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Researches on the ring stiffness condition in radial–axial ring rolling
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 2570–2575
journal homepage: www.elsevier.com/locate/jmatprotec
Researches on the ring stiffness condition in radial–axial ring rolling Lin Hua ∗ , Libo Pan, Jian Lan School of Materials Science and Engineering, Wuhan University of Technology, Wuhan 122#, Luoshi Road, Wuhan 430070, China
a r t i c l e
i n f o
a b s t r a c t
Article history:
The radial–axial ring rolling technology is used mainly to manufacture large rings in many
Received 10 July 2007
industrial fields. During rolling process the ring may be collapsed or deformed unexpectedly
Received in revised form
under the pressure of guide roll, and in this case the ring is wasted. To prevent ring collapsing
22 May 2008
is necessary for smooth rolling. On the basis of rolling theory, the forces exerted to ring in
Accepted 7 June 2008
radial–axial ring rolling process were analyzed, and the stiffness model of ring was proposed. The stiffness condition was derived and the influencing factors of ring stiffness condition were explained. The ring stiffness condition in radial–axial ring rolling is related to the
Keywords:
factors of ring size, rolling ratio, position angel of guide roll, rolls sizes, position of cones,
Radial–axial ring rolling
radial and axial feed speed and friction condition. The effect laws of some factors on ring
Ring
stiffness condition were revealed by FEM simulation. The stiffness condition can be used as
Stiffness condition
a rule of design and manufacturing for radial–axial ring rolling. © 2008 Elsevier B.V. All rights reserved.
Finite element
1.
Introduction
Ring rolling is a kind of advanced technology to manufacture rings in many industrial fields such as machine, automobiles, trains and airplanes, etc. Radial–axial ring rolling technology is primarily used to manufacture large ring. In radial–axial ring rolling, two axial cone rolls are added to roll the axial section of ring comparing to the radial ring rolling. In the rolling process the radial and axial sections of ring are rolled at the same time. During rolling, radial thickness and axial height of ring decease and ring diameter increases. Two guide rolls play an important role in radial–axial ring rolling. They not only help to keep smooth rolling process, but also have great effect on the forming quality of product. Occasionally, during rolling process the ring may be deformed or collapsed unexpectedly under the pressure of guide rolls because the ring’s stiffness is not satisfied, which will result in unqualified ring product. We wonder how many parameters can affect the ring’s stiffness in radial–axial ring rolling. Hua et al. (2001)
∗
Corresponding author. Tel.: +86 2787168391; fax: +86 2787168391. E-mail address: [email protected] (L. Hua). 0924-0136/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2008.06.002
have researched the stiffness condition theoretically in radial ring rolling by constructing the mechanical model. Choi et al. (1992) presented and analyzed the dynamic characteristics of radial–axial ring rolling processes. The rolling technology according to a certain rolling machine was introduced by Guan et al. (1995). Jiang et al. (1994) analyzed the allowable rolling region of radial–axial ring rolling mill from theoretical aspect. However, above documents related to radial–axial ring rolling technology have not involved the ring’s stiffness. In this paper, the ring’s stiffness condition in radial–axial ring rolling is intensively investigated, and the factors influencing the ring’s stiffness are revealed.
2.
Force analysis in radial–axial ring rolling
It is known that the axial rolls are cones, which are different from the radial rolls of cylinder. Suppose the acting points of the forces between ring and cones lie in the center of
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 2570–2575
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Nomenclature B F
axial height of ring radial project of force acting on the ring Fx sum of forces in x direction Fy sum of force in y direction h feed per turn of idle roll ha feed per turn in axial pass H radial thickness of ring during rolling L projected contact length between roll and ring La projection of axial contacting arc in radial direction M bending moment in the cross-section of ring Ms yield bending moment in the cross-section of ring Pix , T1x components of pressures in x direction, i = 1–4 Piy , T1y , components of forces in y direction, i = 1–4 R outer radius of ring during rolling R1 radius of driver roll radius of idle roll R2 Ra1 rolling radius in axial pass Rm middle radius of ring during rolling S distance between the center of deforming zone in ring axial section and the top point of axial cone v feed rate of idle roll va feed rate in axial pass W anti-bend section modulus of the annulus beam Greek letters ˛1 contacting arc angle of driver roll and ring contacting arc angle of idle roll and ring ˛2 ˇ contacting arc angle between cones and ring half of cone angle 2 friction factor in axial pass friction factor between driver roll and ring position angle of guide roll s yield stress of ring material
Fig. 2 – The mechanical model in axial pass.
cone. From geometry of Fig. 1 it is known that Ra = S sin . The mechanical model of ring in axial pass is illustrated in Fig. 2. In the practical rolling process, the two cone rolls are both driven rolls, and their sizes and rotary parameters are the same. The contact friction condition between two cone rolls and ring is supposed to accord with Coulomb’s friction law, and 2 is the friction coefficient here. The normal and tangential forces of cone roll acting upon ring 5 are P and T, respectively, the acting points of the forces are supposed to lie in the midpoint of contacting arcs between cone rolls and ring, respectively. In practical rolling ˇ is very small, the radial project of P can be ignored. So it can be obtained that F = 22 p cos
ˇ 2
The force analysis in radial–axial ring rolling is illustrated in Fig. 3, the normal and tangential forces of driven roll 1 to ring 5 are P1 and T1 , respectively, and the contact friction condition between driven roll and ring is supposed to com-
deforming zone in axial section of ring, which is illustrated in Fig. 1. Ra is the rotating radius of axial cone at central acting point of the force, S is the distance from the center of deforming zone in axial section of ring to the top point of axial
Fig. 1 – Relative placement of ring and rolls in axial pass.
(1)
Fig. 3 – Forces analysis to ring in rolling.
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ply to Coulomb’s friction law, then we have T1 = 1 P1 . Idle roll 2 and guide rolls are driven to rotate, they cannot afford any friction moments, then the frictions of them on ring are supposed to be zero in rolling for simplicity. Therefore, there is only normal force P2 between idle roll and ring, and the normal forces between two guide rolls and ring are P3 and P4 . In this model, to simplify the computation, the acting points of P1 , T1 and P2 are supposed to lie in the midpoint of respective contacting arc with ring as demonstrated (Hua and Zhao, 1997). In this model, L is the projected contact length between roll and ring, and it is approximately equal to the length of the contact arc. As presented (Hawkyard et al., 1973), the projected contact length between driver roll and ring can be considered to be equal to that between idle roll and ring. The force in x direction of the axial cone rolls to ring is Fx1 . As analyzed above, F is radial project of forces to the ring, then Fx1 = F. The origin of the coordinate system O lies in the center of ring. According to the equilibrium condition of ring and expression (1), we have
Fx = P1x + T1x + P2x + P3x + P4x − Fx1 = −P1 sin
˛1 ˛1 ˛2 + 1 P1 cos − P2 sin − P3 cos 2 2 2
+P4 cos − 22 P cos
ˇ =0 2
(2)
Fy = P1y + T1y + P2y + P3y + P4y = −P1 cos +P2 cos
˛1 ˛1 − 1 P1 sin 2 2
˛2 − P3 sin − P4 sin = 0 2
Fig. 4 – The mechanical model of ring stiffness condition.
3.
Model of ring stiffness condition
The ring is collapsed or deformed under the pressure of guide roll can be regarded as that an annulus beam fixed in the juncture of driver roll and idle roll is deformed plastically under the pressure of guide roll. Then the mechanical model of ring’s stiffness condition can be built as shown in Fig. 4. The crosssection area of annulus beam is equal to the section area of ring rolled. The material model is supposed to be the ideal elastic–plastic model. According to the theory of beam bending, when the maximum normal stress in the cross-section exceeds to the yield stress of material, the beam will deform plastically. So the stiffness condition expression of the ring is that
(3) M ≤ Ms = Ws
(6)
From expression (2) and (3) we have
P1 1 cos
˛1 + ˛2 − sin 2
−P3 cos −
˛2 2
˛1 + ˛2 2
+ P4 cos +
− 22 P cos
˛2 2
=0
The bending moment and the anti-bend section modulus could be given by
ˇ ˛2 cos 2 2 (4)
In practical ring rolling, R1 > R2 L, Ra1 La , then we have sin ˛1 /2 → 0, cos ˛1 /2 → 1, cos ˇ/2 → 1, the expression (4) can be simplified as follows 1 P1 cos
˛2 ˛2 ˛2 − 22 P cos = P3 cos − 2 2 2
˛2 − P4 cos + 2
(7)
3(1 P1 − 22 P)Rm ≤ s BH2 tan tan(˛2 /2)
s BH2 tan tan(˛2 /2) 3(1 P1 − 22 P)
Substitute Rm = R − H/2 into expression above, we have
˛2 ˛2 = 2P3 sin sin 2 2
1 P1 − 22 P . 2 sin tan(˛2 /2)
BH2 6
Then the following expression can be obtained from above expressions (6) and (7)
Rm ≤
The force of guide roll exerting to ring is obtained as expression (5) P3 =
W=
From the expression above we have
Usually, two guide rolls in rolling machine are controlled under the same condition, so it could be assumed that P3 = P4 , then (1 P1 − 22 P) cos
M = P3 Rm cos ,
(5)
R ≤ Rmax =
s BH2 tan tan(˛2 /2) H + 2 3(1 P1 − 22 P)
where Rmax is the permitting maximum of R. As shown (Hua et al., 2001), the radial rolling force can be expressed as P1 = ns BL. The axial rolling force can be deduced in the same way and expressed as P = n s HLa , here the coefficient n is 4.5. And tan ˛2 /2 ≈ L/2R2 . So it can be known that R ≤ Rmax =
H2 L tan H + 2 27R2 (1 L − (22 H/B)La )
(8)
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 2570–2575
√ It is known L ≈ 2h/((1/R1 ) + (1/R2 )) which was demonstrated by Hawkyard et al. (1973) and La = S ha tan which was given by Hua et al. (2001). Substituting them into expression (8) we have R≤Rmax =
H2 tan √ 27R2 (1 −((2 H)/B h) (2/R1 )+(2/R2 ) S ha tan) +
H 2
(9)
According the expression (9), the outer radius of ring R should not exceed the maximum radius Rmax required by ring’s stiffness condition in radial–axial ring rolling.
4.
Analysis and discussion
According to expression (9), the ring stiffness condition in radial–axial ring rolling is related to its dimension and rolling ratio, and also have relation with position angel of guide roll, dimensions of rolls, position of cone rolls, radial and axial feed speed and friction condition. Then the following conclusions can be drawn: (1) When the position angle of guide roll increases, Rmax will increase, which is helpful to the ring stiffness condition. But if is too large, there will be interference between guide roll and driven roll. Under the premise of not interfering, increasing is favorable to stiffness condition of ring, which is the essential reason why is in the range of 60◦ to 70◦ as presented (Hua et al., 2001). (2) Increasing of radial friction coefficient 1 will result in decreasing of Rmax and is harmful to the ring stiffness condition. But 1 increasing is available for ring to be bitten into the radial pass. To ensure ring’s being bitten, the radial friction coefficient cannot be too large, which is the main reason why the product roundness will be not ideal when 1 is too large in practice. (3) If the axial friction coefficient 2 increases, Rmax will increase, which is advantageous to the ring stiffness condition. Since 2 increasing is available for ring to be bitten to the axial pass, the radial friction coefficient is favorable to both the axial biting condition and the ring stiffness condition. (4) Increasing of h will result in decreasing of Rmax , which is unfavorable to the stiffness condition. Increasing of h is available for ring to be plastically penetrated and not available for ring to be bitten. Under the premise that ring is plastically penetrated and bitten, h cannot be too large. (5) Increasing of ha will result in increasing of Rmax , which is favorable to the stiffness condition. Increasing of ha is not available for ring to be bitten into axial pass and available for ring to be plastically penetrated. Under the premise that ring is plastically penetrated and bit, ha increasing is helpful to the ring stiffness condition. (6) If the distance from the center of deforming zone in axial section of ring to the top point of axial cone S increases, Rmax will increase, which is advantageous to the ring stiff-
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Table 1 – The experimental parameters Parameters Outer diameter of ring blank (mm) Inner diameter of ring blank (mm) Height of ring blank (mm) Outer diameter of ring product (mm) Inner diameter of ring product (mm) Height of ring product (mm) Diameter of drive roll (mm) Diameter of idle roll (mm) Rotational speed of drive roll (rad/s)
Dimension Ø 680 Ø 400 120 Ø 1000 Ø 800 100 Ø 800 Ø 240 2
ness condition. Then during rolling the position of cone roll can be regulated to ensure S be larger. (7) When the ratio of H/B increases, Rmax will increase, which is available to the ring stiffness condition. So the axial height cannot be too large when designing ring blank. Under the condition of same rolling ratio, B increasing means decreasing of H, then the value of H/B will decrease, which is harmful to the ring stiffness condition. The analysis accords with the condition that the axial height of ring blank cannot be too large in the practical rolling manufacture as stated (Hua et al., 2001). In addition, it is apparent that H increasing is favorable to ring stiffness. Furthermore, Rmax decreases with increasing of R1 and R2 , which is disadvantageous to ring stiffness. However, according to expression (9), the influence of R1 and R2 on the stiffness condition is not apparent. And the top angle of cone 2 increasing is helpful to the ring stiffness condition. In general, the top angle of cone roll is determined by rolling machine, when certain rolling machine is chosen, the angle will be decided.
5.
Simulation of Fem and analysis
The FEM is a main effective method to analyze radial–axial ring rolling technology and provide conducts. Kim et al. (1990) simulated ring rolling process using three-dimension finite element method. Hu et al. (1994) also presented a 3D FE modeling measure of ring rolling. The movement of guide roll was simulated by using finite element method as shown (Forouzan et al., 2003). Hence, the FEM was used to analyze the rolling process in this paper. The following is the FE simulation on a specified ring rolling experiment. The cross-section of ring blank is rectangle, and the parameters about the rolling experiment are showed in Table 1. The effect rules of the parameters in radial pass such as 1 , and h on the ring stiffness condition are similar to that in radial ring rolling technology as demonstrated (Hua et al., 2001), then, the simulating experiment of FEM in this paper will involve mainly the parameters of 2 , ha and S. In the present investigation, the explicit dynamic and rigid-plastic finite element methods are used to avoid long computation time and make improvement of computation accuracy. The simulating model is constructed as Fig. 5. In this experiment, the ring material used is aluminum alloy 7050. The rolling temperature is about 470 ◦ C. Since the rolling time
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j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 2570–2575
Fig. 5 – The simulating model of rolling.
is long, ring should be reheated during practical rolling. For simplification in this model, the properties with temperature 470 ◦ C are given to material during rolling ignoring change of temperature. A uniform mesh with 8-noded first-order reduction integration continuum elements is used and the adaptive remeshing method is employed. The rolls are treated as rigid bodies. The guide rolls revolve around some fixed axes with increasing of ring diameter in rolling according to the practical movement condition. It is assumed that the frictions in radial and axial passes meet the law of Coulomb friction. And the radial friction coefficients are given 0.4. The rotational speed of axial cones should match that of driver roll to keep rotational linear speed equal. The radial feed rate is shown in Fig. 6 in the experiment. The processes are simulated under different parameters. The ring stiffness is determined by pressure of guide roll, here the simulation was mainly to verify the influencing rules of parameters on ring stiffness condition, so the pressure curves of guide roll in different rolling process are analyzed. Since ring dimension are the same during different rolling processes, high pressure of guide roll will be more harmful to the ring stiffness.
Fig. 6 – The curve of radial feed rate.
5.1.
Simulating experiment with different 2
The rolling processes with three different axial friction coefficients 2 are simulated, and the other parameters and rolling condition are the same. The three different 2 are 0.1, 0.15 and 0.2. The axial feed rate feed rate va is 0.2 mm/s, and the distance from the center of deforming zone in axial section of ring to the top point of axial cone roll S is 210 mm. Then the pressure curves of guide roll in these three processes are shown in Fig. 7. From the results, when 2 is 0.1, the maximum pressure of guide roll is about 330 kN; when 2 is 0.15, the maximum value goes to about 285 kN; when 2 goes to 0.2, it is about 215 kN. Hence, with the value of axial friction coefficient increases, the average pressure of guide roll decreases, which is helpful to the ring stiffness. The results accord with above theoretical analysis.
5.2.
Simulating experiment with different va
The rolling processes with three different axial feed rates va are simulated. The rotational speed of driven roll is constant,
Fig. 7 – The pressure curves of guide roll with different 2 .
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point of axial cone roll S are 160 mm, 185 mm and 210 mm. The axial friction coefficient 2 is 0.15, and the axial feed rate va is 0.2 mm/s. Then the pressure curves of guide roll in these three processes are shown in Fig. 9. According to the curves, the maximum pressures of guide rolls in three rolling processes are, respectively, 395 kN, 330 kN and 285 kN. The average pressure of guide roll also decreases with the distance increasing. So S increasing is helpful to the ring stiffness, which is accorded with above theoretical analysis.
6.
Fig. 8 – The pressure curves of guide roll with different va .
and different feed rates mean different feeds per rotation ha . In this experiment axial feed rate decreases, axial feed per rotation will decrease accordingly. The three different va are 0.18 mm/s, 0.2 mm/s and 0.22 mm/s. The axial friction coefficient 2 is 0.15, and the distance from the center of deforming zone in axial section of ring to the top point of axial cone S is 210 mm. The pressure curves of guide roll in three processes are shown in Fig. 8. When the axial feed rate va is 0.18 mm/s, the maximum pressure of guide roll during rolling is about 300 kN; when va is 0.2 mm/s, the maximum pressure during rolling is about 285 kN; when va is 0.22 mm/s, the value goes to about 260 kN. According to these data and curves, the axial feed rate increases, the axial feed per rotation will increase accordingly, and the pressure of guide roll will decreases, which is favorable to the ring stiffness. The results verify the theoretical analysis above.
5.3.
Simulating experiment with different S
The rolling processes with three different placements of axial cone rolls are simulated. The three different distances from the center of deforming zone in axial section of ring to the top
Fig. 9 – The pressure curves of guide roll with different S.
Conclusions
The ring keeps stiffness during rolling is one of the essential conditions to obtain stable rolling process and final qualified product. The model of the ring stiffness in radial–axial ring rolling was proposed, and the stiffness condition of ring in rolling was derived. The results showed that the ring stiffness condition in radial–axial ring rolling is related to its dimension and rolling ratio, and also have relation with position angel of guide roll, dimensions of rolls, position of cones, radial and axial feed rate and friction condition. The results are proved by FEM simulation. In the axial pass, the increasing of 2 , ha and S are all favorable to the ring stiffness condition during radial–axial ring rolling, which can be used as a guideline for radial–axial ring rolling technology.
Acknowledgement The authors would like to thank the support of National Natural Science Foundation of China Key Project (No. 50335060).
references
Choi, H.D., Cho, H.S., Lee, J.S., 1992. On the dynamic characteristics of radial–axial ring rolling processes. J. Eng. Industry 114, 188–195. Forouzan, M.R., Salimi, M., Gadala, M.S., 2003. Guide roll simulation in FE analysis of ring rolling. J. Mater. Process. Technol. 142, 213–223. Guan, H.Y., Chen, Zh.G., Song, T., 1995. The rolling technology of radial–axial ring rolling machine. Forging Stamp. Technol. 3, 37–40 (in Chinese). Hawkyard, J.B., Johnson, W., Kirkland, J., 1973. Analysis for roll force and torque in ring rolling with some supporting experiment. Int. J. Mech. Sci. 15, 873–893. Hu, Z.M., Pillinger, I., Hartley, P., et al., 1994. Three-dimension finite-element modeling of ring rolling. J. Mater. Process. Technol. 45, 143–148. Hua, L., Zhao, Z.Z., 1997. The extremum parameters in ring rolling. J. Mater. Process. Technol. 69, 273–276. Hua, L., Huang, X.G., Zhu, C.D., 2001. Ring Rolling Theory and Technology. China Machine Press, Beijing, pp. 2–16 (in Chinese). Jiang, R.D., Li, C.X., Wang, Y.G., 1994. Allowable rolling region of radial–axial ring rolling mill. Forging Stamp. Technol. 1, 33–37 (in Chinese). Kim, N., Machida, S., Kobayshi, S., 1990. Ring rolling process simulation by the three-dimension finite element method. Int. J. Machine Tools Manuf. 30, 569–577.
journal homepage: www.elsevier.com/locate/jmatprotec
Researches on the ring stiffness condition in radial–axial ring rolling Lin Hua ∗ , Libo Pan, Jian Lan School of Materials Science and Engineering, Wuhan University of Technology, Wuhan 122#, Luoshi Road, Wuhan 430070, China
a r t i c l e
i n f o
a b s t r a c t
Article history:
The radial–axial ring rolling technology is used mainly to manufacture large rings in many
Received 10 July 2007
industrial fields. During rolling process the ring may be collapsed or deformed unexpectedly
Received in revised form
under the pressure of guide roll, and in this case the ring is wasted. To prevent ring collapsing
22 May 2008
is necessary for smooth rolling. On the basis of rolling theory, the forces exerted to ring in
Accepted 7 June 2008
radial–axial ring rolling process were analyzed, and the stiffness model of ring was proposed. The stiffness condition was derived and the influencing factors of ring stiffness condition were explained. The ring stiffness condition in radial–axial ring rolling is related to the
Keywords:
factors of ring size, rolling ratio, position angel of guide roll, rolls sizes, position of cones,
Radial–axial ring rolling
radial and axial feed speed and friction condition. The effect laws of some factors on ring
Ring
stiffness condition were revealed by FEM simulation. The stiffness condition can be used as
Stiffness condition
a rule of design and manufacturing for radial–axial ring rolling. © 2008 Elsevier B.V. All rights reserved.
Finite element
1.
Introduction
Ring rolling is a kind of advanced technology to manufacture rings in many industrial fields such as machine, automobiles, trains and airplanes, etc. Radial–axial ring rolling technology is primarily used to manufacture large ring. In radial–axial ring rolling, two axial cone rolls are added to roll the axial section of ring comparing to the radial ring rolling. In the rolling process the radial and axial sections of ring are rolled at the same time. During rolling, radial thickness and axial height of ring decease and ring diameter increases. Two guide rolls play an important role in radial–axial ring rolling. They not only help to keep smooth rolling process, but also have great effect on the forming quality of product. Occasionally, during rolling process the ring may be deformed or collapsed unexpectedly under the pressure of guide rolls because the ring’s stiffness is not satisfied, which will result in unqualified ring product. We wonder how many parameters can affect the ring’s stiffness in radial–axial ring rolling. Hua et al. (2001)
∗
Corresponding author. Tel.: +86 2787168391; fax: +86 2787168391. E-mail address: [email protected] (L. Hua). 0924-0136/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2008.06.002
have researched the stiffness condition theoretically in radial ring rolling by constructing the mechanical model. Choi et al. (1992) presented and analyzed the dynamic characteristics of radial–axial ring rolling processes. The rolling technology according to a certain rolling machine was introduced by Guan et al. (1995). Jiang et al. (1994) analyzed the allowable rolling region of radial–axial ring rolling mill from theoretical aspect. However, above documents related to radial–axial ring rolling technology have not involved the ring’s stiffness. In this paper, the ring’s stiffness condition in radial–axial ring rolling is intensively investigated, and the factors influencing the ring’s stiffness are revealed.
2.
Force analysis in radial–axial ring rolling
It is known that the axial rolls are cones, which are different from the radial rolls of cylinder. Suppose the acting points of the forces between ring and cones lie in the center of
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 2570–2575
2571
Nomenclature B F
axial height of ring radial project of force acting on the ring Fx sum of forces in x direction Fy sum of force in y direction h feed per turn of idle roll ha feed per turn in axial pass H radial thickness of ring during rolling L projected contact length between roll and ring La projection of axial contacting arc in radial direction M bending moment in the cross-section of ring Ms yield bending moment in the cross-section of ring Pix , T1x components of pressures in x direction, i = 1–4 Piy , T1y , components of forces in y direction, i = 1–4 R outer radius of ring during rolling R1 radius of driver roll radius of idle roll R2 Ra1 rolling radius in axial pass Rm middle radius of ring during rolling S distance between the center of deforming zone in ring axial section and the top point of axial cone v feed rate of idle roll va feed rate in axial pass W anti-bend section modulus of the annulus beam Greek letters ˛1 contacting arc angle of driver roll and ring contacting arc angle of idle roll and ring ˛2 ˇ contacting arc angle between cones and ring half of cone angle 2 friction factor in axial pass friction factor between driver roll and ring position angle of guide roll s yield stress of ring material
Fig. 2 – The mechanical model in axial pass.
cone. From geometry of Fig. 1 it is known that Ra = S sin . The mechanical model of ring in axial pass is illustrated in Fig. 2. In the practical rolling process, the two cone rolls are both driven rolls, and their sizes and rotary parameters are the same. The contact friction condition between two cone rolls and ring is supposed to accord with Coulomb’s friction law, and 2 is the friction coefficient here. The normal and tangential forces of cone roll acting upon ring 5 are P and T, respectively, the acting points of the forces are supposed to lie in the midpoint of contacting arcs between cone rolls and ring, respectively. In practical rolling ˇ is very small, the radial project of P can be ignored. So it can be obtained that F = 22 p cos
ˇ 2
The force analysis in radial–axial ring rolling is illustrated in Fig. 3, the normal and tangential forces of driven roll 1 to ring 5 are P1 and T1 , respectively, and the contact friction condition between driven roll and ring is supposed to com-
deforming zone in axial section of ring, which is illustrated in Fig. 1. Ra is the rotating radius of axial cone at central acting point of the force, S is the distance from the center of deforming zone in axial section of ring to the top point of axial
Fig. 1 – Relative placement of ring and rolls in axial pass.
(1)
Fig. 3 – Forces analysis to ring in rolling.
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ply to Coulomb’s friction law, then we have T1 = 1 P1 . Idle roll 2 and guide rolls are driven to rotate, they cannot afford any friction moments, then the frictions of them on ring are supposed to be zero in rolling for simplicity. Therefore, there is only normal force P2 between idle roll and ring, and the normal forces between two guide rolls and ring are P3 and P4 . In this model, to simplify the computation, the acting points of P1 , T1 and P2 are supposed to lie in the midpoint of respective contacting arc with ring as demonstrated (Hua and Zhao, 1997). In this model, L is the projected contact length between roll and ring, and it is approximately equal to the length of the contact arc. As presented (Hawkyard et al., 1973), the projected contact length between driver roll and ring can be considered to be equal to that between idle roll and ring. The force in x direction of the axial cone rolls to ring is Fx1 . As analyzed above, F is radial project of forces to the ring, then Fx1 = F. The origin of the coordinate system O lies in the center of ring. According to the equilibrium condition of ring and expression (1), we have
Fx = P1x + T1x + P2x + P3x + P4x − Fx1 = −P1 sin
˛1 ˛1 ˛2 + 1 P1 cos − P2 sin − P3 cos 2 2 2
+P4 cos − 22 P cos
ˇ =0 2
(2)
Fy = P1y + T1y + P2y + P3y + P4y = −P1 cos +P2 cos
˛1 ˛1 − 1 P1 sin 2 2
˛2 − P3 sin − P4 sin = 0 2
Fig. 4 – The mechanical model of ring stiffness condition.
3.
Model of ring stiffness condition
The ring is collapsed or deformed under the pressure of guide roll can be regarded as that an annulus beam fixed in the juncture of driver roll and idle roll is deformed plastically under the pressure of guide roll. Then the mechanical model of ring’s stiffness condition can be built as shown in Fig. 4. The crosssection area of annulus beam is equal to the section area of ring rolled. The material model is supposed to be the ideal elastic–plastic model. According to the theory of beam bending, when the maximum normal stress in the cross-section exceeds to the yield stress of material, the beam will deform plastically. So the stiffness condition expression of the ring is that
(3) M ≤ Ms = Ws
(6)
From expression (2) and (3) we have
P1 1 cos
˛1 + ˛2 − sin 2
−P3 cos −
˛2 2
˛1 + ˛2 2
+ P4 cos +
− 22 P cos
˛2 2
=0
The bending moment and the anti-bend section modulus could be given by
ˇ ˛2 cos 2 2 (4)
In practical ring rolling, R1 > R2 L, Ra1 La , then we have sin ˛1 /2 → 0, cos ˛1 /2 → 1, cos ˇ/2 → 1, the expression (4) can be simplified as follows 1 P1 cos
˛2 ˛2 ˛2 − 22 P cos = P3 cos − 2 2 2
˛2 − P4 cos + 2
(7)
3(1 P1 − 22 P)Rm ≤ s BH2 tan tan(˛2 /2)
s BH2 tan tan(˛2 /2) 3(1 P1 − 22 P)
Substitute Rm = R − H/2 into expression above, we have
˛2 ˛2 = 2P3 sin sin 2 2
1 P1 − 22 P . 2 sin tan(˛2 /2)
BH2 6
Then the following expression can be obtained from above expressions (6) and (7)
Rm ≤
The force of guide roll exerting to ring is obtained as expression (5) P3 =
W=
From the expression above we have
Usually, two guide rolls in rolling machine are controlled under the same condition, so it could be assumed that P3 = P4 , then (1 P1 − 22 P) cos
M = P3 Rm cos ,
(5)
R ≤ Rmax =
s BH2 tan tan(˛2 /2) H + 2 3(1 P1 − 22 P)
where Rmax is the permitting maximum of R. As shown (Hua et al., 2001), the radial rolling force can be expressed as P1 = ns BL. The axial rolling force can be deduced in the same way and expressed as P = n s HLa , here the coefficient n is 4.5. And tan ˛2 /2 ≈ L/2R2 . So it can be known that R ≤ Rmax =
H2 L tan H + 2 27R2 (1 L − (22 H/B)La )
(8)
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√ It is known L ≈ 2h/((1/R1 ) + (1/R2 )) which was demonstrated by Hawkyard et al. (1973) and La = S ha tan which was given by Hua et al. (2001). Substituting them into expression (8) we have R≤Rmax =
H2 tan √ 27R2 (1 −((2 H)/B h) (2/R1 )+(2/R2 ) S ha tan) +
H 2
(9)
According the expression (9), the outer radius of ring R should not exceed the maximum radius Rmax required by ring’s stiffness condition in radial–axial ring rolling.
4.
Analysis and discussion
According to expression (9), the ring stiffness condition in radial–axial ring rolling is related to its dimension and rolling ratio, and also have relation with position angel of guide roll, dimensions of rolls, position of cone rolls, radial and axial feed speed and friction condition. Then the following conclusions can be drawn: (1) When the position angle of guide roll increases, Rmax will increase, which is helpful to the ring stiffness condition. But if is too large, there will be interference between guide roll and driven roll. Under the premise of not interfering, increasing is favorable to stiffness condition of ring, which is the essential reason why is in the range of 60◦ to 70◦ as presented (Hua et al., 2001). (2) Increasing of radial friction coefficient 1 will result in decreasing of Rmax and is harmful to the ring stiffness condition. But 1 increasing is available for ring to be bitten into the radial pass. To ensure ring’s being bitten, the radial friction coefficient cannot be too large, which is the main reason why the product roundness will be not ideal when 1 is too large in practice. (3) If the axial friction coefficient 2 increases, Rmax will increase, which is advantageous to the ring stiffness condition. Since 2 increasing is available for ring to be bitten to the axial pass, the radial friction coefficient is favorable to both the axial biting condition and the ring stiffness condition. (4) Increasing of h will result in decreasing of Rmax , which is unfavorable to the stiffness condition. Increasing of h is available for ring to be plastically penetrated and not available for ring to be bitten. Under the premise that ring is plastically penetrated and bitten, h cannot be too large. (5) Increasing of ha will result in increasing of Rmax , which is favorable to the stiffness condition. Increasing of ha is not available for ring to be bitten into axial pass and available for ring to be plastically penetrated. Under the premise that ring is plastically penetrated and bit, ha increasing is helpful to the ring stiffness condition. (6) If the distance from the center of deforming zone in axial section of ring to the top point of axial cone S increases, Rmax will increase, which is advantageous to the ring stiff-
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Table 1 – The experimental parameters Parameters Outer diameter of ring blank (mm) Inner diameter of ring blank (mm) Height of ring blank (mm) Outer diameter of ring product (mm) Inner diameter of ring product (mm) Height of ring product (mm) Diameter of drive roll (mm) Diameter of idle roll (mm) Rotational speed of drive roll (rad/s)
Dimension Ø 680 Ø 400 120 Ø 1000 Ø 800 100 Ø 800 Ø 240 2
ness condition. Then during rolling the position of cone roll can be regulated to ensure S be larger. (7) When the ratio of H/B increases, Rmax will increase, which is available to the ring stiffness condition. So the axial height cannot be too large when designing ring blank. Under the condition of same rolling ratio, B increasing means decreasing of H, then the value of H/B will decrease, which is harmful to the ring stiffness condition. The analysis accords with the condition that the axial height of ring blank cannot be too large in the practical rolling manufacture as stated (Hua et al., 2001). In addition, it is apparent that H increasing is favorable to ring stiffness. Furthermore, Rmax decreases with increasing of R1 and R2 , which is disadvantageous to ring stiffness. However, according to expression (9), the influence of R1 and R2 on the stiffness condition is not apparent. And the top angle of cone 2 increasing is helpful to the ring stiffness condition. In general, the top angle of cone roll is determined by rolling machine, when certain rolling machine is chosen, the angle will be decided.
5.
Simulation of Fem and analysis
The FEM is a main effective method to analyze radial–axial ring rolling technology and provide conducts. Kim et al. (1990) simulated ring rolling process using three-dimension finite element method. Hu et al. (1994) also presented a 3D FE modeling measure of ring rolling. The movement of guide roll was simulated by using finite element method as shown (Forouzan et al., 2003). Hence, the FEM was used to analyze the rolling process in this paper. The following is the FE simulation on a specified ring rolling experiment. The cross-section of ring blank is rectangle, and the parameters about the rolling experiment are showed in Table 1. The effect rules of the parameters in radial pass such as 1 , and h on the ring stiffness condition are similar to that in radial ring rolling technology as demonstrated (Hua et al., 2001), then, the simulating experiment of FEM in this paper will involve mainly the parameters of 2 , ha and S. In the present investigation, the explicit dynamic and rigid-plastic finite element methods are used to avoid long computation time and make improvement of computation accuracy. The simulating model is constructed as Fig. 5. In this experiment, the ring material used is aluminum alloy 7050. The rolling temperature is about 470 ◦ C. Since the rolling time
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j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 2570–2575
Fig. 5 – The simulating model of rolling.
is long, ring should be reheated during practical rolling. For simplification in this model, the properties with temperature 470 ◦ C are given to material during rolling ignoring change of temperature. A uniform mesh with 8-noded first-order reduction integration continuum elements is used and the adaptive remeshing method is employed. The rolls are treated as rigid bodies. The guide rolls revolve around some fixed axes with increasing of ring diameter in rolling according to the practical movement condition. It is assumed that the frictions in radial and axial passes meet the law of Coulomb friction. And the radial friction coefficients are given 0.4. The rotational speed of axial cones should match that of driver roll to keep rotational linear speed equal. The radial feed rate is shown in Fig. 6 in the experiment. The processes are simulated under different parameters. The ring stiffness is determined by pressure of guide roll, here the simulation was mainly to verify the influencing rules of parameters on ring stiffness condition, so the pressure curves of guide roll in different rolling process are analyzed. Since ring dimension are the same during different rolling processes, high pressure of guide roll will be more harmful to the ring stiffness.
Fig. 6 – The curve of radial feed rate.
5.1.
Simulating experiment with different 2
The rolling processes with three different axial friction coefficients 2 are simulated, and the other parameters and rolling condition are the same. The three different 2 are 0.1, 0.15 and 0.2. The axial feed rate feed rate va is 0.2 mm/s, and the distance from the center of deforming zone in axial section of ring to the top point of axial cone roll S is 210 mm. Then the pressure curves of guide roll in these three processes are shown in Fig. 7. From the results, when 2 is 0.1, the maximum pressure of guide roll is about 330 kN; when 2 is 0.15, the maximum value goes to about 285 kN; when 2 goes to 0.2, it is about 215 kN. Hence, with the value of axial friction coefficient increases, the average pressure of guide roll decreases, which is helpful to the ring stiffness. The results accord with above theoretical analysis.
5.2.
Simulating experiment with different va
The rolling processes with three different axial feed rates va are simulated. The rotational speed of driven roll is constant,
Fig. 7 – The pressure curves of guide roll with different 2 .
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 2570–2575
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point of axial cone roll S are 160 mm, 185 mm and 210 mm. The axial friction coefficient 2 is 0.15, and the axial feed rate va is 0.2 mm/s. Then the pressure curves of guide roll in these three processes are shown in Fig. 9. According to the curves, the maximum pressures of guide rolls in three rolling processes are, respectively, 395 kN, 330 kN and 285 kN. The average pressure of guide roll also decreases with the distance increasing. So S increasing is helpful to the ring stiffness, which is accorded with above theoretical analysis.
6.
Fig. 8 – The pressure curves of guide roll with different va .
and different feed rates mean different feeds per rotation ha . In this experiment axial feed rate decreases, axial feed per rotation will decrease accordingly. The three different va are 0.18 mm/s, 0.2 mm/s and 0.22 mm/s. The axial friction coefficient 2 is 0.15, and the distance from the center of deforming zone in axial section of ring to the top point of axial cone S is 210 mm. The pressure curves of guide roll in three processes are shown in Fig. 8. When the axial feed rate va is 0.18 mm/s, the maximum pressure of guide roll during rolling is about 300 kN; when va is 0.2 mm/s, the maximum pressure during rolling is about 285 kN; when va is 0.22 mm/s, the value goes to about 260 kN. According to these data and curves, the axial feed rate increases, the axial feed per rotation will increase accordingly, and the pressure of guide roll will decreases, which is favorable to the ring stiffness. The results verify the theoretical analysis above.
5.3.
Simulating experiment with different S
The rolling processes with three different placements of axial cone rolls are simulated. The three different distances from the center of deforming zone in axial section of ring to the top
Fig. 9 – The pressure curves of guide roll with different S.
Conclusions
The ring keeps stiffness during rolling is one of the essential conditions to obtain stable rolling process and final qualified product. The model of the ring stiffness in radial–axial ring rolling was proposed, and the stiffness condition of ring in rolling was derived. The results showed that the ring stiffness condition in radial–axial ring rolling is related to its dimension and rolling ratio, and also have relation with position angel of guide roll, dimensions of rolls, position of cones, radial and axial feed rate and friction condition. The results are proved by FEM simulation. In the axial pass, the increasing of 2 , ha and S are all favorable to the ring stiffness condition during radial–axial ring rolling, which can be used as a guideline for radial–axial ring rolling technology.
Acknowledgement The authors would like to thank the support of National Natural Science Foundation of China Key Project (No. 50335060).
references
Choi, H.D., Cho, H.S., Lee, J.S., 1992. On the dynamic characteristics of radial–axial ring rolling processes. J. Eng. Industry 114, 188–195. Forouzan, M.R., Salimi, M., Gadala, M.S., 2003. Guide roll simulation in FE analysis of ring rolling. J. Mater. Process. Technol. 142, 213–223. Guan, H.Y., Chen, Zh.G., Song, T., 1995. The rolling technology of radial–axial ring rolling machine. Forging Stamp. Technol. 3, 37–40 (in Chinese). Hawkyard, J.B., Johnson, W., Kirkland, J., 1973. Analysis for roll force and torque in ring rolling with some supporting experiment. Int. J. Mech. Sci. 15, 873–893. Hu, Z.M., Pillinger, I., Hartley, P., et al., 1994. Three-dimension finite-element modeling of ring rolling. J. Mater. Process. Technol. 45, 143–148. Hua, L., Zhao, Z.Z., 1997. The extremum parameters in ring rolling. J. Mater. Process. Technol. 69, 273–276. Hua, L., Huang, X.G., Zhu, C.D., 2001. Ring Rolling Theory and Technology. China Machine Press, Beijing, pp. 2–16 (in Chinese). Jiang, R.D., Li, C.X., Wang, Y.G., 1994. Allowable rolling region of radial–axial ring rolling mill. Forging Stamp. Technol. 1, 33–37 (in Chinese). Kim, N., Machida, S., Kobayshi, S., 1990. Ring rolling process simulation by the three-dimension finite element method. Int. J. Machine Tools Manuf. 30, 569–577.