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Predicting polygonal-shaped defects during hot ring rolling using a rigid-viscoplastic finite element method
ARTICLE IN PRESS
International Journal of Mechanical Sciences 50 (2008) 306–314 www.elsevier.com/locate/ijmecsci
Predicting polygonal-shaped defects during hot ring rolling using a rigid-viscoplastic finite element method Ho Keun Moona, Min Cheol Leeb, Man Soo Jounc,d, a
R&D Center, FAG Bearings Korea Ltd., Changwon 641-020, Republic of Korea Department of Mechanical Engineering, Gyeongsang National University, GyeongNam 660-701, Republic of Korea c School of Mechanical and Aerospace Engineering, Gyeongsang National University, Gazwa-dong 900, Jinju-City, GyeongNam 660-701, Republic of Korea d Research Center for Aircraft Parts Technology, Gyeongsang National University, GyeongNam 660-701, Republic of Korea b
Received 11 September 2006; received in revised form 19 June 2007; accepted 22 June 2007 Available online 28 June 2007
Abstract We applied a rigid-viscoplastic finite element method to investigate polygonal-shaped defects that occur during ring rolling and developed an improved analysis model with relatively fine finite elements near the roll gap to reduce the computational time, along with a scheme to minimize the volume change. We also simulated the formation of a polygonal-shaped defect during hot ring rolling of a ball bearing outer race. The results were in good qualitative agreement with actual experimental phenomena. r 2007 Elsevier Ltd. All rights reserved. Keywords: Ring rolling; Finite element method; Computer simulation; Polygonal-shaped defect; Ball bearing race
1. Introduction Ring rolling is used to manufacture a wide variety of mechanical components, ranging from small parts in bearings to large-scale parts used in power generation plants, aircraft engines, and large cylindrical vessels [1,2]. In recent years, ring rolling has been applied to the mass production of hollow axisymmetric mechanical components. The advantages of ring rolling include a short production time, uniform quality, smooth surface, close tolerance, and considerable savings in material cost. However, the process design for ring rolling can be quite complex. Controlling the dimensional accuracy is not only a difficult problem, but design failures also cause frequent fractures of the mandrel, which decrease the productivity of the process. A considerable amount of time is required to develop a new ring-rolling process due to the trial-andCorresponding author. School of Mechanical and Aerospace Engineering, Gyeongsang National University, Gazwa-dong 900, Jinju-City, GyeongNam 660-701, Republic of Korea. Tel.: +82 55 751 5316; fax: +82 55 751 5316. E-mail addresses: [email protected] (H.K. Moon), [email protected] (M.C. Lee), [email protected] (M.S. Joun).
0020-7403/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2007.06.004
error nature of the process design. Therefore, analysis tools, including computer simulation technology, can greatly aid in the process design. Most conventional ring-rolling research has been based on empirical methods [3–6], and the few attempts [7–9] to achieve analytical solutions regarding ring-rolling processes were based on numerous assumptions, thus restricting their applicability. Over the last two decades, various attempts have sought to apply numerical techniques, such as finite element methods, to ring rolling. Kim et al. [10] proposed a two-mesh approach to reduce the computational time and presented a useful technique to reduce the volume change in simulations. Yang et al. [11] performed a three-dimensional (3D) finite element analysis of a small material segment near the roll gap with the other regions removed. However, computational limitations have prevented these methods from being applied to real problems. Hu et al. [12] proposed a hybrid mesh approach and Lim et al. [13] used the approach to simulate a profile ringrolling process using an elastic–plastic finite element method. Joun et al. [14] predicted the deformed shapes of ring cross sections using an approximation approach based on a two-dimensional (2D) finite element method and
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applied the approach to the ring-rolling design process. Hamzah and Sta˚hlberg [15] described an optimal initial material for a large ring-rolled product and Xie et al. [16] applied a rigid-viscoplastic dynamic explicit finite element method to analyze ring-rolling processes. Joun et al. [17] performed a 3D rigid-plastic finite element analysis of a ring-rolling process for a tapered roller bearing race. Davey and Ward [18,19] used an arbitrary Lagrangian–Eulerian formulation and successive preconditioned conjugate gradient method to analyze ring-rolling processes. Yea et al. [20] predicted the spread and contact regions in ringrolling processes using a 3D rigid-plastic finite element method with geometric updating algorithm. Utsunomiya et al. [21] simulated a cold ring-rolling process using a 2D implicit elastic–plastic finite element method while Koo et al. [22] analyzed an aluminum ring-rolling process using a rigid-thermoplastic finite element method. Pauskar et al. [23] applied a static implicit dual mesh and a dynamic explicit finite element method to a few cold ring-rolling examples. Although several studies have attempted to analyze ringrolling processes using various numerical methods, they have not been able to provide full details on the polygonalshaped defects that occur during ring rolling. In this study, a rigid-viscoplastic finite element method with numerical schemes to reduce the computational time and minimize the volume change was applied to predict the polygonalshaped defects formed during hot ring rolling. The predicted results were compared with experiments.
2. Finite element formulation and solution scheme 2.1. Flow problem formulation A common plastic flow analysis problem in metal forming requires us to find the velocity field vi that satisfies the following boundary value problem. The material or workpiece is denoted as domain V with boundary S. The boundary S can be divided into a velocity-prescribed boundary Svi , where the velocity is given as vi ¼ v¯ i ; a traction-prescribed boundary Sti , where the stress vector is ¯ðnÞ given as tðnÞ i ¼ ti ; and a die–workpiece interface Sc, where the no-penetration condition, vn ¼ v¯ n , must be maintained when the interface is in compression. It is assumed that the material is incompressible (vi,i ¼ 0), isotropic, and rigidviscoplastic, and obeys the Huber–von Mises yield criterion and its associated flow rule, s0ij ¼
2s¯ _ij , 3_¯
(1)
where s0ij and _ij are the deviatoric stress tensor and strainrate tensor, respectively. The flow stress s¯ in Eq. (1) is assumed to be a function of the effective strain ¯ and the effective strain rate _¯, i.e., s¯ ¼ s即 ; _¯Þ. It is also assumed that the effect of acceleration and gravity on the force equilibrium is negligible and the process is isothermal.
307
When the penalty method for the incompressibility condition is employed, the weak form of the above boundary value problem can be written as Z Z s0ij oij dV þ K _ii ojj dV V V Z Z X ¯ ti oi dS þ mkgðvt Þot dS ¼ 0, ð2Þ S ti
Sc
where oij is defined as 12ðoi;j þ oj;i Þ and the weighting function oi is arbitrary except that it vanishes on S vi and that on ¼ 0 on Sc. In the equation, K is a large positive constant, known as the penalty constant, which approximately maintains the incompressibility condition and has a meaning of K _ii ¼ sjj =3. The subscripts n and t indicate the normal and tangential components, respectively, k and m are the shear yield stress and friction factor, respectively, and g(vt) is a function reflecting the effect of the relative velocity between the tool and the workpiece on friction. The following function proposed by Chen and Kobayashi [24] is used to obtain gðvt Þ: 2 vt v¯ t gðvt Þ ¼ tan1 , (3) p a where vt and v¯ t are the tangential components of the workpiece and die velocities, respectively, and a is a small positive constant related to j¯vt j. 2.2. Analysis model for the ring-rolling process Ring rolling is a special rolling process in which a ringshaped workpiece is drawn into the gap between the work roll (or the profile forming roll) and the mandrel (or the pressure roll) and caused to expand in diameter and decrease in thickness. In the process, the workpiece is rolled repeatedly into the gap, one rotation following another, while its thickness is gradually reduced, similar to multipass rolling. Fig. 1 shows the ring-rolling machine that we studied, which was developed for hot ring rolling of relatively large bearing races of more than 150 mm in diameter. While ring rolling by the machine shown in Fig. 1(b), the rolling mandrel, which is freely mounted, is fixed and the work roll not only transmits the forming power but it also controls the reduction in thickness. The guide rolls help maintain the stability of the ring-rolling process and improve the product roundness. These are located outside of the workpiece, as shown in Fig. 1, and thus the guide-roll forces are exerted from outside the ring toward the work roll. It is difficult to simulate ring-rolling processes. First, the rotational speed of the mandrel is unknown and is determined by the frictional forces induced by both the workpiece and rotating components. It therefore cannot be easily determined, even though the friction between the workpiece and mandrel can be modeled. Thus, we assumed that the mandrel was frictionless. Second, it is difficult to achieve a moment balance because the contact region of
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Initial shape Final shape
Profile forming roll Workpiece Mandrel Guide roll
Profile forming roll
Workpiece Guide roll
Guide roll cylinder
Guide roll cylinder
Size-control device
Fig. 1. Ring rolling machine used in this paper: (a) schematic description and (b) experimental apparatus.
the roll gap is so small that an insufficient number of finite elements may be in contact with both the mandrel and the work roll, given computational limitations. Therefore, we also assumed that the workpiece and the work roll around the maximum pressure point rotated without slippage in the circumferential direction and that the interface obeyed a constant shear frictional law in the axial direction. The fact that the guide rolls are forcecontrolled also makes ring-rolling simulations difficult. Many studies [10–17,19–23] have neglected the guide rolls during simulations because the force exerted by a guide roll is relatively small compared to the rolling force exerted on the workpiece by the mandrel. We also excluded the guide rolls in our study. Computational time is also a major concern in ringrolling simulations. We adopted the hybrid mesh approach proposed by Hu et al. [12] to reduce the computational time and proposed a new nodal point updating scheme. We employed two meshes: a reference mesh and its related analysis mesh, as shown in Fig. 2. Both meshes were composed of several subdomains defined by the same key sections. Each subdomain in the reference mesh was divided into relatively fine hexahedral elements, and each subdomain in the initial analysis mesh was divided into relatively coarse hexahedral elements. The final analysis mesh was generated by replacing the coarse finite elements in the subdomains near the roll gap of the initial analysis mesh with the fine finite elements in their associated subdomains of the reference mesh. As a result, we could obtain a realistic finite element mesh system composed of fine finite elements in the roll gap and coarse finite elements in the other regions.
During the simulations, the reference and analysis meshes helped each other by passing geometrical information and finite element solutions between them. At each solution step, the nodal points and state variables of the reference mesh were updated using the simulation results of the analysis mesh. The nodal coordinates and state variables, including the strain in the main deforming zone and on key sections, were shared by the two meshes, and the other nodal coordinates and state variables of the reference mesh were estimated from the results of the analysis mesh. Geometrical discrepancies between the two meshes were inevitable, resulting in errors during the data transfer or interpolation when estimating the nodal coordinates and state variables. Therefore, an appropriate scheme was necessary to reduce the error during these processes. For this purpose, Hu et al. [12] calculated the updated nodal points by interpolating the incremental displacements for the nodes outside of the main deforming zone. However, this scheme may also increase the error during interpolation because some nodes of the reference mesh do not inherently exist inside the solution domain defined by the analysis mesh. To minimize the interpolation error or numerical volume change, we calculated the nodal coordinates and state variables of the reference mesh using an elastic finite element analysis of the subdomains that did not belong to the main deforming zone. For this purpose, we assumed that the material was incompressible and elastic. The nodes on the key sections were treated as displacement-prescribed boundaries where the displacement components were obtained from a rigid-viscoplastic finite element analysis of the analysis mesh at each solution
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Fig. 2. A typical analysis mesh and its related reference mesh: (a) a reference mesh and (b) an analysis mesh.
Coarsened subdomain
Refined subdomain copied from the reference mesh Unchanged subdomains
Fig. 3. Concept of remeshing: (a) analysis mesh just before remeshing, (b) reference mesh just before remeshing, and (c) remeshed analysis mesh.
step. Fig. 3 shows the problem definition used to analyze the reference mesh in which each subdomain was defined by the free surfaces and key sections. All the nodal displacement components of the key sections, already
calculated in the plastic flow analysis of the analysis mesh, were prescribed. When the coarse finite elements of the analysis mesh approached the main deforming zone, a new finite element
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mesh system was automatically generated; i.e., remeshing was performed with the help of the reference mesh. Fig. 4 shows the concept of remeshing the analysis mesh. All the required information about the analysis mesh was the same as that of the reference mesh since all the nodal points of the former were contained in the finite element mesh system of the latter. The above remeshing scheme was
Free surface
essentially the same as the hybrid mesh approach [15]. Detailed solution procedures of this approach are summarized in Fig. 5. 2.3. Node updating scheme for the analysis mesh According to Davey and Ward [19], nodal points inevitably tend to move outward from the center of rotation if they are on a rotating body and their nodal velocities are assumed constant between two solution steps. In ring rolling, such a tendency causes excessive volume changes during the computer simulation, posing a problem that must be resolved. We separated the mean or rigid-body translation component of the nodal ring velocity from the other components. The mean velocity was then calculated based on the mean value of the nodal velocities of the reference mesh, and was updated directly between positions. The remaining nodal velocity, defined as VN for node Nt, was further divided into the deformation component (VD, which was used to update the material directly node-by-node) and the rigid-body rotation component
Free surface A subdomainfor elastic finite element analysis Key sections where nodal displacements are prescribed
Fig. 4. Definition of the elastic finite element analysis problem for the reference mesh.
Input process geomatries and process conditions
Set i = 1 Generate initial analysis mesh (AM) and reference mesh (RM)
Carry out rigid-viscoplastic finite element analysis for the current analysis mesh at the i-th solution step
Update the analysis mesh by the presented new node updating scheme
Copy all nodal coordinates of AM into their corresponding nodal coordinates of RM
Carry out elastic finite element analysis of each section of RM to update the nodal coordinates except the already updated nodes in the previous step
Check stop criterion
Yes
Stop
No Check need of remeshing
Yes Remesh by the presented remeshing scheme
No i=i+1
Fig. 5. Detailed solution procedures of the presented approach.
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N′ N t+Δt VD
DD VN
VR
DR n
r
VN Δt
DN
r
Nt
Nt
n
ω Δt Axis of rotation
Axis of rotation
O
O
Fig. 6. Node updating scheme: (a) velocity and (b) displacement.
Fig. 7. Three polygonal shapes made by hot ring rolling: (a) pentagon, (b) hexagon, and (c) 16 corner polygon.
(VR, which was obtained by rotating the entire material at its calculated angular velocity around its central axis or the axis of rotation), as shown in Fig. 6. The deformation velocity VD was calculated from
Table 1 Material properties and process conditions Flow stress (MPa)
s¯ ¼ 68:0_¯
VD ¼ VN VR ,
Initial material (workpiece) Mandrel diameter Work roll diameter Initial temperature Work roll rotational speed Work roll feed rate Friction factor
+45.0 87.3 27.0 mm +40.0 mm +434.0 mm 1150 1C N ¼ 50 rpm, 1136.0 mm/s As described in Fig. 8 m ¼ 0.3
(4)
with VR ¼ ro n, where o is the angular velocity and n and r are the unit vector and the radius from the axis of rotation, respectively, as defined in Fig. 6. Thus, a new point Nt+Dt due to the deformation and rigid-body rotation for the time increment Dt could be obtained from moving DD ¼ VDDt after the rigidbody rotation with respect to the axis of rotation by degree oDt, as shown in Fig. 6. If the node Nt is moved to a new point Nt+Dt by simply updating the vector VNDt, as shown in Fig. 6, then excessive volume changes may take place.
0:18
[26]
3. Computer simulation showing the formation of polygonalshaped defects Bearing races less than 150 mm in diameter are typically cold ring rolled while those greater than 150 mm in
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diameter are hot ring rolled. A wide range of polygonalshaped defects can occur if the hot-rolled processes are illconditioned. Fig. 7 shows three typical examples. All the 5
Feed speed (mm/sec)
4
3
2
1
0 0
1
2 3 Time (sec)
Fig. 8. Feed rate curve of the work roll.
4
5
polygonal-shaped defects were formed experimentally by the ring-rolling machine shown in Fig. 1(b) when the feed rate exceeded the critical value, even though the guide rolls were operated normally. To prevent the occurrence of defects, we increased the guide-roll forces, but this caused the ring to become plastically deformed by the guide rolls so that normal ring rolling could not be performed. We used a finite element analysis based on the above approach to determine why the polygonal shape of Fig. 7(a) was produced. The material was SAE 52100 bearing steel at 1150 1C. Details of the material behavior can be found in Ref. [26]. The process was assumed to be isothermal, and all necessary information to simulate the problem is summarized in Table 1. From the symmetry of the process, only one-half of the workpiece was analyzed. The work roll and mandrel were modeled as rigid objects. The workpiece was modeled using hexahedral elements. The finite element mesh system for the reference mesh consisted of 3000 elements with 4500 nodes, while that for the analysis mesh consisted of 1200 elements with 1800 nodes. Using the feed rate of the work roll given in Fig. 8, we obtained convergent results after 16,500 solution steps corresponding to 15 revolutions of the workpiece. During the simulation, 225 remeshings were performed.
Fig. 9. Analysis result of polygonal shape formation in hot ring rolling.
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Fig. 9 shows the finite element results for formation of the polygonal-shaped defect. An initial small defect generated in the early stages of the process grew gradually, leading to a large polygonal-shaped defect. The initial small defect caused periodic similar shape defects. We believe that the initial defect was due to the intrinsic instability of the force balance between the friction pulling force and the roll pushing force when the reduction ratio was large. A comparison between Figs. 7(a) and 9 shows a good qualitative agreement between our analysis and the experimental results. The main reason for the polygonalshaped defect formation shown in Fig. 9 is the wobbling motion depicted in Fig. 10, which occurs when the feed rate exceeds a critical value. Various types of guide rolls are used by industry to prevent wobbling, but if the guide rolls
do not support the workpiece with the proper force, polygonal-shaped defects can occur. Even though the guide-roll force can be normal for a certain feed rate, a defect can occur if the feed rate increases while the guideroll force remains fixed. Thus, polygonal-shaped defect formations are related to the guide-roll force and the feed rate. Higher feed rates require higher guide-roll forces, but abnormally high guide-roll forces can decrease the geometric accuracy of the ring. According to Hua and Zhao [25], minimum and maximum feed rates exist for successful ring-rolling operations. Therefore, we performed a finite element analysis under the same conditions except we reduced the initial feed rate from 4.1 to 3.5 mm/s. Fig. 11(a) shows the predicted results, which contained no polygonal-shaped defects. Fig. 11(b) shows the experimental results obtained
Fig. 10. Wobble motion inducing polygonal ring.
Fig. 11. An improved process: (a) predictions and (b) experiments.
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under the improved process conditions. A comparison between the two figures also indicates a good qualitative agreement between our predictions and the experiments. 4. Conclusions We applied a 3D finite element method to determine why polygonal-shaped defects form during hot ring rolling of bearing races. We used a rigid-viscoplastic finite element method and proposed an improved analysis model and a volume change minimization scheme. We compared the simulated results with experiments and found them to be in good qualitative agreement with the actual phenomena. Our study demonstrated that a polygonal-shaped defect could form if the feed rate relative to the rotational speed of the work roll, or the reduction ratio, exceeds a critical value. The actual polygonal shape and number of polygonal angles may differ if the working conditions are slightly different, suggesting that polygonal-shaped defects comprise a type of instability problem. Acknowledgements This work was supported by Grant no. RTI04-01-03 from the Regional Technology Innovation of the Ministry of Commerce, Industry and Energy (MOCIE) of Korea. References [1] Eruc E, Shivpuri R. A summary of ring rolling technology—I. Recent trends in machines, processes and production lines. International Journal of Machine Tools and Manufacture 1992;32(3):379–98. [2] Eruc E, Shivpuri R. A summary of ring rolling technology—II. Recent trends in process modeling, simulation, planning and control. International Journal of Machine Tools and Manufacture 1992;32(3):399–413. [3] Johnson W, MacLeod I, Needham G. An experimental investigation into the process of ring or metal type rolling. International Journal of Mechanical Sciences 1968;10:455–68. [4] Hawyard JB, Johnson W, Kirkland J, Appleton E. Analysis for roll force and torque in ring rolling with some supporting experiments. International Journal of Mechanical Sciences 1973;15:873–93. [5] Mamalis AG, Hawkyard JB, Johnson W. Cavity formation in rolling profiled rings. International Journal of Mechanical Sciences 1975;17:669–72. [6] Mamalis AG, Hawkyard JB, Johnson W. Spread and flow patterns in ring rolling. International Journal of Mechanical Sciences 1976;18:11–6. [7] Ryoo JS, Yang DY, Johnson W. The influence of process parameters on torque and load in ring rolling. Journal of Mechanical Working Technology 1986;12:307–21. [8] Doege E, Aboutour M. Simulation of ring rolling process. In: Advanced technology of plasticity, proceedings of the second international conference on the technology of plasticity, vol. 2, Stuttgart, Germany, August 24–28, 1987. p. 817–24.
[9] Toya Y, Ozawa T, Nishiguchi M. Analysis of simulation in ring rolling. In: Third international conference on rotary metalworking processes, Kyoto, Japan, September 8–10, 1984. p. 279–84. [10] Kim NS, Machida S, Kobayashi S. Ring rolling process simulation by the three-dimensional finite element method. International Journal of Machine Tools and Manufacture 1990;30(4):569–77. [11] Yang DY, Kim KH, Hawkyard JB. Simulation of T-section profile ring rolling by the 3-D rigid-plastic finite element method. International Journal of Mechanical Sciences 1991;33(7):541–50. [12] Hu ZM, Pillinger I, Hartley P, McKenzie S, Spence PJ. Threedimensional finite-element modeling of ring rolling. Journal of Materials Processing Technology 1994;45(1–4):143–8. [13] Lim T, Pillinger I, Hartley P. A finite-element simulation of profile ring rolling using a hybrid mesh model. Journal of Materials Processing Technology 1998;80&81(1):199–205. [14] Joun MS, Chung JH, Shivpuri R. An axisymmetric forging approach to preform design in ring rolling using a rigid-viscoplastic finite element method. International Journal of Machine Tools and Manufacture 1998;38(10&11):1183–91. [15] Hamzah S, Sta˚hlberg U. A study of pore closure in the manufacturing of heavy rings. Journal of Materials Processing Technology 1998;84(1–3):25–37. [16] Xie CL, Dong XH, Li SJ, Huang SH. Rigid-viscoplastic dynamic explicit FEA of the ring rolling process. International Journal of Machine Tools and Manufacture 2000;40(1):81–93. [17] Moon HK, Joun MS, Chung JH, Park CN. Prediction of defect formation in ring rolling by the three-dimensional rigid-plastic finite element method. Transactions of the Korean Society of Mechanical Engineers A 2004;28(10):1492–9. [18] Davey D, Ward MJ. The practicalities of ring rolling simulation for profiled rings. Journal of Materials Processing Technology 2002;125&126(9):619–25. [19] Davey K, Ward MJ. A practical method for finite element ring rolling simulation using the ALE flow formulation. International Journal of Mechanical Sciences 2002;44(1):165–90. [20] Yea YS, Ko YS, Kim NS, Lee JC. Prediction of spread, pressure distribution and roll force in ring rolling process using rigid-plastic finite element method. Journal of Materials Processing Technology 2003;140(1–3):478–86. [21] Utsunomiya H, Saito Y, Shinoda T, Takasu I. Elastic-plastic finite element analysis of cold ring rolling process. Journal of Materials Processing Technology 2002;125&126(9):613–8. [22] Koo SW, Lee JC, Yoon SJ, Kim NS. Analysis of aluminum ring rolling process using thermo-rigid-plastic finite element method. Transactions of the Korean Society of Mechanical Engineers A 2003;27(5):815–22. [23] Pauskar PM, Sawamiphakdi K, Jin DQ. Static implicit vs. dynamic explicit finite element analysis for ring rolling process modeling. In: Eighth international conference on numerical methods in industrial forming processes (NUMIFORM), Columbus, OH, USA, June 13–17, 2004. p. 412–17. [24] Chen CC, Kobayashi S. Rigid-plastic finite element analysis of ring compression-applications of numerical methods to forming processes. ASME AMD 1978;28:163–74. [25] Hua L, Zhao ZZ. The extremum parameters in ring rolling. Journal of Materials Processing Technology 1997;69(1–3):273–6. [26] Moon HK, Lee JS, Yoo SJ, Joun MS, Lee JK. Hot deformation behavior of bearing steels. ASME Journal of Engineering Materials and Technology 2007;129(3):349–55.
International Journal of Mechanical Sciences 50 (2008) 306–314 www.elsevier.com/locate/ijmecsci
Predicting polygonal-shaped defects during hot ring rolling using a rigid-viscoplastic finite element method Ho Keun Moona, Min Cheol Leeb, Man Soo Jounc,d, a
R&D Center, FAG Bearings Korea Ltd., Changwon 641-020, Republic of Korea Department of Mechanical Engineering, Gyeongsang National University, GyeongNam 660-701, Republic of Korea c School of Mechanical and Aerospace Engineering, Gyeongsang National University, Gazwa-dong 900, Jinju-City, GyeongNam 660-701, Republic of Korea d Research Center for Aircraft Parts Technology, Gyeongsang National University, GyeongNam 660-701, Republic of Korea b
Received 11 September 2006; received in revised form 19 June 2007; accepted 22 June 2007 Available online 28 June 2007
Abstract We applied a rigid-viscoplastic finite element method to investigate polygonal-shaped defects that occur during ring rolling and developed an improved analysis model with relatively fine finite elements near the roll gap to reduce the computational time, along with a scheme to minimize the volume change. We also simulated the formation of a polygonal-shaped defect during hot ring rolling of a ball bearing outer race. The results were in good qualitative agreement with actual experimental phenomena. r 2007 Elsevier Ltd. All rights reserved. Keywords: Ring rolling; Finite element method; Computer simulation; Polygonal-shaped defect; Ball bearing race
1. Introduction Ring rolling is used to manufacture a wide variety of mechanical components, ranging from small parts in bearings to large-scale parts used in power generation plants, aircraft engines, and large cylindrical vessels [1,2]. In recent years, ring rolling has been applied to the mass production of hollow axisymmetric mechanical components. The advantages of ring rolling include a short production time, uniform quality, smooth surface, close tolerance, and considerable savings in material cost. However, the process design for ring rolling can be quite complex. Controlling the dimensional accuracy is not only a difficult problem, but design failures also cause frequent fractures of the mandrel, which decrease the productivity of the process. A considerable amount of time is required to develop a new ring-rolling process due to the trial-andCorresponding author. School of Mechanical and Aerospace Engineering, Gyeongsang National University, Gazwa-dong 900, Jinju-City, GyeongNam 660-701, Republic of Korea. Tel.: +82 55 751 5316; fax: +82 55 751 5316. E-mail addresses: [email protected] (H.K. Moon), [email protected] (M.C. Lee), [email protected] (M.S. Joun).
0020-7403/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2007.06.004
error nature of the process design. Therefore, analysis tools, including computer simulation technology, can greatly aid in the process design. Most conventional ring-rolling research has been based on empirical methods [3–6], and the few attempts [7–9] to achieve analytical solutions regarding ring-rolling processes were based on numerous assumptions, thus restricting their applicability. Over the last two decades, various attempts have sought to apply numerical techniques, such as finite element methods, to ring rolling. Kim et al. [10] proposed a two-mesh approach to reduce the computational time and presented a useful technique to reduce the volume change in simulations. Yang et al. [11] performed a three-dimensional (3D) finite element analysis of a small material segment near the roll gap with the other regions removed. However, computational limitations have prevented these methods from being applied to real problems. Hu et al. [12] proposed a hybrid mesh approach and Lim et al. [13] used the approach to simulate a profile ringrolling process using an elastic–plastic finite element method. Joun et al. [14] predicted the deformed shapes of ring cross sections using an approximation approach based on a two-dimensional (2D) finite element method and
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applied the approach to the ring-rolling design process. Hamzah and Sta˚hlberg [15] described an optimal initial material for a large ring-rolled product and Xie et al. [16] applied a rigid-viscoplastic dynamic explicit finite element method to analyze ring-rolling processes. Joun et al. [17] performed a 3D rigid-plastic finite element analysis of a ring-rolling process for a tapered roller bearing race. Davey and Ward [18,19] used an arbitrary Lagrangian–Eulerian formulation and successive preconditioned conjugate gradient method to analyze ring-rolling processes. Yea et al. [20] predicted the spread and contact regions in ringrolling processes using a 3D rigid-plastic finite element method with geometric updating algorithm. Utsunomiya et al. [21] simulated a cold ring-rolling process using a 2D implicit elastic–plastic finite element method while Koo et al. [22] analyzed an aluminum ring-rolling process using a rigid-thermoplastic finite element method. Pauskar et al. [23] applied a static implicit dual mesh and a dynamic explicit finite element method to a few cold ring-rolling examples. Although several studies have attempted to analyze ringrolling processes using various numerical methods, they have not been able to provide full details on the polygonalshaped defects that occur during ring rolling. In this study, a rigid-viscoplastic finite element method with numerical schemes to reduce the computational time and minimize the volume change was applied to predict the polygonalshaped defects formed during hot ring rolling. The predicted results were compared with experiments.
2. Finite element formulation and solution scheme 2.1. Flow problem formulation A common plastic flow analysis problem in metal forming requires us to find the velocity field vi that satisfies the following boundary value problem. The material or workpiece is denoted as domain V with boundary S. The boundary S can be divided into a velocity-prescribed boundary Svi , where the velocity is given as vi ¼ v¯ i ; a traction-prescribed boundary Sti , where the stress vector is ¯ðnÞ given as tðnÞ i ¼ ti ; and a die–workpiece interface Sc, where the no-penetration condition, vn ¼ v¯ n , must be maintained when the interface is in compression. It is assumed that the material is incompressible (vi,i ¼ 0), isotropic, and rigidviscoplastic, and obeys the Huber–von Mises yield criterion and its associated flow rule, s0ij ¼
2s¯ _ij , 3_¯
(1)
where s0ij and _ij are the deviatoric stress tensor and strainrate tensor, respectively. The flow stress s¯ in Eq. (1) is assumed to be a function of the effective strain ¯ and the effective strain rate _¯, i.e., s¯ ¼ s即 ; _¯Þ. It is also assumed that the effect of acceleration and gravity on the force equilibrium is negligible and the process is isothermal.
307
When the penalty method for the incompressibility condition is employed, the weak form of the above boundary value problem can be written as Z Z s0ij oij dV þ K _ii ojj dV V V Z Z X ¯ ti oi dS þ mkgðvt Þot dS ¼ 0, ð2Þ S ti
Sc
where oij is defined as 12ðoi;j þ oj;i Þ and the weighting function oi is arbitrary except that it vanishes on S vi and that on ¼ 0 on Sc. In the equation, K is a large positive constant, known as the penalty constant, which approximately maintains the incompressibility condition and has a meaning of K _ii ¼ sjj =3. The subscripts n and t indicate the normal and tangential components, respectively, k and m are the shear yield stress and friction factor, respectively, and g(vt) is a function reflecting the effect of the relative velocity between the tool and the workpiece on friction. The following function proposed by Chen and Kobayashi [24] is used to obtain gðvt Þ: 2 vt v¯ t gðvt Þ ¼ tan1 , (3) p a where vt and v¯ t are the tangential components of the workpiece and die velocities, respectively, and a is a small positive constant related to j¯vt j. 2.2. Analysis model for the ring-rolling process Ring rolling is a special rolling process in which a ringshaped workpiece is drawn into the gap between the work roll (or the profile forming roll) and the mandrel (or the pressure roll) and caused to expand in diameter and decrease in thickness. In the process, the workpiece is rolled repeatedly into the gap, one rotation following another, while its thickness is gradually reduced, similar to multipass rolling. Fig. 1 shows the ring-rolling machine that we studied, which was developed for hot ring rolling of relatively large bearing races of more than 150 mm in diameter. While ring rolling by the machine shown in Fig. 1(b), the rolling mandrel, which is freely mounted, is fixed and the work roll not only transmits the forming power but it also controls the reduction in thickness. The guide rolls help maintain the stability of the ring-rolling process and improve the product roundness. These are located outside of the workpiece, as shown in Fig. 1, and thus the guide-roll forces are exerted from outside the ring toward the work roll. It is difficult to simulate ring-rolling processes. First, the rotational speed of the mandrel is unknown and is determined by the frictional forces induced by both the workpiece and rotating components. It therefore cannot be easily determined, even though the friction between the workpiece and mandrel can be modeled. Thus, we assumed that the mandrel was frictionless. Second, it is difficult to achieve a moment balance because the contact region of
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Initial shape Final shape
Profile forming roll Workpiece Mandrel Guide roll
Profile forming roll
Workpiece Guide roll
Guide roll cylinder
Guide roll cylinder
Size-control device
Fig. 1. Ring rolling machine used in this paper: (a) schematic description and (b) experimental apparatus.
the roll gap is so small that an insufficient number of finite elements may be in contact with both the mandrel and the work roll, given computational limitations. Therefore, we also assumed that the workpiece and the work roll around the maximum pressure point rotated without slippage in the circumferential direction and that the interface obeyed a constant shear frictional law in the axial direction. The fact that the guide rolls are forcecontrolled also makes ring-rolling simulations difficult. Many studies [10–17,19–23] have neglected the guide rolls during simulations because the force exerted by a guide roll is relatively small compared to the rolling force exerted on the workpiece by the mandrel. We also excluded the guide rolls in our study. Computational time is also a major concern in ringrolling simulations. We adopted the hybrid mesh approach proposed by Hu et al. [12] to reduce the computational time and proposed a new nodal point updating scheme. We employed two meshes: a reference mesh and its related analysis mesh, as shown in Fig. 2. Both meshes were composed of several subdomains defined by the same key sections. Each subdomain in the reference mesh was divided into relatively fine hexahedral elements, and each subdomain in the initial analysis mesh was divided into relatively coarse hexahedral elements. The final analysis mesh was generated by replacing the coarse finite elements in the subdomains near the roll gap of the initial analysis mesh with the fine finite elements in their associated subdomains of the reference mesh. As a result, we could obtain a realistic finite element mesh system composed of fine finite elements in the roll gap and coarse finite elements in the other regions.
During the simulations, the reference and analysis meshes helped each other by passing geometrical information and finite element solutions between them. At each solution step, the nodal points and state variables of the reference mesh were updated using the simulation results of the analysis mesh. The nodal coordinates and state variables, including the strain in the main deforming zone and on key sections, were shared by the two meshes, and the other nodal coordinates and state variables of the reference mesh were estimated from the results of the analysis mesh. Geometrical discrepancies between the two meshes were inevitable, resulting in errors during the data transfer or interpolation when estimating the nodal coordinates and state variables. Therefore, an appropriate scheme was necessary to reduce the error during these processes. For this purpose, Hu et al. [12] calculated the updated nodal points by interpolating the incremental displacements for the nodes outside of the main deforming zone. However, this scheme may also increase the error during interpolation because some nodes of the reference mesh do not inherently exist inside the solution domain defined by the analysis mesh. To minimize the interpolation error or numerical volume change, we calculated the nodal coordinates and state variables of the reference mesh using an elastic finite element analysis of the subdomains that did not belong to the main deforming zone. For this purpose, we assumed that the material was incompressible and elastic. The nodes on the key sections were treated as displacement-prescribed boundaries where the displacement components were obtained from a rigid-viscoplastic finite element analysis of the analysis mesh at each solution
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Fig. 2. A typical analysis mesh and its related reference mesh: (a) a reference mesh and (b) an analysis mesh.
Coarsened subdomain
Refined subdomain copied from the reference mesh Unchanged subdomains
Fig. 3. Concept of remeshing: (a) analysis mesh just before remeshing, (b) reference mesh just before remeshing, and (c) remeshed analysis mesh.
step. Fig. 3 shows the problem definition used to analyze the reference mesh in which each subdomain was defined by the free surfaces and key sections. All the nodal displacement components of the key sections, already
calculated in the plastic flow analysis of the analysis mesh, were prescribed. When the coarse finite elements of the analysis mesh approached the main deforming zone, a new finite element
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mesh system was automatically generated; i.e., remeshing was performed with the help of the reference mesh. Fig. 4 shows the concept of remeshing the analysis mesh. All the required information about the analysis mesh was the same as that of the reference mesh since all the nodal points of the former were contained in the finite element mesh system of the latter. The above remeshing scheme was
Free surface
essentially the same as the hybrid mesh approach [15]. Detailed solution procedures of this approach are summarized in Fig. 5. 2.3. Node updating scheme for the analysis mesh According to Davey and Ward [19], nodal points inevitably tend to move outward from the center of rotation if they are on a rotating body and their nodal velocities are assumed constant between two solution steps. In ring rolling, such a tendency causes excessive volume changes during the computer simulation, posing a problem that must be resolved. We separated the mean or rigid-body translation component of the nodal ring velocity from the other components. The mean velocity was then calculated based on the mean value of the nodal velocities of the reference mesh, and was updated directly between positions. The remaining nodal velocity, defined as VN for node Nt, was further divided into the deformation component (VD, which was used to update the material directly node-by-node) and the rigid-body rotation component
Free surface A subdomainfor elastic finite element analysis Key sections where nodal displacements are prescribed
Fig. 4. Definition of the elastic finite element analysis problem for the reference mesh.
Input process geomatries and process conditions
Set i = 1 Generate initial analysis mesh (AM) and reference mesh (RM)
Carry out rigid-viscoplastic finite element analysis for the current analysis mesh at the i-th solution step
Update the analysis mesh by the presented new node updating scheme
Copy all nodal coordinates of AM into their corresponding nodal coordinates of RM
Carry out elastic finite element analysis of each section of RM to update the nodal coordinates except the already updated nodes in the previous step
Check stop criterion
Yes
Stop
No Check need of remeshing
Yes Remesh by the presented remeshing scheme
No i=i+1
Fig. 5. Detailed solution procedures of the presented approach.
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N′ N t+Δt VD
DD VN
VR
DR n
r
VN Δt
DN
r
Nt
Nt
n
ω Δt Axis of rotation
Axis of rotation
O
O
Fig. 6. Node updating scheme: (a) velocity and (b) displacement.
Fig. 7. Three polygonal shapes made by hot ring rolling: (a) pentagon, (b) hexagon, and (c) 16 corner polygon.
(VR, which was obtained by rotating the entire material at its calculated angular velocity around its central axis or the axis of rotation), as shown in Fig. 6. The deformation velocity VD was calculated from
Table 1 Material properties and process conditions Flow stress (MPa)
s¯ ¼ 68:0_¯
VD ¼ VN VR ,
Initial material (workpiece) Mandrel diameter Work roll diameter Initial temperature Work roll rotational speed Work roll feed rate Friction factor
+45.0 87.3 27.0 mm +40.0 mm +434.0 mm 1150 1C N ¼ 50 rpm, 1136.0 mm/s As described in Fig. 8 m ¼ 0.3
(4)
with VR ¼ ro n, where o is the angular velocity and n and r are the unit vector and the radius from the axis of rotation, respectively, as defined in Fig. 6. Thus, a new point Nt+Dt due to the deformation and rigid-body rotation for the time increment Dt could be obtained from moving DD ¼ VDDt after the rigidbody rotation with respect to the axis of rotation by degree oDt, as shown in Fig. 6. If the node Nt is moved to a new point Nt+Dt by simply updating the vector VNDt, as shown in Fig. 6, then excessive volume changes may take place.
0:18
[26]
3. Computer simulation showing the formation of polygonalshaped defects Bearing races less than 150 mm in diameter are typically cold ring rolled while those greater than 150 mm in
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diameter are hot ring rolled. A wide range of polygonalshaped defects can occur if the hot-rolled processes are illconditioned. Fig. 7 shows three typical examples. All the 5
Feed speed (mm/sec)
4
3
2
1
0 0
1
2 3 Time (sec)
Fig. 8. Feed rate curve of the work roll.
4
5
polygonal-shaped defects were formed experimentally by the ring-rolling machine shown in Fig. 1(b) when the feed rate exceeded the critical value, even though the guide rolls were operated normally. To prevent the occurrence of defects, we increased the guide-roll forces, but this caused the ring to become plastically deformed by the guide rolls so that normal ring rolling could not be performed. We used a finite element analysis based on the above approach to determine why the polygonal shape of Fig. 7(a) was produced. The material was SAE 52100 bearing steel at 1150 1C. Details of the material behavior can be found in Ref. [26]. The process was assumed to be isothermal, and all necessary information to simulate the problem is summarized in Table 1. From the symmetry of the process, only one-half of the workpiece was analyzed. The work roll and mandrel were modeled as rigid objects. The workpiece was modeled using hexahedral elements. The finite element mesh system for the reference mesh consisted of 3000 elements with 4500 nodes, while that for the analysis mesh consisted of 1200 elements with 1800 nodes. Using the feed rate of the work roll given in Fig. 8, we obtained convergent results after 16,500 solution steps corresponding to 15 revolutions of the workpiece. During the simulation, 225 remeshings were performed.
Fig. 9. Analysis result of polygonal shape formation in hot ring rolling.
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Fig. 9 shows the finite element results for formation of the polygonal-shaped defect. An initial small defect generated in the early stages of the process grew gradually, leading to a large polygonal-shaped defect. The initial small defect caused periodic similar shape defects. We believe that the initial defect was due to the intrinsic instability of the force balance between the friction pulling force and the roll pushing force when the reduction ratio was large. A comparison between Figs. 7(a) and 9 shows a good qualitative agreement between our analysis and the experimental results. The main reason for the polygonalshaped defect formation shown in Fig. 9 is the wobbling motion depicted in Fig. 10, which occurs when the feed rate exceeds a critical value. Various types of guide rolls are used by industry to prevent wobbling, but if the guide rolls
do not support the workpiece with the proper force, polygonal-shaped defects can occur. Even though the guide-roll force can be normal for a certain feed rate, a defect can occur if the feed rate increases while the guideroll force remains fixed. Thus, polygonal-shaped defect formations are related to the guide-roll force and the feed rate. Higher feed rates require higher guide-roll forces, but abnormally high guide-roll forces can decrease the geometric accuracy of the ring. According to Hua and Zhao [25], minimum and maximum feed rates exist for successful ring-rolling operations. Therefore, we performed a finite element analysis under the same conditions except we reduced the initial feed rate from 4.1 to 3.5 mm/s. Fig. 11(a) shows the predicted results, which contained no polygonal-shaped defects. Fig. 11(b) shows the experimental results obtained
Fig. 10. Wobble motion inducing polygonal ring.
Fig. 11. An improved process: (a) predictions and (b) experiments.
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under the improved process conditions. A comparison between the two figures also indicates a good qualitative agreement between our predictions and the experiments. 4. Conclusions We applied a 3D finite element method to determine why polygonal-shaped defects form during hot ring rolling of bearing races. We used a rigid-viscoplastic finite element method and proposed an improved analysis model and a volume change minimization scheme. We compared the simulated results with experiments and found them to be in good qualitative agreement with the actual phenomena. Our study demonstrated that a polygonal-shaped defect could form if the feed rate relative to the rotational speed of the work roll, or the reduction ratio, exceeds a critical value. The actual polygonal shape and number of polygonal angles may differ if the working conditions are slightly different, suggesting that polygonal-shaped defects comprise a type of instability problem. Acknowledgements This work was supported by Grant no. RTI04-01-03 from the Regional Technology Innovation of the Ministry of Commerce, Industry and Energy (MOCIE) of Korea. References [1] Eruc E, Shivpuri R. A summary of ring rolling technology—I. Recent trends in machines, processes and production lines. International Journal of Machine Tools and Manufacture 1992;32(3):379–98. [2] Eruc E, Shivpuri R. A summary of ring rolling technology—II. Recent trends in process modeling, simulation, planning and control. International Journal of Machine Tools and Manufacture 1992;32(3):399–413. [3] Johnson W, MacLeod I, Needham G. An experimental investigation into the process of ring or metal type rolling. International Journal of Mechanical Sciences 1968;10:455–68. [4] Hawyard JB, Johnson W, Kirkland J, Appleton E. Analysis for roll force and torque in ring rolling with some supporting experiments. International Journal of Mechanical Sciences 1973;15:873–93. [5] Mamalis AG, Hawkyard JB, Johnson W. Cavity formation in rolling profiled rings. International Journal of Mechanical Sciences 1975;17:669–72. [6] Mamalis AG, Hawkyard JB, Johnson W. Spread and flow patterns in ring rolling. International Journal of Mechanical Sciences 1976;18:11–6. [7] Ryoo JS, Yang DY, Johnson W. The influence of process parameters on torque and load in ring rolling. Journal of Mechanical Working Technology 1986;12:307–21. [8] Doege E, Aboutour M. Simulation of ring rolling process. In: Advanced technology of plasticity, proceedings of the second international conference on the technology of plasticity, vol. 2, Stuttgart, Germany, August 24–28, 1987. p. 817–24.
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