Lin 1996

Journal of Manufacturing Systems Wol. 15/No. 4 1996

Application of Fuzzy Set Theory and Back-Propagation Neural Networks in Progressive Die Design Zone-Ching Lin and Ho Chang, National Taiwan Institute of Technology, Taipei, Taiwan

Abstract Fuzzy set theory is introduced to process the design

details of the uncertainty portion of die design and assist the designer in transforming design items with fuzziness into definite and reasonable design attributes. For design parameters in die design that possess intermediate features, fuzzy cluster analysis is used to obtain design attributes. For theoretical or empirical formulas possessing uncertainty coefficients ranges or preference design parameters, the fuzzy weighted average method is adopted to obtain the feature parameters that conform to die design requirements. For the design of uncertain operation stations, such as the installation of an idle station, the linguistic fuzziness or linear membership function is matched with the network recall value and learning pattern through neural network learning, and then the designer can decide whether it is necessary to install such an operation station.

Finally, this study establishes an expert system prototype to combine the uncertainty problems in three kinds of die design and help the designer obtain a definite design strat- egy while faced with uncertain design items.

Keywords: Fuzzy Sets, Neural Networks, Progressive Dies, Expert System

Introduction The techniques of pressworking concentrate on

the aspects of die design. The die itself is the major deciding factor of the quality and price of products. Other than the personal experiences of the designer, there are also numerous design rules and empirical data for reference in die design handbooks ~ and press die design publications, z-4

Lin, Hsu, and Yao 5 and Lin and Huang 6 published studies on the expert systems of shearing cutting die design s and deep-&awing press die design. 6 For research on uncertainty, Hwang and Liang 7 used the certainty factor method to process the problem of uncertainty in the expert system and designed a con- veyor expert system with an analytic hierarchy process as the systematic strategic method. This expert system consists of( l ) a hierarchy setup module, (2) a hierar-

chy analysis module, (3) a knowledge base module, (4) a certainty factor module, and (5) an interface module. With this system, factories can easily and effectively choose conveying equipment suited to their needs. Seilar a focused on the expert system of project management to examine the inference of its uncertainty. In other way, Tansel and Mclaughlin 9 used the learning feature of neural networks to replace the AR time sequence model and achieved faster detection of tool breakage by means of the modal distinction of cutting signals. Hyun and Cho 1° resorted to the prediction of forming pressure curves during the hydroforming process of neural networks, while Fang u conducted a cutting experiment to understand the influential factors related to finishing turning and then used the integrated fuzzy set model of machinability parameters to predict total machining performance. However, the uncertainty problem present in die design was never mentioned or explored. In general, there are four types of methods used to handle the uncertainty problem, including Bayesian probability theory, certainty factor theory, evidence theory, and fuzzy set theory. The first three types are suitable for dealing with the uncertainty derived randomly. Fuzzy set theory can be used to handle transitional objects and search for the broad definition.

Die design is mostly done by skilled engineers with considerable experience. Due to the huge quantity of uncertainty information and factors to be considered and selected, die design is affected by individual subjectivity to a great extent. These uncertainty data often affect the structure of the die itself and the reliability of sheet parts. The above discussions indicate that the application of fuzzy set theory to handle the problem of uncertainty in die design is a very suitable approach.

Therefore, this paper focuses on the design items with uncertainty features in die design and catego-

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rizes them into three types of uncertainty characteristics. Then fuzzy set theory and back-propagation neural networks are used to study and process these items. The three types of design items with fuzziness (uncertainty) are listed below:

1. During the selection of a die part from die design handbooks, the intermediate features of the design parameters or requirements result in the difficulty of making a definite choice between two standard part sizes. But the appropriate choice between the two standard sizes with uncertainty in size attributes must be made according to practical design viewpoints. For this kind of fuzziness problem, this study used fuzzy cluster analysis 1~ as the processing method, that is, using a dialog to establish a characteristic matrix for the important parameters of quantity, blank type, blank thickness, and tolerance. Also, the membership value symbol- izes the degree of fuzziness of the design parameters with intermediate features, and the group technology graphic number for each parameter is obtained through the cluster analysis of the fuzzy similarity relation, completing the certain choice of die parts at the same time.

2. Uncertainty coefficient values can be present in theoretical or empirical formulas, or uncertainty can result from the design scope of parameters preferred by the designer. For this type of fuzziness problem, the fuzzy weighted average 13 is introduced through interval analysis and cx-cut (alpha-cut) to obtain the feature parameters con- forming to design rules. If the feature parameters cannot satisfy the design rules, only the o~- cut of the uncertainty parameters needs to be corrected to derive the parameter design preferred by the designer that also meets the requirements of the design rules.

3. During the design of press operation stations for a progressive die, the uncertainty of the optional availability of certain operation stations--such as an idle station or pilot position--is processed with artificial neural networks, which assist the designer in deciding whether to install the operation stations. For the operation, the linguistic fuzziness value or linear membership function value is taken as the input value of the neural network, and the sigmoid function is taken as the

neuron's transfer function. The network output value "0" means that the operation station is not necessary, and "1" means that it is necessary. After the error convergence is acquired through the revision of weight value, the recall value from the input patterns and learning patterns are matched, and the mapping input patterns from the a-cut selection are employed to decide whether to install the operation station or not.

Framework of the Expert System Due to the countless variations of pressworking

products and the fact that uncertainty design details of a progressive die have increased with product complexity, it is very difficult to process the uncertainty design details of all types of progressive dies with proper models. Therefore, this study only con- siders progressive die products that possess simple inner and outer contours, such as the piercing and shearing-cut progressive die shown in Figure 1, or that possess only the processing types of piercing, blanking, and bending, under the condition that the completion of pressworking operations requires no more than five operation stations.

This study also establishes an expert system prototype to combine the above uncertainty die design items to assist the designer in making the preliminary design. The system flowchart is shown in Figure 2. In this system, die design parameters can be obtained through the man-machine interface communication between the system program and the user. The system then stores the data in the black- board model and enters the inference engine to

Die block

Guide post

I ~ | _ 1 D i e b l o c k top view

1 piercing I ~ u___up ' ~ 12 location I ~ ].._ ~,' ]3cuto.

opennng

U-die ~ 1 3 12 [ 1 4cutoff

Product Strip

Figure 1 Piercing and Shearing-Cut Progressive Die

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I user

selectitemseXecutive I

I input design feature and I design requirement

[ certain dataeenScYeSte m in fer - 1

t [ design feature inquire I

input uncertain design parameter ]

I fuzzy cluster neural networks subsystem I ] fuzzy weighted ~, average subsystem subsystem

I uncertain data system inference I

die parts group technology graphic number

I graphic display I

.

.

ance angle, guide post, guide bushing, and punch plate. Uncertainty of design coefficients: The fuzzy weighted average (FWA) method is used to derive the die clearance coefficient, stripper coefficient, and minimum outside radius of the guide bushing type of the lower die. Uncertainty of operation station installation for progressive die design: The back-propagation artificial neural network method is employed to help designers decide whether it is necessary to install an idle station.

Introduction of Fuzzy Set Theory and Neural Networks

This section will introduce some portions of fuzzy set theory and neural networks used in this study. The practical uncertainty module of die design will be discussed in detail in the next section.

The fuzzy set in a fuzzy universal set is usually transformed into a crisp set by means of an a-cut. The a-cut of a fuzzy set A is defined as follows:

A,~ = {x e X~ p~A(X) >-- 0~} (1)

Figure 2 Flowchart of Die Design Expert System Prototype

obtain the group technology graphic number through forward inference. In terms of execution, this system conducts the inquiry and inference of the data with certainty first. Then the system is divided into three subsystems according to the details of uncertainty design to conduct the inquiry and inference of uncertainty data. The methods used and the items capable of solving the die design for each part are described below (see Figure 2):

1. Inference of certainty: The knowledge sources in this portion come from the compilation of die expert experiences and the data in die design handbooks. Design contents to be solved include die material, appropriateness of adopting a progressive die, punch penetration rate, pilot, circu- lar radius, width of blank scrap, blank oiliness, and so on.

2. Choice of uncertainty parts: Fuzzy cluster analysis is used to derive the selection of die parts, including die block thickness, die clear-

If R is the fuzzy relation of X and Y and S is the fuzzy relation of Y and Z, then the composition of R and S, R o S, is also a fuzzy relation. If the composition of R and S is C, then its membership function can be written as:

Ixc (x,z) = iXRo s(X,Z) = Vy(txR(x,y) /k lxs(y,z)), e E z (2)

The previous section mentioned that this study used fuzzy cluster analysis to solve the uncertainty problem of the selection of parts in die design. Only its similarity relation will be explained in this section. The detailed procedures will be discussed in the next section.

A crisp binary relation R(X,, X) with reflectivity, symmetry, and transitivity is denoted as an equivalence relation} 416

A fuzzy binary relation with reflectivity, symmetry, and transitivity is denoted as a similarity relation.

To address the second type of die design uncertainty mentioned in the previous section, this paper introduces the fuzzy weighted average (FWA) method; the basic concepts of its algorithm are described as follows. 13

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If the algebraic expression of something contain- ing N number of variables is:

y =J(x,, x~, x3, ..., x,) (3)

For x~, x2, ..., x,, after the a-cut is taken, the mapping interval for each of them is, respectively, [at, b0, ..., [a,, b,]. Then the endpoints of 2N can have Z v number of distinct combinations; that is, x~ can be substituted with al or bi, x2 can be substituted with a: or b2, and so on.

After the different e~-cuts are taken for the variables, the membership function value o fy =J(xb x2, ..., x,) is obtained.

The previous section also mentioned the uncertainty problem of whether to install the operation stations in a progressive die, which can be solved with the neural networks proposed here. The input patterns of network learning and network design will be discussed in detail in the next section. This section provides a brief explanation of the neural networks.

The neural networks consist of an input layer, a hidden layer, and an output layer, which are three- layer back-propagation networks. 17as

Generally speaking, the selection of input patterns during the network learning process will influence the convergence of the network. Among the countless variations of progressive dies, the difficulty in evalu- ating whether the network design is satisfied or not rests in how to obtain better input patterns from die experts and handbooks for network learning. This will be discussed in greater detail in the next section.

The back-propagation neural networks can use the sigmoid function or other differentiable or nonreductive functions, such as sine and hyperbolic tangent functions, as the transfer function. This study used the sigmoid function.

Uncertainty Parameters Module in Die Progressive Design

Uncertainty of Part Selection The selection of die parts from die design hand-

books or tables often causes confusion because of intermediate features. Design parameters with intermediate features bear application significance only if they are discussed from the viewpoint of practical engineering applications. The important parameters considered in this study are described below.

Die Block Thickness The die block must be able to withstand the pres-

sure during pressworking. Therefore, die block thickness is related to blank thickness. According to die design handbooks and practical experiences, 3 die block thickness for low-carbon steel can be obtained from Table 1. In Table 1, when the blank thickness is 1.6 mm, 3.2 ram, and 4.5 ram, there is the greatest fuzziness for choosing die block thickness. The area between + 0.05 mm can be regarded as the intermediate area from a practical point of view. Figure 3 denotes the membership values when the choice of die block thickness is 24 mm or 28 mm and the blank thickness is between the intermediate area. In Figure 3, the linear function is used to denote the membership of the intermediate area. In Figure 4, the trapezoid function is used as the membership function when the choice of die block thickness is 28 ram. The choice of membership function is more or less subjective, which is the difficulty in the application of the fuzzy set theory. For die design, from the practical point of view, if the proper intermediate area [a, b] is chosen, then the linear membership function can obtain satisfactory results? a2

Table 1 Selection Table of Die Block Thickness

Blank Thickness Die Block Thickness Group Technology (ram) (ram) Graphic Number

0 - 1.6 24 101201 1.6 - 3.2 28 101202 3.2 - 4.5 35 101203 4.5 - 6 40 101204

~=

._~

0.5

Die block thickness 24 mrn

j/•lnterrnediate area

~ _ ~ / I , ~ ' L Bie bl°ck t h i c k n e s s 2 8 mm

t 1.55 1.65

1,6

Blank thickness

Figure 3 Membership Functions of Die Block Thickness at 28 mm and 24 mm

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r~ .

1.

0.5

/ \

1.55 1.65 3.15 3.25 1.6 3.2

Blank thickness

Figure 4 Trapezoid Membership Function of Die Block Thickness at 28 mm

# ( x ) - a = a < x < b - a

0 x<a

p(y) b - x = a < x < b

b - a 0 x>b

(4)

( 5 )

The reasons why blank thickness falls in the intermediate area are: (1) the difference in value as a result of the different lots of strips produced by the steel companies; (2) the difference in the upper and lower limit of the blank tolerance itself; (3) the required fine blanking thickness according to design specifications; (4) the die handbook uses the viewpoint of overdesign to make the part selection table. In Figure 3, the membership value at the blank thickness of 1.6 mm is 0.5, meaning that it possesses the greatest fuzziness in the intermediate area and has to be corrected with a minute value to deviate from the greatest fuzziness. The consideration factor for correction will be based on the choice made in the design between the weight reduction (production cost) and the die structural safety (endurance). If the minute correction amount is 0.01, then 0.49 or 0.51 will substitute for the original value of 0.5. The uncertainty membership inference result can thus be avoided.

Table 2 Selection Table for Die Clearance Angle

Blank Thickness t Die Clearance Angle Group Technology (ram) Graphic Number

t <- 1.6 1/4 ° 101205 1.6 -< t -< 4.5 1/2 ° 101206

4.5 -< t 3/4 ° 101207

Clearance F Intermediate area

angle at 1/4 ° |

==

.3

0.5 1/

1.55 1.65 1.6

Clearance angle at 1/2 °

Blank thickness

Figure 5 Membership Function of Die Clearance Angle at 1/4 ° and 1/2 °

Die Clearance Angle Similar to die block thickness, the design for die

clearance angle is also related to blank thickness. The selection is shown in Table 2. It has the greatest fuzziness when the blank thickness is 1.6 mm and 4.5 mm. Its consideration factors for the selection of minute correction amount are product precision and die production cost. The intermediate area is 0.05 for the blank thickness of 1.6 mm and 4.5 mm, and membership functions are shown in Figure 5.

Selection o f Die Sets The die sets with guide post and flat guide bush-

ings can withstand stronger sideways pressworking force. The more guide posts, the better the guiding effects. Dies with general precision should use die sets with four guide posts with fiat guide bushings. Dies with medium production or more and demanding medium precision should use die sets with four guide posts with roller guide bushings. Those with small production and demanding medium precision should use die sets with two guide posts with flat guide bushings. The selection standards are con-

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Table 3 Die Sets Selection Table

Quantity Q Q < 20000 20000 ~ Q < 50000 50000 -< Q

Tolerance 0.025 -< 0.025 -< 0.025 T(mm) 0.1 -< T T-- < 0.1 T-- < 0.025 0.1 --< T T-- < 0.1 T--<0.025 0.1 --< T 0 .025- < T-- < 0.1 T-< 0.025

Die set types use die use die use die set without sets with sets with guide post two guide four guide

posts with posts with flat guide fiat guide bushings bushings

use die use die use die sets with sets with sets with

two guide four guide four guide posts with posts with posts with

roller guide roller guide fiat guide bushings bushings bushings

use die use die use die sets with sets with sets with

two guide four guide Ibur guide posts with posts with posts with

roller guide roller guide fiat guide bushings bushings bushings

117201 117202 117203 117204 117205 117203 117204 117205 117203

cluded as shown in Table 3. Take those with a production quantity under 20,000 as an example. The fuzziness is the greatest when the tolerance value is 0.1 mm and 0.025 mm from Table 3. Let the intermediate value be -_-0.005 (from a practical point of view), and its membership function is shown in Figure 6. Its consideration factors for the selection of minute correction amount during the greatest fuzziness are blank precision demand and die production cost. Table 3 is the die sets selection table for individual dies. For progressive dies, the die sets with four guide posts with roller guide bushings should be substituted for those with two guide posts.

For the aforementioned uncertainty of part selection, this expert system combines fuzzy cluster analysis and the group technology parts code to choose a definite parts design from the design pro- posals. First, design parameters are used to establish a characteristic matrix of the design proposal. Each row of parameters represents a design proposal, and each element represents the modular group technology graphic number or design characteristic. The characteristic matrix is similar to the frame with the slot for the object. The value of the slot is the inferred result of the design. Each design proposal of the characteristic parameters matrix can derive a similar matrix after the similarity transformation. The purpose of the similarity transformation is to find the degree of similarity among the design pro- posals. This study uses cosine angle to derive the similarity: 12

s ( x , , =

[(£kLll'Lk(Xik)2)(ZkP=ll"lk(Xjk)2)] ~

(6)

.o

0.5

Die sets with four guide posts with

flat guide bushings are used / Intermediate area

/

/ ] Die sets with two ../. ~ guide posts with flat T l guide bushings are

used / t Precision tolerance

0.02 0.03 0,025

Figure 6 Membership Function of Die Sets with Four Guide Posts with Flat

Guide Bushings and Those with Two Guide Posts

where g(xik) represents the membership value of the design characteristic parameters k in design proposal i. The similarity matrix itself already satisfies reflectivity and symmetry, that is:

(1) S(X,, X,) = 1

(2) S(X~, Xj) = S(Xi, X~)

To test whether the similarity matrix possesses transitivity, composition computation is conducted for the similarity matrix:

R = S o S (7)

If R C S, then S has transitivity. Otherwise, conduct composition computation for R until the condition of transitivity is satisfied:

R ~ _ R ° R ° R o R o . . . ° R (8)

273

Once R satisfies reflectivity, symmetry, and transitivity, then R is called a fuzzy equivalence matrix. The equivalence matrix is transformed into a cluster matrix with 0 and 1 as symbols through the oL-cut. The definite (only) design proposal that satisfies design parameters can then be inferred after the cluster analysis.

The design parameters represented by every slot in the design characteristic matrix are: (1) production < 20,000; (2) production >- 20,000; (3) no holes inside the blanks; (4) with holes inside the blanks but no need for blanking; (5) with holes inside the blanks and the need for blanking; (6) blank thickness t (ram), t <- 1.6; (7) 1.6 -< t <- 3.2; (8) 3.2 <- t -< 4.5; (9) 4.5 <- t < 6; (10) t -< 1.6; (11) 1.6 -< t -< 4.5; (12) t > 4.5; (13)tolerance T (mm), 0.1 --< T; (14) 0.025 --< T-- < 0.1; (15) T-- < 0.025; (16) 0.1 <- T; (17) 0.025 -< T --< 0.1; (18) T --< 0.025; (19) punch diameter 0-8 (mm); (20) punch diameter 8-11 (mm).

In addition, the group technology graphic number of the parts consists of six digits.

Traditional binary logic is used to indicate the membership degree of the design parameters in the system, which are definite and without fuzziness. 1 in the design proposal characteristic matrix indicates that it meets the design characteristics of such an item, and 0 indicates otherwise.

Uncertainty of Design Equation

Die Clearance The die clearance value in shearing and cutoff

processing is an important factor in deciding product precision. The appropriate choice of clearance value will result in the consistency of the cracks appearing on the knife points of the punch and die and a better cross section of the cutoff opening. Therefore, the clearance value is related to the shearing strength & of the material. The empirical equation of the relation is as below9 9

- = (9) t

Q.

where c is the single die clearance, t is the strip thickness, & is the material shearing strength, and k is the clearance coefficient whose scale is k = 0.008-0.01. How to select a proper value for deriving a better clearance value is the problem studied in

the uncertainty of design equation. During the preliminary design, the fuzziness of these design parameters can be indicated with a membership function. In general, the triangular function can be used for indication. Because the consideration is the preliminary design, the designer only has to decide the three points in the triangular function, such as the membership function of the three types of triangular function styles. A membership function value of the left extreme of 0 indicates that the left-extreme design border is regarded by the designer as the least preferred or most unlikely. The peak value of the membership function is 1, which indicates highest preference by the designer or the ideal value with highest feasibility. The situation of the right extreme can be explained in a similar fashion.

From a practical viewpoint, the clearance value will increase slightly after the die receives shearing friction. Therefore, the membership function for the selection of k is designed by taking the minimum clearance value as the preferred membership value 1, that is, without the right-extreme value. Next, the membership function of the die clearance and strip thickness is found. Here, this study uses the fuzzy weighted average (FWA) method to derive, after computation, those shown in Figure 7.

Mapping the largest expected design parameter value with the largest expected output value (whose membership value is 1) in Figure 7, one gets a c/t of 0.040, which indicates that 0.040 can be used as the

I , , , , i i i

0.035 0.040 0.045 0.050 C l e a r a n c e / t h i c k n e s s (C/ ' / ) , S tee l 0 . 1 % C

1.0

0,9 t

0,8 l 0.7 0.6 0.5 0.4 0.3 0.2

0.1 t 0.0 . . . .

0.030


i I i i I I ]

0 . 0 5 5 0 . 0 6 0

Figure 7 Membership Function of Ratio of Die Clearance and Strip Thickness

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optimal ratio of clearance and strip thickness. But the largest expected output value has to be tested to see if it satisfies the functional requirement of design. The c/t value from the design handbook has to fall between 0.03 and 0.06. The value of 0.040 satisfies the requirement and falls in the left part of the interval. Therefore, the membership function of the original coefficient k does not need to be revised. The value of 0.008 can serve as the ideal clearance value.

Stripping Pressure Ps The value of Ps depends on the size of die clear-

ance, smoothness of punch surface, and nature of the material. The computation equation i s : 19

eS =KtS, (10) L

where L is the circumference of the blank, t is the strip thickness, & is the material shearing strength, and K is the uncertainty stripping coefficient K = 0.02 - 0.20? 9 For the selection of the value of stripping coefficient K in the design equation of stripping pressure, the middle value between 0.02 and 0.20 can be chosen as the expected design parameter, and 0.02 and 0.20 can be chosen as the design limitation values of the left and right extreme, respectively. The triangular membership function is shown in Figure 8.

Die with Guide Bushing While pressing smaller blanks or while piercing,

a guide bushing is commonly used for the lower die. During the shearing process, the inner wall of the

guide bushing bears the radial compressive force P0 because of deformation of the materials. This pressure cannot be easily derived from theoretical analysis. In general, 33% of the blanking or piercing force is used as the applicable value? 9 From the theory of elasticity, the theoretical equation of the tension stress in the tangential direction of circumference caused by the inner pressure of the cylinder is:

(b' / _ aef 1+7]_ Go b 2 _ a z (11)

where f is the average pressure on the inner hole wall, f = Po/rr dTh(mma/kg), d -- 2a, P0 = 0.33P, where P is the blanking or piercing force and P -- LtS,, t is the strip thickness (ram), and L is the circumference length (mm). In Eq. (11), the largest value of tr0 is the stress of the material's tensile strength S, on the location of r = a on the inner hole wall. Then Eq. (11) can be written as:

s _ f ( a Z +bZ) b e _a 2

(a is the inner radius, b is the outer radius) and it can be simplified as:

~a2(~ + f ) b = S , - f

substitute f - P0 _ 0.33P _ 0.33LtS s

(12)

lrd T h rc 2 a T h rc 2 a T h

g

0.02 0.11 0.20

Stripping coefficient K

Figure 8 Experted Membership of Stripping Coefficient K

into Eq. (12) and get:

a 2 S t + ~2aT h b=

~ -~t a33LtS, ~2at

(13)

Eq. (13) is the equation of the smallest outer radius of the guide bushing required in which the guide bushing height Th, strip thickness t, and guide bushing inner radius a can be regarded as uncertain design parameters by the designer's subjective decision. The aforementioned method uses the triangular membership function to indicate the preferred

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design values of strip thickness and guide bushing height. The strip thickness, under the consideration of blanking pressure and die structural stiffness, is not allowed to be larger than the expected value of strip thickness t. Therefore, there is no right extreme of the thickness value in the membership function. If the products are pressworking blanks with a radius of 4.0 mm, the ratio of the strip thickness and clearance in Figure 7 and the membership function of strip thickness can derive the expected membership function value of the clearance value with the FWA method. As for the expected membership value of the bushing inner radius, a production tolerance is considered as the boundary value of the left and right extreme. Let the tolerance interval be +_0.05 mm, then the expected membership function value.

In Eq. (13), if the values ofS = 25 kg/mm, S = 32 kg/mm, L = 2 "rr(4.0) = 25.12 mm are all fixed values. Through the FWA method computation is found the membership value of the guide bushing minimum outer radius as shown in Figure 9. The value of b with the maximum membership value in Figure 9 is 4.088, which means that the outer diameter 2b is 8.176 mm. From the design handbook, it is known that 2b must be larger than 8.2 mm. Therefore, the design parameter expected value has to be revised. The revision is to take the same off-peak value from the output parameter and design parameter. The b value in Figure 9 that meets the design handbook is 4.11 mm. Its mapping o~-cut is 0.7. The design parameters with a membership value of 0.7 in t, Th, and c are the revised ideal design values; that is, let t be 1.7

ram, Th be 17.9 ram, and a be 4.015 ram, which would be the design parameter values in Eq. (13). Another revision is to find the more influential toward output parameter from the uncertainty design parameters and take an preferred design value whose tx value is smaller than 0.7 as the design value.

For the three uncertainty design parameters t, Th, and a in Eq. (13), if the expected value of a = 1 is entered in two of the parameters and only one parameter remains a fuzzy input value, then three sets of output parameter values are obtained as shown in Figure 10. The figure shows that the influence (or degrees of importance) of design parameters a and t on Eq. (13) is far smaller than that of design parameter Th. Therefore, the less influential uncertain parameters can be processed with crisp value in the preliminary design, that is, take the design expected membership function value of et -- 1 as the certain design parameter value.

Uncertainty of Installation of Operation Stations for Progressive Die Design

The installation of an idle station is mostly for increasing the space of die installation in the operation station. In a progressive die operation, if a neighboring operation station has a larger die cavity, it is better to use an idle station to increase die space and at the same time relieve the strip's degree of stretching. If the processing involves pressing strip into die for bending operations, one may install an idle station to relieve and avoid overstretching part of the material. When plastic deformation results in

. . . . . . . . . b lmm)

3.95 3.98 4.00 4.03 4.05 4.06 4.10 4.13 4.15 4.18 4.20

1.00

o.go

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.00

.2

Figure 9 Membership Value of Minimum Outer Radius b

.2

1.00 7

0.904 , ~ a,T, tareuncea~d7

t I ~ oldy a is unce~dnly 0.80 ~ on~ nsuneer~ O{lly t is uncertainty 0.70d 0< \

°4°1 t 0.20--I

0.10-]

0.00 ] . . . . , . . . . . . . . . . . . . , 3.95 3.98 4.(20 4.03 4.05 4.08 4.10 4.13 4.15 4.18 4.20

Minimum b (ram)

Figure 10 Comparison Among Fuzziness of Design Parameters T, t, and a

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partial backup or flanges on strips during pressworking, one may also install an idle station to fiat- ten the deformed portion. The installation of the idle stations not only increases the number of operation stations but also adds to the complexity of the die structure. Therefore, when no concern of die strength arises, the number of operation stations should be cut to reduce die production cost and make maintenance easier. In this case, whether it is necessary to install idle stations becomes uncertain. The influential factors are concluded and given proper membership function as described below:

1. If the lower die has a larger die cavity, then it is necessary to install an idle station to increase die space. The design of the lower die is related to the feeding length of the strip layout. The deter- mination of feeding length in tm'n depends on the blank contour length (CO for selection of the appropriate interval. For example, if the blank contour length is between 75 and 150 ram, then the feeding length should be CI + 2.1 t * 2; if the blank contour length is between 150 and 250 mm, then the feeding length is Cl + 2.4 t * 2. In general, if the contour length of pressworking products is more than 150 mm, they are considered large parts. Their neighboring die cavity dis- tance is only 2.4 t * 2. Therefore, this study pre- supposes that when strip thickness is smaller than 2.0 mm and blank contour length is more than 150 ram, then one should consider installing an idle station; when blank contour length is more than 250 mm, then it is necessary to install an idle station3'439; when blank contour length is between 150 and 250 mm, then the decision is indicated with the linear membership function:

CI - 150 p - 150 < CI <_ 250 (14)

250-150

2. When strip undergoes bending operations, which results in overstretching of part of the material, one may install an idle station to relieve the situation and at the same time avoid the effects on the processing precision of the connecting operation stations due to excessive tension. The degree of stretching of part of the material gets larger as the bend angle increases. During deformation, the inside of the material is squeezed by compressive stress, and the outside stretches due

.

.

to tensile stress, which makes the material thickness gradually become thinner at the bend area and makes the neutral axis gradually move toward the inner surface. Therefore, the extent of thickness variation can indicate the strength of stretching on the material; that is to say that the ratio of the minimum bend radius can be used, under the condition that no cracking appears in the product bend area, and blank thickness can be used as an evaluation standard. For low-carbon steel, the ratio of the minimum bend radius to blank thickness Rmi, / t is 0.5 (see Ref 19); the ratio of blank thickness before and after deformation to / t is 0.4. But when R/ t is 5, to / t is 1, which indicates that blank thickness has not changed. Therefore, the R/t ratio between the boundary of near cracking (0.5) and no change in blank thickness (5) is defined as the membership function of the installation of idle stations3,4,~9:

5 - R / t I . t - - - 0 .5< R / t < 5 (15)

5-0.5

If product processing involves the operation station for cutting scratch, then the cutting scratch needs to be flattened before continuing on to the next station. Therefore, the necessity of installing an operation idle station for cutting scratch is obvious. This can be indicated by the crisp set, with 0 indicating no installation of the idle station and 1 indicating that the installation of the idle station is necessary. The installation of idle stations also directly adds to die production cost. Generally, die production cost of large pressworking parts takes up most of the entire production cost. Therefore, one also has to consider the factor of cost for the installation of idle stations. Its membership function cannot be represented by a linear membership function because of the variations in the designer's needs and viewpoint. It is thus determined by the designer's subjective linguistic fuzziness value. The fuzzy index is shown as below: 16

~t = 0.0-0.2 indicates a strong affirmative. Only the cost factor is considered. It is not necessary to install the idle station.

-- 0.2-0.4 indicates an affirmative. The cost factor is more important than the factor of the idle station installation.

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p =0.4-0.6 indicates an unknown or optional answer. The cost is as important as the idle station installation.

~t = 0.6-0.8 indicates a negative. The cost factor is less important than the need to install the idle station.

kt = 0.8-1.0 indicates a strong negative. The cost is not a valid consideration. The idle station must be installed.

The above four factors affecting the installation of an idle station are used to establish a neural network. The design of the network is described below.

There are four input vectors in the input layer and eight output vectors in the output layer. In general, the number of neurons in the first hidden layer is three to four times the number in the input vector. 14 There are 32 neurons in the hidden layer in this study. The membership functions of the characteristics and uncertainty for each input vector have been described in previous sections.

During the network's learning process, the network input vector uses, according to publications on dies, design handbooks, and the experiences of die design specialists, 0 or 1 as the input value. A total of eight sets of data are used as the patterns for network learning. For the design of the expected vector in the output layer, only one neuron is 1, and all the others are 0, which indicates the need to install an idle station. If all neurons are 0, the installation of the idle station is not necessary. The network completes learning when the network errors converge. At this time, the weight value is at the optimal value, that is to say that it has satisfied the weight value of eight patterns simultaneously.

System Operation Example The flowchart of the prototype progressive die

design expert system is shown in Figure 2. After the user has entered the system, the dialog between the user and the system is listed as below. First, the dialog of die design features and the inference of certainty data is conducted. Then the user enters three subsystems to conduct the inquiry and inference of the features of uncertainty data. Finally, the system infers the group technology graphic numbers of the required design die parts based on all of the input data. Then the graphic numbers are transferred and

display the shape of the die part through the drawing mechanism. The underlined portion is input by the user.

1. Run the fuzzy and neural networks progressive die design expert system

2. Display the rules base

3. Update the rules base

4. Exit the system

Please input your choice

1 (ENTER)

Choice 1 indicates system execution. Then the design characteristics inquiry follows, whose dialog is as follows:

Please input the quantity of the products

.... If you have any problem, please input

A. less than 18,000

B. between 18,000 and 24,000

C. between 24,000 and 32,000

D. more than 32,000

......... please input your choice

D (ENTER)

Please choose a kind of material

A. Cu

B. A1

C. Stainless steel

D. Low-carbon steel

......... Please input your choice

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D (ENTER)

The following results can be obtained after the system inference

(THE RESULT IS

RULES PLATES3 SAYS C 14203)

(THE RESULT IS

RULES PILOT3 SAYS C16103)

The above inference comes from the certainty design parameters. For the uncertain design parameters, the system will continue to converse with the user and let the user input the important characteristic parameters of strip thickness, tolerance value, and blank type as below:

Please input strip thickness

1.62 (ENTER)

Please input PRODUCT TOLERANCE

0.022 (ENTER)

Please input PUNCH DIAMETER

8.08 (ENTER)

Whether the blank has inner hole

A. Yes

B. No


A (ENTER)

Whether the blank needs shearing cut

A. Yes

B. No

......... Please input your choice?

A (ENTER)

The ideal design proposal is inferred after computation by the fuzzy cluster subsystem. The coding graphic numbers are shown below:

The uncertainty die parts coding No. is •

130004

101202

101206

117203

103201

Because the coefficients in the design equation are still undecided, the system asks the user to input the preferred design coefficient. The die clearance coefficient and single-edge clearance value are derived after the FWA subsystem operation, as shown below:

The uncertainty coefficients include

A. Die clearance coefficient (K)

B. Stripper coefficient (K)

C. Tube lower die coefficient (T, t, a)

.......... Please input your choice?

A (ENTER)

Please input your preference die clearance coefficient (0.008 - 0.01)

0.008 (ENTER)

The clearance/thickness ratio is - 0.040

The die clearance is - 0.065

After the design coefficients are decided, the uncertainty of the idle station installation is considered. The user inputs design parameters to serve as the input vector of the network. The conversation is shown below:

Please input maximum diagonal length of outside contour (mm)

220 (ENTER)

Please input the ratio of bending radius to strip thickness R/t

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5 (ENTER)

Whether the blank has cutting scratch design

A. Yes

B. No


B (ENTER)

The membership value derived from the input parameters is the network input value:

0.7 0 0 0.5

The output value derived from network computation is:

Result: 0 0 0 0 0 7 . 12e-006 0.91 0

The oL-cut value of 0.9 taken by the design from the matching of network's recall value and input pattern, according to the design experiences or neural networks' identifying feature the output value is:

0 0 0 0 0 0 1 0

After taking the a-cut, the output value is a-cut, which is the same with the seventh set data in the eight sets of input patterns, that is to say that one neuron is 1, indicating that the installation of idle station is required.

After the system completes the design, it enters the translation program to display a diagram on the screen from the graphic coding number of the inferred parts.

Conclusion In this study, part of the design details with uncer-

tainty in progressive die design are processed with fuzzy set theory and back-propagation neural networks.

In conclusion, the discussions in the previous four sections are summarized as follows:

1. The degree of membership of the uncertainty design parameters in die design is indicated with membership functions. This is different from the traditional binary logic of using 0 or 1 subjec- tively by the designer to indicate all the certainty and uncertainty design values, which thereby

minimizes the human factor in design and reduces design risk.

2. There is also the design uncertainty caused by a single factor, such as the case in which strip thickness may result in the uncertainty of the choice of die thickness under some circum- stances; and the flexible value of the clearance coefficient, which may cause uncertainty in clearance value. These problems are often processed with the fuzzy set theory, that is, using fuzzy cluster analysis to derive definite design attributes. The fuzzy weighted average method can test whether the expected value given sub- jectively by the designer in preliminary design satisfies the requirements of design regulations, and the method can revise the design when it fails to satisfy such requirements.

3. There is also uncertainty caused by multiple factors such as the installation of an idle station, which has to consider four design factors: size of lower die cavity, partial overstretching of the material, cutting scratch, and cost factor. Because the neural network is suitable for the identifying task that must satisfy a large amount of constraints, related factors are established in the form of a matrix and input into the network, which learns about the design patterns. When a set of design parameters is input into the network, it can immediately respond from the patterns learned previously, match these patterns with the output value, and let the designer give an oL-cut value from experiences to determine whether to install an idle station or not.

4. The uncertainty design parameters in die design are integrated by an expert system to improve the past disadvantage of deriving inference only from definite rules. Design problems of certainty as well as uncertainty can then be processed all at once.

References 1. K. Lang, Handbook of Metal Forming (New York: Mc-Graw-Hill

Book Co., 1985). 2. D.E. Ostergaard, Basic Die Making (National Tool, Die & Precision

Machining Association, 1975). 3. J.R. Paguin, Die Design Fundamentals (Reading, MA: Addison-

Wesley, 1984). 4. W.W. Frank, ed., Die Design Handbook, 2nd ed. (ASTME Books,

published by McGraw-Hill, 1972). 5. Zone-Ching Lin, Chaug-Liang Hsu, and Kau-Hsiong Yao, "Planning

and Building an Integrated Design Software of Blanking and Piercing Dies (ESSCP) with a Micro Expert System as a Main Structure," Journal of the Chinese Society of Mechanical Engineers (vl0, n2, 1989), ppl01-120.

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6. Zone-Ching Lin and Chi-Hui Huang, "An Investigation of an Expert System Employing Chinese for Deep-Drawing of a Press Die Design," Journal of the Chinese Society of Mechanical Engineers (v14, n2, 1993), pp63-77.

7. Wen-Sue Hwang and Gau-Rong Liang, "The Design of AHP Conveyors Selection Expert System with Uncertainty Factor," Proceedings of the 3rd National Workshop on Automation Technology (1990).

8. R.K. Seller, "Reasoning About Uncertainty in Certain Expert System: Implications for Project Management Applications," Project Management (v8, nl, Feb. 1990), pp51-59.

9. I.N. Tansel and C. Mclaughlin, "Detection of Tool Breakage in Milling Operations pi: The Neural Network Approach," International Journal of Machine Tools & Manufacture (v33, n4, 1994), pp545-558. 10. B.S. Hyun and H.S. Cho, "Prediction of Forming Pressure Curve for Hydroforming Process Using Artificial Neural Network," Institute of Mechanical Engineers Proceedings, Part I--Journal of Systems & Control Engineering (v208, n12, 1994), pp109-121. 11. X.D. Fang and I.S. Jawahir, "Predicting Total Machining Performance in Finish Turning Using Integrated Fuzzy-Set Models of the Machinability Parameters," International Journal of Production Research (v32, n4, Apr. 1994), pp833-849. 12. X. Haiping and H.P. Wang, "Part Family Formation for GT Applications Based on Fuzzy Mathematics," International Journal of Production Research (v27, n9, 1989), pp1637-1651.

13. W.M. Dong and ES. Wong, "Fuzzy Weighted Averages and Implementation of the Extension Principle," Fuzzy Sets and Systems (v21, 1987), pp183-199. 14. L.A. Zadeh, "Fuzzy Set," Information and Control (v8, 1965), pp338- 353. 15. L.A. Zadeh, "Outline of a New Approach to the Analysis of Complex System and Decision Process," IEEE Transactions on Systems, Man and Cybernetics (vSMC-3, 1973), pp28-44. 16. J.K. George and A.E Tina, Fuzzy Sets, Uncertain~, and Information (Englewood Cliffs, N J: Prentice-Hall, 1988). 17. A. Lapedes and R. Farber, How Neural Networks, Biological Cybernetics (1988). 18. Neural Work Explorer, Neuralware, Inc. (1989). 19. Y-J. Da, Press Working and Die Design (Shin Luh Book Co., 1985).

Authors' Biographies Zone-Ching Lin received his PhD from Purdue University and is a pro-

fessor in the Department of Mechanical Engineering at the National Taiwan Institute of Technology. His research interests are in metalcutting and forming, intelligent manufacturing, and computer-integated manufacturing.

Ho Chang received his master's degree in 1991 from the Department of Mechanical Engineering at the National Taiwan Institute of Technology. He is a lecturer in that department and specializes in manufacturing processes and application of fuzzy theory.

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