5, pressure angle 20, profile shift coefficient 0.4, and clearance ratio 0.25. The variations of gear geometry represented by AR T/G of Eq. 1 and forming load predictions according to the total teeth number are shown in Fig. 5. As can be seen in this figure, AR T/G decreased as the total teeth number increased. This was due to the fixed basic specifications other than the total teeth number, which led to the expansion of R r and R o, while the size of a tooth was almost identical. The forming load increased almost linearly with the number of total teeth according to Fig. 5 b. Fig. 4 Comparison of the cross-sectional grid distortions between FE simulations and experiments available in the literature: a extrusion without a mandrel 10 and b extrusion with a mandrel 11 Figure 6 shows the deformed shapes and distributions of effective strain for the cases of 7 and 18 teeth. The central free end surface of the deformed workpiece was formed to be rounded. Such a roundedness was formed due to the difference of material flow between the center and tooth regions of the gear. Because of the friction effect, the material flow was slower in the tooth die cavity region compared to the central region of the gear. Because of such a difference in material flow, complete filling in the tooth die cavity region was not guaranteed for all cases. In the gear extrusion case with 7 teeth, complete filling of the die cavity was achieved when the punch stroke reached 14.9 mm. However, in the case of 18 teeth, it was not obtained for the punch stroke of 35.1 mm. The effective strain level in the workpiece center region was higher in the case of 7 teeth compared to that of the 18 teeth. It can be also seen that the distribution patterns at the tooth hole were similar to each other and the highest value of the effective strain occurred at the root edge of the tooth. From simulations using the aformentioned basic specifications, it was found that complete filling of the teeth was possible for gears with up to 9 teeth, as shown in Fig. 5 a. To examine the effect of other basic specifications, the total teeth number was fixed as 14 while other specifications were varied as listed in Table 3. This table also shows the variation of 258 Õ Vol. 126, MAY 2004 Transactions of the ASME

Fig. 6 The overall deformed shapes and distributions of the effective strain in cold forward extrusion of a spur gear with the a 7and b 18 teeth number Fig. 5 The variation of a the ratio of total teeth area to gear area and b predicted forming loads according to teeth number AR T/G, required forming load, and completeness of filling according to various combinations. It was found that the required load was greatly influenced by the variation of modules, but AR T/G was almost nearly constant irrespective of the module values. On the other hand, variations in the basic specifications of pressure angle, profile shift coefficient, and clearance ratio affected AR T/G values since the variation of a gear tooth geometry in terms of the aspect ratio between the height and thickness dimensions was relatively larger in comparison with the variation of gear sizes. In particular, it was found out that AR T/G was mainly dependent on the clearance ratio. However, underfilling was not prevented in all the cases when the total teeth number was 14. From these observations, it was construed that the total number of teeth among other gear basic specifications is the most important parameter in governing the filling status of a gear tooth. 4.3 Effect of Forming Variables. When a gear tooth was incompletely formed, the appropriate modification of the current forming conditions should be made. Since there are many influential forming variables on the filling status of the die cavity for a gear tooth, a major governing parameter should be identified. Therefore, in the present investigation, the effect of the forming variables defined in Fig. 3 on the formation of a gear tooth was investigated for the extrusion process of a gear with 18 teeth in the following. 4.3.1 Effect of Die Entrance Angle. The effect of die entrance angle on gear forming was investigated with the variation of entrance angles of 30, 45, and 60. Figure 7 shows the predicted distributions of the effective strain. In this figure the effective strain level at the surrounding of the entrance of a die cavity for a tooth increased according to the increase of die entrance angle. This phenomenon resulted from the expansion of the entrance of a tooth hole leading to severe material deformation. In the same figure, the level of underfilling increased as the die entrance angle increased. However, underfilling at the tip of a tooth occurred in all simulation cases. From this, it was found that the increase of die entrance angle did not solve the underfilling phenomenon in gear forming of 18 teeth. 4.3.2 Effect of Shear Friction Factor. In order to investigate the relationship between the friction and gear formation, the values of constant shear friction factors were varied as 0.1, 0.2, 0.3, Table 3 Required forming load and completeness of filling obtained from FE simulations with the variations of module, pressure angle, profile shift coefficient, and clearance ratio Module Pressure angle 10 15 15 25 Load MN 47.350 108.72 11.400 11.750 AR T/G 0.287 0.287 0.282 0.293 Area of teeth/ area of a gear Completeness of filling No No No No Profile shift coefficient Clearance ratio 0 0.8 0.6 1.0 Load MN 11.300 12.670 11.910 12.155 AR T/G 0.299 0.262 0.351 0.420 Area of teeth/ area of a gear Completeness of filling No No No No Journal of Manufacturing Science and Engineering MAY 2004, Vol. 126 Õ 259

Fig. 8 Contour plots of the velocity distribution in the z direction for various shear friction factors of a 0.1, b 0.2, c 0.3, and d 0.6 these simulations, it was construed that the friction condition directly connected with the lubricant condition might be insignificant to the filling status. 4.3.3 Effect of Initial Billet Radius. In order to investigate the effect of the billet radius on gear formation, simulations were conducted by changing the radius of the cylindrical billet from 73.4 to 59.9 mm with the die entrance angle of 45. Figure 9 shows profiles of the deformed workpiece when the center point of the deformed workpiece reached the value of 35.1 mm in the Fig. 7 The predicted distributions of the effective strain in forward extrusion with entrance angles of a 30, b 45, and c 60 and 0.6 with the die entrance angle of 45. Figure 8 shows the contour plots of the velocity distributions in the z direction when the punch stroke arrived at 35.1 mm. In this figure the velocity of the material flow at the initial position of the entrance region was suddenly reduced due to the increase of friction. However, in the region of the die cavity of a tooth, the overall velocity fields were not greatly affected. As can be seen in this figure, the underfilling phenomenon was also observed in all simulation cases. From Fig. 9 Comparison of deformed shapes of the workpiece obtained from FE simulations with variations of the billet radius 260 Õ Vol. 126, MAY 2004 Transactions of the ASME

Table 4 The punch stroke at the final step, completeness of filling and required forming load obtained from FE simulations Radius of mandrel mm 0 20 30 35 Stroke mm 35.1 33.0 24.3 21.4 Load MN 17.4 19.3 20.3 22.3 AR T/G 0.23 0.28 0.40 0.53 Area of teeth/ area of a gear Completeness of filling No No No Yes mm, 30 mm, and 35 mm. Table 4 summarizes the punch stroke at the final step, required load calculated, and completeness of filling of a tooth obtained from the FE analyses. As shown in this table, it was found that forming load increased as the size of mandrel increased. Complete filling at the tip of a tooth cavity was achieved when the radius of the mandrel was 35 mm in which AR T/G was 0.53. In other cases, underfilling occurred as shown in Fig. 10. Also, the level of underfilling decreased with the increase of the mandrel radius. Thus, prevention of material flow at the center region by the mandrel might lead to improvement of filling in a tooth die cavity. The distributions of the effective strain according to the variation of the mandrel radius are also shown in Fig. 10. As shown in this figure, maximum values of the effective strain were different due to the difference of punch stroke. However, higher levels of the effective strain concentrated on the roots of the teeth for all cases. Fig. 10 Distributions of the effective strain obtained from FE simulations with variations of the mandrel radius of a 20 mm, b 30 mm, and c 35 mm z coordinate. As can be seen in this figure, deformed shapes of the workpiece were almost similar irrespective of the initial billet radii. The predicted forming load was reduced to 8.1 MN in the current case from 17.4 MN for the case of the initial billet radius of 73.4 mm. Underfilling was not solved by decreasing the initial billet radius. 4.3.4 Effect of Mandrel Radius. As the final forming variable, the size of the mandrel was varied for the forward extrusion of a spur gear with 18 teeth as shown in Fig. 10. The simulations were conducted with the variation of the radius of a mandrel of 20 4.4 Determination of a Major Design Parameter. Through FE simulations, it was found that the geometry of a gear including the mandrel radius was most important in determining the filling status. Thus, the usefulness of area ratios of AR T/G and AR G/R was investigated to predict the filling status of a gear. Figures 11 a and b show the relationship between the filling status and area ratios of AR T/G and AR G/R, respectively. In this figure, the solid or hollow circles indicate simulation cases where the basic gear specifications were fixed with a module of 5, pressure angle of 20, profile shift coefficient of 0.4, and clearance ratio of 0.25. As shown, in the extrusion cases without a mandrel, an increase of the total teeth number led to the occurrence of underfilling with decreasing values of AR T/G or AR G/R. However, when a mandrel with an appropriate radius was used, complete filling was achieved for higher values of the total teeth number. More specifically, as can be seen in Figs. 11 a and 11 b, complete formation of a gear tooth was obtained when the value of AR T/G and AR G/R was higher than 0.4 and 1.7, respectively. However, when other basic specifications were changed, these area ratios were not able to properly predict the occurrence of underfilling. These cases are represented by triangles and square in Fig. 11. Square indicates the simulation example when the clearance ratio was changed to 1.0 while the other basic specification was maintained as previously shown in Table 3. The triangles indicate the cases where the gear basic specification was set to be module of 5, pressure angle of 25, profile shift coefficient of 0.2, and clearance ratio of 0.7. On the other hand, ER r/o defined by Eq. 3 was found to correctly apply for these triangle and square simulation cases. As shown in Fig. 12, the triangle and square cases were properly moved to the underfilling zone. Therefore, the limiting value of ER r/o that prevented the likeliness of underfilling was determined to be 2.6 under the present investigation conditions. From these observations, ER r/o was determined as the major design parameter that can be effectively used to predict the occurrence of underfilling in this study. Journal of Manufacturing Science and Engineering MAY 2004, Vol. 126 Õ 261

Table 5 Comparison of required loads obtained from the three-dimensional FE simulation, the equivalent axi-symmetric FE simulation, and empirical equation Number of teeth R e ER e 3D FE simulation MN Axi-symmetric FE simulation MN Empirical equation, Eq. 6 MN 10 26.29 2.96 7.51 7.20 7.62 12 31.37 2.78 9.62 9.20 9.73 14 36.44 2.65 11.50 11.20 12.12 16 41.46 2.57 13.62 14.10 14.79 18 46.48 2.5 17.40 16.60 17.73 The result obtained from the current investigation will be very useful in developing a process design system for the extrusion process of gears. 4.5 Determination of an Equivalent Axi-Symmetric Geometry for Load Prediction. Table 5 shows the comparisons of the required loads obtained from the modified empirical equation defined by Eq. 6, the axi-symmetric simulations by using the equivalent radius, and three-dimensional simulations. According to this table it can be seen that these approaches gave similar results, and the current approximation seemed to be reasonable. Therefore, by using an axi-symmetric analysis or the modified empirical formula of Eq. 6, the required load can be efficiently predicted as a first approximation without performing the threedimensional analysis for the actual complex gear geometry. Fig. 11 FE simulation results for complete filling of a tooth according to the geometric ratio of a spur gear: a the area ratio between the total teeth and a gear and b the area ratio between a gear and root Fig. 12 Relationship between the determined design parameter ER rõo and complete filling status of a tooth The suggested major design parameter ER r/o was in good agreement with the FE simulation results in the following aspects: 1 The parameter is composed of merely geometric parameters of a spur gear and the radius of the billet does not influence this value. 2 The increase of the mandrel radius or the decrease of root radius increases the value of ER r/o. This means the reduction of having underfilling risk. 5 Conclusions In this study, the major design parameter in manufacturing spur gears was determined for cold forward extrusion without or with a mandrel by three-dimensional finite element investigations. The following conclusions were arrived at from the current study: 1 Geometry of a product and the radius of the mandrel are important in forming underfilling in a die cavity among the classified process variables investigated. 2 The major design parameter ER r/o was defined as the ratio between extrusion ratios calculated based on the radii of addendum, gear root and mandrel. This parameter describes well the risk of having underfilling in gear extrusion when its value was below than 2.6. 3 In the prediction of the required load, an axi-symmetric FE simulation for the cross-section having an equivalent area of a product provides a similar result in comparison with the three-dimensional FE simulation. A simplified empirical equation was determined for load calculation as a first approximation for a design engineer. Acknowledgment The authors wish to thank the Components and Materials Technology Development project from the Ministry of Commerce, Industry and Energy and WooShin Engineering & Mechatronics Corporation for the grant without which this work would not have been possible. Nomenclature A G cross-sectional area of gear A O the area of the initial billet A R cross-sectional area of root A T cross-sectional area of total gear teeth AR G/R area ratio of gear area to root area AR T/G area ratio of total teeth area to gear area C shape factor in the modified empirical equation for load prediction ER e extrusion ratio based on the equivalent radius ER o extrusion ratio based on the radius of addendum 262 Õ Vol. 126, MAY 2004 Transactions of the ASME

ER r extrusion ratio based on the radius of root ER r/o. ratio between extrusion ratios defined by addendum and root radii including P forming load R radius of billet container R b radius of base R do outer radius of billet container R dr radius of die relief R e equivalent radius of the cross-sectional geometry of a gear R m radius of mandrel R o radius of addendum R p radius of pitch R r radius of root a e entrance angle of die a r relief angle of die c clearance ratio h dc height of billet container h l height of die land n teeth number m module p profile shift coefficient pressure angle effective strain effective stress yield stress o References 1 Chitkara, N. R., and Bhutta, M. A., 1996, Near-Net Shape Forging of Spur Gear Forms: An Analysis and Some Experiments, Int. J. Mech. Sci., 38 8 9, pp. 891 916. 2 Osman, F. H., and Bramley, A. N., 1995, Preform Design for Forging Rotationally Symmetric Parts, CIRP Ann., 44 1, pp. 227 230. 3 Szentmihali, V., Lange, K., Tronel, Y., Chenot, J. L., and Ducloux, R., 1994, 3-D Finite-Element Simulation of the Cold Forging of Helical Gears, J. Mater. Process. Technol., 43, pp. 279 291. 4 Mamalis, A. G., Manolakos, D. E., and Baldoukas, A. K., 1996, Simulation of the Precision Forging of Bevel Gears Using Implicit and Explicit FE Techniques, J. Mater. Process. Technol., 57, pp. 164 171. 5 Rahman, A. R. O. A., and Dean, T. A., 1981, The Quality of Hot Forged Spur Gear Forms. Part II: Tooth Form Accuracy, Int. J. Mach. Tool Des. Res., 21 2, pp. 129 141. 6 Sadeghi, M. H., and Dean, T. A., 1994, Precision Forging Straight and Helical Spur Gears, J. Mater. Process. Technol., 45, pp. 25 30. 7 Doege, E., and Bohnsack, R., 2000, Closed Die Technologies for Hot Forging, J. Mater. Process. Technol., 98, pp. 165 170. 8 Kondo, K., and Ohga, K., 1995, Precision Cold Die Forging of a Ring Gear by Divided Flow Method, Int. J. Mach. Tools Manuf., 35 8, pp. 1105 1113. 9 Kim, S. Y., and Im, Y. T., 2000, Three-Dimensional Finite Element Simulations of Shape Rolling of Bars, Int. J. Form. Proc., 3 3 4, pp. 253 278. 10 Han, C. H., and Yang, D. Y., 1988, Further Investigation Into Extrusion of Trocoidal Gear Sections Considering Three-Dimensional Plastic Flow, Int. J. Mech. Sci., 30 1, pp. 13 30. 11 Yang, D. Y., Kim, H. S., Lee, C. M., and Han, C. H., 1990, Analysis of Three-Dimensional Extrusion of Arbitrarily Shaped Tubes, Int. J. Mech. Sci., 32 2, pp. 115 127. 12 Altan, T., Oh, S. I., and Gegel, H. L., 1983, Metal Forming: Fundamentals and Applications, American Society for Metals, Ohio. 13 Kwak, D. Y., Cheon, J. S., and Im, Y. T., 2002, Remeshing for Metal Forming Simulations Part I: Two-Dimensional Quadrilateral Remeshing, Int. J. Numer. Methods Eng., 53, pp. 2463 2500. 14 Kwak, D. Y., and Im, Y. T., 2002, Remeshing for Metal Forming Simulations- Part II: Three-Dimensional Hexahedral Mesh Generation, Int. J. Numer. Methods Eng., 53, pp. 2501 2528. 15 Lee, G. A., Kwak, D. Y., Kim, S. Y., and Im, Y. T., 2002, Analysis and Design of Flat-Die Hot Extrusion Process 1. Three-Dimensional Finite Element Analysis, Int. J. Mech. Sci., 44, pp. 915 934. 16 Lee, G. A., and Im, Y. T., 2002, Analysis and Design of Flat-Die Hot Extrusion Process 2. Numerical Design of Bearing Lengths, Int. J. Mech. Sci., 44, pp. 935 946. 17 Lawry, M. H., 2000, I-DEAS Master Series 2.0 Student Guide, SDRC. 18 International Cold Forging Group, 1996, Cold Forgeable Steels. Journal of Manufacturing Science and Engineering MAY 2004, Vol. 126 Õ 263