Research in Engineering Design (1996) 8:207-216 © 1996Springer-Verlag London Limited

Research in

El gineering es gn

Upper and Lower Limits for 'The Principles of Design' Stephan Rudolph Institute of Statics and Dynamicsof Aerospace Structures, Stuttgart University,Stuttgart, Germany

Abstract. Until now, the independence axiom and the information axiom stated in axiomatic design are based on inductive conclusions. These inductive conclusions are justified by design case experience and a claim of the non-existence of any known counter-example. For the independence axiom, a theoretical framework and proof are provided under the assumption that the recently stated evaluation hypothesis applies. This evaluation hypothesis postulates that 'any minimal description in the sense of the Pi-theorem is an evaluation', thus using the similarity theory of physics to establish an evaluation model for the purpose of engineering design analysis. For the information axiom a proof is also provided which leads to a different interpretation. The fact that engineering design case experiences coincide with proofs which naturally result from the evaluation hypothesis provides another piece of evidence for its validity. In this respect, the evaluation hypothesis appears to be a further step towards the establishment of a physically justifiable future science base for engineering design purposes.

Keywords. Axiomatic design; Design evaluation; Evaluation hypothesis; Pi-theorem.

1. Introduction

The book on the axiomatic design approach by Suh [I] has been recently commented on and been put into a general design perspective by others (see Mistree [2]). In addition to that, the focus in this work is on the findings when the recently stated evaluation hypothesis (see Rudolph [3, 4]) based on dimensional analysis is applied to the axiomatic design approach. However by doing so, it is not intended in this paper to share any of the beliefs or disbeliefs concerning the general and unlimited validity claimed by the axiomatic design approach. In this respect, the Correspondence and offprint requests to; S. Rudolph, Institute of Statics and Dynamics of Aerospace Structures, Stuttgart University,Pfaffenwaldring27,D-70550Stuttgart, Germany;e-mail: [email protected].

interpretation of all results presented hereafter is strictly limited to the range of validity of dimensional analysis which is the area of physics. Dimensional analysis has already been applied to engineering design problems in the past [5-7], where it was used to ease the conceptual modeling and helped to gain a deeper understanding of the functional behaviour of the design object. Other works using dimensional analysis as a basis for the technique of qualitative reasoning [8, 9] about design objects have originated from the field of artificial intelligence and have once more underlined the usefulness and versatility of this technique. The main emphasis of this work is to show how the assumption of the evaluation hypothesis, which is based on the technique and the notation of dimensional analysis, relates to 'the principles of design' [1], i.e. the mapping between the design parameters and the functional requirements of the design. To do so, it is first briefly described why dimensional analysis solves the design evaluation problem [3, 4] of the technical aspects of engineering design objects. The theoretical analysis of the principles of axiomatic design is considered to be important, since these principles serve as a guideline for creating a simplified model of the more complex process of design synthesis and analysis. Since design synthesis and analysis themselves are not yet fully understood in all their consequences, it is necessary to investigate how design methods and methodologies proposed as such guidelines in the design process behave and can be validated in areas of 'precise' knowledge, as is done in some areas of physics. Since physics is a part of every design effort, all design methodologies should include physics at least as a special case, and are not allowed to violate its laws. By using this type of reasoning, one might obtain a better insight and understanding of the advantages and drawbacks of a specific design method or methodology in known and already well established areas of physics before trying to extend or to generalize it to other areas outside of physics.

208

It is therefore the purpose of this work to analyse some of the basic properties of the design axioms when applying them to an area of precise knowledge and to show how they relate and conform with 'exact' physics. This is done by the use of dimensional analysis which represents the mathematical foundation of the recently stated evaluation hypothesis (see Rudolph [3, 4]). Once more it is pointed out that dimensional analysis is limited to physics and therefore all of the following is limited to the area of physics. No attempt is made to extend what is going to be stated beyond this original range of validity. The practitioner might therefore find the following, somewhat theoretical considerations, not very helpful in solving his practical design problems right away. However, the hopefully successful establishment of a future science-base for engineering design as envisaged and sought after by Suh [1] among others requires the embedding of any future design methodology in a sound theoretical framework. It is hoped that the work presented here helps to clarify some of the important issues of such a theoretically sound basis of a future science-base for engineering design purposes. The outline of the paper is as follows: this section gives a general overview and definition of terminology used; the second section describes the basic mathematical framework of differential error analysis [10] and contains a short introduction to the evaluation hypothesis [3, 4]. It also reviews the foundation of the axiomatic design approach [1] in the form of its two basic axioms. The third section presents the derivation of the two proofs as the lower and upper limits of the independence and the information axioms. This section shows that axiomatic design is a special case of an entire class of behaviours dictated by the laws of physics, for which the upper and lower limits are presented. As will be shown the axiomatic design approach is identical to the presented lower limit. The fourth section briefly reflects a published design example of the Mit Rim moulding machine [1, 11], in which the close relationship of both the approaches of axiomatic design and the evaluation hypothesis are demonstrated. The paper then concludes with a brief summary. 1.1. Definition of Terms

In later parts of the paper, the terms dimensional homogeneity, dimensionally homogeneous function equation, dimension and dimensionless product will be of major importance for the understanding of the

S. R u d o l p h

subsequent derivations using the technique of dimensional analysis [3, 4, 12]. The term dimensional homogeneity simply means that in any physical function equation, the functional relationship of the function parameters xi, x i applies to the physical dimensions of the parameters (usually expressed in SI units)just like it does to their numerical values (e.g. from F -- ma follows IN] = [kg]-[m/s z] and for exemplary numerical values 6 = 2-3). The principle of dimensional homogeneity guarantees that in every possible and correct physical equation the dimensions on the left hand side of the equal sign are identical to those on the right hand side of the equal sign. The general validity of this principle could be epistemologically justified with the commonly known statement that one cannot compare apples with oranges, which would be the case otherwise. All function equations in physics therefore belong to the class of

dimensionally homogeneous function equations. Therefore the term dimension sometimes has different or even multiple meanings in mathematics and physics. A vector x = {x 1, x2, x3} T is called a three-dimensional vector, since it has three components. If however, the vector components are the three physical variables of the previous example (i.e. x = {m, a, F}T), each of the components additionally has physical dimensions. The term dimensionless product stands for a special class of monomial expressions in the form X j I ~ n= 1 x'-'Ji which have no physical dimensions (i.e. are dimensionless) and are formed out of a (sub)set of physical variables x i, xj. These physical variables x~, xj are again elements of the set of functional parameters in an existing dimensionally homogeneous function equation f ( x ~ , . . . , x,) = 0. An example of a dimensionless product is derived in the Appendix.

2. M a t h e m a t i c a l

Framework

In the later sections, the properties of the real-valued continuously differentiable vector function q~: x ~ y with y = ~o(x) (1) will be referred to. It maps the n-dimensional space x s R~ onto the m-dimensional space y e ~'~. The corresponding vector notations are x = {xl . . . . . x,} T

n ~ N+

(2)

y = { y l . . . . . y~}T

men +

(3)

When evaluating the function y = p(x) in a small neighborhood ~ of some point x, the absolute error

Upper and Lower Limits for 'The Principles of Design'

209

in each vector component can be defined as Axi = xi - xi

i = (1 . . . . . n)

while neglecting higher order terms leads to &0j(x)

(4) i=1

Ayj = ~0j(~) - q~j(x)

j = ( 1 , . . . , m)

xi

~0i(x) ex'

Oxl

(5)

Expanding the function q~ into a Taylor-series at the point x and neglecting second and all higher order terms for each component Ay~ leads to the approximation

= ~ 2j~x~

j = (1 . . . . . m)

(10)

i=l

This approximation result of Eq. (10) can be written by introducing the following definition of the matrix A. Again it is important to note that the

A y j = ~oj(~) - ~oj(x)

er=

2 i=1

~xi

aq~j(x) A x i i= 1

g ,.

j

(1,..

m)

(6)

~xi

This approximation result of Eq. (6) can be written by introducing the following definition of the operator matrix D(x). It is important to note that the operator

Ay=

~(]9I(X) c3xa

~(Pl(X)"

&Ore(X)

~q,.,(x) ~, Ax.J

" Aym)

,~

Axl

I

c3x~ = D(x) Ax

(7)

matrix D(x) has ( m x n) elements. D(x) is generally of a rectangular form, since it depends on the dimensional properties of the spaces R~ and N% only.

2.1. Differential Error Analysis Differential error analysis [10] makes extensive use of Eq. (7) to analyse the sensitivity of algorithms to disturbances due to numerical calculations involving finite precision. Finite numerical precision in computation means that the originally symbolic value x~ of each vector component has to be represented numerically by a real and thus finite machine word xi, with Xi ~ Xi" According to the absolute errors in Eqs (4) and (5), the relative error e~,, and %, of each component can be defined as e~ = (2i - xi)/xi

i = ( 1 , . . . , n)

(8)

eyj = (cpi(~) - ~oj(x))/~pj(x)

j = (1 . . . . . m)

(9)

When evaluating the function y = ~o(x) at some point x and expanding it once again into a Taylor-series

" i'm1

" "'"

J" n

-- Ae x

(11)

k ex,J

matrix A has ( m x n) elements. Again A is generally of a rectangular form, since this depends on the dimensional properties of the spaces N~ and N% only. In the chapter on differential error analysis which is partially reproduced and adapted here for the needs of the later sections, Stoer [10] stresses that Eq. (11) in comparison to Eq. (7) possesses some important advantages. Due to the normalization, the factors 2j~ (also called conditional numbers) are independent of the scaling of x and y and can therefore be interpreted as size-independent measures of the mapping properties. If the values of the factors 2~i are large, the problem is said to be ill-conditioned. In that case, relatively small relative errors e~ in the input will cause large relative errors ey in the output.

2.2. Evaluation Hypothesis During the design synthesis, the designer creates a more and more detailed description of the design object, i.e. the description graph [3, 4] as shown on the right hand side of Fig. 1. Then, during design analysis, the designer analyses the current state of his design by mapping the design object's descriptions onto his goal hierarchy, i.e. the evaluation graph [3, 4] on the left hand side of Fig. 1. This mapping is achieved with transfer functions [3, 4]. This theoretical concept of design evaluation is shown in Fig. 1 and discussed in detail in Refs 3, 4 and 13. The involvement of a human being leads to the central question of to what extent an evaluation reflects also subjective components, such as social and personal beliefs, the cultural background, etc. However, in order to achieve a repeatable and objective evaluation, one should be able to separate the 'objective' part of the design knowledge from the 'subjective' beliefs of the designer and formalize it in some way. This means that an objective evaluation should be a unique property o f the design object only.

210

S. Rudolph

evaluation I J~""-., evaluation

1

I t va' ati°°l "

mapping

The motivation of this idea in creating an objective evaluation method and investigating its principal feasibility and limitations has led to the establishment of the evaluation hypothesis presented in the following section. The evaluation hypothesis states that 'any minimal description in the sense of the Pi-theorem is an evaluation'. This result is obtained through a rigorous analysis of the evaluation problem using the following epistemological reasoning (see Refs 3, 4 and 13 for more details):

Fig. 1. Description, mapping and evaluation.

an equation F of only m < n dimensionless quantities n i can be shown f ( x 1. . . . . x.) = 0

(12)

F(Tzl. . . . . xm) = 0

(13)

where r = n - m is the rank of the dimensional matrix constructed by the x i and with dimensionless quantities (dimensionless products or parameters) nj of the form

rcj = x~ f i xi-~j'

(t4)

/=1

• a reproducible and objective evaluation can only exist if it is based on and derived from some type of law which has to be dimensionally homogeneous, • under the assumption that such an objective evaluation procedure exists, it must not depend on the arbitrarily chosen definitions of physical units and therefore has to be dimensionless, • an evaluation method should turn into exact physics and should be consistent over all hierarchical evaluation levels in the case of complete physical knowledge about a certain design object, and • an evaluation method should consist of a minimal set of evaluation criteria only. If not, some partial evaluation criteria will distort the overall evaluation result when taken multiply into account. Once the above five emphasized epistemological requirements are accepted as meaningful assumptions, it can then be shown that a universal method exists to construct the required dimensionless evaluation quantities from dimensionally homogeneous function equations by means of the Pi-theorem. This means that the above hypothesis is restricted to those areas of physics where such descriptions in the appropriate implicit mathematical form of a functional relationship f of physical variables exist and can be stated (see Refs 3 and 4 for more details)• In the following, the Pi-theorem will be briefly introduced• The Pi-theorem [I2] guarantees, that from the existence of a dimensionally homogeneous and complete equation f of n physical quantities xi, the existence of

with j = 1,.. •, m e N and the eji e ~ as constants. The definition of the dimensional matrix is given on the left part of Fig. 2 and an example is shown in Fig. A. 1 in the Appendix. The right part of Fig. 2 shows the straightforward construction of the dimensionless products nj according to Eq. (14) through the determination of the ~i by means of rank preserving matrix operations. This means that multiples of matrix columns may be added to each other or matrix rows may be interchanged to obtain an upper diagonal form of the dimensional matrix. As a consequence of these rank preserving operations on the dimensional matrix, the dimensional representation of the design parameters x 1. . . . . x, using the standard dimensional basis of the SI-unit system (currently k = 7 known

TFl,lTl'$2 . . . . .

I

I

Xl X2

t

mlm 2 .....

mk

Xl X2

m

1 1

eij

"7~1 Xn

Xra Fig. 2. Definition of dimensional matrix.

1

Upper and Lower Limits for 'The Principles of Design'

211

qo H

X

q

F ~j .......

l: (x:,...

Fig. 3. Description X, mapping 99 and evaluation II.

Fig. 4. Domains and mappings (according to [t, 15]).

independent physical dimensions with m 1 (mass), m 2 (length), m 3 (time), m4 (temperature), m5 (current), m 6 (amount of substance) and m 7 (light intensity)) is transformed into an equivalent dimensional system representation in the form of m] . . . . , m~, with r < k. This and the computation of a dimensionless product are shown in the Appendix• To ease the understanding of the suggested model, the following terminology will be consistently used from now on. This is shown in a mathematically abstracted form in Fig. 3, which is essentially the same diagram as shown in Fig. 1. The Xl . . . . , x, and X represent the description of the parts and of the entire object, while n : , . . . , n m and H represent the corresponding evaluations. The associated mappings of the description onto the evaluation are represented by (Pj and ~0, while f and F are the appropriate aggregation functions (see Refs 3 and 4 for an in-depth discussion). Without further consequences for the main arguments on the dimensionality of the spaces N~ and R"+ in this paper, it is mentioned that the existence proof of solutions of Eq. (14) is constructive but not unique. Since the dimensionless products in Eq• (14) form a free abelian group, all possible solutions to Eq. (13) are of the form

as the customer domain (consisting of customer attributes, CA), the functional domain (consisting of functional requirements, FR), the physical domain (consisting of design parameters, DP) and the process domain (consisting of process variables, P V), as illustrated in Fig. 4. Concerning the mappings in Fig. 4, going from a domain on the fight relative to a domain on the left represents what the designer wants to achieve, i.e. the mappings q0M, (PD and (pc- On the other hand, going from a domain on the left to a domain on the fight represents how the designer proposes to achieve these goals, i.e. the mappings (Pc 1, (p~ 1 and (PM1. The design process with its goal of creating a physical embodiment for a design object is thus the 'zig-zagging' through these four domains by means of these mappings [15]. Consequently, the synthesis process consists of the mappings (pc l, (p~l and q ~ l , while the analysis process consists of the mappings (pu, (PD and (Pc-

j=l

with k = 1. . . . , m ~ t~ and fl~;~ ~. The general solution may thus consist of any arbitrary combinations of the m original nj which also satisfy the condition of structural independence [14]. This means that the square matrix II with elements fikj has to be of full rank to guarantee the equivalence of both dimensionless parameter sets rc = {rq . . . . . n,,} and

2.3. Axiomatic Design In the most detailed descriptions of his work, Suh [1, t 5] defines the design world of the axiomatic design approach as a thought construct of mappings between four different domains. These four domains are defined

2.3.1. Design Matrix To help the designer understand the nature of these mappings during the design task, the following design equation is proposed for the mapping (PD (see Ref. 1): {FR} = [A]{DP}

(16)

For a sensitivity analysis of the proposed design, the design equation may be used in differentiated form (see Ref. 1):

" dFRI IdFR1 t

(. d FR m)

c~FR1-

~DP1

~DPn

~?FRm

¢3FRm

~DP,

3DPo

1

Depending on the general form of the various functional requirements, certain elements ajl -- OFRi/ODP~ of the matrix A may or may not be zero. For m = n three qualitatively distinct cases can occur and are named (see Ref. 1):

• Coupled design. In general, the matrix A is a full matrix (the aji :~ 0 for all i,j). Such a design is called

212

S. Rudolph

coupled design, since a small change in one design variable may affect all other functional requirements. • Decoupled design. If the matrix A is either an upper or a lower triangular matrix (the a~ # 0 for i < j, and ai~ = 0 for i > j, or vice versa), the design is called a decoupled design. The design variables can then be independently changed, if a certain order in these changes is observed. • Uncoupled design. If the matrix A is a diagonal matrix (the aii # 0 and a i i = 0 for i # j), the design is called an uncoupled design. Any change in one design variable will only affect one functional requirement at a time. Suh [1] concludes from these observations that any acceptable design should therefore be uncoupled or at least decoupled, thus eliminating or minimizing the worries of the designer about the coupling of the design parameters. This is expressed by the independence axiom. 2.3.2. Independence Axiom The declarative (or procedural) form of the independence axiom is (see Ref. 1): Maintain the independence of FRs. Then, in order to choose the best design among all acceptable designs, the so-called information content of the design is introduced. This information content can be interpreted as the relation between the specifications of the designer (the set of design parameters DP i, also denoted as xg) and the manufacturing capabilities (expressed in tolerances of the design parameters _+(Axe/2)).

For the ease of manufacturing, a designer should therefore choose among all acceptable designs the design with the minimum information content. The axiomatic design approach and its basic two axioms rely on the fundamental assumption that there are generalizable principles that govern the design process and that these in the form of the design axioms are general principles or self-evident truths that cannot be derived or proven to be true except that there are no counter-examples or exceptions (adapted from Ref. 15). However, it is shown in Section 3 that such a general proof for the establishment of a general mapping ~0D in Fig. 4 exist and can be derived.

3. Proofs The theoretical background of the independence axiom will be examined first using the approach of the evaluation hypothesis [3, 4], which is based on similarity theory. The results obtained are then interpreted and discussed. According to the evaluation hypothesis, the evaluation problem consists of three parts: identifying the description space X = {x 1. . . . , x,} T of the physical design object, the establishment of an appropriate mapping ~oo into the evaluation space, and the identification of the design goals H = {~1. . . . . ~m}T in the evaluation space. Comparing the properties of both the evaluation hypothesis and the axiomatic design approach, the space of the design parameters D P is matched by the space X of the physical variables. Whereas the space of the functional requirements FR is matched by the space of dimensionless groups H associated with these physical variables. This is written as

2.3.3. InJ'brmation Axiom The declarative (or procedural) form of the information axiom is (see Ref. 1): Minimize the information content of the design. The measure of information content 1 of a design is defined as 1 = ~ In ( 1 ~ = _ ~ In (Axi') i=1

\Pi/

i= 1

(18)

\ Xi /

where p~ represents the probability of success that a design variable x~ lies within a certain specified tolerance of _+(Axl/2), i.e. the probability of success that a measured length x~ will lie within the tolerance of the actual length is given by p~ = Axi/xv

D e - {xl . . . . . x,} T

(19)

FR =_ {u 1. . . . . U.,}T

(20)

The fact that the evaluation hypothesis can be applied (i.e. that the assumptions of Eqs (19) and (20) are correct) can be verified by inspection of Eqs (6.27), (6.28) and (6.29) on p. 228 of Suh's book [1]. It will be shown in the following that these equations can be derived as a special case of the evaluation hypothesis. Since the ns are defined by Eq. (14), writing the derivatives in respect to all variables yields

= ~?xj

i= 1 ~

dxi

j = 1. . . . . m

(21)

Upper and Lower Limits for "The Principles of Design"

213

and leads to a differential design matrix form similar to the one shown in Eq. (t7). It is equal to dTlt =

m

~-1X1

Xr - - O;2r 7~2

~21~2

- - ~m

0

Xr

x 1

I,l~m-1

- - O;m- l , r ~ m - I

0

0

...

o

~--~-

0

.-"

0

Xr+l

Xr+2

0

...

0

Xr

Xt

- - O~mrgra Xr

XI

~-~ Xr+ra- i

0

0

""

0

0 ~ Xr+m

dxl dxr (22)

x dxr + 1

dxr + m and, for better distinction, the labeling of x~ with j = (1 . . . . , m) in Eqs (14) and (21), has been modified to Xr+l,.--,Xr+,, of the very same variables xj in Eq• (22). The following important conclusions can now easily be drawn by comparing equations (17) and (22): • There does exist a fixed relationship between the dimensions n and m of both the evaluation space II = {rq . . . . , n,,} T and the space X = { x l , . . . , x,} T of the (physical) design parameters, with m = n - r. This fact is independent of the choice of a specific (physical) process or function and is due to the general principle of dimensional homogeneity, which underlies all modeling of functional relationships of physical variables as shown by the Pi-theorem. • The matrix form in Eq. (22) representing the general mapping q~D: X ~ H in Fig. 4 is therefore generally rectangular and not square, since r >_ 1 is always valid, as shown by the Pi-theorem. This rectangular matrix therefore represents the true upper limit• • In order to obtain a square matrix, exactly r variables need to be held constant in Eq. (22). This means that the corresponding columns of the matrix are deleted• This reduction by r design variables is theoretically always possible• • Generally, it is impossible to predetermine whether it is advantageous in any design situation to keep r of the n design parameters constant right from the beginning and vary the remaining m design parameters only. This is because some design variables, like material properties (Youngs modulus

E, density p) or fundamental physical constants may be difficult or even impossible to alter, while some geometric length 1 of the design object might be more easily alterable• • Assuming that just the design parameters (x 1. . . . . xr) are kept constant, the matrix in Eq. (22) turns into a square and purely diagonal form. This represents a special case which could be interpreted as uncoupled design and consequently represents the lower limit. • Assuming that any other set of r design variables is kept constant in order to achieve a square matrix, any arbitrary matrix behaviour between a full matrix and a diagonal matrix may be obtained. Thus all the behaviours described in Section 2.3.1 as uncoupled, decoupled and coupled design matrices are special cases of Eq. (22), when the first r design parameters (i.e. x l , . . . , xr) are held constant. The behaviour of Eq. (22) based on similarity theory can thus exhibit all the behaviours described in Section 2.3.1 as uncoupled, decoupled and coupled design matrices. Therefore, this is considered as a proof that the design axioms are special cases of the more general formulation of the evaluation hypothesis. Now the so-called information axiom [1] will be analysed here in more detail using the approach demonstrated in the previous section• In the first step, this is done using the analytical technique of differential error analysis [10]. In a second step, the results obtained are then compared and interpreted• Assuming once again the validity of the evaluation hypothesis, one can define FR = {rq . . . . , re,,} and D P - {xl . . . . . x,} according to Eqs (20) and (19), Now writing the design matrix equations in the form of Eq. (11) leads to

I

_o~11

--oq~

1

0

--(z21

--~2r

0

1

:

:

-- O~m-1, 1 --

O~ml

x

0

--•

O-

0

'--

0

".

:

-~_,,~

0

"'"

0

1

0

-a~

0

0

•••

0

1

(23)

214

S. Rudolph

Comparing both Eqs (23) and (18), and taking into account that, according to Eqs (4) and (8) e~, --

Ax i

design matrix: dZ

i = ( 1 , . . . , n)

[_-3K3P-11/SDl[4

¼KaP-3/SD -3/4

(24)

(27)

Xl

is valid, the information axiom as stated in Eq. (18) can be understood as some non-linear form of a norm of the vector ex on the right hand side of Eq. (23). However, the cross-coupling inherent in the physics of the design object and expressed by the rectangular form of the matrix (either in Eqs (23) or (11)), is not fully captured by the estimate of the information axiom. This apparent difference can be attributed to the fact that the same magnitude of relative disturbances ~, in some design variables may have a quite higher impact on the overall relative design goals than other design variables due to the cross-coupling factors c~j~ in Eq. (23).

dD

Rewriting and normalizing Eq. (27) by dividing by Q or Z and using the equalities d(~ = dQ/Q = dQ/KEP1/2D 2 and dZ = d Z / Z = dZ/K3p-3/SD 1/* one obtains

dZ =

(28)

l|(dD)

3 8P

4D..J

Setting P = 1 and D = 1 and rewriting Eq. (28) a last time, one obtains the dimensionless design matrix as stated in Suh's book [1] ([P] and [D] denote the dimensions of P and D)

lIdlj

dZ

-3

¼

dD[

4. Example Since the purpose of this paper is to demonstrate how the inductive approach of axiomatic design and the deductive approach of the evaluation hypothesis are formally related, the application example is taken from Sub's book [1]. This way, the reader can profit from the detailed description of the design problem already published and can compare both views of the same problem, while the focus in this presentation can be restricted to the more theoretical aspects. The design example of the MIT RIM-moulding machine [1, 11] consists of some of the physical design considerations necessary to achieve improved performance of a reaction injection moulding machine. For the two design performance measures of delivering a certain mould flow rate Q while maintaining a certain mix quality Z, the following approximate relationships are identified by means of numerous experiments [11] to be sufficiently precisely described by Q

(25)

= K2P1/2D 2

Z = K 3 P-

3/8D

1/4

Using now the evaluation hypothesis and applying dimensional analysis to the above design equations (25) and (26) as shown in the Appendix, the following two dimensionless quantities, the dimensionless flow rate nl and the dimensionless mix quantity n2, defined as

Q

(30)

1~1 - - K 2 P 1 / 2 D 2

7~2 - -

Z K3p_3/SD1/*

(31)

are obtained. According to the general context of the evaluation hypothesis as shown in Eqs (19) and (20), this leads to a dimensionless expression of the design goals, i.e. the functional requirements, and is according to Eq. (22) equal to Sdrq'~ =

K2

(dTz2J

0

2P

D

Q 0

7~2

3~z2

TO2

K3

8P

4D

(26)

where P is the system pressure and D is the orifice diameter; K 2 and K 3 are numerical constants determined in the experiments. In terms of axiomatic design, Q and Z constitute two of the functional requirements, while P and D represent the design variables. This leads to the following coupled

ldK3[ xldPIdzdQdD

(32)

Upper and Lower Limits for 'The Principles of Design'

215

Considering now the special case of all designs with constant evaluations, i.e. 7z~= const and thus dTzi = 0, and considering K2 and K3 as true constants leads to

=

[TJ

2P 8P

7{;'}

(33)

£0

Rewriting this a last time leads to a dimensionless design matrix

;1 7

(34)

It is very important and strikingly interesting to note that Eqs (30), (3t), (33) and (34) are already derived and used in Sub's book [1] for the description of various design properties (i.e. of the so-called reangularity and semangularity). However, no further explicit statement about a possible and more general use of dimensionless groups outside their traditional applications in engineering is contained therein. An example for a novel application of dimensional analysis is the mapping of the design description onto the design evaluation in terms of the existence of the evaluation hypothesis. The identical notation of Eqs (28) = (6.27) and (29) = (6.29) derived by Suh [I] on p. 228 as a special case of the evaluation hypothesis [3, 4] in Eqs (33) and (34) however suggests that both approaches match effortlessly. As a further hint this statement accounts for the fact that for Eq. (29) P and D are set constant (i.e. to unity) in Ref. 1. Considering K 2 and K 3 to be constant as well, exactly ( n - 1) design parameters are kept constant in both (different) design equations in the form of f ( x 1. . . . . x,) = 0. The initial assumption of the special case of a constant design evaluation with drcj = 0 for both associated sets of dimensionless groups is thus fulfilled and justified.

5. Summary This work has shown how the axiomatic design approach is related to the theoretical framework provided by the evaluation hypothesis. Based on this framework and the notation of dimensionless groups, both the independence and the information axiom of axiomatic design appear to be special cases inside this theoretical framework.

The evaluation hypothesis is restricted to the validity of the method of dimensional analysis and relies on a deductive approach based on epistemological considerations. The fact that the notations of the inductive axiomatic design approach which are based on design case experiences comply with the notations of the evaluation hypothesis provides further evidence for the validity of both approaches.

Acknowledgements I am grateful to Professor Dr Steven Dubowsky for inviting and hosting me at the Massachusetts Institute of Technology (MIT) as well as for bringing the axiomatic design approach to my attention. I thank Professor Dr.-Ing. Bernd Kr6plin, Head of the Institute of Statics and Dynamics of Aerospace Structures (ISD), Stuttgart University, Germany, for encouraging this work. Comments of my colleague Dipl.Ing. Gunter Emrich and cand.aer. Sean Hetterich as well as of four anonymous referees were helpful improving the clarity of the text. The financial support of this work by the Deutsche Forschungsgemeinschaft (DFG) in the form of the interdisciplinary research group Forschergruppe im Bauwesen (FOGIB) 'Ingenieurbauten-Wege zu einer ganzheitlichen Betrachtung' (Construction in Engineering--Paths to an Holistic Approach), is acknowledged.

References 1. Sub, N. (1990) The Principles of Design, Oxford Press, New York. 2. Mistree, F. (1992) Book review: the principles of design, Research in Engineering Design, 3, 243-246. 3. Rudolph, S. (1995) Eine Methodik zur systematischen Bewertung yon Konstrucktionen, PhD dissertation, Universit/it Stuttgart, VDI Fortschrittsberichte, Reihe 1, Nummer 251, D/isseldorf. 4. Rudolph, S. (1995) A methodology for the systematic evaluation of engineering design objects, PhD dissertation, translation of the original german PhD thesis [3] into English language. A copy of this translated PhD thesis is available on request by email from: [email protected], ISD Verlag, Number 02-94, Stuttgart University. 5. Dolinskii, I. (1990) Use of dimensional analysis in the construction of mechanical assemblies for optical instruments, Soviet Journal of Optical Technology, 57(8), 512-514. 6. Kloberdanz, H. (t991) Rechnerunterst/itzte Baureihenentwicklung, VDI Fortschrittsberichte, Reihe 20, Nummer 40, VDI-Verlag, Dtisseldorf. 7. Middendorf, W. (1980) The use of dimensional analysis in present day design environment, IEEE Transactions on Education, 29(4), 190-195. 8. Bhaskar, R.; Nigam, A. (1990) Qualitative physics using dimensional analysis, Artificial Intelligence, 45, 73-111. 9. Sycara, K.; Navinchandra, D. (1989) Integrating case-based reasoning and qualitative reasoning in engineering design, Proceedings Applications of At in Engineering, 231-250. 10. Stoer, J. (1989) Numerische Mathematik, Band 1, Springer, Berlin.

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11. Tucker, C. (t978) Reaction injection molding of reinforced polymer parts, PhD dissertation Massachusetts Institute of Technology, Department of Mechanical Engineering. 12. Bridgman, P. (1922) Dimensional Analysis, Yale University Press, New Haven, CN. 13. Rudolph, S. (1995) On a symbolic CAD-front-end for design evaluation based on the Pi-theorem, Preprints of the IFIP WG 5.2 Workshop on Formal Design Methods for Computer Aided Design, Gero, L and Sudweeks, F. (Editors), Mexico City, June 13-16, 189--203. 14. Pawlowski, J. (197t) Die A_hnlichkeitstheorie in der physikalisch-technischen Forschung. Grundlagen und Anwendungen, Springer, Berlin. 15. Suh, N. (1994) Axiomatic design of mechanical systems, Preprint, Invited Paper on the 50th Anniversity of the Machine Design Division of ASME.

I ml P D 1(2 Q

1 -1/2

m3 It SI-Unit

m2 -1 1 3/2 3

kg m -a s -2

-2

m

kg-a/2 rn3/2 -1

m3 s-1

Meaning pressure diameter constant flow rate

Fig. A.1. Dimensional matrix RIM machine.

t

P D

K2 Q

ml 1 -1/2

m2

~/~3

7rb I

-1 1 3/2 3

-2

1

t

m 2

t

TFg3

1 -1

1/2

2

1 1

Appendix Fig. A.2. Dimensional matrix calculations. Dimensional analysis allows the derivation of dimensionless groups from the relevance list of physical variables of some function f. Using Eq. (25) of the reaction injection machine with Q = K2Pt/zo2

as an example, the implicit notation of the function f ( P , D, K2, Q) = 0 leads to the establishment of a dimensional matrix as shown in Fig. A.I. The elements e~j of this dimensional matrix represent the dimension exponent of the ith physical variable in the j t h physical dimension of the SI-unit system. Using rank preserving operations on the columns of that dimensional matrix as indicated in Fig. 2, the following modified dimensional matrix with an upper diagonal form is obtained, see Fig. A.2. The rank of the dimensional matrix appears to be equal to r = 3; thus with n = 4, m = n - r = 1, one dimensionless product can be constructed according to Eq. (14). Since the sum of the remaining exponents in each matrix

the remaining exponents in each matrix column needs to add up to zero to obtain a dimensionless product, the signs of the elements a~ in the row of Q -= xj with j = 1 in the modified dimensional matrix just need to be inverted to obtain the appropriate exponents of the above so-called basis variables. This leads to the following dimensionless product 7q = Q P - t / 2 D - 2 K ~ I

Q K2pI/2D z

Using the same procedure, the result for the second dimensionless product ~z is obtained when applied to the second design equation (26). For a more indepth coverage of the computation of dimensionless products or an introduction to the technique of dimensional analysis see Refs 12 or 14.