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A study on the forging of external spur gears: upper-bound analyses and experiments
International Journal of Machine Tools & Manufacture 38 (1998) 1193–1208
A study on the forging of external spur gears: upper-bound analyses and experiments J.C. Choia,*, Y. Choib a
Dept. of Mechanical Design Engineering, ERC for NSDM at Pusan National University, 30 Changjeon-dong, Kumjeong-Ku, Pusan 609-735, Korea b Dept. of Mechanical Design Engineering, Graduate School, Pusan National University, 30 Changjeon-dong, Kumjeong-Ku, Pusan 609-735, Korea Received 2 June 1997; in final form 24 November 1997
Abstract In this study, the forging processes of spur gears have been investigated. The forging processes of spur gears have been classified into two types of operations, guiding one and clamping one in this investigation. Two types of forging have been analyzed by using upper bound method. The relative average punch pressure have been calculated and compared with each others for various modules and numbers of teeth on the forging of spur gears. The forging experiments were carried out with a commercial aluminum alloy. The predicted forging loads obtained by the present upper bound methods are shown to approximate to the experimental results at final filling-up stage. The forged parts obtained through the guiding type forging were compared with those obtained through the clamping type forging. 1998 Elsevier Science Ltd. All rights reserved. Keywords: Spur gear forging; Guiding type forging; Clamping type forging; Forging experiments
Notation r,,z E·T N M rb ri
Cylindrical coordinates Total power dissipation Number of teeth Module Radius of base circle Radius of mandrel
* Corresponding author. 0890-6955/98/$19.00 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 9 8 ) 0 0 0 0 9 - 1
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rinv() rm rrt rs(t) Pav ⑀ ⑀¯ ¯
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Radius of involute curve Partition radius Radius of root circle Radius of free surface when height of workpiece is t Average forging pressure Partition height Effective strain Effective stress
1. Introduction Of all the many types of machine elements that exist today, gears are among the most commonly used. In recent years, there has been an increased interest in the production of gears by the precision forming technique. The process of precision gear forging has been developed recently because of its advantages of giving high production rates, improved strength and surface finish. This is due to their inherent advantages compared with conventional cutting methods. Studies on the forming by extrusion of a gear have been practised [1,2]. Process for the cold extrusion of spur and helical gears was presented by Samanta [1]. Choi et al. proposed a new extrusion process for helical gears. They analyzed the new process by using the upper-bound method [2]. In comparison with the extrusion process, the advantage of the forging process lies in its high productivity. In the forming of gears with low height, forging is more useful. In the forging of spur gears, the way to complete filling up materials into a die cavity is regarded as the most important for improving the dimensional accuracy and strength of forged gears. An important aspect of the gear forging process is the load requirement. Two processes of gear forging, are (a) guiding type and (b) clamping type [3,4]. Yang et al. practised forging experiments respectively for spur gears and compared guiding and clamping type forging [4]. Chitkara and Kim [5] have classified the forging process of external spline gears into two methods, an upset forging method and a side extrusion method. The upset forging and the side extrusion method are the same operations as the guiding type and clamping type forging, respectively. In the guiding type forging process, the punch and the die insert are tooth-shaped. The clamping type forging is an operation in which the product is constrained to extrude sideways through an orifice in the container wall. Grover and Juneja [6] studied the metal flow in the clamping type forging of gear-like components. Abdul and Dean [7] investigated the clamping type forging of spur gear forms whose teeth profile shapes were assumed parallel to the centerline of teeth. Kondo and Ohga [8–11] developed the process, utilizing the divided flow technique for precision die forging. The process was based on the guiding type forging and was applied to some machine parts such as a spline, a regular polygon and a spur gear. The authors studied the guiding type forging of spur and helical gears by using the forging experiments and upper-bound method [12–14]. A study of clamping type forging of helical gears has been carried out by the authors [15]. In this paper, the two types of spur gears forging have been analyzed using the upper bound method and axisymmetric parts forgings have been analyzed using the finite element code DEFORM. The authors [12,13] have proposed the kinematically admissible velocity fields for the
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guiding type forging of spur gears. Those for the clamping type forging of spur gears can be obtained from the kinematically admissible velocity fields for the clamping type forging of helical gears [15], for the helix angle is zero. Numerical calculations have been carried out to investigate the effects of various parameters, such as modules, numbers of teeth and friction factors on the forging of spur gears and some forging experiments were carried out with commercial aluminum alloy. The forged parts obtained through the guiding type forging were compared with those obtained through the clamping type forging.
2. Theoretical analysis Two processes of gear forging, (a) guiding type and (b) clamping type are shown in Fig. 1 [3,4]. In the guiding type forging process, the punch and the die insert are tooth-shaped however, the cylindrical punch is used in the clamping type forging. Guiding type forging is a sort of upset forging process. The material, which is restricted by the die wall, is upset by the tooth-shaped punch. Clamping type forging is an operation in which the product is constrained to extrude sideways through an orifice in the container wall. 2.1. FEM analyses of two types of forgings of axisymmetric parts The two types of forgings of axisymmetric parts have been analyzed using a commercially available finite element program, DEFORM, for comparisons. The program works according to the rigid-plastic material-behavior model. Figure 2 shows a comparison of cavity filling between the guiding and clamping type forging.
Fig. 1. Die design schematics for the gear forging [5].
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Fig. 2. Comparison of cavity filling performance.
2.2. The guiding type forging of the spur gears In guiding type forging, the punch and the ejector are tooth-shaped, as is the die insert [see Fig. 1(a)]. The punch compresses a cylindrical billet placed in a die insert. As a consequence the billet material flows into the tooth region. Figure 3 shows the assumed deformation zones during guiding type forging of spur gear with teeth having involute profiles. The kinematically admissible velocity fields for the guiding type forging of spur gears can be seen in Table 1 [12]. 2.3. The clamping type forging of the spur gears The clamping type forging is a sort of side or radial or lateral extrusion. The punch is cylindrical and has the same dimensions as that of the root circle of the gear, which can be seen in Fig. 1(b). Figure 4 shows the assumed deformation zones during clamping type forging of spur gear with teeth having involute profiles. The kinematically admissible velocity fields for the clamping type forging of the spur gears can be obtained from the those for the clamping type forging of helical gears [15], for the helix angle is zero (in Table 2).
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Fig. 3.
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Assumed deformation zones during guiding type forging of a spur gear.
2.4. Upper bound solution The total power dissipation can be calculated numerically from Table 1 and Table 2. (i) The power dissipation due to deformation The power dissipation due to deformation is given by
冕
·· Ep,i = ¯ ⑀¯ dV
(1)
V
where, i represents each zone. (ii) Shear power dissipation The shear power dissipation at the surface of velocity discontinuity is given by · ES =
冕
¯ 兩⌬V兩dS √3
(2)
S
where, S represents the surface of the velocity discontinuity. (iii) Frictional power dissipation The frictional power dissipation at the tool/material interface is given by · Ef =
冕
m¯ 兩⌬V兩dA √3
(3)
Af
where, Af represents frictional interface. Detailed expressions for the plastic, shear and frictional power are not given here because of their complexity. Therefore the upper-bound solution of relative average punch pressure is given by
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Table 1 Kinematically admissible velocity fields of guiding type forging of spur gears u r2 ⫺ r2i 2t r
Region 1 0⭐⭐a ri ⭐ r ⭐ rm
Ur =
Region 2 0 ⭐ ⭐ b rm ⭐ r ⭐ rrt
Ur =
u r2 ⫺ r2i r2rt ⫺ r2i (r ⫺ rm)2 ⫺ 2t r rrt (rrt ⫺ rm)2
U =
u r2rt ⫺ r2i (3r2 ⫺ 4rrm + r2m) 2t rrt (rrt ⫺ rm)2
Ur =
u r2 ⫺ r2i r2rt ⫺ r2i (r ⫺ rm)2 b + 2t r rrt (rrt ⫺ rm)2 ␣ ⫺ b
U =
␣⫺ u r2rt ⫺ r2i (3r2 ⫺ 4rrm + r2m) b 2t rrt (rrt ⫺ rm)2 ␣ ⫺ b
Region 4 b ⭐ ⭐ a rrt ⭐ r ⭐ rb
Ur =
b r2rt ⫺ r2i u r2 ⫺ r2i + 2t r ␣ ⫺ b r
Region 5 b ⭐ ⭐ (t) rb ⭐ r ⭐ rinv()
Ur =
u r2 ⫺ r2i b r2rt ⫺ r2i + 2t r ␣ ⫺ b r
U =
b r2rt ⫺ r2i √r2 ⫺ r2b u r2 ⫺ r2i + 2t rb ␣ ⫺ b r b r
Ur =
u r2 ⫺ r2i 1 u √(r2 ⫺ r2b)3 u r2rt ⫺ r2i b + + 2t r ␣ ⫺ (t) 2t 3rrb 2t r ␣ ⫺ b
U = 0
Region 3 b ⭐ ⭐ a rm ⭐ r ⭐ rrt
再
冎
再
冉
冊
冉 冉
冊 冊
冎
U = 0
Region 6 (t) ⭐ ⭐ a rb ⭐ r ⭐ rs(t)
+
b u r2rt ⫺ r2i 2t rb ␣2 ⫺ {b + (t)}␣ + b(t)
U =
冉
b r2rt ⫺ r2i √r2 ⫺ r2b ⫺ ␣ u r2 ⫺ r2i + 2t rb ␣ ⫺ b r b r (t) ⫺ ␣
Axial velocity component of Regions 1–6 Uz = ⫺
u z t
冊
冦
√r2 ⫺ r2b rb √r2 ⫺ r2b ⫺ tan ⫺ 1 r r rb
冧
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Fig. 4. Assumed deformation zones during clamping type forging of spur gear.
· Et Pav = ¯ ¯ Au
(4)
· where, A is punch/workpiece contact area and Et is the total power dissipation given by the summation of all the various powers derived for the deformation zones. · · · · (5) Et = Ep + Es + Ef
3. Forging experiments The forging experiments were carried out with a commercial aluminum alloy. The material was annealed. To determine the stress–strain relation of the material, a series of ring compression
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Table 2 Kinematically admissible velocity fields of clamping type forging of spur gears. Region 1 0⭐⭐a ri ⭐ r ⭐ rt Z(r) ⭐ z ⭐ t
Ur = 0 U = 0 Uz = ⫺ u
Region 2 0⭐⭐a ri ⭐ r ⭐ rm 0 ⭐ z ⭐ Z(r)
Ur =
Region 3 0 ⭐ ⭐ b rm ⭐ r ⭐ rrt 0⭐z⭐H
u r2 ⫺ r2i 1 2 r Z(r)
U = 0 Uz =
r2 ⫺ r2i 1 ∂Z(r) u z ( ⫺ 2) 2 Z(r) r Z(r) ∂r
Ur =
u r2 ⫺ r2i r2rt ⫺ r2i (r ⫺ rm)2 ⫺ 2H r rrt (rrt ⫺ rm)2
U =
u r2rt ⫺ r2i (3r2 ⫺ 4rrm + r2m) 2H rrt (rrt ⫺ rm)2
再
Uz = ⫺ u
Region 4 b ⭐ ⭐ a rm ⭐ r ⭐ rrt 0⭐z⭐H
z H
再
Ur =
b u r2 ⫺ r2i r2rt ⫺ r2i (r ⫺ rm)2 + 2H r rrt (rrt ⫺ rm)2 ␣ ⫺ b
U =
␣⫺ u r2rt ⫺ r2i (3r2 ⫺ 4rrm + r2m) b 2 2H rrt (rrt ⫺ rm) ␣ ⫺ b
Uz = ⫺ u
Region 5 b ⭐ ⭐ a rrt ⭐ r ⭐ rb 0⭐z⭐H
冎
Ur =
z H
冉
冊
冉 冉
冊 冊
b r2rt ⫺ r2i u r2rt ⫺ r2i + 2H r ␣ ⫺ b r
U = 0 Uz = 0
Region 6 b ⭐ ⭐ (t) rb ⭐ r ⭐ rinv() 0⭐z⭐H
Ur =
b r2rt ⫺ r2i u r2rt ⫺ r2i + 2H r ␣ ⫺ b r
U =
u r2rt ⫺ r2i b r2rt ⫺ r2i √r2 ⫺ r2b + 2H rb ␣ ⫺ b r b r
Uz = 0
冎
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Table 2 Continued. Region 7 (t) ⭐ ⭐ a rb ⭐ r ⭐ rs(t) 0⭐z⭐H
Ur =
冉
冊 冉
√r2 ⫺ r2b rb √r2 ⫺ r2b ⫺ tan ⫺ 1 r r rb U =
冊
冦
1 u r2rt ⫺ r2i r2rt ⫺ r2i b u r2rt ⫺ r2i r2rt ⫺ r2i b + + + . 2H r r ␣ ⫺ b 2H rb rb ␣ ⫺ b ␣ ⫺ (t)
冉
冧
冊
b r2rt ⫺ r2i √r2 ⫺ r2b ⫺ ␣ u r2rt ⫺ r2i + 2H rb ␣ ⫺ b rb r (t) ⫺ ␣
Uz = 0
tests were performed. From the results of the tests, the flow stress and friction factor can be calculated by using the method proposed by Osakada [16], and the friction-area-divided method [17], respectively. 3.1. Ring compression test The outside diameter:inside diameter:thickness ratio of the ring is 6:3:2. The MoS2 was sprayed onto the forgings as a lubricant. The results of the tests can be seen in Table 3. From Table 3 the flow stress and friction factor have been calculated using the method proposed by Osakada [16], and the friction-area-divided method [17], respectively. The stress–strain relation is
¯ = 504.1⑀¯ 0.16 (MPa)
(6)
and the friction factor is 0.13. Table 3 Result of ring compression test Outer diameter Inner diameter Height, t (mm) ⌬t (mm) (mm) (mm)
Load (ton)
Decrease in inner diameter (%)
Reduction in height (%)
15.35 17.275 17.925 19.113 21.038 22.2
5.0 7.5 10.0 12.5 17.5 20.0
⫺1.173 ⫺9.173 ⫺7 ⫺5.173 ⫺5.173 ⫺4.667
5 25 32 42 55 60
7.588 8.188 8.025 7.888 7.888 7.85
4.75 3.75 3.4 2.9 2.25 2.0
0.25 1.25 1.6 2.1 2.75 3.0
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3.2. Gear forging The diameter of the punch for the guiding type forging is less than the cavity of the die, because the punch is moving down along the surface of the die. In this investigation, if the addendum modification coefficient of the punch is ⫺0.05, then addendum and deddendum are less than those of die by as much as 0.2 mm. In the clamping type forging, the clamping die is required for restriction of backward flow. The diameters of the cylindrical punch and hole of the clamping die are less than that of the root circle of the die. The die has been used for two types of forging. A gear shaped die cavity was made by WEDM (wire-cut EDM machine). The die was shrink fitted before W-EDM to prevent deformation which causes gear errors and excessive tensile circumferential stresses, which may result in die fracture during forging. The gear forging experiments were performed on a 300-ton hydraulic press and carried out at room temperature. The spur gear specification is shown in Table 4.
4. Results and discussion 4.1. The results of FEM analyses of the two types of forgings The results of FEM analyses are seen in Figs 2 and 5. Figure 2 shows the deformed shape of the two types of forgings. For the final stage of the forgings, Fig. 5 shows distribution of effective strain and contact pressures on the punch and the die for the guiding type forging [Fig. 5(a)], and the clamping type forging [Fig. 5 (b)]. For the analyses, the material is a commercial aluminum alloy, the stress–strain relation of the material is Eq. (1), and the friction factor is 0.13. Distribution of effective strain in the guiding type forging [Fig. 5 (a)], is almost homogeneous across the cross-section but the distribution in the clamping type forging [Fig. 5 (b)], shows a big difference along the axis. The average punch pressure is almost the same on the die for each process. In the guiding type forging, the maximum and average punch pressure is lower than in the clamping type forging. From the results it is considered that the guiding type is suitable for forging of spur gears.
Table 4 Specification of spur gear die No. of teeth Module Pressure angle Modification coefficient Standard pitch circle diameter. Base circle diameter
18 1.5 20° 0.0 27.0 mm 25.3667 mm
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Fig. 5. The results of FEM analysis for the forging of axisymmetric parts.
4.2. The relative punch pressure In the following figures, the results of upper-bound analyses of the spur gear forging have been calculated by using the kinematically admissible velocity fields (Tables 1 and 2), for clamping and guiding type forging, respectively. Figure 6 shows the relative average punch pressure of the two types of spur gears forging. The values of relative average punch pressure are determined for the spur gears with a pressure angle 20°. The height of the initial billet is 15 mm. The diameter of the mandrel is 30% of the root diameter of a spur gear. The results are found by varying the reduction in height for spur gears with 18, 20, 25 and 30 teeth and each module 1.0, 1.5, 2.0 and 3.0 at frictional condition m = 0.1, as shown in Fig. 6.
Fig. 6. Variation of the relative pressure with respect to reduction in height for different numbers of teeth and module.
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For the guiding forging, variation of the relative average pressure with respect to reduction in height has been plotted and can be seen in Fig. 6(a), that of clamping type forging is shown in Fig. 6(b). The relative average pressures increase abruptly near the final filling-up stage. The relative average punch pressures of the clamping type forgings are higher than that of the guiding type forgings. For two types of forgings, the relative average punch pressures are almost identical for different modules. At the final filling-up states, the relative average pressure a increases slightly with increasing number of teeth for the guiding type forging, and a decreases slightly for the clamping type forging, respectively. This means that the guiding type forging process remains correct in the forging of spur gears, regardless of the magnitude of module. 4.3. Forging load and pressure The variation between that of the predicted forging loads and those obtained form experiment has been plotted in Figs 7 and 8. In Figs 7 and 8, the height of the forged spur gear is 10 and 15 mm, respectively. The predicted forging loads obtained by the present upper bound methods have an approximate tendency to agree with the experimental results at final filling-up stage. Because the material of the punch is AISI D2 (Hrc-62) of which the yield stress is 2150 MPa [18], the average punch pressure values were controlled, to be 1800 MPa at the final state of the forgings. Therefore, the forged spur gears have some different filling-up states. These points are discussed in following section. For the guiding type forging, the predicted forging loads are lower than those of the clamping type forging. More energy is required for clamping type forging. 4.4. Comparisons of forged spur gears Figs 9–12 show the forged spur gears. Right sides of the products are pressing directions. In Figs 9 and 10, the forged spur gear was obtained by the guiding type forging. Figs 11 and 12
Fig. 7. Comparison of the experimental results with the upper-bound solutions (h = 10 mm).
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Fig. 8. Comparison of the experimental results with the upper-bound solutions (h = 15 mm).
Fig. 9.
Cold forged spur gear (guiding type, h = 10 mm). (a) R.H.: 25.35%; (b) R.H.: 26.76%; (c) R.H.: 27.80%.
show the results of the clamping type forging. In Figs 9–12, the height of billets are 14.2 and 21.3 mm, respectively. In Figs 9 and 10, the filling-up state of the portions of the pressing direction on the right side, is better than that of the other, because the friction force on the die wall is applied to the forged gear in a reverse direction to the punch movement. In Fig. 9(c), the filling-up state of the lower workpiece thickness is better than that of the other in Fig. 10(c). Also, for the clamping type forgings (Figs 11 and 12), the filling-up state of the lower workpiece thickness in Fig. 11(c) is better than that of the other in Fig. 12(c). However, the filling-up state of the portion near the ejector is better than that at the pressing direction, because the material which is compressed by the punch flows in a perpendicular direction to that of punch movement.
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Fig. 10.
Cold forged spur gear (guiding type, h = 15 mm). (a) R.H.: 24.52%; (b) R.H.: 27.70%; (c) R.H.: 28.30%.
Fig. 11. Cold forged spur gear (clamping type, h = 10 mm). (a) R.H.: 25.00%; (b) R.H.: 27.10%; (c) R.H.: 27.80%.
Therefore, spur gears with lower thickness are suitable for the guiding and clamping type forging process. Also, it is difficult to obtain a forged spur gear that has a good filling-up state of the portion near the ejector in guiding type forging. Conversely, complications arise near the punch in the clamping type. Comparison of Fig. 9 with Fig. 11 and Fig. 10 with Fig. 12 show that the filling-up state of the forged spur gear can be better obtained through guiding type forging at the same forging punch pressure. 5. Conclusions Through guiding and clamping type forging experiments forged spur gears have been obtained. The forging processes have been analyzed for the axisymmetric parts and spur gears by using
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Fig. 12. Cold forged spur gear (clamping type, h = 15 mm). (a) R.H.: 22.87%; (b) R.H.: 25.70%; (c) R.H.: 26.65%.
the finite element method and upper bound method, respectively. The comparisons for the forging load and pressure have been discussed. The conclusions are follows: 1. The results of 2D-FEM analysis for the two types of forgings show that the average contact pressure of guiding type forging is lower than that of clamping type forging. 2. The predicted forging loads obtained by the present upper bound methods have an approximate tendency to agree with the experimental results at final filling-up stage. From the results of the upper bound analyses, it is clear that the guiding type forging process is correct for the forging of spur gears. 3. Spur gears with a lower thickness are suitable for the two types of forging processes. Also, it is difficult to obtain a forged spur gear that has a good filling-up state of the portion near the ejector in guiding type forging. On the other hand, difficulties arise near the punch in the clamping type.
Acknowledgement This study is supported by the Korean Ministry of Education through Research Fund (ME96-E14).
References [1] S.K. Samanta, Helical gear: a noble method of manufacturing it, Proceedings 4th North American Metalworking Research Conf., Battele’s Columbus Labs, OH, USA, Society of Manufacturing Engineers, 1976, pp. 199–205. [2] J. Choi, H. Cho, H. Kwon, A new extrusion process for helical-gears: experimental study, Journal of Materials Processing Technology 43 (1994) 35–53.
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[3] Y.B. Park, D.Y. Yang, Finite element analysis for precision cold forging of helical gear using recurrent boundary conditions (in Korean), Proceedings of the KSME (I) Spring (1995) 296–301. [4] S.W. Shim, K.S. Song, D.S. Yoon, Y.B. Park, D.Y. Yang, Precision cold forging for reverse idle gear of automobile (in Korean), 2nd Forging Symposium, Pusan, Korea, May, 1995, pp. 145–157. [5] N.R. Chitkara, Y. Kim, An analysis of external spline gear forming by an upper bound energy method, International Journal of Mechanical Science 38 (7) (1996) 777–789. [6] O.P. Grover, B.L. Juneja, Analysis of closed-die forging of gear-like element, Advanced Technology of Plasticity II (1986) 888–893. [7] N.A. Abdul, T.A. Dean, An analysis of the forging of spur gear forms, International Journal of Machine Tool Design Research 26 (2) (1986) 113–123. [8] K. Ohga, K. Kondo, T. Jitsunari, Research on precision die forging utilizing divided flow, Bulletin of the JSME 28 (244) (1985) 2451–2459. [9] K. Kondo, K. Ohga, Development of precision cold die forging process utilizing divided flow, Journal of the JSTP 27 (300) (1986) 121–131. [10] K. Ohga, K. Kondo, K. Hori, Investigation into forming processes of various spur gears, Advanced Technology of Plasticity 2 (1987) 1089–1096. [11] K. Ohga, K. Kondo, Research on application of the precision cold die forging utilizing divided flow to thick products, Advanced Technology of Plasticity II (1993) 1239–1244. [12] Y. Choi, A study on the closed-die cold forging of spur gears, (in Korean), M.S. thesis, Pusan National University, 1995. [13] J.C. Choi, Y. Choi, K.D. Hur, C.H. Kim, A study on the forging of spur gears, International Journal of Mechanical Science 38 (12) (1996) 1333–1347. [14] J.C. Choi, Y. Choi, S.J. Tak, The forging of helical gears (i): experiments and upper-bound analysis, International Journal of Mechanical Science 40 (4) (1998) 325–337. [15] J.C. Choi, Y. Choi, S.J. Tak, The forging of helical gears (ii): comparisons of the forging process, International Journal of Mechanics Science (submitted). [16] K. Osakada, M. Shiraishi, S. Muraki, M. Tokuoka, Measurement of flow stress by the ring compression test, JSME International Journal Series A 34 (3) (1991) 312–318. [17] Y.-T. Lin, J.-P. Wang, An investigation into the constant friction factor with the upper-bound flow function elemental technique, Journal of Materials Processing Technology 36 (1992) 57–68. [18] I.C.F.G., General aspect of Tool Design and Tool Materials for Cold Forging of Steel, No. 4/70, No. 4/82.
A study on the forging of external spur gears: upper-bound analyses and experiments J.C. Choia,*, Y. Choib a
Dept. of Mechanical Design Engineering, ERC for NSDM at Pusan National University, 30 Changjeon-dong, Kumjeong-Ku, Pusan 609-735, Korea b Dept. of Mechanical Design Engineering, Graduate School, Pusan National University, 30 Changjeon-dong, Kumjeong-Ku, Pusan 609-735, Korea Received 2 June 1997; in final form 24 November 1997
Abstract In this study, the forging processes of spur gears have been investigated. The forging processes of spur gears have been classified into two types of operations, guiding one and clamping one in this investigation. Two types of forging have been analyzed by using upper bound method. The relative average punch pressure have been calculated and compared with each others for various modules and numbers of teeth on the forging of spur gears. The forging experiments were carried out with a commercial aluminum alloy. The predicted forging loads obtained by the present upper bound methods are shown to approximate to the experimental results at final filling-up stage. The forged parts obtained through the guiding type forging were compared with those obtained through the clamping type forging. 1998 Elsevier Science Ltd. All rights reserved. Keywords: Spur gear forging; Guiding type forging; Clamping type forging; Forging experiments
Notation r,,z E·T N M rb ri
Cylindrical coordinates Total power dissipation Number of teeth Module Radius of base circle Radius of mandrel
* Corresponding author. 0890-6955/98/$19.00 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 9 8 ) 0 0 0 0 9 - 1
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rinv() rm rrt rs(t) Pav ⑀ ⑀¯ ¯
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Radius of involute curve Partition radius Radius of root circle Radius of free surface when height of workpiece is t Average forging pressure Partition height Effective strain Effective stress
1. Introduction Of all the many types of machine elements that exist today, gears are among the most commonly used. In recent years, there has been an increased interest in the production of gears by the precision forming technique. The process of precision gear forging has been developed recently because of its advantages of giving high production rates, improved strength and surface finish. This is due to their inherent advantages compared with conventional cutting methods. Studies on the forming by extrusion of a gear have been practised [1,2]. Process for the cold extrusion of spur and helical gears was presented by Samanta [1]. Choi et al. proposed a new extrusion process for helical gears. They analyzed the new process by using the upper-bound method [2]. In comparison with the extrusion process, the advantage of the forging process lies in its high productivity. In the forming of gears with low height, forging is more useful. In the forging of spur gears, the way to complete filling up materials into a die cavity is regarded as the most important for improving the dimensional accuracy and strength of forged gears. An important aspect of the gear forging process is the load requirement. Two processes of gear forging, are (a) guiding type and (b) clamping type [3,4]. Yang et al. practised forging experiments respectively for spur gears and compared guiding and clamping type forging [4]. Chitkara and Kim [5] have classified the forging process of external spline gears into two methods, an upset forging method and a side extrusion method. The upset forging and the side extrusion method are the same operations as the guiding type and clamping type forging, respectively. In the guiding type forging process, the punch and the die insert are tooth-shaped. The clamping type forging is an operation in which the product is constrained to extrude sideways through an orifice in the container wall. Grover and Juneja [6] studied the metal flow in the clamping type forging of gear-like components. Abdul and Dean [7] investigated the clamping type forging of spur gear forms whose teeth profile shapes were assumed parallel to the centerline of teeth. Kondo and Ohga [8–11] developed the process, utilizing the divided flow technique for precision die forging. The process was based on the guiding type forging and was applied to some machine parts such as a spline, a regular polygon and a spur gear. The authors studied the guiding type forging of spur and helical gears by using the forging experiments and upper-bound method [12–14]. A study of clamping type forging of helical gears has been carried out by the authors [15]. In this paper, the two types of spur gears forging have been analyzed using the upper bound method and axisymmetric parts forgings have been analyzed using the finite element code DEFORM. The authors [12,13] have proposed the kinematically admissible velocity fields for the
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guiding type forging of spur gears. Those for the clamping type forging of spur gears can be obtained from the kinematically admissible velocity fields for the clamping type forging of helical gears [15], for the helix angle is zero. Numerical calculations have been carried out to investigate the effects of various parameters, such as modules, numbers of teeth and friction factors on the forging of spur gears and some forging experiments were carried out with commercial aluminum alloy. The forged parts obtained through the guiding type forging were compared with those obtained through the clamping type forging.
2. Theoretical analysis Two processes of gear forging, (a) guiding type and (b) clamping type are shown in Fig. 1 [3,4]. In the guiding type forging process, the punch and the die insert are tooth-shaped however, the cylindrical punch is used in the clamping type forging. Guiding type forging is a sort of upset forging process. The material, which is restricted by the die wall, is upset by the tooth-shaped punch. Clamping type forging is an operation in which the product is constrained to extrude sideways through an orifice in the container wall. 2.1. FEM analyses of two types of forgings of axisymmetric parts The two types of forgings of axisymmetric parts have been analyzed using a commercially available finite element program, DEFORM, for comparisons. The program works according to the rigid-plastic material-behavior model. Figure 2 shows a comparison of cavity filling between the guiding and clamping type forging.
Fig. 1. Die design schematics for the gear forging [5].
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Fig. 2. Comparison of cavity filling performance.
2.2. The guiding type forging of the spur gears In guiding type forging, the punch and the ejector are tooth-shaped, as is the die insert [see Fig. 1(a)]. The punch compresses a cylindrical billet placed in a die insert. As a consequence the billet material flows into the tooth region. Figure 3 shows the assumed deformation zones during guiding type forging of spur gear with teeth having involute profiles. The kinematically admissible velocity fields for the guiding type forging of spur gears can be seen in Table 1 [12]. 2.3. The clamping type forging of the spur gears The clamping type forging is a sort of side or radial or lateral extrusion. The punch is cylindrical and has the same dimensions as that of the root circle of the gear, which can be seen in Fig. 1(b). Figure 4 shows the assumed deformation zones during clamping type forging of spur gear with teeth having involute profiles. The kinematically admissible velocity fields for the clamping type forging of the spur gears can be obtained from the those for the clamping type forging of helical gears [15], for the helix angle is zero (in Table 2).
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Fig. 3.
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Assumed deformation zones during guiding type forging of a spur gear.
2.4. Upper bound solution The total power dissipation can be calculated numerically from Table 1 and Table 2. (i) The power dissipation due to deformation The power dissipation due to deformation is given by
冕
·· Ep,i = ¯ ⑀¯ dV
(1)
V
where, i represents each zone. (ii) Shear power dissipation The shear power dissipation at the surface of velocity discontinuity is given by · ES =
冕
¯ 兩⌬V兩dS √3
(2)
S
where, S represents the surface of the velocity discontinuity. (iii) Frictional power dissipation The frictional power dissipation at the tool/material interface is given by · Ef =
冕
m¯ 兩⌬V兩dA √3
(3)
Af
where, Af represents frictional interface. Detailed expressions for the plastic, shear and frictional power are not given here because of their complexity. Therefore the upper-bound solution of relative average punch pressure is given by
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Table 1 Kinematically admissible velocity fields of guiding type forging of spur gears u r2 ⫺ r2i 2t r
Region 1 0⭐⭐a ri ⭐ r ⭐ rm
Ur =
Region 2 0 ⭐ ⭐ b rm ⭐ r ⭐ rrt
Ur =
u r2 ⫺ r2i r2rt ⫺ r2i (r ⫺ rm)2 ⫺ 2t r rrt (rrt ⫺ rm)2
U =
u r2rt ⫺ r2i (3r2 ⫺ 4rrm + r2m) 2t rrt (rrt ⫺ rm)2
Ur =
u r2 ⫺ r2i r2rt ⫺ r2i (r ⫺ rm)2 b + 2t r rrt (rrt ⫺ rm)2 ␣ ⫺ b
U =
␣⫺ u r2rt ⫺ r2i (3r2 ⫺ 4rrm + r2m) b 2t rrt (rrt ⫺ rm)2 ␣ ⫺ b
Region 4 b ⭐ ⭐ a rrt ⭐ r ⭐ rb
Ur =
b r2rt ⫺ r2i u r2 ⫺ r2i + 2t r ␣ ⫺ b r
Region 5 b ⭐ ⭐ (t) rb ⭐ r ⭐ rinv()
Ur =
u r2 ⫺ r2i b r2rt ⫺ r2i + 2t r ␣ ⫺ b r
U =
b r2rt ⫺ r2i √r2 ⫺ r2b u r2 ⫺ r2i + 2t rb ␣ ⫺ b r b r
Ur =
u r2 ⫺ r2i 1 u √(r2 ⫺ r2b)3 u r2rt ⫺ r2i b + + 2t r ␣ ⫺ (t) 2t 3rrb 2t r ␣ ⫺ b
U = 0
Region 3 b ⭐ ⭐ a rm ⭐ r ⭐ rrt
再
冎
再
冉
冊
冉 冉
冊 冊
冎
U = 0
Region 6 (t) ⭐ ⭐ a rb ⭐ r ⭐ rs(t)
+
b u r2rt ⫺ r2i 2t rb ␣2 ⫺ {b + (t)}␣ + b(t)
U =
冉
b r2rt ⫺ r2i √r2 ⫺ r2b ⫺ ␣ u r2 ⫺ r2i + 2t rb ␣ ⫺ b r b r (t) ⫺ ␣
Axial velocity component of Regions 1–6 Uz = ⫺
u z t
冊
冦
√r2 ⫺ r2b rb √r2 ⫺ r2b ⫺ tan ⫺ 1 r r rb
冧
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Fig. 4. Assumed deformation zones during clamping type forging of spur gear.
· Et Pav = ¯ ¯ Au
(4)
· where, A is punch/workpiece contact area and Et is the total power dissipation given by the summation of all the various powers derived for the deformation zones. · · · · (5) Et = Ep + Es + Ef
3. Forging experiments The forging experiments were carried out with a commercial aluminum alloy. The material was annealed. To determine the stress–strain relation of the material, a series of ring compression
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Table 2 Kinematically admissible velocity fields of clamping type forging of spur gears. Region 1 0⭐⭐a ri ⭐ r ⭐ rt Z(r) ⭐ z ⭐ t
Ur = 0 U = 0 Uz = ⫺ u
Region 2 0⭐⭐a ri ⭐ r ⭐ rm 0 ⭐ z ⭐ Z(r)
Ur =
Region 3 0 ⭐ ⭐ b rm ⭐ r ⭐ rrt 0⭐z⭐H
u r2 ⫺ r2i 1 2 r Z(r)
U = 0 Uz =
r2 ⫺ r2i 1 ∂Z(r) u z ( ⫺ 2) 2 Z(r) r Z(r) ∂r
Ur =
u r2 ⫺ r2i r2rt ⫺ r2i (r ⫺ rm)2 ⫺ 2H r rrt (rrt ⫺ rm)2
U =
u r2rt ⫺ r2i (3r2 ⫺ 4rrm + r2m) 2H rrt (rrt ⫺ rm)2
再
Uz = ⫺ u
Region 4 b ⭐ ⭐ a rm ⭐ r ⭐ rrt 0⭐z⭐H
z H
再
Ur =
b u r2 ⫺ r2i r2rt ⫺ r2i (r ⫺ rm)2 + 2H r rrt (rrt ⫺ rm)2 ␣ ⫺ b
U =
␣⫺ u r2rt ⫺ r2i (3r2 ⫺ 4rrm + r2m) b 2 2H rrt (rrt ⫺ rm) ␣ ⫺ b
Uz = ⫺ u
Region 5 b ⭐ ⭐ a rrt ⭐ r ⭐ rb 0⭐z⭐H
冎
Ur =
z H
冉
冊
冉 冉
冊 冊
b r2rt ⫺ r2i u r2rt ⫺ r2i + 2H r ␣ ⫺ b r
U = 0 Uz = 0
Region 6 b ⭐ ⭐ (t) rb ⭐ r ⭐ rinv() 0⭐z⭐H
Ur =
b r2rt ⫺ r2i u r2rt ⫺ r2i + 2H r ␣ ⫺ b r
U =
u r2rt ⫺ r2i b r2rt ⫺ r2i √r2 ⫺ r2b + 2H rb ␣ ⫺ b r b r
Uz = 0
冎
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Table 2 Continued. Region 7 (t) ⭐ ⭐ a rb ⭐ r ⭐ rs(t) 0⭐z⭐H
Ur =
冉
冊 冉
√r2 ⫺ r2b rb √r2 ⫺ r2b ⫺ tan ⫺ 1 r r rb U =
冊
冦
1 u r2rt ⫺ r2i r2rt ⫺ r2i b u r2rt ⫺ r2i r2rt ⫺ r2i b + + + . 2H r r ␣ ⫺ b 2H rb rb ␣ ⫺ b ␣ ⫺ (t)
冉
冧
冊
b r2rt ⫺ r2i √r2 ⫺ r2b ⫺ ␣ u r2rt ⫺ r2i + 2H rb ␣ ⫺ b rb r (t) ⫺ ␣
Uz = 0
tests were performed. From the results of the tests, the flow stress and friction factor can be calculated by using the method proposed by Osakada [16], and the friction-area-divided method [17], respectively. 3.1. Ring compression test The outside diameter:inside diameter:thickness ratio of the ring is 6:3:2. The MoS2 was sprayed onto the forgings as a lubricant. The results of the tests can be seen in Table 3. From Table 3 the flow stress and friction factor have been calculated using the method proposed by Osakada [16], and the friction-area-divided method [17], respectively. The stress–strain relation is
¯ = 504.1⑀¯ 0.16 (MPa)
(6)
and the friction factor is 0.13. Table 3 Result of ring compression test Outer diameter Inner diameter Height, t (mm) ⌬t (mm) (mm) (mm)
Load (ton)
Decrease in inner diameter (%)
Reduction in height (%)
15.35 17.275 17.925 19.113 21.038 22.2
5.0 7.5 10.0 12.5 17.5 20.0
⫺1.173 ⫺9.173 ⫺7 ⫺5.173 ⫺5.173 ⫺4.667
5 25 32 42 55 60
7.588 8.188 8.025 7.888 7.888 7.85
4.75 3.75 3.4 2.9 2.25 2.0
0.25 1.25 1.6 2.1 2.75 3.0
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3.2. Gear forging The diameter of the punch for the guiding type forging is less than the cavity of the die, because the punch is moving down along the surface of the die. In this investigation, if the addendum modification coefficient of the punch is ⫺0.05, then addendum and deddendum are less than those of die by as much as 0.2 mm. In the clamping type forging, the clamping die is required for restriction of backward flow. The diameters of the cylindrical punch and hole of the clamping die are less than that of the root circle of the die. The die has been used for two types of forging. A gear shaped die cavity was made by WEDM (wire-cut EDM machine). The die was shrink fitted before W-EDM to prevent deformation which causes gear errors and excessive tensile circumferential stresses, which may result in die fracture during forging. The gear forging experiments were performed on a 300-ton hydraulic press and carried out at room temperature. The spur gear specification is shown in Table 4.
4. Results and discussion 4.1. The results of FEM analyses of the two types of forgings The results of FEM analyses are seen in Figs 2 and 5. Figure 2 shows the deformed shape of the two types of forgings. For the final stage of the forgings, Fig. 5 shows distribution of effective strain and contact pressures on the punch and the die for the guiding type forging [Fig. 5(a)], and the clamping type forging [Fig. 5 (b)]. For the analyses, the material is a commercial aluminum alloy, the stress–strain relation of the material is Eq. (1), and the friction factor is 0.13. Distribution of effective strain in the guiding type forging [Fig. 5 (a)], is almost homogeneous across the cross-section but the distribution in the clamping type forging [Fig. 5 (b)], shows a big difference along the axis. The average punch pressure is almost the same on the die for each process. In the guiding type forging, the maximum and average punch pressure is lower than in the clamping type forging. From the results it is considered that the guiding type is suitable for forging of spur gears.
Table 4 Specification of spur gear die No. of teeth Module Pressure angle Modification coefficient Standard pitch circle diameter. Base circle diameter
18 1.5 20° 0.0 27.0 mm 25.3667 mm
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Fig. 5. The results of FEM analysis for the forging of axisymmetric parts.
4.2. The relative punch pressure In the following figures, the results of upper-bound analyses of the spur gear forging have been calculated by using the kinematically admissible velocity fields (Tables 1 and 2), for clamping and guiding type forging, respectively. Figure 6 shows the relative average punch pressure of the two types of spur gears forging. The values of relative average punch pressure are determined for the spur gears with a pressure angle 20°. The height of the initial billet is 15 mm. The diameter of the mandrel is 30% of the root diameter of a spur gear. The results are found by varying the reduction in height for spur gears with 18, 20, 25 and 30 teeth and each module 1.0, 1.5, 2.0 and 3.0 at frictional condition m = 0.1, as shown in Fig. 6.
Fig. 6. Variation of the relative pressure with respect to reduction in height for different numbers of teeth and module.
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For the guiding forging, variation of the relative average pressure with respect to reduction in height has been plotted and can be seen in Fig. 6(a), that of clamping type forging is shown in Fig. 6(b). The relative average pressures increase abruptly near the final filling-up stage. The relative average punch pressures of the clamping type forgings are higher than that of the guiding type forgings. For two types of forgings, the relative average punch pressures are almost identical for different modules. At the final filling-up states, the relative average pressure a increases slightly with increasing number of teeth for the guiding type forging, and a decreases slightly for the clamping type forging, respectively. This means that the guiding type forging process remains correct in the forging of spur gears, regardless of the magnitude of module. 4.3. Forging load and pressure The variation between that of the predicted forging loads and those obtained form experiment has been plotted in Figs 7 and 8. In Figs 7 and 8, the height of the forged spur gear is 10 and 15 mm, respectively. The predicted forging loads obtained by the present upper bound methods have an approximate tendency to agree with the experimental results at final filling-up stage. Because the material of the punch is AISI D2 (Hrc-62) of which the yield stress is 2150 MPa [18], the average punch pressure values were controlled, to be 1800 MPa at the final state of the forgings. Therefore, the forged spur gears have some different filling-up states. These points are discussed in following section. For the guiding type forging, the predicted forging loads are lower than those of the clamping type forging. More energy is required for clamping type forging. 4.4. Comparisons of forged spur gears Figs 9–12 show the forged spur gears. Right sides of the products are pressing directions. In Figs 9 and 10, the forged spur gear was obtained by the guiding type forging. Figs 11 and 12
Fig. 7. Comparison of the experimental results with the upper-bound solutions (h = 10 mm).
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Fig. 8. Comparison of the experimental results with the upper-bound solutions (h = 15 mm).
Fig. 9.
Cold forged spur gear (guiding type, h = 10 mm). (a) R.H.: 25.35%; (b) R.H.: 26.76%; (c) R.H.: 27.80%.
show the results of the clamping type forging. In Figs 9–12, the height of billets are 14.2 and 21.3 mm, respectively. In Figs 9 and 10, the filling-up state of the portions of the pressing direction on the right side, is better than that of the other, because the friction force on the die wall is applied to the forged gear in a reverse direction to the punch movement. In Fig. 9(c), the filling-up state of the lower workpiece thickness is better than that of the other in Fig. 10(c). Also, for the clamping type forgings (Figs 11 and 12), the filling-up state of the lower workpiece thickness in Fig. 11(c) is better than that of the other in Fig. 12(c). However, the filling-up state of the portion near the ejector is better than that at the pressing direction, because the material which is compressed by the punch flows in a perpendicular direction to that of punch movement.
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Fig. 10.
Cold forged spur gear (guiding type, h = 15 mm). (a) R.H.: 24.52%; (b) R.H.: 27.70%; (c) R.H.: 28.30%.
Fig. 11. Cold forged spur gear (clamping type, h = 10 mm). (a) R.H.: 25.00%; (b) R.H.: 27.10%; (c) R.H.: 27.80%.
Therefore, spur gears with lower thickness are suitable for the guiding and clamping type forging process. Also, it is difficult to obtain a forged spur gear that has a good filling-up state of the portion near the ejector in guiding type forging. Conversely, complications arise near the punch in the clamping type. Comparison of Fig. 9 with Fig. 11 and Fig. 10 with Fig. 12 show that the filling-up state of the forged spur gear can be better obtained through guiding type forging at the same forging punch pressure. 5. Conclusions Through guiding and clamping type forging experiments forged spur gears have been obtained. The forging processes have been analyzed for the axisymmetric parts and spur gears by using
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Fig. 12. Cold forged spur gear (clamping type, h = 15 mm). (a) R.H.: 22.87%; (b) R.H.: 25.70%; (c) R.H.: 26.65%.
the finite element method and upper bound method, respectively. The comparisons for the forging load and pressure have been discussed. The conclusions are follows: 1. The results of 2D-FEM analysis for the two types of forgings show that the average contact pressure of guiding type forging is lower than that of clamping type forging. 2. The predicted forging loads obtained by the present upper bound methods have an approximate tendency to agree with the experimental results at final filling-up stage. From the results of the upper bound analyses, it is clear that the guiding type forging process is correct for the forging of spur gears. 3. Spur gears with a lower thickness are suitable for the two types of forging processes. Also, it is difficult to obtain a forged spur gear that has a good filling-up state of the portion near the ejector in guiding type forging. On the other hand, difficulties arise near the punch in the clamping type.
Acknowledgement This study is supported by the Korean Ministry of Education through Research Fund (ME96-E14).
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[3] Y.B. Park, D.Y. Yang, Finite element analysis for precision cold forging of helical gear using recurrent boundary conditions (in Korean), Proceedings of the KSME (I) Spring (1995) 296–301. [4] S.W. Shim, K.S. Song, D.S. Yoon, Y.B. Park, D.Y. Yang, Precision cold forging for reverse idle gear of automobile (in Korean), 2nd Forging Symposium, Pusan, Korea, May, 1995, pp. 145–157. [5] N.R. Chitkara, Y. Kim, An analysis of external spline gear forming by an upper bound energy method, International Journal of Mechanical Science 38 (7) (1996) 777–789. [6] O.P. Grover, B.L. Juneja, Analysis of closed-die forging of gear-like element, Advanced Technology of Plasticity II (1986) 888–893. [7] N.A. Abdul, T.A. Dean, An analysis of the forging of spur gear forms, International Journal of Machine Tool Design Research 26 (2) (1986) 113–123. [8] K. Ohga, K. Kondo, T. Jitsunari, Research on precision die forging utilizing divided flow, Bulletin of the JSME 28 (244) (1985) 2451–2459. [9] K. Kondo, K. Ohga, Development of precision cold die forging process utilizing divided flow, Journal of the JSTP 27 (300) (1986) 121–131. [10] K. Ohga, K. Kondo, K. Hori, Investigation into forming processes of various spur gears, Advanced Technology of Plasticity 2 (1987) 1089–1096. [11] K. Ohga, K. Kondo, Research on application of the precision cold die forging utilizing divided flow to thick products, Advanced Technology of Plasticity II (1993) 1239–1244. [12] Y. Choi, A study on the closed-die cold forging of spur gears, (in Korean), M.S. thesis, Pusan National University, 1995. [13] J.C. Choi, Y. Choi, K.D. Hur, C.H. Kim, A study on the forging of spur gears, International Journal of Mechanical Science 38 (12) (1996) 1333–1347. [14] J.C. Choi, Y. Choi, S.J. Tak, The forging of helical gears (i): experiments and upper-bound analysis, International Journal of Mechanical Science 40 (4) (1998) 325–337. [15] J.C. Choi, Y. Choi, S.J. Tak, The forging of helical gears (ii): comparisons of the forging process, International Journal of Mechanics Science (submitted). [16] K. Osakada, M. Shiraishi, S. Muraki, M. Tokuoka, Measurement of flow stress by the ring compression test, JSME International Journal Series A 34 (3) (1991) 312–318. [17] Y.-T. Lin, J.-P. Wang, An investigation into the constant friction factor with the upper-bound flow function elemental technique, Journal of Materials Processing Technology 36 (1992) 57–68. [18] I.C.F.G., General aspect of Tool Design and Tool Materials for Cold Forging of Steel, No. 4/70, No. 4/82.