03BasicConcept I[2]

7/28/2019 03BasicConcept I[2]

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1POSTECH ME PCM Chapter 03 Basic concept I

Basic Concept of Axiomatic Design II

Prof. W. HwangDept. of Mechanical Engineering

Postech


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1. The Second Axiom

The Information Axiom

The design that has the smallest information contents is the best.


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2.1 Information Contents

Information content Ii for a given FRi is definedin terms of the probability Pi of satisfying FRi.

The information is given in units of bits.

Ex: Cutting a shaft length of 40.1 m

Probability = P = 2(0.1 m)/(4 m) = 1/20Information = I = log2(1/P) = log2 (20) = 4.32 bits

)(log1

log 22 yprobabilitisPP

P

I iii

i


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In the general case of m FRs,

All FRs are statistically independent,

All FRs are not statistically independent,

Where Pi|{j} is the conditional probability of satisfying FRi

given that all other correlated {FRj}j=1,, i-1 are also satisfied

)(log }{}{2 yprobabilitntjoitheisPPI mmsys

m

i

im PP1

}{ m

i

i

m

i

isys PII1

2

1log

m

i

jim PP1

}{|}{}1,,2,1{}{ ijfor

m

i

jisys PI

1

}{|2log


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RangeSystem

RangeCommonP

CR

SR

PI 22 log

1log

Figure.3.1. FR vs. Probability Density

,rangedesigninincludedisrangesystemWhen

,rangedesignofoutisrangesystemWhen

0I

I


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Example 1.1 Cutting Rod to a Length

Two kinds of cutting condition

Rod A to 1 0.000001 meters and Rod B to 1 0.1 meters

The nominal length of the rod is 30 meters rather than 1 meter

How is the probability of success?

The answer depends on the cutting equipment available for the job!

However, generally

So, PA is lower than PB.

lengthnalnomi

tolerance

fP


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Example 1.2 Cutting of the Rod with a Hacksaw

The PDF of hacksaw

Rod A -> Probability 0 (For One Micron) -> I =

Rod B -> Probability = 1 (For Ten Centimeters) -> I = 0

Figure 3.2. The design ranges and the system pdf for hacksaw


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3. Complexity

Def. : A measure ofuncerta intyin achieving the specified FRs

Complexity is related to information content.

A design is called complex when its probability of success is low.

The greater the information required to achieve the FRs of a design, the

greater is the information content, and thus the complexity.


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4. Advantage of information axiom

Need not additional weighting factors

Select the best among those designs that are acceptable

Provide theoretical basis for design optimization and robust

design

The intention of the designer and the importance assigned to

each FR by the designer are represented by the design range.


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5.1 Robust design

How can we get robust design?

Elimination of bias

Reduction of variance


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5.2.1 Elimination of bias

The distance between the target value and the mean of the systempdf is called bias.

Figure 3.3. Design range, system range, common range

and system pdf for a FR


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5.2.2 Elimination of bias

In one-FR design, the bias can be reduced or eliminated bychanging the appropriate DP.

However, in multi-FRs design, the bias can not eliminated when the

design is coupled.

When the design is decoupled or uncoupled design, the bias can be

eliminated by changing DPs.


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5.3 Reduction of variance

Variance : A statistical measure of the variability of a pdf.

Variability is caused by a number of factors.

Noise

Coupling

Environment

Random variations in design parameters

The variance must be minimized.


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5.3.1 Reduction of stiffness

Suppose there is only one FR that is related to its DP as

FR1=(A11)DP1

stiffness

In a linear design, the allowable

tolerance for DP1, given the specified

design range for FR1, depends on the

magnitude of A11, i.e. the stiffness.

The smaller the stiffness, the larger is

the allowable tolerance of DP1. Figure 3.4. Allowable variation of DP as afunction of stiffness


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Example 2.1 Automobile Wheel Cover(hub cap)

Sheet metal is pressed to make a decorative cover to hide the lug

nuts that hold the rim of the wheel assembly onto the car.

Holes are punched in the rim of the wheel and metallic clip springs

are welded in the wheel cover. To attach the cover on the rim, the

springs are pushed into the holes in the rim.

FR1 = Prevent the cover from falling off the wheel when car goes over a bump

FR2 = Make the mounting of the rim easy when the tires are replaced.

Design Range is from 30N to 38N

Figure 3.5. A hub cap for cars


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Because of slight misplacement of the spring during welding andthe wear of punching dies during fabrication of the rim, it was

found that the force is not always in this range.

Figure 3.6. Percentage of parts vs. retention force


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The DP is the deflection of the spring that is determined by the

relative radial position of the spring with respect to the punched-hole

location.

To reduce the system range, which is associated with the variance,

so that the system range is inside the design range, we need to usesprings with softer stiffness.

FR1=Retention force= A11(DP1)=( ) (rrim-rspring)springofstiffness


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When we use a soft spring,

the allowable DR range is

larger than when we use a

stiff spring.

Then, we can reduce the

variance.

Figure 3.7. Spring force as a function of the requireddeflection of the spring


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5.3.2.1 Immune to variation

When there are many DPs that affect a given FR, design should be

done so that the FR will be immune to variation of all these other

DPs except the one specific DP chosen to control the FR.

Consider the one-FR design problem.

FR = f(DPa, DPb, DPc) or

FR = AaDPa + AbDPb + AcDPc

In this case, we make FR immune to DPb and DPc changes, which can be doneeither if Ab and Ac are small or if DPb and DPc are fixed so that they remain

constant. Then, we can reduce the variation of the design.


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5.3.2.2 Immune to variation

Now consider the case of the multi-FR ideal design.

For reducing variation from errors in DPs, the coefficients A11, A22

and A33 should be small, but large enough to exceed the requiredsignal-to -noise ratio.

3

2

1

33

22

11

3

2

1

00

00

00

DP

DP

DP

A

A

A

FR

FR

FR


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Example 3.1 Windshield Wiper-Robust Mounting Design

Eliminate squeaking sounds under all conditions

The windshield-wiper assembly

consists of the pivot of the wiper

blades, itself and the mounting

bracket with three-mounting holes

The location of the three holes

defines the angle of the pivot of

the wiper-blade assembly when it

is mounted on the car.Figure 3.8. Windshield and windshield

-wiper assembly


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The source of noise is the chatterof the rubber blade due to

frictional force and the elastic

nature of the blade.

The physics of chatter

Force apply -> Exert the force on

the mass-> The spring

compression -> The spring

stretching -> Frictional forcedecrease -> The mass stop ->

Repeat

Figure 3.9. Modeling of chatter


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When the direction of the wiper reverses at the end of the stroke and

goes from right to left, the blade will experience the reversal of the

loading direction- elastic unloading first, followed by the change in

the direction of the applied force.

Figure 3.10. Attack angle of bladeFigure 3.11. History of tangential force

acting on the blade


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The condition for flipping is :

tan

sincos

or

FhFh

wiperthetoappliedforcecontactanttresultheisF

Figure 3.12. Condition for flipping of the blade

Figure 3.13. System pdf and designrange of the attack angle


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The attack angle is a function of the position of the windshield.

Where, X1 = pivot axis

X2 = position of the twist arm

X3 = vector normal to the windshield

Expanding the attack angle about its mean position

)()()(),,( *333

*

22

2

*

11

1

*

3

*

2

*

1

* XXX

XXX

XXX

XXX

),,( 321 XXX

),,(*

3

*

2

*

1 XXX


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The mean shift in the attack angle is given by

The variance of the mean shift

)()()( *333

*

22

2

*

11

1

* XXX

XXX

XXX

2

3

3

2

2

2

2

1

1

2)(

X

XX

XX

X


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The last two term on the right-hand side of the variance may be assumed to be negligible by properinstallation of the windshield and the accurate manufacture of the twist arm.

Then, the variance of the mean shift

The deviation of the attack angle due to the pivot axis may expressed in terms of the error in the

windshield assembly as received and the errors due to deviations of mounting point coordinates for

three holes, which may be expressed as

When is (d /dMi) Mi the smallest, the attack angle is immune to the variation of the mounting

position Mi.

2

1

1

2)(

X

X

holesthreeforscoordinatentpoi

mountingtheareMwhere i

3

1

1

1

1

1 i

i

i

receivedasM

MX

XX

X


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The net hole(solid circle) eliminates three DOF.The elongated hole allows in-and-out motion along

one direction thus using 2 DOF.

The loose hole only restricts vertical motion.

When the net hole is at Position #3, the meanshift and the standard deviation of the attack angle

were the smallest.

Once the #3 hole is fixed, the sensitivity was not

very high, regardless of whether #1 hole or #2 hole

was the elongated hole, although the results

favored making #1 the elongated hole.Figure 3.14. Location of three

mounting holes.


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Problem

Meaning of I/O , U/D and F/A

motions

Role and Principal of mounting

hole

Goal of mounting

Figure ex.2.4.h Location of Mounting Holes


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1. I/O , U/D and F/A

U/D : up and down motion

I/O : in and out motion

F/A : forward and aft motion

From Rotational motion

of 3 DOF


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1-1. Motion 1(I/O and F/A)

Plate

Wiper arm

F=T/L

N

Jointmotion

I/O and F/A motion: FromYawing Motion

Motion

I/O

F/A

Motion

Net

hole


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1-2. Motion 2-1(U/D)

U/D Motion:FromPitching and Vibration

MotionU : velocity of

air(140km/h over)

MotionVibration From

roadMotion

Net

hole


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1-2. Motion 2-2 (U/D)

U

U

U/D Motion : From Rolling Motion


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2-1. Roles of Mounting Points

Net hole:Eliminating 3DOF

Elongated hole:

Allows 2 motion(U/D, either I/O

or F/A)

Loose hole:

restricts the vertical motion(U/D)

Figure ex.2.4.h Location of Mounting Holes


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2-2. Principle mounting point

A : Nethole C : Loose

hole

B : Elongated

hole

1.Axis

between 2

point

2.Axis of

perpendicular

between 2point

A :

3DOF(x,y,z) motion is fixed

A and B :

Rotational motion of 2Axis

direction is fixed (1DOF)

B :Yawing motion(1DOF) control

and Vertical motion (rotational

motion of 1Axis direction

1DOF) allows

C :vertical motion control

6DOF motion of Plate is

controlled

Slot


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3. Goal of Mounting

Meaning :

Fix and control rigid body motion

of wiper linkage system

Keeping optimal attack angle

through to prevent vibration

problems during wiping

Figure ex.2.4.a Windshield and Windshield-

Wiper Assembly


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5.3.3.1 Fix the values of extra DPs

When the design is a redundant design the variance of the FRs can

be reduced by identifying the key DPs and preventing the extra DPs

from variation,

i.e., fixing the values of these extra DPs.


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5.3.3.2 Fix the values of extra DPs

Consider a multi-FR design.

By fixing DP4, DP5 and DP6 or by making the coefficient of these DP

zeros, we can make the design an ideal and uncoupled design.

The variation can also be reduced by setting A14, A15, A25, A26, A34 and

A36 to zero so that the FRs will be immune to changes of in DP4, DP5

and DP6.

6

000

000

000

5

4

3

2

1

363433

262522

151411

3

2

1

DPDP

DP

DP

DP

DP

AAA

AAA

AAA

FR

FR

FR


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5.3.4 Minimize the random variation of DPs and PVs

One way of reducing the variance of the FRs is to reduce therandom variation of input parameters since they contribute to the

total random variation of the FRs. The variance of FR may be

expressed as

By reducing the variance of any of the DPj, we can reduce the

contributions to the variance of FRi

. Moreover, if some of the DPs

are in dependent of one another, the relevant covariance terms

disappear from the above equation, further reducing the

contributions to the variance of FRi.

n

j

j

k

kjikij

n

j

DPjijFRi DPDPCovAAA1

1

11

222 ),(2


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5.3.5 Increase the design range

In some special cases, the design range can be increased

without jeopardizing the design goals.

Then, the system range may be inside the design range.( I = 0 )


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Example 4.1 Re-evaluation of buying a House

FR1 = Commuting time must be in the range of 15 to 30 minutes.FR2 = The quality of the high school must be good ,i.e.,more than 65% of the high

school graduates must go to reputable colleges.

FR3 = The quality of air must be good over340 days a year.(original)

FR4 = The price of the house must be reasonable, i.e., a four bedroom house with

3,000 square feet of heated space must be less than $650,000.

Town FR1=Commutetime[min]

FR2=Quality ofschool[%]

FR3=Quality ofair[days]

FR4=Price

[103$]

A 20 to 40 50 to 70 300 to 320 $450 to 550

B 20 to 30 50 to 75 340 to 350 $450 to 650

C 25 to 45 50 to 80 350 and up $600 to 800


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Town I1[bits] I2[bits] I3[bits] I4[bits] I

A 1.0 2.0 Infinite 0 Infinite

B 0 1.32 0 0 1.32

C 2.0 1.0 0 2.0 5.0

Figure 3.15. Probability distribution ofcommuting time

Figure 3.16. Probability distribution of thequality of schools


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If the range of air quality is expended from 340 days a year to 300days a year,

Town A is acceptable, but town B st i l l is best choice.

Town I1[bits] I2[bits] I3[bits] I4[bits] I

A 1.0 2.0 0 0 3

B 0 1.32 0 0 1.32

C 2.0 1.0 0 2.0 5.0

Then,


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5.3.6.1 Integration of DPs

The information contents can be made small by reducing the

likelihood of introducing errors when many physical parts are

assembledorby making the manufacturing operation simple.


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5.3.6.2 Integration of DPs

Examples)

Figure 3.17. Beverage can Figure 3.18. Bottle and can opener


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Example 5. Faucet design(for integration)

Figure 3.19. Three valves Figure 3.20. Two valves

Figure 3.21. Two valves Figure 3.22. One valve


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6.1. Common mistake by Designers

Coupling due to insufficient number of DPsIn an ideal design, the number of FRs and the number of DPs are the same.

Not recognizing a decoupled design

The decoupled design must be changed according to the proper sequence.

Having more DPs than FRs

The redundant design must fix extra DPs.


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6.2. Common mistake by Designers

Not creating a robust design- Not minimize the Information Content through

elimination of bias and reduction of variance

Concentrating on symptoms rather than cause- Importance of establishing and concentrating on FRs


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Example 6.1. Hood Lock-and-Release Mechanism

Make undesirable sound.

Analyze current design.

Design an improved design


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Original Design


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FR1 = Hold the pin (attached to the hood) in the locked positionFR2 = Release pin from locked position to an open position

C1 = Noise level should not exceed 50 dB.

C2 = Manufacturing cost cannot be more than current cost.

DP1 = Mechanical locking mechanism

DP2 = Release mechanism

2

1

2

1

0

0

DP

DP

X

X

FR

FR


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FR1 = Hold the pin (attached to the hood) in the locked position

FR11 = Locate the pin (attached to the hood) at the locked position.

FR12 = Lock the pin.

DP11 = A cam plate that provides a dead-stop positionDP12 = Rotating cam plate with a slot for the pin and a cam profile to

engage a spring-loaded ratchet mechanism

(to keep the ratchet spring loaded against the cam surface)

12

11

12

11

0

0

DP

DP

X

X

FR

FR


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FR2 = Release pin from locked position to an open position

FR21 = Release the pin

FR22 = Put the pin and the rotating disk at the normally open position

DP21 = Ratchet-removing mechanismDP22 = Spring force at the hinge of the hood to pull the pin and rotate

the rotating locking disk out of the locked positions

(lets put a spring on the hood hinge)

22

21

22

21

0

0

DP

DP

X

X

FR

FR


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FR22 = Put the pin and the rotating disk at the normally open position

FR221= Put the rotating disk at normally open position.

FR222= Put the pin at its normally open position.

DP221 = Soft spring of the latch (that replaces the heavy spring)DP222 = Equilibrium position determined by the spring force on the hood

hinge and the weight of the hood

This design is a completely uncoupled design.

222

221

222

221

0

0

DP

DP

X

X

FR

FR


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The original design differs from thisproposed design in that it relies on the

heavy spring to unlock and push the

hood upward.

To provide enough energy to

accelerate the hood upward, a heavy

spring was used.

Then a stopper had to be placed on

the lock plate to stop the cam.


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Original design

FR221 = Accelerate the pin (and hood) out of its closed position.

FR222= Put the cam plate at normally open position.

DP221 = Heavy spring/cam plateDP222 = Stopper

The original design is a coupled design, but noise is made

because of the conversion of mechanical energy to acoustic

energy by the stopper.

222

221

222

2210

DP

DP

XX

X

FR

FR


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7.1 Summary

The field of AD is a broad one that transcends specificengineering fields and encompasses such areas as

management and business.

The basic concept of AD theory are presented as a scientific

base.

The basic concepts and methodologies of AD include the

concepts of domains, mapping, the two design axioms,

decomposition, hierarchy, and zigzagging.


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7.2 Summary

Several key terms, such as functional requirement (FR), designparameter (DP), and process variable (PV) are defined. The

acceptance of these definitions is a pre-requisite in applying the

axiomatic principles for design.

Mapping between the domains generates design equations anddesign matrices. The design equation models the relationship

between the design objectives (whatthe design is trying to

achieve) and the design features (howthe design goals are to be

satisfied).

Uncoupled and decoupled designs are shown to satisfy the

Independence Axiom and thus are acceptable. Coupled designs

do not satisfy Independence Axiom and thus are unacceptable.


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7.3 Summary

The Independence Axiom states that the functional requirementsmust be maintained independent of one another by choosing

appropriate design parameters.

The Information Axiom deals with information content, the

probability of satisfying the FRs, and complexity.

To be able to satisfy the functional requirements, the designer

must always think in terms of FRs before any solution is sought.

Robust design is a design that satisfies the functionalrequirements easily, although large tolerances are given to DPs

and PVs.

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