Design for manufacturing: a dimensioning aspect

Journal of Materials Processing Technology 91 (1999) 121 – 127

Design for manufacturing: a dimensioning aspect P. Ji *, K.H. Lau Department of Manufacturing Engineering, The Hong Kong Polytechnic Uni6ersity, Hung Hom, Kowloon, Hong Kong Received 23 December 1997

Abstract This paper discusses the dimensioning problem in Design for Manufacturing (DFM) and Concurrent Engineering (CE). In a DFM/CE environment, a product and it’s process plan is initially designed, then finalized. In the final stage of the product’s design, a dimension of the product may be modified, which requires the manufacturing side to change the corresponding dimension without delay: consequently with the information on the manufacturing side, the product designer can make a correct decision quickly. This paper presents a mathematical approach to tackle this dimensioning problem in DFM/CE. Two cases are discussed: one is the size change of a design dimension, whilst the other is the datum surface change of a design dimension. The corresponding actions in manufacturing dimensions for the two cases are analyzed in detail in this paper. The approach is very simple and fast, and the product designer can view the response of the manufacturing side almost simultaneously when a design dimension is changed. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Design for manufacturing; Dimensioning; Process planning; Tolerance charting

1. Introduction

2. Background

In a design for manufacturing (DFM) and concurrent engineering (CE) environment, an initial design for a product is created, then it is finalized. To cope with this product-design practice in the DFM/CE environment, the process plan for the product in the manufacturing side, i.e. the procedure to make the product, is proposed to be designed in two phases [1]: in the first phase, a rough process sequence is initialized, and in the second phase, the process sequence is finalized and a complete process plan is generated. In the final stage of product design, a dimension (its size or its datum surface) of the product is often revised. The situation that is encountered now is: once a dimension of the product is changed, the dimensions in the manufacturing side, i.e. in the process plan, have to be changed without delay, so the product designer can make a correct decision quickly. This paper will extend the digraphic method [2] for tolerance charting to solve this practical DEM/CE problem.

The procedure of a detailed mechanical product design involves the specification of the structure, the dimensions and the tolerances. The structure or the form of the product is decided from its functional performance; the tolerances are specified according to it’s performance and the relationship with other components, i.e. the mating fits; and its dimensions or geometry are determined from its strength requirements, etc. In the final stage of the product design, the structure is seldom modified because the functional requirements are already clarified, but the dimensions may be changed, particularly in a DFM/CE environment, by use of the parametric technique. Such a design dimensional change may modify the size (the nominal value), or alter the datum surface of a dimension. In the prototype system that the authors are developing for process planning in a DFM/CE environment, the Pro/Engineer package is used for product-design purposes. In order to cope with the manufacturing dimensions in a process plan, the tolerance-charting technique [3–5] has been computerized. The models and methods we used for computerizing tolerance charting can be found elsewhere [2,6]. Some results

* Corresponding author. Tel.: +852-2766-6596; fax: +852-23625267.

0924-0136/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 9 8 ) 0 0 4 2 9 - 4

122

P. Ji, K.H. Lau / Journal of Materials Processing Technology 91 (1999) 121–127

from other researchers are also available [7 – 10]. A state-of-the-art review on tolerance charting was reported recently [11]. A simple product, shown in Fig. 1(a), will be used to illustrate the DFM/CE dimensioning problem. Fig. 1(b) is a process plan to make the product. All working dimensions (Xs) in the process plan are unknown. Making use of the tolerance-charting technique, the solution for these unknowns are shown in Fig. 2, i.e. X1=0.979, X2 = 1.994, X3 =3.003, X4 = 4.031, X5 = 1.008, X6=X7 =X8 =1.000, X9 =4.00. In the final

design stage, if the dimension Y3 is changed to 3.2, i.e. DY3= 0.2, then the product designer would like to know which manufacturing dimensions will be affected by this design change. Obviously, the tolerance chart can be re-calculated in order to obtain the solution for the problem, which was the way used in the past. Another problem is if the datum surface of the design dimension Y3 is changed to another surface, what will happen to the manufacturing dimensions in the process plan? DFM/CE requires the manufacturing side to make an appropriate response to these design changes

Fig. 1. (a) Showing a product and (b) it’s process plan.

P. Ji, K.H. Lau / Journal of Materials Processing Technology 91 (1999) 121–127

123

Fig. 2. The tolerance chart for the process plan.

quickly. Re-calculating a tolerance chart obviously takes a long time and is not suitable in the DFM/CE environment. This paper presents a new approach to solve the problem. The approach extends the digraphic method [2] for tolerance charting, uses the dimensional relationship between design and manufacturing to analyze the design dimensional changes, and provides a fast manufacturing solution to the changes.

3. The digraphic method and the reverse dimensional chains In order to handle the dimensioning problem in DFM/CE, the dimensional relationship between design and manufacturing should be constructed. The digraphic method [2] can be applied easily for this purpose. The key concepts in the digraphic method are the three directed trees, and the reverse and forward dimen-

sional chains. A detailed description of the digraphic method is available [2]. In general, if a surface of a product is considered as a node (vertex) in graph theory, and a dimension or stock removal as an arc, three digraphic trees can be generated from a tolerance chart [2]. The three trees are called the blueprint dimension tree, the blueprint dimension and stock removal tree (the Y tree), and the working dimension tree (the X tree), respectively. Figs. 3–5 showing the three directed trees for the example product, respectively. The direction of an arc in these trees is in accordance with the positive direction of the x-axis in the tolerance chart (Fig. 2). By use of these trees, a reverse dimensional chain is introduced as follows: if an arc (an X) in the X tree is ‘added’ to the Y tree, then a unique loop can be obtained, this unique loop being called a reverse dimensional chain. Furthermore, according to the digraphic method, a reverse dimensional chain can be

P. Ji, K.H. Lau / Journal of Materials Processing Technology 91 (1999) 121–127

124

Fig. 3. The blueprint dimension tree

represented as a linear mathematical equation. Since there are m arcs (working dimensions) in the X trees (for example, nine arcs in Fig. 5), m reverse dimensional chains can be constructed. The nine reverse dimensional chains for this example are as follows: X1 =Y1 − Y8+Y9 X2 =Y2 − Y7+Y9 X3 =Y3 − Y6+Y9 X4 =4 + Y5+Y9 X5 =Y3 + Y4+Y6

Fig. 5. The working dimension tree (the X tree).

X6 =Y3 + Y4 X7 = −Y2 + Y3 X8 = −Y1 + Y2 X9 =Y4

(1)

Similarly, the m forward dimensional chains (in general, they are just referred to as dimensional chains, and

shown in a tolerance chart, as illustrated in Fig. 2) can also be obtained if an arc (a Y) in the Y tree is ‘added’ to the X tree. The nine forward dimensional chains for this example are as follows: Y1= − X6−X7− X8+ X9 Y2= − X6− X7+ X9 Y3= − X6+X9 Y4= X9 Y5= − X3+ X4− X5 Y6= X5− X6 Y7= − X2+X3+ X5− X6− X7 Y8= − X1+X3+ X5− X6− X7− X8 Y9= X3+ X5− X9

(2)

4. The structure matrix and invariance

Fig. 4. The blueprint dimension and stock removal tree (the Y tree).

Eq. (1) for the reverse dimensional chains can be rewritten in a matrix format, which is:

P. Ji, K.H. Lau / Journal of Materials Processing Technology 91 (1999) 121–127

0 0 0 0 0 0 −1 1ÇÆY1Ç Æ X1ÇÆ 1 Ã ÃÃ ÃÃ Ã X2 0 1 0 0 0 0 −1 0 1 Y2 Ã ÃÃ ÃÃ Ã X3 0 0 1 0 0 −1 0 0 1 Y3 Ã ÃÃ ÃÃ Ã X4 0 0 0 1 1 0 0 0 1 Y4 Ã ÃÃ ÃÃ Ã X5 = 0 0 −1 1 0 1 0 0 0 Y5 Ã ÃÃ ÃÃ Ã X6 0 0 −1 1 0 0 0 0 0 Y6 Ã ÃÃ ÃÃ Ã X7 0 −1 1 0 0 0 0 0 0 Y7 Ã ÃÃ ÃÃ Ã X8 −1 1 0 0 0 0 0 0 0 Y8 Ã ÃÃ ÃÃ Ã X9 0 0 0 1 0 0 0 0 0 Y9 Ã Ã È Ã É È Ã ÉÃ È ÉÃ

In general, the above equation set is: X =QY

(3)

where:

125

has a relationship with the forward structure matrix P: Q= P − 1

(5)

From Eq. (3), it is known that the element values in Q are determined by the coefficients, irrespective of the values of either Xi (i= 1, 2,..., m) or Yj ( j=1, 2,..., m). The forward structure matrix P also has this invariance property. In other words, both Q and P are determined by the structure of the product and the process plan to make the product. If both the structure of the product and the process plan are fixed, then Q and P are invariant to the design dimensions and stock removals, Yj ( j=1, 2,..., m), and the working dimensions, Xi (i= 1, 2,..., m).

X = (X1, X2, ..., Xm )T Y = (Y1, Y2, ... Ym )T

5. Manufacturing’s response to design dimension change

Q =[qij ],

The invariance property of the above structure matrix Q or P can be applied to the area of DFM/CE since they represents the dimensional relationship between design and manufacturing mathematically. The first application, the size change of a design dimension, will be discussed in the next sub-section.

is the m ×m coefficient matrix, and

qij = Á1 Í −1 Ä0

if Yj is an increasing constituent link of Xi ; if Yj is a deceasing constituent link of Xi ; 0 otherwise.

i, j= 1, 2, ..., m

5.1. Size change of a design dimension

Similarly, Eq. (2) in a matrix format is: ÁY1ÂÁ0 0 0 ÃY2ÃÃ0 0 0 ÃY3ÃÃ0 0 0 ÃY4ÃÃ0 0 0 ÍY5Ì=Í0 0 −1 ÃY6ÃÃ0 0 0 ÃY7ÃÃ0 −1 1 ÃY8ÃÃ −1 0 1 ÄY9ÅÄ0 0 1

0 0 0 0 1 0 0 0 0

0 0 0 0 −1 1 1 1 1

−1 −1 −1 0 0 −1 −1 −1 0

−1 −1 0 0 0 0 −1 −1 0

−1 0 0 0 0 0 0 −1 0

1 ÂÁX1Â 1 ÃÃX2Ã 1 ÃÃX3Ã 1 ÃÃX4Ã 0 ÌÍX5Ì 0 ÃÃX6Ã 0 ÃÃX7Ã 0 ÃÃX8Ã − 1ÅÄX9Å

i.e. Y = PX

In the DFM/CE environment, when a product is initially designed, the process plan to make the product will be determined. Thus, the working dimensions in the process plan can be obtained by means of Eq. (3). In the final stage of design, some sizes of design dimensions in the product will be modified. This can be implemented easily by a computer aided design (CAD) packages, such as Pro/Engineer. If Y. and X. represent the new sizes for the design dimensions Y and working dimensions X, respectively, and the changes between previous and current design dimensions are Dy, i.e. Dy = Y. −Y, then according to Eq. (3), we have: X. = QY. = QY+QDy = X+QDy

where: P= [pij ], is the m × m coefficient matrix, and pi, j = Á1 Í−1 Ä0

if Xj is an increasing constituent link of Yi ; Xj is a deceasing constituent link of Yi ; otherwise.

i, j= 1, 2, ..., m The coefficient matrix Q is determined by the structure (the nodes and the relationships between the nodes, i.e. the arcs) of the Y tree. Thus, the coefficient matrix Q can be called a structure matrix, or a reverse structure matrix, in order for it to be distinguished from another coefficient matrix P, the forward structure matrix. Obviously, the reverse structure matrix Q

Fig. 6. Datum surface change

126

P. Ji, K.H. Lau / Journal of Materials Processing Technology 91 (1999) 121–127

X. 5 =X5− DY3=1.008 −0.2 =0.808

Y. 3 (in general, Y. j ) is the closed link while Y1 is a decreasing constituent link, and Y3 is an increasing constituent link. In general, Yj is always an increasing constituent link because Y. j is always the closed link and Y. j is going to replace Yj. In general, if there are n design (blueprint) dimensions in a product, Eq. (7) can be represented as follows:

X. 6 =X6− DY3=1.000 −0.2 =0.800

Y. = Yj + CY

X. 7 =X7+ DY3=1.000 +0.2 =1.200

where C= [ck ], is a 1× m row vector, and

The above calculation is very simple if only one design dimension is changed. For example, if DY3 = 0.2, i.e. Y3 is changed from 3.00 to 3.20 in the example product, then from the structure matrix Q, we can obtain: X. 3 =X3+ DY3=3.003 +0.2 =3.203

(8)

Á1 if Yk is an increasing constituent link of Y. j ; ck = Í −1 if Yk is a deceasing constituent link of Y. j ; Ä0 otherwise All other Xs are not affected by this design alteration. Since Y3 is changed, so the above X values are obtained simply by adding or subtracting DY3, depending on the values of the third column in Q. In general, if only one dimension Yj is changed by DYj (which can be negative), then the following equation can be obtained: X. = X+Qj DYj

(6)

where Qj is Q’s jth column vector. In practice, the design dimensions can be modified one by one. By using Fig. 6, once a design dimension is changed, the corresponding working dimensions in manufacturing are modified almost simultaneously in this prototype system.

5.2. Datum surface change of a design dimension If the datum surface of a design dimension is changed, the problem becomes more complicated. Suppose the datum surface of the design dimension Y3 is changed from current surface E to D in the blueprint, i.e. from E0 to D0 in the tolerance chart, and the new dimension is denoted as Y. 3. Since Y. 3 is a design dimension, it can be ‘added’ to the blueprint dimensional tree as illustrated in Fig. 6, and a unique loop in the graph can then be obtained, which can be described by a mathematical equation as follows:

k= 1, 2...n, k"j k= 1,2...n, k"j k= 1,2...m

Obviously, the jth element in C is zero since Yj will be replaced by Y. j, and all elements from n+ 1 to m are also zero since they are stock removals. In this example, C= (− 1, 0, 0, 0, 0, 0, 0, 0, 0). Eq. (7) can also be rewritten as: Y3= Y. 3+Y1

(9)

because Y3 will be replaced by Y. 3. Take Eq. (9) into Eq. (1), then the following equation set can be obtained: X1= Y1−Y8+ Y9 X2= − Y2+ Y7+ Y9 X3= − Y. 3+ Y6− Y9

+ Y1

X4= − Y4+ Y5+ Y9 X5= Y. 3+Y4+ Y6

−Y1

X6=Y. 3− Y4

−Y1

X7= −Y2− Y. 3

+ Y1

X8= − Y1− Y2 X9= Y4

(10)

Let Y. represent the new values for Y, and for example, Y. = (Y1, Y2, Y. 3, Y4,…., Y9)T in this product, and D=[dk ]T, an m× 1 column vector, and dk =

!

1 if k= j 0 otherwise

Y. 3+Y1 − Y3=0

k= 1,2…m

or:

In this example, D= (0, 0, 1, 0, 0, 0, 0, 0, 0)T. Then Eq. (10) can be written in a matrix format:

Y. 3=Y3 − Y1

(7)

Eq. (7) can be very complicated if the structure of the product is complex, such as a jet engine shaft as in the Appendix (see Figure A-1), so a blueprint dimensional tree is necessary to obtain the above mathematical equation. Furthermore, Eq. (7) can be considered as a dimensional chain (a design dimensional chain), then

X= QY. − QDCY. =QY. − Qj CY. =Q. Y.

(11)

where: Q. = Q− QDC=Q− Qj C

(12)

Here, Qj is Q’s jth column vector. Thus, Q. can be simply updated as follows: if Yk (k"j ) is a decreasing

P. Ji, K.H. Lau / Journal of Materials Processing Technology 91 (1999) 121–127

constituent link in the design dimensional chain (Eq. (7)) of Y. j, then all elements in the kth column of Q should be added by the jth column; and if Yk (k " j ) is an increasing link, then all elements in the kth column should be subtracted by the jth column. On the other hand, the forward structure matrix P is required to be updated from this datum surface change due to the tolerance control reason [2]. Obviously, P can be updated by use of Eq. (5), i.e. the inverse relationship between the reverse and the forward structure matrix. However this requires the calculation of the inverse of a matrix. For a complex product, such as that in Appendix A (Figure A-1), it will take some time to do. If Eq. (8) is rewritten in matrix format, then Y. = Y+ DCY= PX +DCPX = P. X

127

Appendix A

(13)

so: P. = P+ DCP

(14)

As a matter of fact, P is updated by the following rule: if Yk (k"j ) is a decreasing constituent link in the design dimensional chain (Eq. (7)) of Y. j, then all elements in the jth row of P should be subtracted by the kth row; and if Yk (k "j ) is an increasing link, then all elements in the jth row should be added by the kth row. It is noted that only the first n columns in the reverse structure matrix Q are affected by a datum surface change of the design dimension Yj (1 5 j5 n), and only the jth row in the forward structure matrix P is updated by the datum surface change. This phenomenon can be called a semi-invariance of the structure matrices.

6. Conclusions This paper presented a dimensioning problem in DFM/CE. By using the digraphic method and the reverse dimensional chains, the manufacturing side can quickly take an action in responce to a design dimensional change during the product-design phase. Whatever the size or datum surface change of a design dimension in a product, all manufacturing dimensions can respond quickly by the use of Eqs. (6) and (11), so that a product designer can view the manufacturing result of his or her design change almost simultaneously. Furthermore, in the case of a datum surface for a product dimension, the relevant matrices P and Q, which represent the dimensional relationship between design and manufacturing, can be updated quickly by means of Eqs. (10) and (14). The approach has been implemented in a parametric design environment (Pro/ Engineer), and is very helpful to the product designer.

.

Fig A-1: a jet engine shaft (simplified) and its blueprint dimensional tree.

References [1] R.S. Ahluwalia, P. Ji, Process planning in concurrent engineering environment, in: The International Industrial Engineering Conference Proceedings, San Francisco, May 20 – 23, 1990, pp. 535– 540. [2] P. Ji, J.Y.H. Fuh, R.S. Ahluwalia, A digraphic approach for dimensional chain identification in design and manufacturing, ASME Trans. J. Manuf. Sci. Eng. 118 (1996) 539 – 544. [3] P. Ji, Computer aided operational dimensions calculation, Masters thesis, Beijing University of Aeronautics and Astronautics, P.R. China, 1984. [4] O.R. Wade, Tolerance Control in Design and Manufacturing, Industrial Press, New York, 1967. [5] O.R. Wade, Tolerance control, in: T.J. Drozda, C. Wick (Eds.), Tool and Manufacturing Engineers Handbook vol. 1, The Society of Manufacturing Engineers, Dearborn, Michigan, 1983, pp. 2:1 – 2:60. [6] P. Ji, A tree approach for tolerance charting, Int. J. Prod. Res. 31 (1993) 1023 – 1033. [7] J.R. He, G.C.I. Lin, Computerised trace method for establishing equations for dimensions and tolerance in design and manufacture, Int. J. Adv. Manuf. Technol. 7 (1992) 210 – 217. [8] S.A. Irani, R.O. Mittal, E.A. Lehtihet, Tolerance chart optimization, Int. J. Prod. Res. 27 (1989) 1531 – 1552. [9] B.K.A. Ngoi, S.L. Fang, Computer-aided tolerancing charting, Int. J. Prod. Res. 32 (1994) 1939 – 1954. [10] T. Xiaoqing, B.J. Davies, Computer aided dimensional planning, Int. J. Prod. Res. 26 (1988) 283 – 297. [11] B.K.A. Ngoi, Y.C. Kuan, Tolerance charting: the state-ofthe-art review, Int. J. Comput. Appl. Technol. 8 (1995) 229– 242.