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Indian J.Sci.Res.1(2) : 410-415, 2014
ISSN:2250-0138(Online)
ISSN : 0976-2876 (Print)
__________________________________ 1Corresponding author
SPRINGBACK REDUCTION IN EXPLOSIVE DEEP DRAWING PROCESS OF A TRAPEZOIDAL CUP
ALI JABBARI1a
aAsistant Professor, Department of Mechanical Engineering, Faculty of Engineering, Arak University, Arak 38156-8-8849, Iran
ABSTRACT
In this paper, a die shape optimization method for reduction of spring back defect due to elastic recovery behavior in explosive deep drawing
process is presented by using the reduced basis technique coupled by finite element and design of experiments methods. The primary
objective of the proposed method is to reduce the enormous number of design variables required to define the initial die shape. The reduced
basis technique is a weighted combination of several basis shapes. The aim of the method is to find the best combination using the weights
for each shape as the design variables. The experimental design of Taguchi method is used to build the approximation model and to perform
optimization. This method is demonstrated on the die shape optimization of deep drawing of a trapezoidal cup.
KEYWORDS: Explosive Deep Drawing Process, Springback, Reduced Basis Technique, Design Of Experiment Method.
Explosive deep drawing process has been used to form
trapezoidal cups from a trapezoidal die shape because of high
deformation rate compared to other production methods,
including casting and machining. However, the earring defect
caused by anisotropy behavior of sheet metal seriously affects
the method performance. Since the earring defect is produced
by the sheet metal anisotropy, it is greatly affected by the
initial die shape. Spring back angle minimization can be
performed using die shape optimization to obtain better stress
distribution and also to reduce the number of production
stages.
The main challenge of current optimization methods is the
number of design variables required for die shape
optimization.
An integrated algorithm is presented in this research and it is
conducted to die shape optimization in a sheet metal Explosive
deep drawing process of a trapezoidal cup. An innovative,
comprehensive way of using an efficient design variables
linking method, termed as reduced basis technique (Jabbari et
al., 2009). is demonstrated for die shape optimization. In the
reduced basis technique, many initial die shapes, called basis
shapes, are combined linearly by assigning weight factors.
Different resultant shapes can be generated by changing their
weight factors. Therefore, the number of design variables
required to define the die shape is reduced to the number of
basis shapes. So, the weights assigned for each basis shapes
are the design variables and the optimization goal is to find the
best possible combination of these weights to minimize spring
back angle.
The algorithm presented in this paper focuses on the Taguchi
design of experiments method which is the combination of
mathematical and statistical techniques used in the empirical
study of relationships and optimization, in which several
independent variables influence a dependent variable or
response.
SPRINGBACK PHENOMENON
This phenomenon is due to the elastic deformation part of
recovery and it affects the final geometry of the output
product. However the analysis of the behavior of matter and
spring back is complicated. Due to the large deformation, high
strain rate, mechanical properties and size of the work piece
interaction interface between the interface and the behavior of
gas bubbles resulting from the explosion of non-linear
dynamics, this issue is very complex. Factors that may be
involved in the return spring blast are: wave transmitter,
increasing thickness, increasing or decreasing the radius of the
punch and matrix spans the width and the behavior of
anisotropic plate. Spring back is an intrinsic property of
deformation which with a proper design, it can be brought to a
minimum.
Figure 1. Spring Back defect.
FINITE ELEMENT ANALYSIS
In this research, sheet metal anisotropy effect in Explosive
deep drawing process has been studied using finite element
ALI JABBARI : SPRINGBACK REDUCTION IN EXPLOSIVE DEEP DRAWING PROCESS OF A TRAPEZOIDAL CUP
Indian J.Sci.Res.1(2) : 410-415, 2014 411
package of ABAQUS. In process modeling, punch, matrix and
die holder considered asrigid parts. Material properties and
size of the die and punch is listed in Table 1. Dynamic/Explicit
Analysis which is appropriate for metal forming problems has
been selected. Shell elements of S4R are used to mesh the die.
The “Partition” command is used to mesh the sheet more
accurate. A view of meshed die is shown in Fig. 2.
Figure 2. A view of meshed die.
Then, the results of Explosive deep drawing process
simulation analyzed. Fig. 3 shows final trapezoidal cup after
Explosive deep drawing process.
Figure 3. Trapezoidal cup due to die deformation.
DIE SHAPE OPTIMIZATION
The generated earrings due to sheet anisotropy is clearly seen
in Figure 3.In order to reduce the height of the sheet shape
earrings, Taguchi design of experiments method is used to
optimize initial die shape. Using Taguchi method to optimize
one or several variables must be examined at several levels,
and a well defined objective function, which is a function of
the height of the earring. The weight ratio of 3 to a maximum
radius ( ), the average radius ( ) minimum radius ( )
were allocated, And the height of the earring in each of the 3
radius, a factor attributed to each of the planes. Table 2 shows
the factors considered. The radius of maximum coefficient is
very low because no amount of uncut sheets and sheets in the
initial state. In other rays of the Rings height, weight
coefficients were considered.
OPTIMIZATION METHOD
Although the reduced basis technique is widely used in metal
forming process, and shape optimization in permanent die
motors (Paul Degarmo., 2003). but it is suggested for die
shape optimization in this work. However, it should be
adopted for shape optimization of Explosive deep drawing
process.
Appropriate starting basis shapes are required to employ the
algorithm to find optimumblank shape design. The problem
can be solved in multiple levels as shown in Fig. 3 in which
the optimization procedure guides the designer progressively
in selecting viable basis shapes. In first Level, the basis shapes
may not be anywhere near to what they are supposed to be, but
by the first set of basis shapes the one can determine a best
combination from the first trial shapes.
Figure 3. Algorithem of design optimization.
CASE STUDY
The initial die shape optimization of trapezoidal cup is
demonstrated in this work. The finite element package
ABAQUS is used to simulate the process and to calculate the
spring back angle in order to conduct DOE. A view of the
trapezoidal cup deep drawing model is shown in Figure 4.
Specifications of the investigated sheet metal and the finial
cup are presented in Table 1.
ALI JABBARI : SPRINGBACK REDUCTION IN EXPLOSIVE DEEP DRAWING PROCESS OF A TRAPEZOIDAL CUP
Indian J.Sci.Res.1(2) : 410-415, 2014 412
Figure 4. A view of Explosive deep drawing process
modeling.
Table 1. Specification of the sheet metal and the final cup.
A B N M MELTING
TEMP
TRANSITION
TEMP
MASS
DENSITY
YOUNG
MODULUS
JOHNSON
COOK
1750 3800.001 0.5 0.32 1811 298 7.89 2000000
D1 D2 D3 D4 D5 MELTING
TEMPERTUR
TRANSITIN
TEMPERTUR
REFERENCE
STRAIN
RATE
JOHNSON
COOK
DAMAGE
-2.2 5.43 -0.47 0.016 0.63 1811 298 1
DETONATI
ON WAVE
SPEED
A B OMEGA R1 R2 MASS
DENSITY
TNT 693000 3737700 37471 0.35 4.15 0.9 1.63
C0 S GAMMA0 MASS
DENSITY
MOHIT 149000 1.79 1.65 1
Figure 5. The selected basis shapes.
2-D FEA simulations of the basis shapes are performed in
ABAQUS software to find the spring back angle for
preliminary analysis as shown in Fig. 5. The spring back
angles of the Basis shapes are 3.81, 2.52, and 0.12 (angel),
respectively. From this preliminary analysis, it can be said that
the Basis 2 is more successful than the other two shapes in
reducing the spring back angle. Therefore, the contribution of
Basis 2 must be more than the other basis shapes, which must
be recognized by the optimizer. Each Basis shape is defined
by one shape variables. These shape variables form the
respective Basis vector. These basis vectors is combined with
the weighting factors, a1, a2and a3 that correspond to each
basis vector based on the following equation
where, and n is number of basis shapes.
ALI JABBARI : SPRINGBACK REDUCTION IN EXPLOSIVE DEEP DRAWING PROCESS OF A TRAPEZOIDAL CUP
Indian J.Sci.Res.1(2) : 410-415, 2014 - 413 -
The reduced basis technique is applied to three basis vectors
and the number of design variables is decreased to three,
which are the weights for each basis vector. By changing these
weights, it is possible to obtain various resultant die shapes for
the optimizer to find the best combinations of these weights.9
DOE points are generated to conduct simulation. All of the
resultant die shapes are scaled to maintain in a limited area.
A DOE/Taguchi approach is used to study the effects of
multiple variables simultaneously. Three factors including
Basis shapes weighting factors will be investigated and their
optimum values will be specified through ANOVA. Based on
known variation of spring back angle with respect to different
factors, each factor is considered to have three levels.
Therefore, an L-9 orthogonal array has been selected to run
the experiments. Table IV shows the factors and their levels
and the layout for the selected array is also presented in Table
V.
Table 2. Weighting factors and their selected value.
FACTOR LEVEL 1 LEVEL 2 LEVEL 3
A1(R=45) 0.2 0.1 0.03
A2(R=46.5) 0.5 0.7 0.9
A3(R=51) 0.3 0.2 0.07
Table 3. A standard L-9 (9 Experiments Runs) array for
three three levels factors.
1 2 3
TRIAL1 1 1 1
TRIAL2 1 2 2
TRIAL3 1 3 3
TRIAL4 2 1 2
TRIAL5 2 2 3
TRIAL6 2 3 1
TRIAL7 3 1 3
TRIAL8 3 2 1
TRIAL9 3 3 2
Angles obtained from experiments described in Table 3 in
Table 4 is given.
Table 4. Angles obtained from experiments.
TRIAL NO. RADIUS
1 47.55
2 47.04
3 46.51
4 47.43
5 46.69
6 47.42
7 46.95
8 47.76
9 48.07
The experiments are carried out using FEA and the spring
back angle obtained from each experiment is listed in Table 5.
Table 5. The values of spring back angle in all 9 trials.
TRIAL NO. SPRING BACK ANGLE (
1 0.44
2 1.62
3 2.52
4 1.23
5 2.31
6 1.24
7 2
8 0.01
9 0.9
Considering spring back angle as cost function, the results are
investigated. The main effects table, which presents the mean
value of spring back angle for each factor at all levels, is
shown in Table 6. ANOVA table for spring back angle is
listed in Table 7.
ALI JABBARI : SPRINGBACK REDUCTION IN EXPLOSIVE DEEP DRAWING PROCESS OF A TRAPEZOIDAL CUP
Indian J.Sci.Res.1(2) : 410-415, 2014 - 414 -
Table 6. The mean value of spring back angle for each
factor at all levels.
FACTOR LEVEL
1
LEVEL
2
LEVEL
3
A1 1.526 1.593 0.969
A2 1.223 1.313 1.553
A3 0.563 1.25 2.276
Table 7. ANOVA table for spring back angle.
FACTOR F S V F PS P(%)
A1 2 0.702 0.351 2.354 0.404 7.173
A2 2 0.174 0.087 0.584 0 0
A3 2 4.461 2.23 14.946 4.162 73.844
OTHER/ERROR 2 0.297 0.148 18.983
TOTAL 9 5.636 100.00%
Predicted optimum combination of factors in this step is
shown in Table 8.
Table 8. Optimum condition for spring back angle.
FACTORS VALUES LEVEL CONTRIBUTION
A1 0.03 3 -0.394
A2 0.5 1 -0.14
A3 0.3 1 -0.8
TOTAL CONTRIBUTION
OF FACTORS
-1.335
GRAND AVERAGE OF
PERFORMANCE
1.363
EXPECTED RESULT AT
OPTIMUM
0.029
As shown in Table 8, it is observed from the analysis of results
that Basis 3 doesn't have considerable effect on the spring
back angle. Since its respective weighting factor in optimum
condition is close to zero. Therefore, the optimum shape of
this process used as a Basis shape for second level.
The optimum values for weighting factor a1, a2 and a3 are
0.05, 0.80, and 0.30, respectively. The spring back angle of the
resultant shape is . It is clear that most of the
contribution is from Basis 2. A comparison of maximum Von
Mises stress for basis shapes and opimal shape is shown in
Fig. 6.
(A=45) (B=46.5) (C=51) (D=47.78)
Figure 6. A comparison of maximum Von Mises stress for
basis shapes and optimal shape.
As shown in Fig. 7, the maximum value of the spring back
angle for Basis shapes are 3.81, 2.52, and 0.12 (degree), which
it has been reduced significantly by this method to .
(B)
(A)
Figure 7. A comparison of initial (A) and
final (B) cup.
.
CONCLUSION
A die shape optimization method for Explosive deep drawing
process is introduced in this paper using the reduced basis
technique. The concept of design optimization process is
introduced, which aids the designer in the selection of
practical basis shapes that will give spring back angle
ALI JABBARI : SPRINGBACK REDUCTION IN EXPLOSIVE DEEP DRAWING PROCESS OF A TRAPEZOIDAL CUP
Indian J.Sci.Res.1(2) : 410-415, 2014 - 415 -
reduction. It is important to mention that if expert knowledge
is available, then practical basis shapes can be selected and the
optimum die shape can be obtained in a single level.
Increasing the number of basis shapes also enables the
designer to obtain a better die shape, but the computation time
also increases to build an approximation model. The reduced
basis method aids in the use of the Taguchi design of
experiments models for optimization. Most die shapes
obtained by this method are practical. The presented algorithm
has been applied on die shape optimization of a trapezoidal
cup as a case study. An optimum die shape has been achieved
by the implemented algorithms, starting from three basis
shapes such as three circular arcs. The spring back angle has
been reduced significantly by this optimization method.
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Shape Optimization of Permanent Die SynchronousMotors
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