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Design of the open-die elongation process using optimization technique
Journal of Materials Processing Technology, 34 (1992) 157-162
157
Elsevier
Design of the open-die elongation process using optimization technique R. Szyndler and B. Klimkiewicz University of Mining and Metallurgy, 30-059 Krakow, Poland
Abstract The method of the design of the open-die elongation process using optimization procedures is described in the paper. Basing on the numerical analysis and experimental results, the function relations between the technological parameters for various shapes of dies are defined. These relations are the base for the formulation of the objective function for the optimization model. The forging ratio, intensity of the process and the products' quality estimated by the local forging's inhomogeneity are chosen as the optimization criterion. The computer system for the design of the open-die elongation process is developed and described in the paper. The final effect of the work is the numerical code for the control of the forging nest. 1. INTRODUCTION The questions concerning quality of products and technical reality for the realization of the process often arise during modelling of metal forming processes. Designer has to decide whether the operation he chooses is optimal regarding technological parameters, quality and costs of manufacturing of products. The answer for the above questions has been sought by an analytical and numencal simulation of the process. Present work focuses on the problem of the ingot open-die forging in flat and shaped dies. 2. ASSUMPTIONS AND CRITERIA FOR THE OPTIMIZATION OF THE INGOT FORGING PROCESS The dimensions of the stock ingot and a maximum linear distance on the cross section of the forging d ~ are the basic information for the design of the optimum preliminary processing (consisting of upsetting and elongation passes) of ingots. The parameters which determine mathematical model of the open-die forging for ingots are: - total elongation ~ calculated as a product of the elongation coefficients in subsequent passes, - the minimum deformation c* defined by the sum of the local effective strains in the subsequent passes operations of upsettin~ and elongation. The external limitations in the optimization procedure are: maximum allowable reduction in the upsetting which is a criterion function of the forged material E f (T, ~ ~) and allowed temperature range ( T p.,~., T p.... ) for the forging process. The objective function for the model can be chosen from the following: uniform strain distribution in the subsequent operations or intensification of the preliminary processing or minimum of forging reduction ratio. Numerical realization of the first objective function leads to the determination of the number n of technological operations necessary to obtain required forging reduction ratio. Further, limitations connected with the boundary conditions f (D ~) and f ( D w) are introduced and parameters describing designed process and guaranteeing uniform deformation are calculated: max
=
0924-0136/92/$05.00 © 1992 ElsevierSciencePublishersB.V. All rightsreserved.
158
2
=Xok?-2~k=~
2
n-2
-A~
Dx~,D~J
(i)
d~
with an assumption X o = X~ = k k. In equation (1) vector of the optimization parameters consists of the diameters after upsetting D ~o, D ~, D ~k and the diameters after elongation D ~, D ~. The second objective function creates the process which is designed in the extreme allowable technological conditions. The maximum reductions in the upsetting are limited by the function [ ( D ~) while the shape in the elongation is limited by the maximum allowed slenderness ratio. Thus, the second objective function can be written as: X_
D~ ~
D2~-2~2 w U~k
(2)
Typical schematic operations for the preliminary processing of ingots obtained by the optimtzation procedure with the first and the second objective functions are shown in Table 1. Third objective function requires the designing path to be proposed. The choice is between the conditions (1) and (2) with an assumption that a number of technological operations r~ is one of the components of the optimization vector, which is the solution of the problem. Criteria for the realization of the objective function are: 6" <- rain Z c,,
(3)
l-1
where l is a number of the subsequent technological operation (upsetting or elongation) and ~ is the effective strain field. Table 1. Optimization schedules for preliminary processing of ingots obtained from the optimization procedure (assuming x = 6). intensification of the process
uniform distribution of reductions no.
type
0 1 2 3 4 5 6
stock upsetting elongation upsetting elongation upsetting elongation
geometrical parameters mm Do =350 D ~421 D z =310 D 3 =421 D 4 =310 D~ =421 D6 =318
H0 =610 H l =420 H z =775 H a --420 H 4 =775 H 5 ---420 H 6 =738
technol, param.
e ~ =0.31 h j =1.85 e z =0.46 h,e =1.85 e3 =0.46 k 2 =1.76
no.
type
0 1 2 3 4 5 6
stock upsetting elongation upsetting elongation upsetting elongation
geometrical parameters mm Do =350 D 1 =495 D e =310 D 3 =438 D 4 =310 D z =345 D6 =318
H o =610 H l =305 H z =775 H 3 =387 H 4 =775 H 5 %26 H 6 =738
technol. param.
e i =0.50 k l =2.54 e 2 =0.50 k 2 =2.00 ~3 =0.19 k a =1.18
In the optimization calculations situation in a single step includes an analysis of several upsetting and forging operations. In order to accelerate the computations, the approximate methods of the simulation of the for~ing process [1,2] are used in the present work. Determined distributions of the effective strain are presented in the initial normalized coordinate system. What allows direct summation ot strains [3]. Schematic pass for the realization of the third objective function is shown in Figure 1.
159 STOCK Do,Lo
[ ~=x~"-a"ak]
I
I
~FII+liE~Q'I>
IF (i+i .EQ .i / > ELONGATION
N•O [
YES
YES
~o
I
SODE~ pass
SIMDI~TION Y~_~ Transformatton£~
P
Transgormatlonl
x
~ransformation
y12
I
n
I=1
I
I
Figt~.re 1. Designing of the preliminary processing of ingots with an assumption of the minimum forging reduction ratio. 3. OPTIMIZATION OF THE ELONGATION PROCESS The model of the elonsation process has been developed with the assumptions and technological limitations gwen below. Three types ofpasses are distinguished depending on the stock material and final product: 1. rectangular to square or rectangular pass, 2. round to square pass, 3. square to roundpass. Pass type 1 can be performed in the flat or shaped dies. Passes type 2 and 3 are performed in the flat dies only. Basic parameters for modelling the elongation process are: - dimensions of the stock material and final product,
160 initial and final for~ing temperature, yield strength and Ume of single reduction during elongation process. Parameters which determine the model are: reduction, feed, elongation coefficient in the last pass type 1 and, eventually for the round ingot, dimension of the square after first pass or, for the round final product, dimension of the square in the last pass. Mathematical model for the pass type 1 uses the method described in [4] and based on the Chile's equation: -
-
k-
1 1 -~(I -[.)
(4)
where f n is the coefficient which depends on the absolute feed-to-width of the billet ratio, c is the reduction. Chile's relationships have been verified analytically and supplemented with experimental data [5] for the shaped dies (passes 2 and 3). The model is extended by an inclusion of the analysis of forces and temperatures, the methods described in [6,7,8] being used for the purpose. Recapitulating, the objective functions of the model are: qualitative objective function J * energetic objective function E" intensification of elongation M" Definition of the qualitative objective function is based on the assumption that the uniform deformation is the main criterion. Empirical experience and industrial practice show that the uniform deformation is achieved for the feed of about 0.5 l u where l a is the length of the deformation zone. In the present work the qualitative objective function is formulated as: -
-
-
1
~l [(x~- xi-l)- (x,.l - x,)] 2 (A~) ~
J*=N-2~-I
(5)
where N is the number of the reductions in the forging process, x, is the coordinate of the centre of the deformation zone in i 'h operation and A x is the average distance between the centres of the two adjacent deformation zones. Realization of the objective function (5) results in the technological parameters which assure the most uniform distances between the centres of the deformation zone in all subsequent reductions necessary to obtain final product. The centres of the zones have been determined after the deformation what means that the elongation is taken into consideration. Figure 2 shows the locations of the centres of the zones in the product forged in three passages with the assumption of constant absolute feed (a), constant relative feed (b), starting each passage with the displacement equal to 0.5 l a with respect to the previous one (c) and using process parameters being the components of the optimization vector (d). The values of the qualitative objective function are also presented m Figure 2. The method of direct search [9] has been applied here in the optimization procedure. The choice is justified by small range of changes of variational parameters. The energetic function is the second option in the optimization. The functional has been defined using simplified relationships which describe work for plastic deformation for a single reduction in the elongation process [6]: h/~ = h/~(O, c, S , l)
(6)
161
ii
A
B
l odd reductions
Ii I ~i li~ 11I
f
ii
li
I even reductions
Fc': 1 1 6 . 5 6
P~ i
i I !Ji Ii oddre~uctions I H! I ! I t evenreductions
II
.
I
IE Fc-- 124.. 76
!iq ii
C
i] odd reductions I
even reductions
Fc= 14-9.33 D
kip il I I1:,I il il ~ !1 t! ii
i
I
.............
pass
I] odd reductions 'i even reductions
F ~: 45 ~qri 1,
p a s s 2,
p a s s :3.
Figure 2. Distribution of the centres of the deformation zones after elongation of the round bar 4, 350 m m , L o = 700 mm into a square (a, = 150 mm). where l is the relative feed, S is the area of the surface of the forging between dies and o = o ( T, e ) is the yield strength. Energetic objective function is defined as: *
N
E = min ~ Id~
(7)
Significant simplifications introduced into calculations of the work for the plastic deformation led to the situation that the yield strength function became the dominating factor which decides about the convergence of the whole solution. In the case of the multi-variable function o = / (T, •) (for example for narrow interval of the admissible forging temperatures or small number of the single reductions N or short time of a single operation), the optimization vector has a tendency towards lower limits of the technological parameters (the smallest allowed reduction and feed). The third choice for the objective function in the optimization procedure is an intensification of the forging process. The solution of this function does not give the optimization vector but the intervals of the variations of the technological parameters which result in the smallest number of reductions necessary to obtain final dimensions, what is written as: M'=minN
(8)
Typical solution of the elongation process for the billet D o = 350 mm and L o = 2500 mm forged to the square cz~ = 150 mm with the application of the third objective function is shown in Figure 3.
162 A 0.1( 0.75
CI r.,,,1
0,19
/
0.28
0.38
0.47
B 0,56
0.6,5
vIMPFSII~LE ////o.,, \
0.56
6~01
L~ 0.57 Lq >
3
,2
~., 0.48 < ,.J
0.48
,~ p.3a
0.38
0.29
0.29
o.2~.4,!
' '0.38 ~: ' 0.19 '0.28 0.47 '0.56 '0.65
!
0.20
Figure 3. Optimization of the elongation process with the intensificationbeing the objective function, D o = 350 ram, Lo = 2500 mm, a k = 150 ram, (A - number of passages necessary to perform the process, B - number of reductions necessary to perform the process). CONCLUSIONS Presented method and optimization procedures are introduced into the computed code which has been developed. The code can be used for technology design and for modelling of technological processes for manufacturing rods, rollers, prisms and pressed forgings (shafts, rolls) with the square, rectangular or round cross sections on presses and hammers in flat and shaper dies [5].Beyond this, the program designs the technology for the preliminary processing of forging ingots and it can be used as a part of the control system for the forging machines. The optimization procedures are developed for the complex objective functions. REFERENCES
1 I.Ya. Tarnovskiy and A.A. Pozdieyev, Teoriya obrabotki metallov davleniyem, Metallurgiya, Moskva, 1963. 2 G.J. Gun, Teoreticheskiye osnovy obrabotki metallov davleniyem, Metallurgiya, Moskva, 1980. 3 B. Klimkiewicz and R. Szyndler, Proc.Conf. Metal Forming'89, Krakow-Krynica, 1989, 139. 4 P. Wasiunyk, Teoria procesow kucia i prasowania, WNT, Warszawa, 1982. 5 R. Szyndler, B. Klimkiewicz, J. Sinczak, S. Szczepanik, M. Lesniewski and B. Szyndler, Research Project, Ministry of Education, no. 18.351.18, AGH Krakow, 1991. 6 Z. Krzekotowski, Technologia kucia swobodnego i polswobodnego, Slask, Katowice, 1964. 7 W.A. Tyurin, A.V. Habarov, A.I. Dubkov and L.P. Belova, Kuzn. Shtamp. Proizvodstvo, (1982) 16. 8 V.L. Kolmogorov and A.I. Golomidov S.V. Karpov and V.P. Fedotov, hv. V.U.Z., Chernaya Metallurgiya, 24 (1981) 40. 9 T. Kreglewski, T. Rogowski, A. Ruszynski and J. Szymanowski, Metody optymalizacji w jezyku FORTRAN, PWN, Warszawa, 1984.
157
Elsevier
Design of the open-die elongation process using optimization technique R. Szyndler and B. Klimkiewicz University of Mining and Metallurgy, 30-059 Krakow, Poland
Abstract The method of the design of the open-die elongation process using optimization procedures is described in the paper. Basing on the numerical analysis and experimental results, the function relations between the technological parameters for various shapes of dies are defined. These relations are the base for the formulation of the objective function for the optimization model. The forging ratio, intensity of the process and the products' quality estimated by the local forging's inhomogeneity are chosen as the optimization criterion. The computer system for the design of the open-die elongation process is developed and described in the paper. The final effect of the work is the numerical code for the control of the forging nest. 1. INTRODUCTION The questions concerning quality of products and technical reality for the realization of the process often arise during modelling of metal forming processes. Designer has to decide whether the operation he chooses is optimal regarding technological parameters, quality and costs of manufacturing of products. The answer for the above questions has been sought by an analytical and numencal simulation of the process. Present work focuses on the problem of the ingot open-die forging in flat and shaped dies. 2. ASSUMPTIONS AND CRITERIA FOR THE OPTIMIZATION OF THE INGOT FORGING PROCESS The dimensions of the stock ingot and a maximum linear distance on the cross section of the forging d ~ are the basic information for the design of the optimum preliminary processing (consisting of upsetting and elongation passes) of ingots. The parameters which determine mathematical model of the open-die forging for ingots are: - total elongation ~ calculated as a product of the elongation coefficients in subsequent passes, - the minimum deformation c* defined by the sum of the local effective strains in the subsequent passes operations of upsettin~ and elongation. The external limitations in the optimization procedure are: maximum allowable reduction in the upsetting which is a criterion function of the forged material E f (T, ~ ~) and allowed temperature range ( T p.,~., T p.... ) for the forging process. The objective function for the model can be chosen from the following: uniform strain distribution in the subsequent operations or intensification of the preliminary processing or minimum of forging reduction ratio. Numerical realization of the first objective function leads to the determination of the number n of technological operations necessary to obtain required forging reduction ratio. Further, limitations connected with the boundary conditions f (D ~) and f ( D w) are introduced and parameters describing designed process and guaranteeing uniform deformation are calculated: max
=
0924-0136/92/$05.00 © 1992 ElsevierSciencePublishersB.V. All rightsreserved.
158
2
=Xok?-2~k=~
2
n-2
-A~
Dx~,D~J
(i)
d~
with an assumption X o = X~ = k k. In equation (1) vector of the optimization parameters consists of the diameters after upsetting D ~o, D ~, D ~k and the diameters after elongation D ~, D ~. The second objective function creates the process which is designed in the extreme allowable technological conditions. The maximum reductions in the upsetting are limited by the function [ ( D ~) while the shape in the elongation is limited by the maximum allowed slenderness ratio. Thus, the second objective function can be written as: X_
D~ ~
D2~-2~2 w U~k
(2)
Typical schematic operations for the preliminary processing of ingots obtained by the optimtzation procedure with the first and the second objective functions are shown in Table 1. Third objective function requires the designing path to be proposed. The choice is between the conditions (1) and (2) with an assumption that a number of technological operations r~ is one of the components of the optimization vector, which is the solution of the problem. Criteria for the realization of the objective function are: 6" <- rain Z c,,
(3)
l-1
where l is a number of the subsequent technological operation (upsetting or elongation) and ~ is the effective strain field. Table 1. Optimization schedules for preliminary processing of ingots obtained from the optimization procedure (assuming x = 6). intensification of the process
uniform distribution of reductions no.
type
0 1 2 3 4 5 6
stock upsetting elongation upsetting elongation upsetting elongation
geometrical parameters mm Do =350 D ~421 D z =310 D 3 =421 D 4 =310 D~ =421 D6 =318
H0 =610 H l =420 H z =775 H a --420 H 4 =775 H 5 ---420 H 6 =738
technol, param.
e ~ =0.31 h j =1.85 e z =0.46 h,e =1.85 e3 =0.46 k 2 =1.76
no.
type
0 1 2 3 4 5 6
stock upsetting elongation upsetting elongation upsetting elongation
geometrical parameters mm Do =350 D 1 =495 D e =310 D 3 =438 D 4 =310 D z =345 D6 =318
H o =610 H l =305 H z =775 H 3 =387 H 4 =775 H 5 %26 H 6 =738
technol. param.
e i =0.50 k l =2.54 e 2 =0.50 k 2 =2.00 ~3 =0.19 k a =1.18
In the optimization calculations situation in a single step includes an analysis of several upsetting and forging operations. In order to accelerate the computations, the approximate methods of the simulation of the for~ing process [1,2] are used in the present work. Determined distributions of the effective strain are presented in the initial normalized coordinate system. What allows direct summation ot strains [3]. Schematic pass for the realization of the third objective function is shown in Figure 1.
159 STOCK Do,Lo
[ ~=x~"-a"ak]
I
I
~FII+liE~Q'I>
IF (i+i .EQ .i / > ELONGATION
N•O [
YES
YES
~o
I
SODE~ pass
SIMDI~TION Y~_~ Transformatton£~
P
Transgormatlonl
x
~ransformation
y12
I
n
I=1
I
I
Figt~.re 1. Designing of the preliminary processing of ingots with an assumption of the minimum forging reduction ratio. 3. OPTIMIZATION OF THE ELONGATION PROCESS The model of the elonsation process has been developed with the assumptions and technological limitations gwen below. Three types ofpasses are distinguished depending on the stock material and final product: 1. rectangular to square or rectangular pass, 2. round to square pass, 3. square to roundpass. Pass type 1 can be performed in the flat or shaped dies. Passes type 2 and 3 are performed in the flat dies only. Basic parameters for modelling the elongation process are: - dimensions of the stock material and final product,
160 initial and final for~ing temperature, yield strength and Ume of single reduction during elongation process. Parameters which determine the model are: reduction, feed, elongation coefficient in the last pass type 1 and, eventually for the round ingot, dimension of the square after first pass or, for the round final product, dimension of the square in the last pass. Mathematical model for the pass type 1 uses the method described in [4] and based on the Chile's equation: -
-
k-
1 1 -~(I -[.)
(4)
where f n is the coefficient which depends on the absolute feed-to-width of the billet ratio, c is the reduction. Chile's relationships have been verified analytically and supplemented with experimental data [5] for the shaped dies (passes 2 and 3). The model is extended by an inclusion of the analysis of forces and temperatures, the methods described in [6,7,8] being used for the purpose. Recapitulating, the objective functions of the model are: qualitative objective function J * energetic objective function E" intensification of elongation M" Definition of the qualitative objective function is based on the assumption that the uniform deformation is the main criterion. Empirical experience and industrial practice show that the uniform deformation is achieved for the feed of about 0.5 l u where l a is the length of the deformation zone. In the present work the qualitative objective function is formulated as: -
-
-
1
~l [(x~- xi-l)- (x,.l - x,)] 2 (A~) ~
J*=N-2~-I
(5)
where N is the number of the reductions in the forging process, x, is the coordinate of the centre of the deformation zone in i 'h operation and A x is the average distance between the centres of the two adjacent deformation zones. Realization of the objective function (5) results in the technological parameters which assure the most uniform distances between the centres of the deformation zone in all subsequent reductions necessary to obtain final product. The centres of the zones have been determined after the deformation what means that the elongation is taken into consideration. Figure 2 shows the locations of the centres of the zones in the product forged in three passages with the assumption of constant absolute feed (a), constant relative feed (b), starting each passage with the displacement equal to 0.5 l a with respect to the previous one (c) and using process parameters being the components of the optimization vector (d). The values of the qualitative objective function are also presented m Figure 2. The method of direct search [9] has been applied here in the optimization procedure. The choice is justified by small range of changes of variational parameters. The energetic function is the second option in the optimization. The functional has been defined using simplified relationships which describe work for plastic deformation for a single reduction in the elongation process [6]: h/~ = h/~(O, c, S , l)
(6)
161
ii
A
B
l odd reductions
Ii I ~i li~ 11I
f
ii
li
I even reductions
Fc': 1 1 6 . 5 6
P~ i
i I !Ji Ii oddre~uctions I H! I ! I t evenreductions
II
.
I
IE Fc-- 124.. 76
!iq ii
C
i] odd reductions I
even reductions
Fc= 14-9.33 D
kip il I I1:,I il il ~ !1 t! ii
i
I
.............
pass
I] odd reductions 'i even reductions
F ~: 45 ~qri 1,
p a s s 2,
p a s s :3.
Figure 2. Distribution of the centres of the deformation zones after elongation of the round bar 4, 350 m m , L o = 700 mm into a square (a, = 150 mm). where l is the relative feed, S is the area of the surface of the forging between dies and o = o ( T, e ) is the yield strength. Energetic objective function is defined as: *
N
E = min ~ Id~
(7)
Significant simplifications introduced into calculations of the work for the plastic deformation led to the situation that the yield strength function became the dominating factor which decides about the convergence of the whole solution. In the case of the multi-variable function o = / (T, •) (for example for narrow interval of the admissible forging temperatures or small number of the single reductions N or short time of a single operation), the optimization vector has a tendency towards lower limits of the technological parameters (the smallest allowed reduction and feed). The third choice for the objective function in the optimization procedure is an intensification of the forging process. The solution of this function does not give the optimization vector but the intervals of the variations of the technological parameters which result in the smallest number of reductions necessary to obtain final dimensions, what is written as: M'=minN
(8)
Typical solution of the elongation process for the billet D o = 350 mm and L o = 2500 mm forged to the square cz~ = 150 mm with the application of the third objective function is shown in Figure 3.
162 A 0.1( 0.75
CI r.,,,1
0,19
/
0.28
0.38
0.47
B 0,56
0.6,5
vIMPFSII~LE ////o.,, \
0.56
6~01
L~ 0.57 Lq >
3
,2
~., 0.48 < ,.J
0.48
,~ p.3a
0.38
0.29
0.29
o.2~.4,!
' '0.38 ~: ' 0.19 '0.28 0.47 '0.56 '0.65
!
0.20
Figure 3. Optimization of the elongation process with the intensificationbeing the objective function, D o = 350 ram, Lo = 2500 mm, a k = 150 ram, (A - number of passages necessary to perform the process, B - number of reductions necessary to perform the process). CONCLUSIONS Presented method and optimization procedures are introduced into the computed code which has been developed. The code can be used for technology design and for modelling of technological processes for manufacturing rods, rollers, prisms and pressed forgings (shafts, rolls) with the square, rectangular or round cross sections on presses and hammers in flat and shaper dies [5].Beyond this, the program designs the technology for the preliminary processing of forging ingots and it can be used as a part of the control system for the forging machines. The optimization procedures are developed for the complex objective functions. REFERENCES
1 I.Ya. Tarnovskiy and A.A. Pozdieyev, Teoriya obrabotki metallov davleniyem, Metallurgiya, Moskva, 1963. 2 G.J. Gun, Teoreticheskiye osnovy obrabotki metallov davleniyem, Metallurgiya, Moskva, 1980. 3 B. Klimkiewicz and R. Szyndler, Proc.Conf. Metal Forming'89, Krakow-Krynica, 1989, 139. 4 P. Wasiunyk, Teoria procesow kucia i prasowania, WNT, Warszawa, 1982. 5 R. Szyndler, B. Klimkiewicz, J. Sinczak, S. Szczepanik, M. Lesniewski and B. Szyndler, Research Project, Ministry of Education, no. 18.351.18, AGH Krakow, 1991. 6 Z. Krzekotowski, Technologia kucia swobodnego i polswobodnego, Slask, Katowice, 1964. 7 W.A. Tyurin, A.V. Habarov, A.I. Dubkov and L.P. Belova, Kuzn. Shtamp. Proizvodstvo, (1982) 16. 8 V.L. Kolmogorov and A.I. Golomidov S.V. Karpov and V.P. Fedotov, hv. V.U.Z., Chernaya Metallurgiya, 24 (1981) 40. 9 T. Kreglewski, T. Rogowski, A. Ruszynski and J. Szymanowski, Metody optymalizacji w jezyku FORTRAN, PWN, Warszawa, 1984.