For obtaining the profile and the projected area of contact zone exactly during alloyed bar rolling by Round-Oval-Round pass sequence, the analytic model for the length contact and contact boundary curve were built firstly by considering the influence of the spread of the outgoing workpiece on the effective height of outgoing workpiece and roll mean radius, and then the contact surface was discretized by finite flow line elements. Moreover, the radius equation and bite angle equation of different flow lines were derived and they were all expressed as the function of the position angle, then any flow line on the contact surface can be determined since the position angle has been given. Finally, since the analytic equation for the projected area of contact surface was hard to be integrated directly, the analytic model was proposed by summing up the area of discretized parts on the contact surface. Based on the analytical model of contact boundary and flow line element, 3-dimension contact surface w as rebuilt by mathematical software, and the validity of analytic model was examined by the bar rolling experiments and the numerical simulation of alloyed bar rolling by rigid-plastic FEM software. Compared with the existing models, the precision of the projecting area of contact zone was improved obviously. So, it can be applied in alloyed bar rolling to predict the projected area of contact zone and velocity of outgoing workpiece exactly.
Compared with products rolled with the oval-square-oval pass sequence, the bar or rod rolled with round-oval-round pass sequence has a better surface quality and mechanical performance, and the round-oval-round pass sequence is the most common roll pass in bar or rod continuous rolling recently. Since the characteristics of non-uniform distribution of stain, stress and flow velocity on the deformation zone, it is difficult to analyze the process of alloyed bar rolling in oval-square-oval pass sequence accurately [
In the past few years, the research on the alloyed bar rolling process was carried out by the simulation methods and experiments. References [
It is not easy to obtain the precise projecting area of contact zone and the accuracy profile of contact boundary since the contour of contact zone is not regular and difficult to be defined. So, at present the geometry and projected area of contact zone is mostly calculated by the simplified methods, such as the empirical equation [
Shinokura and Takai [
The profile of contact boundary and the projecting area of contact zone are correlated to the solution of the rolling force, the velocity field distribution, stress field distribution and strain field distribution directly. So, it is very important to determine the profile of contact boundary and the projected area of contact zone exactly.
As can be seen in
Shinokura and Takai [
The size of oval groove, round groove and corresponding incoming workpiece were shown in
H 0 ¯ = A 0 − A s 0 2 C Z 0 (1)
H m 0 ¯ = A 0 − A s 0 − A h 2 C Z 0 = A e 0 2 C Z 0 (2)
where A 0 is the section area of incoming workpiece, A s 0 , A h , A e 0 are the non-effective reduction area, effective reduction area and effective exit section
area respectively, ( C y 0 , C z 0 ) is the intersection point between the profile of incoming workpiece and the roll pass.
In round-oval pass rolling, A h and A s 0 may be obtained by
A h / 2 = arctan ( C Y 0 / C Z 0 ) ⋅ R a 2 − [ ∫ - C Y 0 C Y 0 ( R 1 2 − y 2 − D z ) d y − C Y 0 ⋅ C Z 0 ] (3)
A s 0 / 2 = arctan ( C Z 0 / C Y 0 ) ⋅ R a − C Y 0 ⋅ C Z 0 . (4)
In oval-round pass rolling, A h and A s 0 may be given by
A h / 2 = π 8 W max ⋅ H p − A s 0 2 − arccos ( C Z 0 / R Y 0 ) ⋅ R g 2 − C Y 0 ⋅ C Z 0 (5)
A s 0 / 2 = arcsin ( C Z 0 / R 1 ) ⋅ R 1 2 − ( D Z + C Y 0 ) ⋅ C Z 0 (6)
where R 1 is the radius of the oval groove, R s is the radius of the curvature of the incoming cross-section, D Z is the distance along the Z-axis direction between the origin coordinate
Then the maximum contact length L max was obtained by
L max = R m 0 ( H 0 ¯ − H m 0 ¯ ) = R m 0 A h 2 C Z 0 (7)
where R m 0 is the mean roll radius at the entrance-section, it can be shown as
R m 0 = R min + H p 2 − H m 0 ¯ 2 . (8)
For obtaining the projected area of contact zone in alloy bar rolling, Shinokura and Takai expressed the contact boundary curve by the function L max 1 − x 2 C y 2 according to the empirical data, and the projected area was shown as
A p = ∫ 0 C y L max 1 − x 2 C y 2 d x = π 2 L max C y . (9)
As shown in
On the base of Equation (9) Y.lee given another contact boundary curve function L max ( 1 − x C y ) 1 / m to modify this equation and the projected area was shown as
A p = ∫ 0 C y L max ( 1 − x C y ) 1 / m d x = 3 2 L max C y ( m = 1 3 ) . (10)
Moreover, as shown in Equation (9) and Equation (10), two hypothetical functions for indicating the contact boundary curve were given directly without any reasoning and any derivation process. Although the error between the results of two equations are not obvious when the size of rolling workpiece is small enough and then the contact length L and Cy is small enough, the absolute error of these two equations will be considerable and it should not be ignored in large diameter bar rolling. So it is not precise enough for these two semi-analytic models to calculate the projected area of multi-pass alloyed bar rolling, then an analytic model should be built to predict the projected area accurately.
In summing up these models for projected area, the Equation (2), which is based on the graphical solving method, does not take the influence of the spread and the contact boundary status of deformed workpiece into account. Therefore, the Graphical solving method can be just used as an estimating value when the spread of deformed workpiece is small enough (
As shown in
L m = R m ( H 0 ¯ − H m ¯ ) = ( R min + H p / 2 − G / 2 − H m ¯ / 2 ) ( H 0 ¯ − H m ¯ ) . (11)
In round-oval pass rolling, the equivalent height of outgoing workpiece was shown as
H m ¯ 2 = A e 4 C y = ∫ − C y C y ( R 1 2 − y 2 − D z ) d y C y = R 1 2 ( 2 C y R 1 cos ( arcsin C y R 1 ) + 2 arcsin C y R 1 ) 4 C y − D z . (12)
In oval-round pass rolling, the equivalent height of outgoing workpiece was
expressed as
H m ¯ 2 = A e 4 C y = arctan ( C y / C z ) R + g 2 C y C z C y . (13)
As can be seen in
The flow plane was defined as a set of eccentric continuous cylindrical surfaces having almost straight generators parallel to the roll axis. Assuming that at any cross section along the roll bite the bar height deformation is uniform, the deformation zone was constituted by a set of flow plane, and the contact surface was constituted by a set of flow line element f α . The flow line element f α is a set of concentric circular arc with different radius R and different bite angle θ α s , which center is attached on the roll axis. Since the roll radius in the roll pass is different and the height of incoming workpiece is different along the y-axis direction, the bite angle θ α s , along the whole contact boundary, is not a constant but a variable which changes with a different position angle α or a different roll radius R .
The radius of flow line f α was expressed as
R = R min + R 1 ( 1 − cos α ) ( 0 ≤ α ≤ arcsin C y R 1 or 0 ≤ α ≤ arccos C z + D z R 1 ) . (14)
The three-dimension coordinates of a random point on the contact surface was shown as
{ x = L − R sin θ ( 0 ≤ θ ≤ θ α s ) y = R 1 sin α z = R ( 1 − cos θ ) + R 1 cos α − D z = − R cos θ + R c . (15)
On the symmetry plane of deformed workpiece, the position angle α is 0 and the bite angle θ α s reaches the maximum value θ max
θ max = arccos ( 1 − H 0 − H p 2 R 0 ) . (16)
At the exit section of deformed workpiece, the bite angle θ α s is 0 and the position angle α reaches the maximum value α max
α max = arccos C z + D z R 1 (17)
According to the Equation (15), the coordinates of points on the contact boundary was expressed as
{ x s = L − R sin θ α s y s = R 1 sin α z s = R ( 1 − cos θ α s ) + R 1 cos α − D z = − R cos θ α s + R c . (18)
In the contact zone, the height of the profile at y = 0 along the x-direction is expressed as
H ( x ) = H 0 − 2 L R 0 x + 1 R 0 x 2 . (19)
The curve equation of contact boundary S can be obtained approximately by interpolating between the point ( 0 , 0 , H 0 ) and the point ( L , C y , C z ) . The coordinate of any point on the contact boundary S is shown as ( x , C y s , C z s ) , and the coordinate C z s can be shown as
2 C z s = H 0 − 2 ( H 0 − 2 C z ) x s L + ( H 0 − 2 C z ) x s 2 L 2 . (20)
Substituting the xs of Equation (18) into Equation (20) yields
2 C z s = ( H 0 − 2 C z ) R 2 sin 2 θ α s L 2 + 2 C z . (21)
According to the equation C z s = z s and sin 2 θ α s = 1 − cos 2 θ α s yields
( H 0 − 2 C z ) − R 2 cos 2 θ α s L 2 + 2 R cos θ α s + ( H 0 − 2 C z ) R 2 L 2 − 2 ( R c − C z ) = 0 . (22)
Solving the Equation (22) yields
θ α b = arccos L 2 + L L 2 + ( H 0 − 2 C z ) [ ( H 0 − 2 C z ) R 2 L 2 − 2 ( R c − C z ) ] R ( H 0 − 2 C z ) . (23)
Substituting Equation (14) into Equation (23) yields
θ α s = arccos L 2 + L L 2 + ( H 0 − 2 C z ) [ ( H 0 − 2 C z ) ( R min + R p ( 1 − cos α ) ) 2 L 2 − 2 ( R c − C z ) ] ( R min + R 1 ( 1 − cos α ) ) ( H 0 − 2 C z ) . (24)
For the oval-round pass sequence, the radius of flow line f α was expressed as
R = R min + R g ( 1 − cos α ) ( 0 ≤ α ≤ arcsin C y R g or 0 ≤ α ≤ arccos C z R g ) . (25)
The three-dimensional coordinates of a random point on the contact surface was shown as
{ x = L − R sin θ ( 0 ≤ θ ≤ θ α s ) y = R g sin α z = R ( 1 − cos θ ) + R g cos α = − R cos θ + R c . (26)
On the symmetry plane of deformed workpiece, the position angle α is 0 and the bite angle θ α s reaches the maximum value θ max
θ max = arccos ( 1 − H 0 − H p 2 R 0 ) . (27)
At the exit section of deformed workpiece, the bite angle θ α s is 0 and the position angle α reaches the maximum value α max
α max = arccos C z R g . (28)
By the same methods as round-oval pass sequence, the bite angle of oval- round pass sequence was
θ α s = arccos L 2 + L L 2 + ( H 0 − 2 C z ) [ ( H 0 − 2 C z ) ( R min + R g ( 1 − cos α ) ) 2 L 2 − 2 ( R c − C z ) ] ( R min + R g ( 1 − cos α ) ) ( H 0 − 2 C z ) . (29)
According to definition of projected area of contact zone, it should be calculated by the integral equation as
A p = 2 ∫ 0 α max R sin θ α s d y = 2 ∫ 0 α max R sin θ α s R 1 cos α d α . (30)
However, it is too complex for Equation (30) to integrate it directly and get a function for projected area. So, the flow line field method based on the discrete law is an effective way to calculate projected area.
Once the position angle of flow line α is given, the roll radius R and the bite angle θ α s against different position angle of flow line can be determined. Then the space position and the length of any flow line on the contact surface can be obtained. If the whole contact zone was discretized into n flow lines with different position angle α i and different bite angle θ α s i , the arc length of any flow line was expressed as
L f i = R i θ α s i = f ( α i ) ( 0 ≤ i ≤ n ) . (31)
The position angle of ith flow line can be shown as
α i = i n α max ( 0 ≤ i ≤ n ) . (32)
The projected length of this flow line on plane xoy can be obtained by
L p i = R i sin θ α s i ( 0 ≤ i ≤ n ) . (33)
Substituting Equation (32) into Equation (14), Equations ((24), (25) and (29)) to replace the position angle α can yield R i and θ α s i in oval pass rolling and round pass rolling respectively, then the projected area of contact zone on the plane xoy can be shown as
A p = 2 ∫ 0 α max R sin θ α s d y = ∑ i = 0 n ( R i sin θ α s i ) C y n . (34)
The alloyed bar rolling experiments had been accomplished in BEIMAN SPICIAL STEEL CO. LTD, the round workpiece of diameter 171 mm were rolled in one oval pass and one round pass of 22-stand Pomini Rolling Mill. The deformation zone of rolling workpices was obtained by stopping the rolling process when the workpiece was rolled in the oval pass and round pass simultaneously. The rolling schedule is shown as
As can be seen in
| Parameter | Roll diameter (mm) | Bite angle (degree) | Motor speed (rpm) | Roll speed (rpm) | Roll gap (mm) | Height of incoming workpiece (mm) | Maximum Height of roll pass (mm) |
|---|---|---|---|---|---|---|---|
| Oval pass | 730 | 24.8 | 822 | 7.8 | 20 | 171 | 112 |
| Round pass | 730 | 28 | 887 | 10.1 | 12 | 197.6 | 136 |
| Parameters | (mm2) | (mm) | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Oval pass | 22,965.83 | 747.89 | 6796.77 | 15,421.17 | 17,274.3 | 99.22 | 93.08 | 142.95 | (77.71, 35.66) | (92.65, 25.95) | 325.39 | 319 | 328.46 | |||
| Round pass | 17,684.15 | 914.81 | 4697.9 | 12,071.44 | 13,409.19 | 123.05 | 117.77 | 170.94 | (49.05, 47.1) | (56.93, 37.19) | 309.17 | 303 | 312.31 | |||
contact zone were listed one by one.
As shown in
As shown in
As can be seen in
Results of contact length and projected area from the novel model, the calculating results from the existing models, the experimental data and the simulation results were all listed in
Since the outgoing workpiece of oval pass rolling will be rolled in next round pass as an incoming workpiece and the section profile at the exit of oval pass influence the contact surface of round pass rolling greatly, the prediction error of round pass is obviously greater than the prediction error of oval pass. Moreover,
| Parameter Pass | Length of contact zone (mm) | Error of contact length (%) | Projected area of contact zone (mm2) | Error of projected area (%) | |
|---|---|---|---|---|---|
| Oval Pass | Graphical solution | 132.8 | +1.45 | 16,635.0 | −9.98 |
| Shinokura Formula | 119.3 | −8.86 | 17,351.9 | −6.10 | |
| Y. Lee Formula | 119.3 | −8.86 | 17,222.5 | −6.80 | |
| Novel model | 128.0 | −2.22 | 18,035.4 | −2.40 | |
| Experimental results | 130.9 | 0 | 18,478.6 | 0 | |
| Simulating results | 126.8 | −3.13 | 17,763.4 | −3.87 | |
| Round pass | Graphical solution | 133.1 | +1.99 | 10,225.0 | −14.14 |
| Shinokura Formula | 121.7 | −6.74 | 10,877.5 | −8.66 | |
| Y.Lee formula | 121.7 | −6.74 | 10,392.6 | −12.73 | |
| Novel model | 128.9 | −1.23 | 12,312.3 | +3.39 | |
| Experimental results | 130.5 | 0 | 11,908.8 | 0 | |
| Simulating result | 127.4 | −2.38 | 11,456.6 | −3.80 | |
results of projected area from the novel analytic model approaches the experimental data and simulation results very well, and its error is less than any existing models.
1) The contact boundary is a complex 3-dimension curve, and its profile is not only concerned with the parameters of pass profile R1, Rg, Rmin, G, Dz and the shape parameters of incoming workpiece H0, Ra, but also influenced by the coordinates of critical point ( C Y , C Z ) ;
2) The modified contact length model is rational because the influence the effective section area of the outgoing workpiece Ae, the critical point ( C Y , C Z ) on the contact boundary, the effective height of outgoing workpiece H m ¯ and the mean roll radius R m , has been taken into account in this model;
3) Based on the different position angle α and bite angle θ α s , the flow line element discretizes the complicated 3-dimension contact surface conveniently and makes it easier to rebuild the contact surface, and it is a good way to analyze the non-uniform stress and strain distribution accurately;
4) The discretizing and summing up method is an efficient way to solve the projected area, and results from this method approach the experimental data and simulating results very well.
Dong, Y.G., Zhu, H. and Song, J.F. (2017) Novel Analytic Model for the Projected Contact Zone Based on the Flow Line Element Method in Alloyed Bar Rolling. Open Access Library Journal, 4: e3247. http://dx.doi.org/10.4236/oalib.1103247