- Email: [email protected]
An upper-bound analysis for the forging of spur gears
Journal of Materials Processing Technology 104 (2000) 67±73
An upper-bound analysis for the forging of spur gears Jongung Choia, Hae-Yong Chob,*, Chang-Yong Joc a Institute of Metal Forming Intzestr. 10 D-52056 Aachen Germany Dept. of Mechanical Engineering, Chungbuk National University, Cheongju, Chungbuk, 361-763, South Korea c High Temperature Materials Group, Korea Institute of Machinery and Materials, Changwon, Kyungnam, 641-010, South Korea b
Received 21 May 2000
Abstract The forging of spur gears has been investigated by means of upper-bound analysis. A kinematically admissible velocity ®eld for the forging of spur gears has been newly proposed; especially, a neutral surface has been introduced into the forging of gears by using hollow billets with a ¯at punch. The half pitch of the gear has been divided into seven deformation regions. By using the kinematically admissible velocity ®eld, the power requirements and suitable conditions for the forging of spur gears were successfully calculated by a numerical method. As a result, (i) for reducing the forging load, the use of a hollow billet is more effective than a solid billet; (ii) for increasing the dimensional accuracy of forged gears, punch with a mandrel is desirable; (iii) the radius of the neutral surface and the inner radius approach zero as the height reduction increases; (iv) a suitable number of teeth for the forging of a spur gear was found to be between 15 and 20; (v) for the dimensional accuracy of forged gears, it is necessary to determine the elastic deformation of die and workpiece, the clearance between the die and punch, etc. # 2000 Elsevier Science S.A. All rights reserved. Keywords: Spur gear; Gear forging; Kinematically admissible velocity ®eld; Involute curve; Neutral surface
1. Introduction Two groups of manufacturing methods for gears, i.e. cutting and non-cutting, are available in engineering industry. The forging of spur gears is one of the non-cutting methods. The process for the forging of gears has been developed recently [1±7]. For the forging of spur gears, the way to complete ®lling up of the material into a die cavity is regarded as the most important aspect for improving the dimensional accuracy of gears. For complete ®lling up, predicting the power requirement is an important feature of the forging process. Grover and Juneja [8] and Kondo and coworkers [9] analyzed the forging of the spur gears by the upper-bound method. They assumed the tooth pro®le to be trapezoidal. Abdul and Dean [4] assumed the tooth pro®le as parallel lines to the centerline of the teeth. Nevertheless, the normal velocity is discontinuous on the shear surface in the analysis done by Juneja and Kondo and the velocity ®eld due to Dean cannot be used to evaluate the condition at the surface of the die.
*
Corresponding author.
This paper describes a mathematical method based on the upper-bound theory which enables the simulation of the forging of spur gears by using solid billets with ¯at punch, and hollow billets with ¯at punch or a punch with a mandrel. Especially, the involute curve has been used to represent the sides of gear tooth and a neutral surface was introduced for forging using the hollow billets with ¯at punch. From the base circle inwards, the tooth pro®le has been assumed as an arc as shown in Fig. 2. Numerical calculations have been done to study the effect of various variables such as the module (M), the number of teeth (N), the ratio of the inner to the outer diameter (Di/Do), etc. The results are compared with experimental results carried out using Al 2024 alloy. 2. Upper-bound analysis 2.1. Kinematically admissible velocity ®eld The cylindrical coordinate system (r, y, z) is used in this analysis. Throughout the analysis, the following assumptions are employed: (1) The shape of the free ¯ow surface is a circle centered from the gear center O. (2) A constant friction factor is considered on the die±workpiece
0924-0136/00/$ ± see front matter # 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 0 ) 0 0 5 2 4 - 0
68
J. Choi et al. / Journal of Materials Processing Technology 104 (2000) 67±73
5. Nomenclature Di Dm Do E_ F E_ P E_ S m M N r, y, z rb rf rn rr R t u Ur, Uy, Uz
inner diameter of billet (mm) diameter of mandrel (mm) outer diameter of billet (mm) frictional power dissipation rates internal power dissipation rates shear power dissipation rates friction constant module of the gear number of teeth cylindrical coordinates system radius of the base circle of the gear (mm) radius of the fillet of the gear (mm) radius of the neutral surface (mm) root circle of the gear (mm) distance from the gear center O to a certain point on the involute curve (mm) instantaneous height of the billet being deformed (mm) velocity of punch (mm/s) velocity components
Greek letters a half pitch angle (rad) e_ strain rates e_ effective strain rates volumetric strain rates e_ V angle of the fillet region (rad) yf j angle between the radial velocity vector and the normal direction to the die surface at the fillet (rad)
Fig. 1. Schematic illustration of spur gear forging.
sub-divided, basically into seven zones of deformation labeled I±VII, where the plastic ¯ow is assumed to take place. The neutral surface assumed as a circle with radius rn as shown in Fig. 2, and for each step it was determined using the Golden section method [10]. Selecting the radius of the neutral surface is important in analyzing the forging of spur gears. For forging using solid billets, the radius rn can be selected as zero, and for forging using hollow billets and a punch with mandrel, we could select rn as the radius of the mandrel. If the punch moves down with unit velocity, the axial velocity, Uz and the axial strain rate are given by the following equation: u Ur ÿ Z t where t represents the current height of the workpiece. 1. For deformation region I (0ya, rirrn)
contact-ing surface. (3) The diameter of the billet is equal to that of the root circle. (4) The material is homogeneous and rigid±work-hardening. The kinematically admissible velocity ®eld of the workpiece should satisfy the volume constancy and boundary condition expressed as @Ur 1 @Uy ; e_ yy Ur ; @r r @y e_ v e_ rr e_ yy e_ zz 0 e_ rr
e_ zz
@Uz ; @z
The boundary condition is that on the surface of the workpiece the material should not ¯ow across the die surface. The process for the forging of spur gears is illustrated schematically in Fig. 1. To analyze the forging of spur gears with N teeth, the gear was divided into 2N deforming units. Fig. 2 shows a generic deforming unit bounded by two plane of symmetry with adjacent units. No metals can cross or shear along a plane of symmetry, hence the deformation within a unit is self-contained, with interaction with adjacent units, throughout the forging process. Each of the deforming units, shown in Fig. 2, is then further
Fig. 2. Deformation zones for a half pitch of the spur gear.
J. Choi et al. / Journal of Materials Processing Technology 104 (2000) 67±73
This region is bounded by a neutral surface as shown in Fig. 2. In this region, it is assumed that the circumferential velocity is zero. The velocity ®eld for this region is given as follows: u r2 r ÿ n ; Uy 0 Ur r 2t 2. For deformation region II (0yyI, rnrrf) In this region, the workpiece makes contact with the die surface FG, so it is assumed that the radial velocity is zero. The velocity ®eld for this region is given as follows: u Ur 0; Uy ry t 3. For deformation region III (yIya, rnrrr) At the boundary of this region, it is assumed that the circumferential velocity UyII UyIII ; where yy I , UyIII 0; where ya and the radial velocity UrIII UrI 0; where rrn. The velocity ®eld for this region is given as follows: u rn2 a ur yI rÿ a ÿ y ; Uy Ur r 2t a ÿ yI t a ÿ yI 4. For deformation region IV (yIyyS, rrrrb) The boundary condition in this region is that the normal velocity to the die surface should be zero. The velocity ®eld for this region is given as follows: ur CIV ur CIV ; Uy cot j Ur r r 2t 2t where cot j should be expressed as a function of radius r as in the following equation: cot j CIV
rr rf 4 ÿ r 2 ÿ rf2 2 ÿ r 2 ÿ rr2 2rf rr 2 ÿ r 2
2 2 rr 2 ÿ r 2 1=2 2 rr rf r ÿ rr 2rf 2 2 ur y1 ur a r n 2t a ÿ y1 2t a ÿ y1
;
1 CII tanÿ1 A tanÿ1 B 2 0q1 2rf r2 ÿ r 2 sinÿ1 @ p A 2 rf rr rf q 1 ÿ r 2 ÿ rr2 2rf rr 2 ÿ r 2 4 rr rf 2 q 2rf rr 2rf rr 2 ÿ r 2 4rf rr rf ; Aÿ rr r 2 ÿ rr2 q 2rf rr 2rf rr 2 ÿ r 2 4rf rr rf Bÿ rr r 2 ÿ rr2 6. For deforming region VI (ySy(ySinv fR), rbrR) The workpiece makes contacts with the die surface JM, which has an involute pro®le as shown in Fig. 2. Thus the velocity normal to the die surface should be zero, Uy cos fRÿUr sin fR0. The angle y on the involute curve can be expressed as a function of radius r only in the following equation: p p r 2 ÿ rb2 r 2 ÿ rb2 ÿ tanÿ1 y yS rb rb Therefore, it can be assumed that the circumferential velocity on the involute curve is a function of radius r only and that the circumferential velocity in the deforming region is a function of radius r and angle y. The velocity ®eld for this region is given as follows: ur CVI ur CVI 2 y ÿ yS fr Ur 2 ; Uy r r 3t 3t where fr tanÿ1 r 2 ÿ rb2 1=2 =rb : CVI
urb3 rb CI00 CIV rb 00 C CV r; 6t a ÿ yS 2t a ÿ yS II
CI00 CI rrb ; CII00 CII rrb
5. For deformation region V (ySya, rrrrb) The boundary condition of this region is that UyV UyIV and Uy0 for yyS and ya, respectively. Thus it can be assumed that the circumferential velocity is distributed as a linear function of angle y. The velocity ®eld for this region is given as follows: ur CI CIV CII CV r 2t 2tr a ÿ y a ÿ yS r a ÿ y ur CIV cot j Uy r a ÿ yS 2t q 1 r 2 ÿ rr2 2rf rr 2 ÿ r 2 CI 2 0q1 2rf rr 2 ÿ r 2 2rf2 sinÿ1 @ p A 2 rf rr rf
69
Ur
(1)
7. For deformation region VII (ySinv fRya, rbrR) The boundary condition of this region is that UyVII UyVI for yySinv fR and Uy0 for ya, where inv fRconstant. Thus it can be assumed that the circumferential velocity is distributed as a linear function of circumferential coordinate y. The velocity ®eld for this region is given as follows: ur ur CVI ÿ 2 Ur CR 2t 6t r u r rb tanÿ1 Rcon ÿ Rcon CR 2r 3t 2 1 Rcon CVII Uy CVI ÿ tanÿ1 Rcon 2 r r r aÿy ur CVI 2 inv fR fr (2) a ÿ yS inv fR 3t r
70
J. Choi et al. / Journal of Materials Processing Technology 104 (2000) 67±73
3. Results and discussion
where p inv fR r 2 ÿ rb2 ; CR ; a ÿ yS inv fR rb 1 CR0 a ÿ yS inv fR 2 1 CI00 CIV CII00 urb CVI ÿ 2 CV ÿ CR CVII 6t a ÿ yS 2t a ÿ yS rb p p R2 ÿ rb2 R2 ÿ rb2 ÿ tanÿ1 inv fR rb rb Rcon
2.2. Upper-bound solution 1. Internal power dissipation rates (E_ P ) The normal strain rates are calculated using Eq. (2). The strain rate for each region is calculated using the following equations: 1 @Uy Uy 1 @Ur ÿ ; e_ ry r 2 @r r @y 1 @Uy 1 @Uz 1 @Ur @Uz ; e_ rz (3) e_ yz @r 2 @z r @y 2 @r _ expression Using in this the effective strain rate, e, becomes rq 2 2 e_ e_ e_ 2yy e_ 2zz 2_e2ry 3 rr In addition, the internal energy dissipation rates are expressed as Z (4) e_ P;i s0 e_ i dV V
Here, i represents each of the regions I±VII. The internal power of deformation can be calculated numerically. 2. Shear power dissipation rates (E_ S ) The shear power dissipated at the surface of velocity discontinuity is given by Z s0 (5) E_ S p jDVj dS 3 S where |DV| is the velocity discontinuity and ``dS'' is the area of the surface. 3. Frictional power dissipation rates (E_ F ) The frictional power dissipated at each die±workpiece interface is given by the following equation: Z ms0 jDvj dA (6) E_ F p 3 FA where dA is the area of the die±workpiece interface, m the frictional constant and |Dv| the velocity discontinuity at each interface.
The forging of spur gears using solid cylindrical billets and hollows billets with a ¯at punch and a punch with a mandrel has been investigated by means of upper-bound analysis. A kinematically admissible velocity ®eld for forging of spur gears has been proposed, wherein an involute curve has been introduced to represent the shape of the die pro®le. The half pitch of the gear has been divided into seven deformation regions. Especially, a neutral surface has been introduced into the forging of spur gears by using hollow billets with a ¯at punch. The neutral surface was assumed as a circle with radius rn as shown in Fig. 2 and for each deforming step the radius rn was determined using the Golden section method in the analysis. To select the radius of the neutral surface is important in the analysis for the forging of gears using a hollow cylindrical billet with a ¯at punch. For forging using solid billets, the radius rn should be selected as zero, and for forging using hollow billets and a punch with a mandrel, a radius rn, or the radius of mandrel, should be assumed. Numerical calculations have been done to study the effect of various variables such as the module, the number of teeth, the friction factor, etc. The numerical values of the relative average punch pressure are determined for gears of modules from 1 to 2 with a pressure angle of 208. For each module, the number of teeth is varied from 10 to 35, the thickness of gear is 10 mm, the height of tooth is 2.25 m and constant friction factors are considered. The ratio of the reduction in height is varied from 0 to 40% and the ratio of inner and outer diameters is varied from 0.1 to 0.4. For a given module of a gear, the variation of relative pressures and forging loads are found by varying the friction factors from 0.0 to 0.1 and the number of teeth. For the given conditions, the calculated results are shown in Fig. 3. The ®gures show the variation of relative forging pressure with reduction in height and the ratio of inner and outer diameters for number of teeth 15, module 2 and friction condition, m0.1. In the ®gures, each curve represents the ratio of the inner and outer diameters. The relative forging pressures and forging loads are similarly varied independently of the ratio of the diameters, thus the ratio 0.35 was selected for the calculations in this study. The relative forging pressures are found for solid and hollow gears by varying the reduction in height for number of teeth 15, module 2 at given friction factors, m0.1 as shown in Fig. 4. The relative forging pressures for hollow gears are smaller than those for solid gears. From the above results, it is found that hollow gears are suitable to decrease the forging load. To inspect the effect of the neutral surface and inner radius, the radii of the neutral surface and the inner radius are plotted versus height reduction as shown in Fig. 5. They approach zero as the height reduction increases. This has been established in the experimental results shown in Fig. 7(b) and it corresponds with previous results. This
J. Choi et al. / Journal of Materials Processing Technology 104 (2000) 67±73
71
Fig. 5. Variation of the neutral surface and the inner radius (N15; M2.0 and m0.1).
which it is concluded that it is not suitable to forge with a number of teeth below 10 or above 25 (Fig. 7). The upper-bound solutions were compared with experimental results, which were obtained using Al 2024 alloy. The lubricant was MoS2. The chemical compositions of Al 2024 alloy are shown in Table 1 and the ¯ow stress of the material is expressed as [11]: s0 358:0 e0:156 MPa
(7)
Fig. 3. Presentation of (a) relative pressures and (b) forging loads, for each inner radii of the hollow billet.
means that a punch with mandrel should be suitable for complete corner ®lling. The relative forging pressures are found by varying the number of teeth with respect to the reduction in height for modules 1.0 and 2.0 at given friction factors, m0.1 as shown in Fig. 6. The relative forging pressures at the ®nal step are minimum around the number of teeth of 15±20 for each module. From these the suitable number of teeth for each module in the forging of gears can be found, from
Fig. 4. Comparison of the relative pressure between a solid and a hollow billet.
Fig. 6. Comparison of relative pressure for different number of teeth: (a) M1.0, m0.1; (b) M2.0, m0.1.
72
J. Choi et al. / Journal of Materials Processing Technology 104 (2000) 67±73 Table 1 Chemical composition (wt.%) of Al 2024 aluminum alloy Al
Cu
Si
Fe
Mn
Mg
Zn
Pb
92.2
4.48
0.60
0.46
0.87
1.12
0.20
0.056
A ¯at punch was used for the forging for gears using a solid and hollow billets in the experimental work. The theoretically calculated loads are in good agreement with the experimental results, as shown in Fig. 8. In the ®gure, the small difference between the theoretical and experimental results could be caused by elastic deformation of the die,
Fig. 7. Billet and forged gears for each step: (a) solid gears; (b) hollow gears.
Fig. 8. Comparison of loads between analysis and experiment for the forging of spur gears: (a) solid billet (N15, M2.0, m0.1); (b) hollow billet (N15, M2.0, m0.1).
Fig. 9. Micro-structures of: (a) the billet; (b) the forged solid gear; (c) the forged hollow gear (200) etched with Keller's solution.
J. Choi et al. / Journal of Materials Processing Technology 104 (2000) 67±73
elastic spring-back of the workpiece, extrusion of the workpiece through the clearance between the punch and die, etc. These factors should be considered in the design of an accurate die for the forging of gears. The billets and the forged gears for each step are shown in Fig. 7. The micro-structures of the billet and the forged gears are shown in Fig. 9. A globular structure can be seen in (a), dislocations and ®ber ¯ow are shown in (b) and (c). It is seen that the structures of the forged gears have smaller grain size than that of the billet material. The hardness of the forged gears for each step shows that hardness of hollow gears is higher than that of solid gears. This means that the hollow gears are more work-hardened than the solid gears. It was found that the results of experiment are in good agreement with the upper-bound solutions. 4. Conclusions Forging of spur gears has been investigated by means of upper-bound analysis. From numerical calculations and experimental testing, the following conclusions are made. 1. A kinematically admissible velocity ®eld for the forging of spur gears has been newly proposed. A neutral surface has been introduced into the forging of hollow gears from hollow billets with a ¯at punch. Load requirements and suitable conditions for the forging of spur gears were successfully calculated by a numerical method. In addition, it was found that the calculated solutions are in good agreement with the experimental results. Therefore, the kinematically admissible velocity ®eld in this study is very useful to predict the load requirement for the forging of spur gears.
73
2. The forging pressure increased with the number of teeth and reduction in height for a given module. The proper number of teeth for forging is 15±20 at modules 1 and 2. 3. The radius of the neutral surface and the inner radius approach zero as the height reduction increases. This has been con®rmed by the experimental results. 4. For reducing the forging load, the use of hollow billet is more effective than the use of a solid billet. References [1] S.K. Samanta, Helical gear: a novel method of manufacturing it, NAMRC (1976) 199±205. [2] F. Dohmann, O. Traudt, Metal ¯ow and tool stress in cold forging of gear components, J. Adv. Technol. Plasticity II, (1986) 1081±1089. [3] A.R.O. Abdul-Rahman, T.A. Dean, The quality of hot forged spur gear forms. Part I: Mechanical and metallurgical properties, Int. J. Mach. Tool Des. Res. 21 (2) (1981) 109±127. [4] N.A. Abdul, T.A. Dean, An analysis of the forging of spur gear forms, Int. J. Mach. Tool Des. Res. 26 (2) (1986) 113±123. [5] S. Fujikawa, H. Yoshioka, S. Shimamura, Cold- and warm-forging applications in the automotive industry, J. Mater. Process. Technol. 35 (1992) 317±342. [6] Moriguchi, Cold forging of gear and other complex shapes, J. Mater. Process. Technol. 35 (1992) 439. [7] Choi, H.Y. Cho, H.H. Kwon, A new extrusion process of helical gears: experimental study, J. Mater. Process. Technol. 44 (1993) 35. [8] O.P. Grover, B.L. Juneja, Analysis of closed-die forging of gear-like elements, J. Adv. Technol. Plasticity II (1984) 888±893. [9] K. Ohga, K. Kondo, T. Jitsunari, Research on precision die forging utilizing divided ¯ow, Bull. JSME 28 (244) (1985) 2451±2459. [10] Vanderplaats, Numerical Optimization Techniques for Engineering Design, McGraw-Hill, New York, 1984, p. 41. [11] Yang, Investigation into non-steady state 3-dimensional extrusion of trochoidal helical gear by rigid plastic ®nite element method, Ann. CIRP 13 (1) (1994) 223±229.
An upper-bound analysis for the forging of spur gears Jongung Choia, Hae-Yong Chob,*, Chang-Yong Joc a Institute of Metal Forming Intzestr. 10 D-52056 Aachen Germany Dept. of Mechanical Engineering, Chungbuk National University, Cheongju, Chungbuk, 361-763, South Korea c High Temperature Materials Group, Korea Institute of Machinery and Materials, Changwon, Kyungnam, 641-010, South Korea b
Received 21 May 2000
Abstract The forging of spur gears has been investigated by means of upper-bound analysis. A kinematically admissible velocity ®eld for the forging of spur gears has been newly proposed; especially, a neutral surface has been introduced into the forging of gears by using hollow billets with a ¯at punch. The half pitch of the gear has been divided into seven deformation regions. By using the kinematically admissible velocity ®eld, the power requirements and suitable conditions for the forging of spur gears were successfully calculated by a numerical method. As a result, (i) for reducing the forging load, the use of a hollow billet is more effective than a solid billet; (ii) for increasing the dimensional accuracy of forged gears, punch with a mandrel is desirable; (iii) the radius of the neutral surface and the inner radius approach zero as the height reduction increases; (iv) a suitable number of teeth for the forging of a spur gear was found to be between 15 and 20; (v) for the dimensional accuracy of forged gears, it is necessary to determine the elastic deformation of die and workpiece, the clearance between the die and punch, etc. # 2000 Elsevier Science S.A. All rights reserved. Keywords: Spur gear; Gear forging; Kinematically admissible velocity ®eld; Involute curve; Neutral surface
1. Introduction Two groups of manufacturing methods for gears, i.e. cutting and non-cutting, are available in engineering industry. The forging of spur gears is one of the non-cutting methods. The process for the forging of gears has been developed recently [1±7]. For the forging of spur gears, the way to complete ®lling up of the material into a die cavity is regarded as the most important aspect for improving the dimensional accuracy of gears. For complete ®lling up, predicting the power requirement is an important feature of the forging process. Grover and Juneja [8] and Kondo and coworkers [9] analyzed the forging of the spur gears by the upper-bound method. They assumed the tooth pro®le to be trapezoidal. Abdul and Dean [4] assumed the tooth pro®le as parallel lines to the centerline of the teeth. Nevertheless, the normal velocity is discontinuous on the shear surface in the analysis done by Juneja and Kondo and the velocity ®eld due to Dean cannot be used to evaluate the condition at the surface of the die.
*
Corresponding author.
This paper describes a mathematical method based on the upper-bound theory which enables the simulation of the forging of spur gears by using solid billets with ¯at punch, and hollow billets with ¯at punch or a punch with a mandrel. Especially, the involute curve has been used to represent the sides of gear tooth and a neutral surface was introduced for forging using the hollow billets with ¯at punch. From the base circle inwards, the tooth pro®le has been assumed as an arc as shown in Fig. 2. Numerical calculations have been done to study the effect of various variables such as the module (M), the number of teeth (N), the ratio of the inner to the outer diameter (Di/Do), etc. The results are compared with experimental results carried out using Al 2024 alloy. 2. Upper-bound analysis 2.1. Kinematically admissible velocity ®eld The cylindrical coordinate system (r, y, z) is used in this analysis. Throughout the analysis, the following assumptions are employed: (1) The shape of the free ¯ow surface is a circle centered from the gear center O. (2) A constant friction factor is considered on the die±workpiece
0924-0136/00/$ ± see front matter # 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 0 ) 0 0 5 2 4 - 0
68
J. Choi et al. / Journal of Materials Processing Technology 104 (2000) 67±73
5. Nomenclature Di Dm Do E_ F E_ P E_ S m M N r, y, z rb rf rn rr R t u Ur, Uy, Uz
inner diameter of billet (mm) diameter of mandrel (mm) outer diameter of billet (mm) frictional power dissipation rates internal power dissipation rates shear power dissipation rates friction constant module of the gear number of teeth cylindrical coordinates system radius of the base circle of the gear (mm) radius of the fillet of the gear (mm) radius of the neutral surface (mm) root circle of the gear (mm) distance from the gear center O to a certain point on the involute curve (mm) instantaneous height of the billet being deformed (mm) velocity of punch (mm/s) velocity components
Greek letters a half pitch angle (rad) e_ strain rates e_ effective strain rates volumetric strain rates e_ V angle of the fillet region (rad) yf j angle between the radial velocity vector and the normal direction to the die surface at the fillet (rad)
Fig. 1. Schematic illustration of spur gear forging.
sub-divided, basically into seven zones of deformation labeled I±VII, where the plastic ¯ow is assumed to take place. The neutral surface assumed as a circle with radius rn as shown in Fig. 2, and for each step it was determined using the Golden section method [10]. Selecting the radius of the neutral surface is important in analyzing the forging of spur gears. For forging using solid billets, the radius rn can be selected as zero, and for forging using hollow billets and a punch with mandrel, we could select rn as the radius of the mandrel. If the punch moves down with unit velocity, the axial velocity, Uz and the axial strain rate are given by the following equation: u Ur ÿ Z t where t represents the current height of the workpiece. 1. For deformation region I (0ya, rirrn)
contact-ing surface. (3) The diameter of the billet is equal to that of the root circle. (4) The material is homogeneous and rigid±work-hardening. The kinematically admissible velocity ®eld of the workpiece should satisfy the volume constancy and boundary condition expressed as @Ur 1 @Uy ; e_ yy Ur ; @r r @y e_ v e_ rr e_ yy e_ zz 0 e_ rr
e_ zz
@Uz ; @z
The boundary condition is that on the surface of the workpiece the material should not ¯ow across the die surface. The process for the forging of spur gears is illustrated schematically in Fig. 1. To analyze the forging of spur gears with N teeth, the gear was divided into 2N deforming units. Fig. 2 shows a generic deforming unit bounded by two plane of symmetry with adjacent units. No metals can cross or shear along a plane of symmetry, hence the deformation within a unit is self-contained, with interaction with adjacent units, throughout the forging process. Each of the deforming units, shown in Fig. 2, is then further
Fig. 2. Deformation zones for a half pitch of the spur gear.
J. Choi et al. / Journal of Materials Processing Technology 104 (2000) 67±73
This region is bounded by a neutral surface as shown in Fig. 2. In this region, it is assumed that the circumferential velocity is zero. The velocity ®eld for this region is given as follows: u r2 r ÿ n ; Uy 0 Ur r 2t 2. For deformation region II (0yyI, rnrrf) In this region, the workpiece makes contact with the die surface FG, so it is assumed that the radial velocity is zero. The velocity ®eld for this region is given as follows: u Ur 0; Uy ry t 3. For deformation region III (yIya, rnrrr) At the boundary of this region, it is assumed that the circumferential velocity UyII UyIII ; where yy I , UyIII 0; where ya and the radial velocity UrIII UrI 0; where rrn. The velocity ®eld for this region is given as follows: u rn2 a ur yI rÿ a ÿ y ; Uy Ur r 2t a ÿ yI t a ÿ yI 4. For deformation region IV (yIyyS, rrrrb) The boundary condition in this region is that the normal velocity to the die surface should be zero. The velocity ®eld for this region is given as follows: ur CIV ur CIV ; Uy cot j Ur r r 2t 2t where cot j should be expressed as a function of radius r as in the following equation: cot j CIV
rr rf 4 ÿ r 2 ÿ rf2 2 ÿ r 2 ÿ rr2 2rf rr 2 ÿ r 2
2 2 rr 2 ÿ r 2 1=2 2 rr rf r ÿ rr 2rf 2 2 ur y1 ur a r n 2t a ÿ y1 2t a ÿ y1
;
1 CII tanÿ1 A tanÿ1 B 2 0q1 2rf r2 ÿ r 2 sinÿ1 @ p A 2 rf rr rf q 1 ÿ r 2 ÿ rr2 2rf rr 2 ÿ r 2 4 rr rf 2 q 2rf rr 2rf rr 2 ÿ r 2 4rf rr rf ; Aÿ rr r 2 ÿ rr2 q 2rf rr 2rf rr 2 ÿ r 2 4rf rr rf Bÿ rr r 2 ÿ rr2 6. For deforming region VI (ySy(ySinv fR), rbrR) The workpiece makes contacts with the die surface JM, which has an involute pro®le as shown in Fig. 2. Thus the velocity normal to the die surface should be zero, Uy cos fRÿUr sin fR0. The angle y on the involute curve can be expressed as a function of radius r only in the following equation: p p r 2 ÿ rb2 r 2 ÿ rb2 ÿ tanÿ1 y yS rb rb Therefore, it can be assumed that the circumferential velocity on the involute curve is a function of radius r only and that the circumferential velocity in the deforming region is a function of radius r and angle y. The velocity ®eld for this region is given as follows: ur CVI ur CVI 2 y ÿ yS fr Ur 2 ; Uy r r 3t 3t where fr tanÿ1 r 2 ÿ rb2 1=2 =rb : CVI
urb3 rb CI00 CIV rb 00 C CV r; 6t a ÿ yS 2t a ÿ yS II
CI00 CI rrb ; CII00 CII rrb
5. For deformation region V (ySya, rrrrb) The boundary condition of this region is that UyV UyIV and Uy0 for yyS and ya, respectively. Thus it can be assumed that the circumferential velocity is distributed as a linear function of angle y. The velocity ®eld for this region is given as follows: ur CI CIV CII CV r 2t 2tr a ÿ y a ÿ yS r a ÿ y ur CIV cot j Uy r a ÿ yS 2t q 1 r 2 ÿ rr2 2rf rr 2 ÿ r 2 CI 2 0q1 2rf rr 2 ÿ r 2 2rf2 sinÿ1 @ p A 2 rf rr rf
69
Ur
(1)
7. For deformation region VII (ySinv fRya, rbrR) The boundary condition of this region is that UyVII UyVI for yySinv fR and Uy0 for ya, where inv fRconstant. Thus it can be assumed that the circumferential velocity is distributed as a linear function of circumferential coordinate y. The velocity ®eld for this region is given as follows: ur ur CVI ÿ 2 Ur CR 2t 6t r u r rb tanÿ1 Rcon ÿ Rcon CR 2r 3t 2 1 Rcon CVII Uy CVI ÿ tanÿ1 Rcon 2 r r r aÿy ur CVI 2 inv fR fr (2) a ÿ yS inv fR 3t r
70
J. Choi et al. / Journal of Materials Processing Technology 104 (2000) 67±73
3. Results and discussion
where p inv fR r 2 ÿ rb2 ; CR ; a ÿ yS inv fR rb 1 CR0 a ÿ yS inv fR 2 1 CI00 CIV CII00 urb CVI ÿ 2 CV ÿ CR CVII 6t a ÿ yS 2t a ÿ yS rb p p R2 ÿ rb2 R2 ÿ rb2 ÿ tanÿ1 inv fR rb rb Rcon
2.2. Upper-bound solution 1. Internal power dissipation rates (E_ P ) The normal strain rates are calculated using Eq. (2). The strain rate for each region is calculated using the following equations: 1 @Uy Uy 1 @Ur ÿ ; e_ ry r 2 @r r @y 1 @Uy 1 @Uz 1 @Ur @Uz ; e_ rz (3) e_ yz @r 2 @z r @y 2 @r _ expression Using in this the effective strain rate, e, becomes rq 2 2 e_ e_ e_ 2yy e_ 2zz 2_e2ry 3 rr In addition, the internal energy dissipation rates are expressed as Z (4) e_ P;i s0 e_ i dV V
Here, i represents each of the regions I±VII. The internal power of deformation can be calculated numerically. 2. Shear power dissipation rates (E_ S ) The shear power dissipated at the surface of velocity discontinuity is given by Z s0 (5) E_ S p jDVj dS 3 S where |DV| is the velocity discontinuity and ``dS'' is the area of the surface. 3. Frictional power dissipation rates (E_ F ) The frictional power dissipated at each die±workpiece interface is given by the following equation: Z ms0 jDvj dA (6) E_ F p 3 FA where dA is the area of the die±workpiece interface, m the frictional constant and |Dv| the velocity discontinuity at each interface.
The forging of spur gears using solid cylindrical billets and hollows billets with a ¯at punch and a punch with a mandrel has been investigated by means of upper-bound analysis. A kinematically admissible velocity ®eld for forging of spur gears has been proposed, wherein an involute curve has been introduced to represent the shape of the die pro®le. The half pitch of the gear has been divided into seven deformation regions. Especially, a neutral surface has been introduced into the forging of spur gears by using hollow billets with a ¯at punch. The neutral surface was assumed as a circle with radius rn as shown in Fig. 2 and for each deforming step the radius rn was determined using the Golden section method in the analysis. To select the radius of the neutral surface is important in the analysis for the forging of gears using a hollow cylindrical billet with a ¯at punch. For forging using solid billets, the radius rn should be selected as zero, and for forging using hollow billets and a punch with a mandrel, a radius rn, or the radius of mandrel, should be assumed. Numerical calculations have been done to study the effect of various variables such as the module, the number of teeth, the friction factor, etc. The numerical values of the relative average punch pressure are determined for gears of modules from 1 to 2 with a pressure angle of 208. For each module, the number of teeth is varied from 10 to 35, the thickness of gear is 10 mm, the height of tooth is 2.25 m and constant friction factors are considered. The ratio of the reduction in height is varied from 0 to 40% and the ratio of inner and outer diameters is varied from 0.1 to 0.4. For a given module of a gear, the variation of relative pressures and forging loads are found by varying the friction factors from 0.0 to 0.1 and the number of teeth. For the given conditions, the calculated results are shown in Fig. 3. The ®gures show the variation of relative forging pressure with reduction in height and the ratio of inner and outer diameters for number of teeth 15, module 2 and friction condition, m0.1. In the ®gures, each curve represents the ratio of the inner and outer diameters. The relative forging pressures and forging loads are similarly varied independently of the ratio of the diameters, thus the ratio 0.35 was selected for the calculations in this study. The relative forging pressures are found for solid and hollow gears by varying the reduction in height for number of teeth 15, module 2 at given friction factors, m0.1 as shown in Fig. 4. The relative forging pressures for hollow gears are smaller than those for solid gears. From the above results, it is found that hollow gears are suitable to decrease the forging load. To inspect the effect of the neutral surface and inner radius, the radii of the neutral surface and the inner radius are plotted versus height reduction as shown in Fig. 5. They approach zero as the height reduction increases. This has been established in the experimental results shown in Fig. 7(b) and it corresponds with previous results. This
J. Choi et al. / Journal of Materials Processing Technology 104 (2000) 67±73
71
Fig. 5. Variation of the neutral surface and the inner radius (N15; M2.0 and m0.1).
which it is concluded that it is not suitable to forge with a number of teeth below 10 or above 25 (Fig. 7). The upper-bound solutions were compared with experimental results, which were obtained using Al 2024 alloy. The lubricant was MoS2. The chemical compositions of Al 2024 alloy are shown in Table 1 and the ¯ow stress of the material is expressed as [11]: s0 358:0 e0:156 MPa
(7)
Fig. 3. Presentation of (a) relative pressures and (b) forging loads, for each inner radii of the hollow billet.
means that a punch with mandrel should be suitable for complete corner ®lling. The relative forging pressures are found by varying the number of teeth with respect to the reduction in height for modules 1.0 and 2.0 at given friction factors, m0.1 as shown in Fig. 6. The relative forging pressures at the ®nal step are minimum around the number of teeth of 15±20 for each module. From these the suitable number of teeth for each module in the forging of gears can be found, from
Fig. 4. Comparison of the relative pressure between a solid and a hollow billet.
Fig. 6. Comparison of relative pressure for different number of teeth: (a) M1.0, m0.1; (b) M2.0, m0.1.
72
J. Choi et al. / Journal of Materials Processing Technology 104 (2000) 67±73 Table 1 Chemical composition (wt.%) of Al 2024 aluminum alloy Al
Cu
Si
Fe
Mn
Mg
Zn
Pb
92.2
4.48
0.60
0.46
0.87
1.12
0.20
0.056
A ¯at punch was used for the forging for gears using a solid and hollow billets in the experimental work. The theoretically calculated loads are in good agreement with the experimental results, as shown in Fig. 8. In the ®gure, the small difference between the theoretical and experimental results could be caused by elastic deformation of the die,
Fig. 7. Billet and forged gears for each step: (a) solid gears; (b) hollow gears.
Fig. 8. Comparison of loads between analysis and experiment for the forging of spur gears: (a) solid billet (N15, M2.0, m0.1); (b) hollow billet (N15, M2.0, m0.1).
Fig. 9. Micro-structures of: (a) the billet; (b) the forged solid gear; (c) the forged hollow gear (200) etched with Keller's solution.
J. Choi et al. / Journal of Materials Processing Technology 104 (2000) 67±73
elastic spring-back of the workpiece, extrusion of the workpiece through the clearance between the punch and die, etc. These factors should be considered in the design of an accurate die for the forging of gears. The billets and the forged gears for each step are shown in Fig. 7. The micro-structures of the billet and the forged gears are shown in Fig. 9. A globular structure can be seen in (a), dislocations and ®ber ¯ow are shown in (b) and (c). It is seen that the structures of the forged gears have smaller grain size than that of the billet material. The hardness of the forged gears for each step shows that hardness of hollow gears is higher than that of solid gears. This means that the hollow gears are more work-hardened than the solid gears. It was found that the results of experiment are in good agreement with the upper-bound solutions. 4. Conclusions Forging of spur gears has been investigated by means of upper-bound analysis. From numerical calculations and experimental testing, the following conclusions are made. 1. A kinematically admissible velocity ®eld for the forging of spur gears has been newly proposed. A neutral surface has been introduced into the forging of hollow gears from hollow billets with a ¯at punch. Load requirements and suitable conditions for the forging of spur gears were successfully calculated by a numerical method. In addition, it was found that the calculated solutions are in good agreement with the experimental results. Therefore, the kinematically admissible velocity ®eld in this study is very useful to predict the load requirement for the forging of spur gears.
73
2. The forging pressure increased with the number of teeth and reduction in height for a given module. The proper number of teeth for forging is 15±20 at modules 1 and 2. 3. The radius of the neutral surface and the inner radius approach zero as the height reduction increases. This has been con®rmed by the experimental results. 4. For reducing the forging load, the use of hollow billet is more effective than the use of a solid billet. References [1] S.K. Samanta, Helical gear: a novel method of manufacturing it, NAMRC (1976) 199±205. [2] F. Dohmann, O. Traudt, Metal ¯ow and tool stress in cold forging of gear components, J. Adv. Technol. Plasticity II, (1986) 1081±1089. [3] A.R.O. Abdul-Rahman, T.A. Dean, The quality of hot forged spur gear forms. Part I: Mechanical and metallurgical properties, Int. J. Mach. Tool Des. Res. 21 (2) (1981) 109±127. [4] N.A. Abdul, T.A. Dean, An analysis of the forging of spur gear forms, Int. J. Mach. Tool Des. Res. 26 (2) (1986) 113±123. [5] S. Fujikawa, H. Yoshioka, S. Shimamura, Cold- and warm-forging applications in the automotive industry, J. Mater. Process. Technol. 35 (1992) 317±342. [6] Moriguchi, Cold forging of gear and other complex shapes, J. Mater. Process. Technol. 35 (1992) 439. [7] Choi, H.Y. Cho, H.H. Kwon, A new extrusion process of helical gears: experimental study, J. Mater. Process. Technol. 44 (1993) 35. [8] O.P. Grover, B.L. Juneja, Analysis of closed-die forging of gear-like elements, J. Adv. Technol. Plasticity II (1984) 888±893. [9] K. Ohga, K. Kondo, T. Jitsunari, Research on precision die forging utilizing divided ¯ow, Bull. JSME 28 (244) (1985) 2451±2459. [10] Vanderplaats, Numerical Optimization Techniques for Engineering Design, McGraw-Hill, New York, 1984, p. 41. [11] Yang, Investigation into non-steady state 3-dimensional extrusion of trochoidal helical gear by rigid plastic ®nite element method, Ann. CIRP 13 (1) (1994) 223±229.