A Generalized Three Dimensional Upper Bound

Metal Forming

A Generalized Three Dimensional Upper Bound Analysis of the Profiled Ring Rolling Process 1)

2)

Ali Parvizi , Karen Abrinia , Mohsen Hamedi

2)

1)

R&D center, Mapna Locomotive Company, P.O. Box 19395-6448, Tehran, Iran, [email protected]; 2) Associate professor, School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran, [email protected], [email protected]

Abstract. A new solution based on upper bound method is presented in this paper to solve the profiled ring rolling process. The ring with rectangular geometrical cross section is rolled between main roll and mandrel and forms the Inner Square Protrusion (ISP) profile on the ring. The profile in the cross section of the ring is divided into four sections and an admissible velocity field and strain rates for each section are derived from the parametric definition of streamlines in the deforming zone. Using the proposed velocity field, the internal, shear and frictional powers are determined and the upper bound power for the ring rolling are obtained, parametrically. Minimizing the upper bound power with respect to neutral point position, the neutral point situation and the rolling force are determined. The present method is validated using experimental results of other investigators. As an engineering tool, this approach can be applied efficiently to the profiled ring rolling process and can provide a useful tool in industrial applications. Keywords: Profiled ring rolling, Admissible velocity field, Upper bound method;

1. INTRODUCTION

Profiled ring rolling is a specialized form of rolling process whose product is a ring with appropriate profile on the lateral, radial or axial surfaces. These profiles may have the rectangular, circular or complicated combined geometrical shape. Inner and outer bearing races, wheel tires for locomotives, flanges for wind power industry and rings used in power generation plants and aircraft engines are some products of profiled ring rolling processes. The upper bound formulation involves finding a kinematically admissible velocity field, which leads to a minimum forming energy and gives detailed information on the material flow. Recently, the upper bound method was used by authors [1] and many other researchers to analyze the metal forming processes. Some complexities of the ring rolling process including the unsteady state nature of the process, the unsymmetrical nature of the geometry, the continuously varying deforming zone, complication of the boundary conditions and the irregular behavior of friction force at the deforming zone, all in all make the upper bound analysis of the ring rolling process complicated. In other words, there is hardly any upper bound analysis of ring rolling process except that given by Ryoo et al. [2] and Yang and Ryoo [3]. They derived the dual two dimensional velocity field for the ring with rectangular cross section and developed an upper bound solution for the ring rolling process. The other researchers such as Hahn and Yang [4], Alfozan and Gunasekera [5] and Ranatunga and Gunasekera [6] proposed an analytical method for the simulation of profiled ring rolling based on the Upper Bound Elemental Technique. In this paper, a new solution based on upper bound method is presented to solve the profiled ring rolling process. The ring with rectangular cross section is rolled between main roll and mandrel and the Inner Square Protrusion (ISP) profile is formed on the ring. The material is assumed to be isotropic, incompressible and follows a rigid–perfect plastic behavior. The elastic deformations are assumed to be negligible. The admissible velocity fields

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for deforming zone are derived based on the parametric definition of streamlines. Using the proposed velocity field, internal, shear and frictional powers are obtained. Minimizing the upper bound power with respect to neutral point position, the neutral point location and the rolling force are determined. The present analysis is validated using experimental results of other researchers.

2. THEORITICAL FORMULATION 2.1. General deforming zone The general geometry for the deforming zone of the profiled ring rolling process is shown in figure 1. It is assumed that the deforming zone consists of such stream surfaces as OAAO within which streamlines such as BB are located. Moreover, a general material point such as P in the deforming zone travels on such streamlines and the proportionality relation OB OA  OB OA holds along the entire path. It can thus be seen that this assumption presents a method to mathematically define the deforming zone in the profiled ring rolling of an arbitrary section. Considering the linear interpolation, the vector equation for any streamline BB is defined as follows     r  r (t )  (1  t )  r1  t  r2 0  t 1 (1) where t is a axial relative parameter varying between 0 and 1 and r1 and r2 are the position vectors of any streamlines at entry and exit with respect to center o , such as B and B . Therefore, the arbitrary point position at the deforming zone such as P could be represented by     (2) r (u, q, t )  f (u, q, t ) i  g (u, q, t ) j  h(u, q, t ) k

Figure 1. General geometry for deforming zone

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83

Metal Forming

where f, g and h are functions defining the x, y and z coordinates and u, q and t are normalized dimensionless parameters which describe the distance in radial direction, the angle and the tangential distance, relatively as follows  OB OP  u ,q ,t ; 0  u, q, t  1 (3) OA 2 n L where n is the number of symmetry repetition. Therefore, the equation (2) defines the geometry of the deforming zone completely while u, q and t vary between 0 and 1. 2.2. Kinematically admissible velocity field Considering a point P in figure 1 which is moving on the stream line BB , The general vector equation for the velocity of this point is given by     0  u, q, t  1 V  V x i  V y j  Vz k (4) Since BB is assumed to be a stream line, the unit tangent vectors of Eq. (2) for the position of point P and Eq. (4) for the velocity of point P coincide and the following relationships could easily be derived [7] Vz  M (u, q, t ) , Vx  f t ht  Vz , V y  g t ht  Vz (5) where M is a general function will be obtained from the incompressibility condition and makes the velocity field admissible, automatically. ft, gt and ht are the derivatives of f, g and h with respect to t. Substituting the velocity components from Eq. (5) into Incompressibility condition Vx x  V y y  Vz z  0 and making some simplification, the following formula for function M is obtained [8] C

M ( fu gq  f q gu ) 

hq ht

( ft gu  fu gt ) 

hu ( f q gt  ft gq ) ht

(6)

where parameter C is a constant could be determined from the boundary conditions of the problem. Therefore, Eq. (5) is solved based on the results of Eq. (6) and the strain rates are obtained by  xx   yy   zz 

V x x V y y

V z z

,

 xy 

,

 xz 

,

 yz

1 V x V y  ( ) x 2 y

1 V x V z  ( ) x 2 z 1 V y V z  (  ) y 2 z

(7)

2.3. Upper Bound Power The total power J required to deform the material in a ring rolling operation is expressed as sum of the internal, shear and frictional powers as follows J  Wi  (W en  W ex )  W f (8)

velocity discontinuity components of total ring rolling power can be expressed as follows at the entry and the exit 1 y  y 1 1 2 2 2  ( x, y )  

Wen 

W ex 

3  Se

y

3  Se

Ven dS en 

Vex dS ex 

(Vx  V y )  dudq  ( u, q )  t  0 3 0 0 

y

1  2 2 2  ( x, y )  ( )  V V   dudq x y (u, q)  t 1 3 0 0 

(10)

1 1

and the energy dissipated due to friction between workpiece and main roll is given by m y 1 1   ( x, z )  W f  dudt (Vz  Vr ) 2  Vx2  Sec  (11)  0 0  (u, t )  q 0 3   where Sec  ( N12  N 22  N 32 ) N 2 , N  r q  r t q0 and Vr is tangential velocity of the main roll. The external power (J*) is the sum of the power imposed by the driving torque of the main roll and the power imposed by the pressing load of the mandrel. Considering the driving torque from equation (17) of our previous study [8], the ring rolling force per unit width of the ring based on the upper bound solution is obtained as follows F  [ J *  m y 3  ( L  2 xn )  R1  n] / v (12)

3. DEFORMING ZONE GEOMETRY

A ring of primary rectangular cross section with uniform thickness 2W and height H is rolled through a pair of main roll and mandrel. Figure 2 shows the coordinates and notations. hi and ho denote the thickness of the ring at the entry and the exit of the roll gap, respectively. It is assumed that the cross section of the radial planes (xy) remain plane during the process. The entry of the deforming zone is located at the first contact of the ring with the main roll. The exit plane is also located at the centers of main roll and the mandrel. The contact length between the ring and the rolls are small as compared with the roll circumference and can be obtained as follows [2] ( R  R2  ho  hi ) 2  R12  R22 2 12 } ] L  [ R12  { 1 (13) 2( R1  R2  ho  hi ) If the slip between the main roll and the ring is neglected, decrease in ring thickness in any rotation is given by [9] h  hi  ho  (v n)  ( R3o R1 ) (14) The friction factor multiplied by the shear yield strength (   mk  m y 3 ) is used to present interface friction between the main roll and the ring.

Using the parametric notation, the plastic deformation power for the ring rolling process is expressed as follows W i  2

y

3

1 1 1

 xx2   yy2   zz2

0 0 0

2



(

1

  xy2   yz2   zx2 ) 2 J du dq dt

(9)

where  ii are strain rates in various directions,  y is the yield stress and |J| is the determinant of the Jacobian for transformation of coordinates from x, y, z to u, q, t. The

84

Figure 2. Ring rolling geometry

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Metal Forming

For simplicity, the friction at the interface of the mandrel and the ring is assumed to be neglected. In passing through the roll gap (figure 2), the ring thickness is steadily reduced and the ring radius progressively increased from entry to exit. Furthermore, it has been assumed that solid grooved mandrel draw the billet ring inside the race and increase the profile height. The main roll surface speed is equal to an intermediate value between the ring entry and exit speeds. The point at which the ring tangential speed and the main roll surface speed become equal is called a neutral point. As the ring geometry is assumed to be symmetrical (figure 2), only a half of the deforming zone suffices for the purpose of this analysis. The entry and exit cross-sections for half of the deforming zone with the ISP profile are shown in figure 3.

Figure 3. Cross sections at the entry and exit of deforming zone with ISP profile Considering the complexity of the deforming zone and the approximate metal flow in the ring with ISP profile, it was decided that a single curve will be insufficient to describe the entire deforming zone. To facilitate the analysis, it was assumed that the deforming zone consists of four smaller zones (figure 3). Therefore, for each of the four sections of the deforming zone, a separate straight line has been formulated.

4. BOUNDARY CONDITIONS

According to figure 3, the boundary conditions for the ring rolling process with ISP profile are as follows: 1) Equality of the ring tangential speed and the main roll surface speed at the neutral point I ) q  0 and t  t n  Z n Z 0  V1z  R1n (15) As the contact between ring and the main roll surface only happen in zone I, this condition is dedicated to this zone. 2) Equality of tangential velocity component (Vz) with rotational velocity of the ring at the entry surface (t=0) Sec I ) V1z t 0  ( R3o  uW tan(1 ))  r Sec II ) V2 z

t 0

 ( R3o  uhi )   r

Sec III ) V3 z

t 0

 ( R3o  u( hi  ti ))  r

Sec IV ) V4 z

t 0

 ( R3o  u( hi  ti ))   r

(16)

5. VELOCITY FIELDS

The entry and exit sections of zone I from the ISP deforming zone cross section are shown in figure 4. The dimensionless parameters u, q and t which define the coordinate of each point in zone I are as follows u  u W , q  1 1 , t  z Z 0 ; 0  u , q, t  1 (17) The coordinate of arbitrary point at the entry and exit planes are as follows www.steelresearch-journal.com

Figure 4. The entry and exit sections of zone I h ˆ  r1  u  W iˆ  (u  W tan  1  ) j  Z 0 kˆ 2  r0  uW ( iˆ tan 1 ˆj )

(18)

Taking into consideration the incompressibility condition (Vi Ai=Vo Ao), the inner radius of the ring at the exit plane is obtained as follows hi ( R3i  hi 2)  W  t i ( R3i  t i 2)  a    (19)  (ho2 2)  W  (t o2 2)  a  R3iEx   ( ho  W  t o  a ) It is also assumed that the particles which are entered the deforming zone from within the perimeter of triangle OP1A exit the deforming zone from the area bound by OP1A . Considering the incompressibility condition for zone I, 1 is calculated as a function of 1 as follows 1 tan  1  [ B1  B12  4C1] 2 3 3 B1   ( R3iEx  ho ), C1  ( R3i  hi ) tan 1  tan 2 1 W W

(20)

Substituting equation (20) into (18) and using the Eq. Sec I of (16), the velocity field at zone I is obtained as 1 (tan  1 ) (1  t n )  t n q 1 R  uW tan(1 ) q 0 V1z  R1  n  3o   (tan   R3o 1) (21) (1  t )  t cos 2  1 q V1x  0

,

V1 y  1 Z 0  [uW  tan 1  tan  1  

h ]  V1z 2

Since one complete rotation of the ring is considered to attain the thickness reduction of h (Eq. (14)), it is necessary to substitute the ring and main roll contact in the deforming zone (Z0) with contact length in one complete rotation of ring ( Z 0 ) which is obtained by Z 0  {(1  t n )  t n  1 1  (tan 1 ) q q0 }  R3o

(22)

Using the same method, the velocity fields in sections II, III and IV of deforming zone are derived. Substituting the velocity fields into Eqs. (9)-(11) and the subsequent results in Eq. (8), the upper bound power for the ring rolling process is obtained as a function of neutral point position (tn=Zn/Z0). Thus, the quantity tn is regarded as a power optimization parameter. When values of tn vary from the entry plane (tn=0) to the exit plane (tn=1), the resulting power (J*) initially decreases, passes through a minimum and then increases. It can be said that the almost actual value of tn is those one which minimizes the power J*, and the correspondent power is nearly the actual one.

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85

Radius Rotational velocity Radius Feed speed Profile width Inner radius Outer radius Width Material

Main roll Mandrel

Ring

R1=4.5 in n=31 rpm R2=1.375 in v=0.019 in/rev a=0.25 in R3i=1.5 in R3o=2.5 in W=1.0 in Al. alloy

Pressing Load (tonf)

200

250

Mean Radius of Ring (mm)

2.5 2

1.5

120

140

160

180

200

220

Mean Radius of Ring (mm)

240

130

150

170

190

210

Mean Radius of Ring (mm)

2

1.5

1

0.5 100

1 0.5

2.5

: Upper Bound Method, v=0.9 mm/s : Experiment, Ryoo and Yang [2-3]

: Upper Bound Method, v=0.7 mm/s : Experiment, Ryoo and Yang [2-3]

: Upper Bound Method, v=1.2 mm/s : Experiment, Ryoo and Yang [2-3]

1

0.5 100

150

200

Mean Radius of Ring (mm)

250

Figure 6. Ring rolling force with respect to mean radius of the ring for different feed speeds.

7. CONCLUSIONS

8. REFERENCES

0 0

: Experiment, Yang et al. [10] : Upper Bound Method

10

20

30

40

Reduction in Thicknesses (%)

50

Table 2. Parameters of ring rolling process [2-3]

86

150

2 1.5

2

6

Case study 2: Considering the zero width for the ring profile, the results of the profiled ring rolling process can be used for the ring with rectangular cross section. The process parameters given in table (2) are adopted based on the Ryoo and Yang [2-3] experimental study. The forces with respect to the current mean radius of the ring are shown in figure 6 for different values of feed speeds and compared with the experimental results of Ryoo and Yang [2-3]. For any quantity of the feed speed, the force increase slowly with increase in the ring mean radius. Furthermore, a much higher force is necessary to maintain the ring rolling process in the case of relatively high feed speeds and large ring mean radius. The slope of the curve for the bigger ring is less than the smaller one.

Ring

1

0.5 100

2.5

4

8

Figure 5. Variation of force with respect to reduction in ring thickness and comparison with experimental results

Mandrel

: Upper Bound Method, v=0.5 mm/s : Experiment, Ryoo and Yang [2-3]

A three dimensional velocity field for upper bound analysis of T-section profiled ring rolling process is derived. Minimizing the upper bound power with respect to neutral point position, the neutral point situation and the rolling force are determined. It is seen that the necessary force for accomplishment of the process increases slowly with increase in the ring mean radius. Furthermore, a much higher force is necessary to maintain the ring rolling process in the case of relatively high feed speeds and large ring mean radius. Considering the zero width for the ring profile, the results of the profiled ring rolling process can be used for the ring with rectangular cross section. Comparison of the presented upper bound results with those one from experimental study of other investigator verify the accuracy of the present analysis. As an engineering tool, this approach can be applied efficiently to the profiled ring rolling process and can provide a useful tool in industrial applications.

The results of present upper bound analysis and experimental studies of Yang et al. [10] are shown in figure 5. The necessary pressing load of the process with respect to the thickness reduction of the ring is presented. It is seen that the force increase slowly with increase in the ring thickness reduction. Some inaccuracies in the numerical solution of the present formula as well as the factual error in the experimental data acquisition (because of the dynamic nature of the process) caused that the present upper bound loads become lower than the experimental ones for reduction in thickness up to 30%.

Main roll

1.5

Load (KN) / y (N/mm2)

Table 1. Parameters of profiled ring rolling process [10]

2

Load (KN) / y (N/mm2)

2.5

Based on the derived velocity fields in the deforming zone, the following two case studies are investigated. Case study 1: Since some experimental study is done by Yang et al. [10] for the ring rolling process with T profile, their process parameters (such as table 1) are taken into consideration to verify the present upper bound analysis.

Load (KN) / y (N/mm2)

6. RESULTS AND DISCUSION

Load (KN) / y (N/mm2)

Metal Forming

Radius Rotational velocity Radius Feed Speed Inner radius Outer radius Width

R1=275 mm n=47 rpm R2=45 mm v=1.2 mm/s R3i=60 mm R3o=107.5 mm W=52 mm

[1]

K. Abrinia, A. Fazlirad: Three-dimensional analysis of shape rolling using a generalized upper bound approach, J. of Mater. Process. Technol., 209 (2009), 3264-3277. [2] J.S. Ryoo, D.Y. Yang, W. Johnson: The influence of process parameters on torque and load in ring rolling, J. of Mech. Work. Technol., 12 (1986), 307-321. [3] D.Y.Yang, J.S.Ryoo: An investigation into the relationship between torque and load in ring rolling, J. Eng. Ind., 109 (1987) 190-196. [4] Y.H. Hahn, D.Y. Yang: UBET analysis of the closed-pass ring rolling of rings having arbitrarily shaped profiles, J. of Mater. Process. Technol., 40 (1994), 451-463. [5] A. Alfozan, J.S. Gunasekera: An upper bound elemental technique approach to the process design of axisymmetric forging by forward and backward simulation, J. Mater. Process. Technol., 142 (2003), 619-627. [6] V. Ranatunga, J.S. Gunasekera: UBET-Based Numerical Modeling of Bulk Deformation Processes, J. of Mat. Eng. Perform., 15 (2006), 47-52. [7] K. Abrinia: A Generalised Upper Bound Solution for Threedimensional Extrusion of Shaped Sections Using Bilinear and Advanced Surface Dies” Ph.D. Thesis, Univ. of Manchester, (1990). [8] A. Parvizi, K.Abrinia, M. Salimi: Slab Analysis of Ring Rolling Assuming Constant Shear Friction, J. of Mater. Eng. and Perform., 20(9) (2011), 1505-1511. [9] Lin H, Zhi ZX: The extremum parameters in ring rolling, J. Mater. Process. Technol., 69 (1997), 273-276. [10] D.Y. Yang, K.H. Kim, J.B. hawkyard: Simulation of T-section profile ring rolling by the 3-D rigid-plastic finite element method, Int. J. Mech. Sci., 33 (7) (1991), 541-550. Acknowledgements. This work was sponsored by the Mapna Group of Iran under grant No. RD-88-19. The advice and financial support of the Mapna Group are gratefully acknowledged.

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