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Unformatted text preview: The Principles of Design & Design Axioms References: Magrab, 1997, Integrated Product and Process Design and
DeveIOprnent, CRC Press. Nam P Sub, 1990, The Principles of Design, Oxford University Press. The deﬁnitions exceptions. Theorem is a pr0position that may not be selfevident but that can be proved from accepted axioms. It is therefore equivalent to a law or principle. A theorem is valid if it
its referent amoms and deductive steps are valid. process to fulﬁll the FRs. Mathematical Representation ofthe Independence Axiom: Design Equation FRI All Aln DP:
FRm Ami H. Amn when m=n, design matrix is square 3 cases: ' ° The W design — simplest case of design:
FR, A“ o 0 DP,
FR: 2 0 .422 0 DP2
FR3 0 0 A33 D133 ' The coupled design  undesirabie FR] A” A12 A13 DP
FR2 = A21 A22 A13 DP2
FRB A3! A32 A33 ' A decoupled design  the design matrix A is a triangular matrix FRi A“ 0 0 DP;
FR? 2 A21 A22 0 DP2
FR3 A31 A32 A33 DPz The independence ofFRs can be assured ifwe adjust the DPS in a particular order. Example: Twoknob Faucet FRs: FRI = Obtain water flow rate FRZ = Obtain water temperature _ This is a coupled design.
DPs: DPl = Means to adJust cold water flow
DPZ = Means to adjust hot water ﬂow FRs: FRI = Obtain water flow rate 457‘
FR2 = Obtain water temperature . . . mtg?”
. . d d s an.
DPS: DPl = Water flow regulating dewce Thls 15 an uncou 16 e 1 as! —’
DP2 = Water temperature regulating device we er Theorem 2: Redundant Designﬁvo. of FR < no. ofDP)
Ps than FRs, the design is either a redundant design or a coupled Theorem 3: [deal Design (no. of FR= no. of DP) When the number of DPS is equal to number of FRs, the design can be uncoupled or
decoupled. Design Axioms There are two design axioms that govern good design. Axiom 1 deals with the
relationship between functions and physical variables, and Axiom 2 deals wrth the
complexity of design. Axiom 1
The Independent Axiom
Maintain the independence of PBS. Axiom 2
The Information Axiom Minimize the information content of the design Axiom 1 states that during the design process, as we go from the FRs in the functional
domain with DPs in the physical domain, the mapping must be such that a perturbation in
a particular DP must affect only its referent FR. 
Axiom 2 states that, among all the designs that satisfy the Independence Axiom, the one
with minimum information content is the best design. Since there are infinite number of designs that can satisfy a given set of FRs. Axioms 1
and 2 can be restated. Axiom 1 The Independence Axiom Alternate Statement 1: An optimal design always maintains the independence of FRs Alternate Statement 2: In an acceptable design, the DPs and the PBS are related in such a way that
speciﬁc DP can be adjusted to satisfy its corresponding FR without affecting
other functional requirements. Axiom 2 The Information Axiom Alternate Statement: The best design is a functionally uncoupled design that
has the minimum information content. Functional coupling should not be confused with physical coupling, which is often
desirable as a consequence of Axiom 2. Integration of more than one function in a Single
part, as long as the functions remain independent, should reduce complexity. Example: Canfbottle opener; constraints: low cost, operated manually.
FRs are PR1: open beverage bottles. FRZ: open beverage cans. ‘
_
— Example:
One of the basic requirements of the intelligent machine is that it must not be a coupled system. Otherwise, the machine will be difficult to control and will require continuous
fine tuning. The brakeforming operation is one of the most commonly used sheet metal
forming pr0cesses. In brakeforming operation a sheet metal is bent into a bend angle 6f
(at a deformation xf), using a die and punch. The punch pushes the sheet metal, which is
supported by the die. This is done by experienced Operator with trial and error process.
It is not an easy process. The sheet must be overbent to an angle :90 (at a deformation x0),
so that when the load is released, it spring back by an angle 60 6}: The unloading follows the dotted line. The difﬁcult in achieving the correct bend anlge is caused by the variation and workhardening characteristics in the material properties.
Prom classical beam theory, we can deduce i.
BEJ‘G’WJ' fi'i’amen1l sew
5F 6d agfe‘l
we
2
6 =t ‘1 5i 1
a an (d) () E1: tan M" =tan x—f
(9049,, (1 If we are able to measure F0 and x0 (and subsequently xf), then We have a means of
controlling the process as stated below. In order to obtain we have consider the following three FRsz
FR; = M0 (Generate moment)
FR; = 60, (Bend and deform metal)
FR; = 9;; (Release to ﬁnal bend metal) The DPs are not explicitly chosen because they are governed by equation (1). Thus, the
design equation becomes M 1 o 0 Fad/2 O a = 0 1 o tan'1(xo/d) (2) 0 6f 0 1 —1 Fad/(Ztan‘IEI) To implement this design equation the following'procedure is employed. The punch is brought down and the plate is subject'to a force Fa, which results in a displacement under
it 'of x0. The punch is removed and xf is measured. From the measured values F0, x0 and
x;, the material preperty El can be determined by equation (I). We now apply a slowly increasing force F0 and x6 until their values produce the desired 6} as computed from
equation (2). 15 Corollaries These corollaries may be more useful in making speciﬁc design decisions, since they can
be applied to actual situations more readily than can the original axioms. They may be
called design rules, and are all derived from the two basic axioms. ' Corollary 1: (Decoupling of coupled design) Decouple or separate parts or aspects of
a solution if FRs are coupled or become interdependent in the design proposed. Corollary 1 states that functional independence must be ensured by decoupling if a
proposed design couples the FRs. Corollary 2: (Minimization of FRs) Minimize the number of Pits and constraints. Corollary 2 states that as the number of Pits and constraints increases, the system
becomes more complex and thus raises the information content. Corollary 3: (Integration of Physical Parts) Integrate design features in a single
physical part if FRs can be independently satisﬁed in the proposed solution. Corollary 3 states that, as long as the F125 are not coupled by the physical integration of
parts, the integration strategy should be followed if it reduces the information content of
the design. It states that the number of physical parts should be reduced in order to
decrease the information content, if this can be done without coupling FRS. Corollary 4: (Use of Standardization) Use standardized or interchangeable parts if
the use of these parts is consistent with the FRs and constraints. Corollary 4 states a well known design rule: use standard parts. In order to reduce
inventory and minimize the information required for manufacturing and assembly,
special parts should not be used if standard parts can fulfill the FRs. Furthermore, the
number of standard parts shouldbe minimized sokas to decrease the inventory costs and
simplify inventory management. Interchangeable parts allow for the reduction of inventory, as well as the simpliﬁcation of manufacturing and service operations; that is.
they reduce the information content. Corollary 5: (Use of Symmetry) Use symmetrical shapes and/or arrangements if
they are consistent with the FRs and constraints. Corollary 5 is self evident. Symmetrical parts requires less information to manufacture and to orient in assembly. Not only should the shape be symmetrical when ever possible, but hole locations and other features should be placed symmetrically to minimize the
information required during manufacture and use. 16 gorollary 6: (Largest Tolerance) Specify the largest allowable tolerance in stating
Rs. Corollary 6 deals with tolerances. Since it becomes increasingly difficult to manufacture
a product as the tolerance is reduced, more information is required to produce parts with
tight tolerances. On the other hand, if the tolerance is too large, then the error in assembly may accumulate such that FRs cannot be satisﬁed. The specification of tolerances should
be made as large as possible, but should remain consistent with the likelihood of producing functionally acceptable parts. The correct tolerance band is that which
minimizes the overall information content. When tolerance band is too large, the
information content will increase since the subsequent manufacturing processes will
require more information. Excess tolerance reduce reliability and thus increase the need
for maintenance; this contributes to the increased information content. Corollary 7: (Uncoupled Design with Less Information) Seek an uncoupled design
that requires less information than coupled designs in satisfying a set of FRs. Corollary 7 states there is always an uncoupled design that involves less information than
a coupled design. If a designer proposes an uncoupled design which has more
information content than a coupled design, then the designer should return to the
"drawing board" to develop another uncoupled or decoupled design having less information content than the couple design. Deﬁnition of information Axiom 2 (the information axiom) deals with the complexity. The final output of the
design process is a block of information for use in subsequent manufacturing or other
operations. Example: It is easy to draw a shaft and Specify its dimensions and tolerances,
but in order to manufacture the shaft we have to choose a set of machines and
manufacturing processes. If we select a wrong machine, the machining task can be quite
complex, and thereby entail a great deal of information. On the other hand, we all know
that if we have the correct tool, the job can be done much more easily. Example: if the shaft has to be measured to be 4 +/ 0.1m, then an ordinary measuring
tape may not be used with any degree of conﬁdence. We can use the tape measure, but
the probability of measuring it to within the accuracy specified by the FR may be very
low. In order to increase the probability, we will need more accurate devices. The notion
of information is very closely related to the probability of achieving the PR. The information content of a design may be defined quantitatively as the logarithm of the
probability of fulfilling the Specified FR. Information content I is defined in terms of
probability as
Information=l=log2(1/p)
where p is the probability of achieving the tolerance objectives. Example: If the FR is to
have a shaft length of 4 +/ 0.1m, then the probability of being within the tolerance
deﬁnes the information. If we assume uniform distribution, the probability p of producing
an acceptable shaft is given by the ration of tolerance to the dimension
p=2*0.1/4 l7 The logarithmic deﬁnition for information is convenient to use. The total information
content of a series of process is the addition of each of the information content, since the total information probability is a multiplication. The information content comes from the
1nformat10n theory developed in 1949 by Shannon and Weaver. Assuming the ratio range/tolerance is uniform throughout the
range, the information content deﬁned above can be written as I Information=log2(rangeftolerance) To generalize, the relationship between the designer's speciﬁcation and the capability of a
manufacturing system 15 illustrated in the ﬁgure below. This is a plot of the probability distribution of a given system parameter versus the absolute value of the design
parameter. A uniform probability density ﬁinction is assumed. £ 1.; Design range I Probability density Ll System range Design parameter The design range is deﬁned as the tolerance associated with the DP Speciﬁed by the designer. The system range is the capability of the manufacturing system given in terms of
tolerances. The common range is the overlapping range of the design range and the sytem range. The definition for the information content is given by I =10g2[ system range ] common range 18 m
A hypothetical factory has three machine tools. The following are the speciﬁcations. S stern Ran e (Machine caabiIities) .a Design Range
Dim Accuracy (urn) Surface Rouhnessuim) soo+soo .m
200 a +200 0  so
0 3 .' '\ ‘r‘  Dim .   P“i:t_\i"iil‘ll} Accuracy If?
, ' 573 7l€ﬂﬂ Yi'u‘Iaé oi: M/c. # i ’ J firﬁ _ ~5:o ‘7' I 76 .  O 41%,in Luge Officde 7'? i
" 9’ _ 10g2 (140/13)+ log; (90/0)=oo log; (l40/13)+ log; (90/0)=m log2 (5/2.5)+ log2 (4.5/2.5)=I.85 Conclusion
Based on the minimum infomation content, machine #3 is the best for the manufacture of I all the 3 component parts. However, only the dimension accuracy and surface ﬁnish have
been considered. If the machining time and labour cost are to be consider, machine #1 or
#2 may be selected for the operation. Exercise: John is looking for a house. He is considering two different towns, which are located
near the town in which his company is located. His decision is based on three factor: (1) land
price, (ii) travelling time, and (iii) environment. He is willing to pay up to $4,000 per sq. ft, to
spend as much as 1.5 hours for transportation, and to live in an environment which meets 80% of
his expectations. The two towns selected offer the following possibilities: Town A Town B
Land price ($ per sq. ft.) 20003000 3500  4000
Travelling time (hours) 1.0  1.2 0.8  1.0
Environment (% of satisfaction) 80 ~ 90 60  30 Which is a better townfor John? 19 ...
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