=Paper=
{{Paper
|id=Vol-2843/paper30
|storemode=property
|title=A variant of solving the optimization problem of finding the order of bending of sheet metal parts with parallel bends in time (paper)
|pdfUrl=https://ceur-ws.org/Vol-2843/paper030.pdf
|volume=Vol-2843
|authors=Fedor Yu. Trapeznikov,Alexandr A. Petunin
}}
==A variant of solving the optimization problem of finding the order of bending of sheet metal parts with parallel bends in time (paper)==
https://ceur-ws.org/Vol-2843/paper030.pdf
A variant of solving the optimization problem of finding
the order of bending of sheet metal parts with parallel
bends in time*
Fedor Yu. Trapeznikov1 and Alexandr A. Petunin2
1
PJSC "MZIK", 18, Kosmonavtov Ave., Ekaterinburg, Sverdlovsk Region, 620017, Russian
Federation
2
Federal State Autonomous Educational Institution of Higher Education “Ural Federal Univer-
sity named after the first President of Russia B.N.Yeltsin”, 19, Mira street, Ekaterinburg,
Sverdlovsk Region, 620002, Russian Federation
bigbratstudio@gmail.com
Abstract. The article proposes a solution to the optimization problem of finding
the bending order with the smallest value of the objective function in time when
bending complex profiles from a sheet metal with parallel bending lines (ac-
cording to the OK12-93 classifier, subgroups 745410; 745420; 745430 and
745440. An algorithm is formulated for finding the most continuous bending
order, formulas are derived that are necessary to calculate the sweep, techno-
logical dimensions for each transition and check the feasibility of subsequent
bends, in the presence of perfect bends, a number of algorithms for compiling
technological transitions to parts with 1, 2, 3 bends are also presented privately.
The creation of this optimization algorithm made it possible to find options for
bending orders that require the least time to manufacture a part and significantly
reduced the time for technological calculations when drawing up a technologi-
cal process.
Keywords: Bending, CNC, Sheet metal, Reducing bending time, Bending se-
quence.
1 Introduction
At the moment, at most enterprises, the calculation of unfolded sheet metal parts and
the order of their bending is carried out manually by drawing each transition in CAD
systems and checking the feasibility of subsequent transitions, if any. All attention is
focused on obtaining a certain order of bending of the part, which provides a given
bend profile, but is not optimal from the point of view of the minimum time spent on
manufacturing the part. At the same time, each large foreign manufacturer of press
brakes has its own expensive software, which is able to find the bending order of a
*
Copyright © 2021 for this paper by its authors. Use permitted under Creative Commons License Attribu-
tion 4.0 International (CC BY 4.0).
part in automatic mode and visualize it, but it is not known whether it is the most
optimal solution for a given part. In addition, this software has a number of disadvan-
tages, among which the most significant are the price, the presence of functions that
are not in great demand at most machine-building enterprises and the lack of domestic
software. In this regard, it becomes necessary to create your own algorithm, and then
software that can generate the correct bending orders in the shortest possible time and
compare them in terms of execution time to find the least time-consuming bending
order.
When developing a bending order, rules of thumb are used. They choose a bending
order suitable for obtaining a given bending profile (that is, the feasibility of all transi-
tions), the required tool for each transition, calculate the possibility of installing
hardly several tools to reduce the total bending time - excluding the time for tool
change. When developing a technological process, a process engineer has problems to
optimize bending transitions. The functions of bending time Tbend (1) and bending
cost Fcost (2) are considered as optimization in these problems:
n n n n
tbend Ti N i M i Vi (1)
i 1 i 1 i 1 i 1
Where:
i=1, 2, …, n
n – the number of bends in the part;
Ti – time spent on placing the workpiece on the table (placing the workpiece on the
table, bringing the workpiece to the stop);
Ni – time spent on changeover for bending to another size (changing the position of
the back stops);
Mi – time spent on changing the tool (for example, to perform 2 or more bends at
once;
Vi – time spent on the tool stroke (performing idle, working and reverse idling).
n n n n
Fcos t Ti Fb N i Fs M i Fch Vi Fm (2)
i 1 i 1 i 1 i 1
Where:
i=1, 2, …, n
n – the number of bends in the part;
Fb – cost of 1 unit of time for placing the workpiece;
Fs – cost of 1 unit of time for changeover;
Fch – cost of 1 unit of time per tool change;
Fm – cost of 1 unit of time per tool stroke.
The components of the bending time Tbend and the cost of bending Fcost are related
to each other by the coefficients (standard hours) laid down in each enterprise and
consisting of fixed and variable costs, such as depreciation of equipment, cost of elec-
tricity, wages of a worker, etc. In this regard, this work will solve the problem of re-
ducing the bending time Tbend of the part, and already on the basis of the gain ob-
tained at each enterprise, it will be possible to estimate the gain in the cost of manu-
facturing each part, by multiplying the time obtained by certain coefficients (standard
hours) and comparing the obtained cost with the cost previously pledged for this part.
To speed up the calculation of the order of bending, which ensures a given bend
profile, it is necessary to empirically compose an algorithm for finding solutions to
this problem. To find the least time-consuming variant of the bending order, it is nec-
essary to derive from each transition the components of the total bending time given
in formula (1), and calculate the time function Tbend and compare them. After compar-
ing and searching for the smallest value of the bending time of a given profile, it will
be possible to conclude that this order is the most advantageous in terms of time.
Automation of this process will also give a significant benefit in the time of techno-
logical preparation for the production of a part and exclude cases of defects in manu-
facturing, due to the appearance of situations of impossibility of making subsequent
bends in the presence of already perfect ones - the specified bend profile will not be
obtained.
The bending order is a sequence of the form:
J=(J1, J2, …, Jn), (3)
Where:
J – bending order function;
J1, J2, …, Jn – sequential numbers of places of bend of a part.
For example, the order of bending of a part can have two solutions:
J=(1, 8, 4, 3, 6, 5, 7, 2) and J*=(1, 2, 5, 6, 8, 7, 3, 4)
This problem is a special case of the traveling salesman problem, which generally
has n! solutions. To filter out the wrong decisions, in this case, there are technological
limitations, derived geometrically and empirically confirmed and causing a given
bend profile to be obtained. As an example, a part is shown that has several difficult
places to calculate the correct bending order. The order of all technological transi-
tions, calculated by an experienced process engineer, to obtain the part taken as an
example and shown in Figure 1, is shown in Figure 2. As can be seen from the scan, it
is not always possible to use the classical bending method according to the rule "from
the periphery to the center" , in this case, the bending order looks like this: J = (1, 8, 4,
3, 6, 5, 7, 2), in this regard, it is necessary to enumerate various options to find the
right solutions, by drawing several bending options to obtain a suitable details, which
takes a lot of time and there is a high probability of miscalculation, which will entail a
marriage during the manufacture of a part in the shop and a recalculation of all transi-
tions. It may also be necessary to design bending equipment due to the impossibility
of using the existing one. Manually enumerating all the options and finding the best
one at the least time-consuming in the presence of 5 options or more (from 120 op-
tions) is not advisable and will take a lot of time.
To eliminate the human factor and significantly reduce the time for technological
calculations, it is necessary to formulate an algorithm and derive the necessary formu-
las for drawing up a program for the automatic calculation of several variants of the
bending transition sequences and checking each option for feasibility using the exist-
ing equipment.
Fig. 1. Bending profile of a part.
Fig. 2. Scheme of marking the involute of the part for bending indicating the bending sequence.
2 Materials and methods
To develop an algorithm for checking the bending orders for feasibility, it is neces-
sary to determine and derive formulas required for calculations and checks for the
feasibility of the subsequent bend in the presence of available bends after each transi-
tion.
List of formulas required for derivation:
─ formula for calculating the length of the involute;
─ formula for calculating the technological dimension required to set up the press for
the next bend, based on the previous bend;
─ formula for calculating all distances to each bend from both ends of the part (Fig-
ure 2);
─ a formula for calculating the determination of the possibility of making a subse-
quent bend in the presence of already perfect bends (projection on the X-axis, Fig-
ures 3 and 4);
─ determining the absence of conflicts between the punch and the part (Figure 5).
Calculating all values of n! can be performed by any combinatorial permutation
search algorithm without repetitions. In reality, in detail, the number of bends rarely
exceeds 8 - a set of errors during bending and a shortage in the length of the sweep on
the last bends affect, while the value of n! with 8 bends and, accordingly, 9 shelves,
the part will equal 362880 variants.
Formula for calculating the length of the unfolded when calculating a bent part
with one bend:
L ( L1 L2 ) [ 2(tg ( 1 )(r1 s )) 1 ( r1 k1 s )] (4)
2 180
If there are 2 bends in the part, the formula looks like this:
L ( L1 L2 ) [2(tg ( 1 )(r1 s )) 1 (r1 k1s )]
2 180
(5)
2 2
( L3 [2(tg ( )(r2 s )) (r2 k2 s )])
2 180
If there are three or more bends in the part (in the general case), the formula looks
like this:
n
1 1
L ( L L ) [2(tg ( 2 )(r s)) 180 (r k s)]
i 3
i 1 2 1 1 1
( Li [ 2(tg ( i 1 )( ri 1 s )) i 1 ( ri 1 ki 1s )]) ... (6)
2 180
... ( Ln [ 2(tg ( n 1 )( rn 1 s )) n 1 ( rn 1 k n 1s )])
2 180
The following formula is designed to calculate the unfolding of the part when
specifying the lengths of the sides to the points of their intersection in the presence of
dimensions of only straight sections, the formula will take on:
n
1 n
L ( L L ) [ 180 (r k s)] ... ( L [ 180 (r k s)])
i 3
i 1 2 1 1 n n n (7)
The formula for calculating the distance from the end of the part to the first bend,
as well as this formula is suitable for calculating the machine setup for making a bend
after the previous bend has already been completed:
n
n 180 ( rn k n s )
L ( Ln (tg ( )( rn s )) (8)
2 2
Formula for calculating the distance from the end to the next one after the extreme
bend in the involute of the part:
L ( L1 L2 ) [2(tg ( 1 )(r1 s)) 1 (r1 k1s )]
2 180
2
(r2 k2 s ) (9)
(tg ( 2 )(r2 s )) 180
2 2
The formula for calculating the distance from the end to the subsequent bends (not
the extreme and not following the extreme) in the scan - if there are 5 bends or more
(formula in general form):
n
1 1
L ( L L ) [2(tg ( 2 )(r s)) 180 (r k s)] L
i 3
i 1 2 1 1 1 i
[ 2(tg ( i 1 )( ri 1 s )) i 1 ( ri 1 ki 1s )]) (tg ( i )( ri s ))
2 180 2
i
180 ( ri ki s ) n 1
... ( Ln [ 2(tg ( )( rn 1 s )) (10)
2 2
n
n 1 n 180 ( rn k n s )
(rn 1 k n 1s )]) (tg ( )( rn s ))
180 2 2
When jumping over several bends, it is necessary to calculate the distance to set up
the machine for the next bend:
L ( Lm 1 Lm ) [2(tg ( m 1 )(rm 1 s)) m 1 (rm 1 km 1s )] ...
2 180
... ( Ln [2(tg ( n 1 )(rn 1 s )) n 1 (rn 1 k n 1s )]
2 180
(11)
n
n 180 (rn k n s)
(tg ( )(rn s))) ,
2 2
Where:
L1; L2; …; Lm; … Ln – the length of the shefs of the bent part;
φ1; φ2; …; φm; …; φn; – bending angles between shelfs L1 and L2; L2 and L3; Lm-1
and Lm; Ln-1 and Ln respectively;
r1; r2; …; rm; … rn – internal bending radii between shelfs L1 and L2; L2 and L3; Lm-1
and Lm; Ln-1 and Ln respectively;
s – the thickness of the sheet material from which the part is made;
k1; k2; …; km; … kn – the coefficients of displacement of the neutral layer (k-
factor) in the places of bending between the straight sections L1 and L2; L2 and L3; Lm-
1 and Lm; Ln-1 and Ln respectively.
The formula for calculating the projection of the curved shelf on the X-axis (Figure
3):
L ( L cos ) ( s cos(90 ))
x 1 1 1
(12)
The formula for calculating the projection of the curved flange on the Y axis:
L y L1 sin 1 (13)
Fig. 3. Projection of curved shelves on the X axis.
When bending the next shelf (making the second bend sequentially) after the ex-
treme shelf, the projection of the end (extreme point) onto the X axis will be calcu-
lated by the formula:
L2 sin(1 )
Lx ( )
sin(180 1 2 )
L2 sin(1 )
sin( 2 ) (14)
sin(180 1 2 ) L
sin(1 2 90)
sin(1 ) 1
Fig. 4 - Projection of 2 curved shelves on the X axis.
Equation for checking the absence of conflicts between the workpiece and the tool
when making the second bend:
2
L s (sin( )
1 2
sin(180 1 ( 2 ))
1
2
B
sin(90 1 2
2
2 L2 ,
(15)
2 2
(cos(90 )) sin
2 2
Where:
B- punch width, mm.
If the above condition is met, then the next bend at a given angle is possible (Fig-
ure 5).
Fig. 5 - Limit location of the bend without a conflict between the workpiece and the tool.
r
The neutral layer displacement coefficient k (k-factor) depends on the ratio and
s
is selected from the table 1.
Table 1. K-factor.
r/s 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0,9
k 0.323 0.34 0.356 0.367 0.379 0.389 0.4 0.417 0,42
r/s 1 1.2 1333 1.4 1.5 1.666 1.7 1.8 1,9
k 0.421 0.426 0.433 0.436 0.441 0.446 0.447 0.449 0,452
r/s 2 2.5 3 3.5 4 4.5 5 5.5 6
k 0.455 0.459 0.463 0.466 0.469 0.473 0.477 0.479 0,48
r/s 6.5 7 7.5 8 8.5 9 9.5 10 and more
k 0.483 0.485 0.488 0.49 0.493 0.495 0.499 0.5
3 Results and Discussion
The next step after deriving the required formulas is to determine and compose an
algorithm for composing the bending order and an algorithm for checking the feasibil-
ity of each bend.
3.1. General algorithm for selecting the order of bending of a part with 3 or more
bends:
─ Calculation of the length of the sweep according to the formula (6) or (7) and all
distances from each of the 2 ends of the part to each bend according to the formu-
las (8), (9), (10);
─ Generation of flexible orders and checking each for feasibility (particular variants
of the algorithms are given below);
─ Revealing the presence of variable terms for calculating the total bending time for
each of the possible options;
─ Calculation of technological dimensions for each of the transitions for the least
time-consuming bending option according to formulas (9), (10).
3.2. General rules for checking bends for satisfiability and simplified algorithms
for special cases:
─ If there is only one bend in the part, that is, L1 <> 0 and L2 <> 0 and L3 = 0,
bending is performed from either end of the part and, therefore, there are only two
options for the order of bending, depending on the size L1 or L2 to be performed. ,
depending on the tolerance for the length of the shelf, the second dimension will be
obtained automatically with the correct calculation of the involute.
─ If there are two bends in the part, that is, L1 <> 0 and L2 <> 0 and L3 <> 0 and L4
= 0, the possible number of options is 4 and the least time-consuming bending or-
der is calculated by searching for shelves with the same length, when condition of
equality of moduli of bending angles. If the values of the angles are not equal or
not equal to the lengths of the shelves, then the order of bending will depend on the
tolerances for the lengths of the shelves, first dimensions are made with narrower
tolerance fields, and then with wider ones in ascending order. It is also necessary to
check the feasibility of bends by the formula (15).
─ When bending a part with three bends (the number of shelves is 4), in the classic
version, bending starts from the extreme bends, then the middle bend is performed.
If L4 <> 0 and L5 = 0, то: 'The length of shelf # 4 is not zero, and the 5th shelf is
missing
If L1 = L2, then:
The order of bending will be: bending of the shelf L4, then L1, then
L2
Else If L1 = L3, then:
The order of bending will be: bending of the shelf L4, then L1, then
L3
Else If L4 = L2, then:
The order of bending will be: bending of the shelf L1, then L4, then
L2
Else If L4 = L3, then:
The order of bending will be: bending of the shelf L1, then L4, then
L3
Else
The order of bending will be: bending of the shelf L1, then L4, then
L2
End If
In general, the bending of a part with three bends is possible, even in the presence
of a closed contour of the part, with a part width of up to 300 mm and provided that
special equipment is used.
3.3 As a rule, in the details, the extreme bends are the first to be made, located di-
rectly first from the ends of the part, to which they are parallel, and then, from the
perfect bends, all the following bends are performed in turn in order from the edge to
the center, but there are cases of exception (Figure 6a), when bending must be done
from the middle to the edge (Figure 6b), otherwise the workpiece will conflict with
the matrix.
а b
Fig. 6. The procedure for bending a profile of this form: a - finished part; b- first transition.
In this case, the bending order search algorithm will look like this:
If φm-1 = φm+1 = (- φm+2) = (- φm+3) and Lm = Lm+2 = Lm+4 (16)
Write to the array the bending order: φm+1; φm+2; φm; φm+3
Else
A general algorithm for searching and checking for the feasibility of bending op-
tions of the bending order is performed.
3.4 However, in cases where the end of the first bendable shelf is not at the level of
the outer surface of the 4th bendable shelf, then bending "from edge to center" is pos-
sible provided that the blank can be placed in the required position on the die for the
next bend. That is, the technological dimension should be greater than or equal to the
sum of the length of the folded shelf number m-2 (when bending the shelf number m)
and the distance from the edge of the matrix to the center of the strand (very often it is
equal to half the width of the matrix):
H matrix
Ltech Lm2 (17)
2
3.5 It is also necessary to take into account the direction of the bend and alternate
bends, which must be done clockwise and counterclockwise. Therefore, when speci-
fying the bending angles, it is necessary to indicate the bending direction with the +
and - signs: “+” - counterclockwise and “-” - clockwise.
3.6 If 2 bends that are next to each other correspond to the condition: φn-1 >0 and φn
>0, then one of these bends must be performed at the last moment due to the fact that
after it has been completed, adjacent bends will be impossible to perform without
special equipment.
3.7 If 3 bends located next to each other correspond to the condition: φn-1 >0 and φn
>0 and φn+1 >0 , then such a part cannot be bent by the universal method without the
use of special equipment - a punch on cantilever suspensions. This punch can bend
parts with a closed loop - when 3 or 4 bends occur in one direction: clockwise or
counterclockwise, provided the length of the part is not more than ≈ 300 mm, depend-
ing on the thickness of the sheet metal due to low rigidity, and in due to this deflec-
tion of the punch.
In this regard, when calculating the order of bending of a part with 4 bends or more,
it is necessary to introduce the above checks into the algorithm.
Below is an algorithm for fast selection of the search for one correct bending order,
which is obtained from the thoughts of a process engineer. This algorithm allows you
to get only one option and this option can be no less time consuming:
‘n-number of shelves in the part, for example, if n = 8, then there are 8 shelves and
7 bends in the part.
For i=1 to i=n-2
If φi >0 and φi+1 >0 and φi+2 >0, then
Message: "the part cannot be manufactured using this method"
Else:
i=i+1 'Repeat check for each triple fold
For i=1 to i=n-1
If Li = Ln and│φi│= │φn-1│ then:
If Li = Li+1, and φi=- φi+1 and Ln ≠ Ln-1, then:
Write to the array of the bend order: φn-1; φi; φi+1
If Ln = Ln-1 and φn-1=- φn-2 and Li ≠ Li+1 then:
Write to the array of the bend order: φi; φn-2; φn-1
If φi= φi+1, then:
Write to the array of the bend order: φi; φn-1; φn-2
Else Write to the array of the bend order: φn-1; φi
i=i+1
n=n-1
Next, it is necessary to check the feasibility of subsequent bends according to the
formula (15).
If one of the bends fails the check, then it is removed from the bend order array and
the check order is repeated.
Repeat the cycle.
In the general case, the search for possible orders of bending is carried out using a
combinatorial algorithm for searching for permutations without repetitions, then,
when each option is found, the possibility of performing this order of bending is
checked according to the algorithm presented above. If the result is positive, this order
is remembered and the next order is generated.
After enumerating all possible options and filtering out unsuitable ones, it is neces-
sary to determine the presence of variable components of the time function in each of
the transitions, namely: the presence of the need to readjust the Ni machine and the
need to change the tool Mi to complete this transition. The components of the time for
setting the workpiece on the table Ti and the time for the tool stroke Vi can be con-
ventionally taken as constant values for a given machine model, which do not depend
on the number of bends in the part.
For the example given in the introduction (Figure 1), using a combinatorial algo-
rithm and subsequent feasibility check, it was possible to find another variant of the
bending order: J'=(8, 6, 5, 7, 4, 3, 1, 2). With this order of bending, the number of
permutations of the tool is equal to 1 - the initial installation of the tool on the press.
The number of readjustments is 7. Let's calculate the Tbend value according to the
formula (1):
'
t bend (8 3,15) (7 4,65) (1 95) (8 4,51) 188,83с 3,14 min (18)
Next, we will compare the obtained bending time with the time for which the exist-
ing bending order J=(1, 8, 4, 3, 6, 5, 7, 2), is performed, shown in Figure 2. With this
bending order, the number of tool permutations is also 1 , the number of changeovers
is also 7.
t bend (8 3,15) (7 4,65) (1 95) (8 4,51) 188,83с 3,14 min (19)
Based on this, we can say that in this case, both orders are flexible and there is no
gain in time. And at the same time, the time spent on the selection of the bending
order J was about 2 hours, and the selection of the bending order J 'took no more than
1 minute.
As you can see from the formula, the most time-consuming operation in bending is
tool change. It takes an average of 95 seconds to change the punch, and 70 seconds to
change the die. The calculation of the number of tool changes must also be taken into
account in the check algorithm. Accordingly, such bending orders will be discontin-
ued, in which the tool change occurs only once in one direction without returning to
the previous tool.
4 Conclusion
Let us formulate the main results obtained in this work:
1. A general optimization algorithm for finding the correct options for bending
parts with parallel bends and identifying among the options found, options with a
minimum time to complete all bends is formulated and described.
2. Basic formulas were derived for the theoretical calculation of technological
dimensions:
─ involute length;
─ technological size required to set up the press for the next bend, based on the pre-
vious bend;
─ all distances to each bend from both ends of the part;
─ determining the possibility of making a subsequent bend in the presence of perfect
(projection on the X axis);
─ no conflict of punch and parts.
3. Algorithms for finding the order of bending of parts are formulated:
─ with 1 bend;
─ with 2 bends;
─ with 3 bends;
─ with 4 bends and more.
4. The resulting algorithms have been tested on several dozen details. During
testing, not a single case of incorrect operation of the algorithms was revealed.
5. The creation of this optimization algorithm has significantly reduced the time
for technological calculations when drawing up a technological process.
6. Thus, having received all the necessary formulas for calculations and algo-
rithms in the future, it is possible to create an application capable of automating the
technological process of calculations with graphical construction of bending transi-
tions and setting the necessary technological dimensions based on AutoCAD and
KOMPAS. In the future, it is planned to start developing this application with an in-
terface understandable for any user to significantly speed up calculations and check
the obtained algorithms, as well as to improve them.
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