A Comparison of Three Experimental Designs For

A COMPARISON OF THREE EXPERIMENTAL DESIGNS

FOR TOLERANCE ALLOCATION

A Thesis Presented to

The Faculty ofthe

Fritz J. and Dolores H. Russ College ofEngineering and Technology

Ohio University

In Partial Fulfillment

Ofthe Requirement for the Degree

Master ofScience

Ayman Eloseily

March, 1998 11

ACKNOWLEDGEMENTS

This thesis is dedicated to my wife Katrin, my mom and dad for their support and patience.

A special thanks to my advisor Dr. Richard Gerth for his help and support with his knowledge and expertise through the thesis.

Also, special thanks to the committee members, Dr. Patrick McCuistion and Dr.

David Koonce, for providing their knowledge and expertise. III

TABLE OF CONTENTS

ACKNOWLEDGEMENTS ii

TABLE OF CONTENTS iii

LIST OF FIGURES vi

LIST OF TABLES viii

CHAPTER

1. INTRODUCTION 1

THESIS OBJECTIVE 4

SCOPE 5

2. LITERATURE REVIEW 7

FULL AND FRACTIONAL FACTORIALS 8

T AGUCHI' S INNER AND OUTER ARRAyS 10

BISGAARD'S PRE- AND POST-FRACTIONATION 13

3. CASE STUDY - AN IDLER WHEEL 16

PRODUCT CHARACTERISTIC...... 18

ASSEMBLY SEQUENCE 22

TOLERANCE SPECIFICATION 30

4. METHOD 32 IV

HLM ANALySIS 33

THE EXPERIMENTAL DESIGNS 34

Standard Fractional Factorial 35

Double fractional factorial 39

Taguchi's Inner and Outer Arrays 43

EVALUATION MEASURES 46

5. RESULTS •••.•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••.••••••••••••.•••.••••••••.•••.•••••••.•••.••.••••.••..• 48

ANOVA RESULTS 48

NUMBER OF SIGNIFICANT INDEPENDENT VARIABLES: 52

NUMBER OF EXPERIMENTS & PARTS REQUIRED 55

Standard Design 55

Double Design (positive & negative designs) 56

Taguchi Inner and Outer Arrays Design 57

Bisgaard's pre andpost-.fractionation 58

6. DISCUSSIONS AND CONCLUSION 60

CONCLUSIONS 62

7. REFERENCES 64

APPENDICES 67

APPENDIX A. STANDARD FRACTIONAL FACTORIAL DESIGN 68

APPENDIX B. POSITIVE FRACTIONAL FACTORIAL DESIGN 69

APPENDIX C. NEGATIVE FRACTIONAL FACTORIAL DESIGN 70 v

APPENDIX D. TAGUCHI'S INNER-OUTER ARRAY WITH DATA 71

APPENDIX E. IDLER WHEEL DIMENSIONS 72 VI

List ofFigures

Figure 1. Taguchi Inner and Outer Array for 14 Variables 11

Figure 2. Idler Wheel Assembly 17

Figure 3. Dependent Variable Position 19

Figure 4. The Fourteen Independent Variables 21

Figure 5. Assembly tree diagram ofidler wheel assembly. . 22

Figure 6. Idler Wheel prior to Assembly 23

Figure 7. Base to Fixture Move Specification Window 24

Figure 8. Assembly after Base is Mounted on the Fixture 24

Figure 9. Assembly ofthe Journal to the Shaft 25

Figure 10. Assembly ofthe Shaft to the Base 26

Figure 11. Assembly ofthe Left Bushing to the Wheel. 27

Figure 12. Assembly ofRight Bushing to the Wheel. 27

Figure 13. Assembly ofthe Wheel Sub-Assembly to the Shaft 29

Figure 14. Assembly ofthe Hanger to the Base and the Shaft 30

Figure 15. Example Tolerance Specification Window. Lower and Upper Tolerances

specified for each Independent Variable 31

Figure 16. Comparison ofStandard Fractional Factorial and Double Fractional

Factorial for the Approximately Linear (a) and Non-Linear (b) Cases 40

Figure 17. Taguchi's Inner and Outer Array 45 vii

Figure 18. Standard and Double Fractional Factorial. 61

Figure 19. Comparison ofStandard Fractional Factorial Array and Taguchi's Inner-

Outer Array Design 62 VIII

List ofTables

Table 1. Independent Variables 20

Table 2. Independent variables and their values (Loose tolerance) 38

Table 3. Standard fractional factorial LI6 design 39

Table 4. L16 Array for Positive Design 41

Table 5. L16 Array for Negative Design 42

Table 6. Tight and Loose Tolerance Levels for Taguchi's Inner-Outer Array 44

Table 7. Pooled ANOVA for Standard Fractional Factorial. 49

Table 8. Pooled ANOVA for Positive Fractional Factorial. 50

Table 9. Pooled ANOVA for Negative Fractional Factorial. 50

Table 10. Pooled ANOVA for Taguchi Mean Response 51

Table 11. Pooled ANOVA for Taguchi Variance Response 51

Table 12. Summary Table ofSums of Squares and Percent Contributions ofthe

Different Designs and the HLM Analysis 53

Table 13. Number ofParts Required in the Standard Fractional Factorial. 56

Table 14. Number ofParts Required in the Double Fractional Factorial. 57

Table 15. Number ofParts Required by the Taguchi Design 58

Table 16. Number ofParts Required by Bisgaard's Prefractionation 59 CHAPTERl

INTRODUCTION

This thesis examines three different experimental designs for their applicability to tolerance analysis. This chapter briefly describes the applied problem that motivated the investigation. The concepts oftolerance design and the inadequacies ofcurrent methods will be explained. Designed experiments are proposed as an alternative to the current methods. However, the best experimental design for attacking the tolerance design problem is yet to be determined. The thesis objective is determining the best ofthree proposed experimental designs. Finally the chapter closes with a briefdescription ofthe chapters to follow.

Mass production is based upon the concept ofinterchangeable components

(Bisgaard, 1993). Interchangeability is the concept that all components ofthe same type, which are within acceptable tolerances, can be interchanged in an assembled product without affecting product performance. Thus, acceptable product performance is a function ofthe individual component tolerances.

Tolerances reflect a designer's intentions regarding product functional and behavioral requirements with corresponding implications for manufacturing and quality control. They specify the acceptable limits within which a component dimension is permitted to vary (ASME, 1994). This determines the choice ofmanufacturing processes 2 and is a factor in the final production cost. Tight tolerances can result in expensive manufacturing processes, increased inspection, and excessive processing cost; while loose tolerances may lead to increased product rejects, warranty returns, and dissatisfied customers. The engineer must design high quality products and processes at low cost, by specifying the allowable component dimension variation (i.e., tolerances) that will result in component interchangeability.

A common tolerance design problem is that the product designer has identified the allowable product performance variation and must determine the allowable component variation, that when combined, will result in an acceptable product. Tolerance design is difficult because the designer must understand how the component variations combine to determine the final product performance variation.

In practice, tolerances are most often assigned as an informal compromise between functionality, quality and manufacturing cost (Gerth, 1997). Thus, traditional tolerancing methods use past designs, handbooks, rules ofthumb, and recently CAD default settings. These methods can be imprecise, not based on relevant data, and/or insufficient to guarantee a cost effective, quality design.

Tolerance design techniques can be classified into two categories: tolerance analysis and tolerance synthesis (Lee and Woo, 1993). Tolerance analysis determines the assembly distribution given the individual component dimension distributions. Common tolerance analysis methods are worst case analysis, statistical tolerancing (Evans, 1974; 3

Gerth, 1992), and Monte Carlo simulation (Evans, 1975; Araj and Ermer, 1989; Bjorke,

1989; Gerth, 1992).

Tolerance allocation determines the individual dimension distributions given the specified assembly distribution. Tolerance allocation methods include standards, uniform and proportional scaling (Chase and Greenwood, 1988), various minimum cost optimization algorithms (Gerth, 94; Sayed and Kheir, 1985; Spotts, 1973), and Taguchi methods (Liou, et. aI., 1993).

All current tolerancing algorithms, with the exception ofTaguchi methods, require the functional relationship between component dimensions and product performance to be known or specified. But, in many industrial cases the relationship is unknown. This often happens when physical models ofphenomena are not adequately understood, such as electromagnetic forces, frictional losses, wear phenomena, etc.

For example, in inkjet printers and important performance characteristic is the skew angle: the angle at which the print is printed on the media. Factors such as width of the media tray, diameter ofthe pick and pinch rollers, and position ofthe chassis can affect the skew angle. Since these factors are geometric in nature, their relationship to the skew angle can be determined, and therefore, their tolerances determined using one ofthe above mentioned techniques. However, other important factors that affect the skew angle are deflection ofthe components under load (the components are not rigid) and roller slippage. Their effect on skew is not known and cannot be analytically determined. In 4 such a case, experimental design can be used to determine an empirical relationship between the component variation and performance measure.

An experiment is a test in which purposeful changes are made to the input variables ofa process or a system so that we may observe and identify the reasons for changes in the output response (Montgomery, 1991). The input variables are called independent variables and the response is called the dependent variable. For tolerance design, the independent variables would be the component variation sources and the dependent variable is the product performance measure.

The number ofexperiments conducted, the order in which they are conducted, and the manner in which in the independent variables are changed is called the experimental design. There are a wide variety ofexperimental designs for different purposes. One of the difficulties in conducting designed experiments is determining the most suitable experimental design.

Thesis objective

The objective ofthis research is to compare three alternative experimental designs as to their applicability to correctly identify the component variations that have a statistically significant effect on product performance. The comparison is based on the number ofcorrectly identified independent variables, number ofcomponent parts required, and number ofexperiments required; fewer parts and experiments is better.

The null hypothesis in this research is that the three experimental designs will give the same results at the 0.10 significance level. 5

The three experimental designs are:

1. A standard fractional factorial.

2. A double fractional factorial consisting oftwo standard fractional

factorials called the positive and the negative fraction.

3. Taguchi's inner and outer arrays.

To determine which design can correctly identify the significant component variations requires a case study where the correct answer can be determined. This is possible ifthe case study can be simulated. The case study selected was an idler wheel.

Thus, a simulation was used to evaluate 3 experimental designs, which ifsuccessful, could be applied to other case studies where simulation cannot be used. The simulation program was 3Dcs (Dimensional Control Systems Inc., 1996). It was used not only to determine the correct sources ofvariation, but also to conduct each experiment instead of constructing physical prototypes.

Scope

This thesis focuses on evaluating the different experimental designs. It is assumed that the product components would be manufactured in a prototype shop to ensure that the components are produced to their upper and lower limits (even though simulation is used in the thesis). This cannot be ensured by production manufacturing processes. Issues related to geometric design and tolerancing (GD&T) are also not addressed. Thus, the design drawings are assumed to be correct and issues such as datum 6 selection are not addressed. Also, process-planning issues are only addressed in as much as they affect the simulation model (see Chapter 3 - Case Study).

The thesis will proceed as follows. In chapter 2, two experimental design methods will be discussed: Taguchi methods and Bisgaard's pre- and post- fractionation.

Taguchi was one ofthe methods that were evaluated, and Bisgaard's method is related to the double fractional factorial.

Chapter 3 presents the idler wheel case study and the 3Dcs software package including the High Low Median (HLM) analysis that is used to determine the correct variance contributions. The idler wheel simulation program is also explained here.

Chapter 4 presents the 3 experimental designs and the steps used to conduct the experiments

Chapter 5 presents the analysis and results. The three designs were compared according to the following parameters:

1. The number ofsignificant independent variables correctly identified

(within ±5%) compared to the 3Dcs HLM analysis. and as secondary measures :

2. The number ofexperiments required for each experimental design

(number oftests / assemblies required).

3. The number ofparts manufactured to run the experiments for each design.

In chapter 6, the differences between the different experimental designs are discussed and conclusions drawn. 7

CHAPTER 2

LITERATURE REVIEW

This chapter briefly describes the few research investigations work conducted in the area ofexperimental design tolerancing assembled products. The most relevant studies conducted on tolerancing using designed experiments are:

1- Taguchi's inner and outer array method (Liou, et al, 1993; Islam, 1995) which

is one ofthe designs used in this thesis, and

2- Bisgaard's pre- and post-fractionation concepts which are also used in the

double fractional factorial, but in a slightly altered form.

Although it is beyond the scope ofthis thesis to explain experimental designs, two

k p fundamental designs, the full factorial and the 2 - fractional factorial will be briefly described. For a more detailed explanation the interested reader is referred to

Montgomery, 1993.

Throughout the discussion, the reader should keep in mind, that the scenario assumes the functional relationship is not known, and that empirical methods are required to estimate the function. These empirical methods are conducted by physically building and assembling prototype products according to some experimental design paradigm.

The product performance measure is then measured. 8

Full and fractional factorials

In a designed experiment, the independent variables are changed from one level to another and the system response observed. The manner in which the independent variables are altered is called the experimental design. For example, ifthere are 5

independent variables, ofwhich each can be changed to one oftwo levels, then there are

25 = 32 possible variable combinations. An experiment that evaluates all possible

variable combinations is called a full factorial. Ifevery variable in the full factorial has

only 2 levels, it is called a 2k full factorial, where k is the number ofindependent variables in the experiment.

Full factorials are popular because they are easily created, and they provide a

wealth ofinformation, in particular about the interaction effects between the variables. In

the 25 example above, there are 5 main effects, 10 second order interactions, 10 third

order interactions, 5 fourth order interactions, and 1 fifth order interaction. The

disadvantage is that the number ofrequired experiments grows exponentially with the

number ofindependent variables. Thus, to investigate 7 variables would require 128

experiments.

Fractional factorial designs are a method ofreducing experimental size for

experiments involving a large number ofvariables. It is based on the experience that

lower order effects, such as main and second order interaction effects, dominate a

system's response compared to the higher order effects. Thus, ifone is willing to assume

that higher order effects are negligible compared to lower order effects, the experimental 9 size can be reduced by sacrificing information on the higher order effects. This is done by running only a fraction ofthe full factorial design. In essence the runs containing the information on the higher order effects are not conducted.

k p Fractional factorials are denoted 2 - , where k is the number ofvariables in the design and p is related to the fraction run. These designs, like full factorials, only exist in

3 1 7 4 powers of2, i.e., 2 - = 4 runs, 2 - = 8 runs, and so on. The maximum level of fractionation that is allowed is where one has as many variables as experimental runs minus one. These are called fully saturated fractional factorial designs. Thus, in the above example, one could investigate either 5 or 7 variables in only 8 experiments.

The major disadvantage is the loss ofinformation in the higher order effects.

Indeed, these higher order effects are actually aliased with the lower order effects. In other words, when statistical significance is indicated, it will be unclear wither it is due to the lower order effect indicated, or due to a higher order aliased effect. Usually, two rules ofthumb are used:

1. Lower order effects are more likely to be significant than higher order

effects.

2. Usually, the higher order interaction will contain significant lower order

effects.

These two rules are used to interpret the results ofthe experiments. 10

Taguchi's Inner and Outer Arrays

Taguchi developed a concept called parameter design. It is used to determine the nominal dimensions ofa product that makes the product insensitive to variation in noise.

Thus, there is a distinction between control variables: the variables for which the nominal dimension is be determined, and noise variables: the variables which cause unknown variation in the product. This is done through the use offractional factorial designs, also

called orthogonal arrays. He denotes them as L4, Ls, L16, and so on, depending on the number ofrequired runs.

The concept ofparameter design is to determine the levels ofthe control variables that will reduce in the minimum variation despite variation in the noise variables. The control variables are laid out in an OA called the inner array, and the noise variables are laid out in an OA called the outer array. These two arrays are then crossed. Thus, every run ofthe inner array is run under varying noise conditions. The structure ofthe inner and outer orthogonal array is shown in Figure 1. 11

Outer Array

Run 1 2 3 ... 16

A -1 +1 -1 ... +1

B -1 -1 +1 ... +1

C +1 -1 -1 ... +1

......

Inner array N -1 -1 +1 ... +1

Run A B C ... N

1 -1 -1 +1 -1 Y1,1 Y 1,2 Y 1,3 ... Y 1,16

2 +1 -1 -1 -1 Y 2,1 Y 2,2 Y 2,3 ... Y 2,16

3 -1 +1 -1 +1 Y 3,1 Y 3,2 Y 3,3 ... Y 3,16

......

16 +1 +1 +1 +1 Y 16,1 Y 16 2 Y 16,3 ... Y 16,16

Figure 1. Taguchi Inner and Outer Array for 14 Variables.

The concept has been adapted to determine the optimal tolerance levels that will result in a design that is least sensitive to the dimensions variation within the tolerances

(Liou, et al, 1993). Islam, (1995) describes Taguchi's inner and outer arrays as follows.

"The inner OA is used to study the effect ofthe independent variables and

the outer OA provides noise to each independent variable. The objective of

the inner OA is to determine the significance of the independent variables 12

and to select the levels of the significant variables that optimize the

performance measure. Optimal performance is one that exhibits minimum

variation around a target value. The variation is considered to be due to

noise factors. The outer OA introduces noise factors in the experiment in

a systematic manner. Thus, it is possible to analyze and select the proper

levels ofthe control factors that are least sensitive to noise and will result

in the minimum variance response.

Each outer OA noise combination is treated as a replication of the inner

OA. Thus, if an outer OA requires Lo runs for each run of an inner OA

that require L, runs, the size of the experiment is L, * L, runs. The inner

array specifies the combination oftolerances (loose = 1, tight = 2) used for

each independent variable (component feature). The outer array specifies

the level of noise combinations; i.e., the direction (upper tolerance = +,

lower tolerance = -) that the tolerances deviate from their nominal

position. Together, they define a unique product, which component

features are at the extremes oftheir tolerances."

Islam concluded that a major disadvantage ofthe inner-outer array method is the large number ofexperiments required. For example, determining the variance

contribution of 14 variables requires both an L 16 inner and outer array resulting in 16*16

= 256 experiments. 13

Bisgaard's pre- and post-fractionation

Bisgaard (1993) developed a new method for determining the component tolerance limits ofassembled products also utilizing fractional factorial designs.

However, he uses cost ofexperimentation as a guide to determining his experimental designs. He states that costs are incurred due to two factors:

1. The number ofcomponents to be manufactured for conducting the experiments, and

2. The number ofproducts assembled and tested.

Bisgaard illustrates his method by the following example. Suppose a product consists ofthree mating component types 1, 2, & 3, each with three dimensions (A, B, C),

(D, E, F), (G, H, J). A full factorial design would require 23 = 8 components ofpart number 1 to be manufactured. Similarly for parts 2 and 3, resulting in 24 total parts.

Now, these 24 parts can be assembled in a total of8*8*8 = 512 different ways. Each of the assemblies then needs to be tested to measure the performance characteristic. The cost ofthe full factorial experiment can be reduced by using fractional factorials to reduce the number ofcomponents built (pre-fractionation) and the number ofassemblies produced (post-fractionation).

For example, to pre-fractionate component 1, 4 components can be built

3 1 according to a 2 - experimental design. The other components can also be pre

fractionated resulting in only 4+4+4=12 components instead ofthe 24 required by a full

factorial design. Ifthese 12 components are assembled in all possible combinations, it

would result in only 4*4*4 = 64 assemblies. 14

The number ofrequired assemblies can be further reduced by post-fractionating the assembly matrix. The post-fractionation process is very complex and requires an understanding offractional factorial designs.

Fractional factorial designs sacrifice information on higher order effects by aliasing them with lower order effects. The alias structure is governed by the defining relationship, I. For example, ifthe ABC interaction is sacrificed in a 23 design, then

I=ABC, and C can be estimated by aliasing it with the AB interaction: C=AB. The alias structure is obtained by multiplying modulo 2 the effect with the defining relationship, where I is the identity. So, in the above example: C*I = ABC2 => C = AB. Another important concept is design resolution. It defines the level ofaliasing present, and is defined as the length ofthe shortest word in the defining relationship. A resolution III design is the lowest acceptable design resolution.

Returning to the example, the prototype components are made according to three

3 1 2 - designs with defining contrast subgroups 11= ABC, 12 = DEF and 13 = GHJ respectively. Thus, the defining contrast for the full assembly design is I = ABC = DEF=

GHJ = ABCDEF = ABC GHJ = DEFGHJ = ABCDEFGHJ (the 3 subcontrasts and their crossproducts). Since the length ofthe shortest word is 3, the design is ofresolution III.

Now, we can clearly see that ifwe fractionate on any single or two-letter word, then the defining relationship will fall below III, which is unacceptable. So, to post fractionate one must select a letter from each ofthe subcontrasts and form a new assembly

subcontrast. For example, 14 = ADG would be acceptable, as would I5=BEH. Using 14 15 the new defining contrast becomes I = ABC = DEF= GHJ = ABCDEF = ABC GHJ ==

DEFGHJ = ABCDEFGHJ = ADG == BCDG == AEFG = ADHJ = BCEF == ABHJ = EFJH

= BCEFHJ. This is still a resolution III design. One can still fractionate further with Is, but not beyond that, because the resolution will drop too low.

Since each subcontrast represents a halffraction, i. e., dividing the number of

experiments by two, the number ofresulting assemblies that must be tested

postfractionating with 14 and Is results in 64/2/2 = 16 assemblies. This reduces the cost of

assembly and testing.

The advantage ofBisgaard's method is that it cuts down prototype parts that need

to be produced and the number ofassemblies that need to be evaluated. This reduces

both the costs ofproducing parts and running the experiments.

In his case study, Bisgaard used the nominal and upper tolerance limits as the

factor levels. This is the same as what will be called the positive fractional factorial

design in Chapter 4. And his pre- and post-fractionation method will be used to

determine the number ofcomponents and assemblies that would be required in the case

study. This result will be compared with the number ofcomponents and assemblies that

are required by the other methods.

In chapter 3, the model chosen for the thesis and the steps required to be

performed before running the experimental design will be discussed. 16

CHAPTER 3

CASE STUDY - AN IDLER WHEEL

The research objective is to evaluate alternative experimental designs as to their applicability to tolerance design when the underlying stackup function is unknown. In order to conduct an evaluation, the results ofthe various designs are to be compared to a known result. Thus, the case study must have a stackup function that can be derived or modeled. In addition it must be an assembled product in order to have a stackup function.

The idler wheel (Figure 2) meets these criteria and was used to demonstrate the three experimental designs. Specifically,

1. a geometric product performance measure was selected so that it is a function

ofgeometric dimensions which can be modeled, and

2. it consists ofseparate components that can be assembled and disassembled,

thus creating a tolerance stack.

The function ofan idler wheel is to apply tension to a moving belt with variable speeds and to keep the belt from slipping out ofplace. The idler wheel consists ofsix parts: base, hanger, shaft, wheel, left bushing, and right bushing. Each component is

manufactured separately and then assembled. 17

Right Bushing

Figure 2. Idler Wheel Assembly.

3Dcs was used to simulate the idler wheel assembly. First, AutoCAD was used to draw a three-dimensional drawing ofthe idler wheel. The 3D drawings were saved as

IGES files and then imported into 3Dcs. 18

Once the part geometries have been specified through the CAD files, constructing a 3Dcs simulation model consists ofthree major steps: constructing the assembly sequence, specifying the tolerances on the individual component dimensions, and defining the product characteristic or measurement dimension. From a programming point ofview, the product characteristic must be defined after the assembly sequence has been determined. However, for ease ofcomprehension, this section will be explained first.

Product Characteristic

The important product characteristic is the sum clearance between the left and right bushings, and base and hanger axle supports (Figure 3). The sum clearance is given by

[1]

Where:

a l = Clearance between left bushing top and base bracket

a 2 = Clearance between left bushing bottom and base bracket.

~l = clearance between right bushing top and hanger bracket.

~2 = clearance between right bushing bottom and hanger bracket.

The clearance should be large enough to permit easy rotation ofthe wheel and two bushings. Ifthe clearance is zero then the two bushings would rub against the base and the hanger axle support creating excessive friction and resulting in product failure. The 19 variation in the sum clearance is affected by fourteen tolerances (Figure 4). The fourteen independent variables are given in Table 1.

Figure 3. Dependent Variable Position. 20

Table 1. Independent Variables.

Label Variable

A Shaft length

B Base Axle Support (perpendicular)

C Base Axle Support hole true position

D Base holes diameter

E Base right hole true position

F Base left hole true position

G Hanger Axle Support (perpendicular)

H Hanger right hole true position

I Hanger Axle Support hole true position

J Hanger holes diameter

K Hanger left hole true position

L Wheel length

M Right Bushing head thickness

N Left Bushing head thickness 21

I ... I J --I D ____ 1 ~_~- G _0 .; -T0fo 0

~~~ H,K

L A E,F

B J------;. ~ ______...

Figure 4. The Fourteen Independent Variables. 22

Assembly Sequence

The assembly sequence is called an assembly tree (see Figure 5). It consists of

individual assembly processes called moves, which specify which objects move to other

objects thereby forming subassemblies. A feature of3Dcs is that it allows one to

graphically view the move statements. The idler wheel prior to assembly is shown in

Figure 6. Each move statement will be explained in detail accompanied by the screen

shot showing the assembly after the move statement has been invoked.

Base ~ Fixture -

Base &Shaft ~ Sub-assembly ---- Shaft - Shaft f-- -. Sub-ssembly

Journal f--- -, Wheel - ----. Final Assembly

<, -. Wheel Left Bushing f-- -.. Sub-assembly ----

Right - bushing

Hanger ~

Figure 5. Assembly tree diagram ofidler wheel assembly 23

~.... ~4

) I

J'"16

Figure 6. Idler Wheel prior to Assembly.

The assembly process is as follows:

1. The base was mounted on the fixture using a three-two-one point move. A

screen shot showing the move statement window is shown in Figure 7. The

three points on the bottom ofthe base specify the primary plane (z-axis), the

centers ofthe two forward holes specify the secondary plane (y-axis), and an

arbitrary point specifies the tertiary plane (x-axis). A total number of 12

points have to be specified, 6 points on the base (object) and 6 matching

points on the fixture (target). Figure 8 shows the assembly after the move is

complete. 24

·• 1 ~ ~!M ' • 1 I•

, i-. l..<';"j ..:'~ .:: ;.".'~" .::::} :;..: _ ,I ~

Figure 7. Base to Fixture Move Specification Window.

Figure 8. Assembly after Base is Mounted on the Fixture. 25

2. The assembly process tree shows a subassembly for the shaft and the journal.

When manufacturing physical prototypes ofthe shaft, both the journal and the

shaft are one part. However, 3Dcs can only accumulate variation through

move statements. To simulate a manufacturing process, where the journal on

the shaft is made, the shaft then refixtured to make the other journal, requires

creating a part called journal and moving it onto the shaft (Figure 9). This was

done using a two-point move along the journal and shaft centers, thus forming

the shaft subassembly.

Figure 9. Assembly ofthe Journal to the Shaft.

3. The shaft was assembled to the base assuming interference fit using a two-

point move. The move utilized the two center points ofthe shaft journal and 26

the base axle support hole. This process forms the base and wheel sub

assembly (Figure 10).

Figure 10. Assembly ofthe Shaft to the Base.

4. Next the bushings were assembled to the wheel using a two-point move. The

center points ofthe busing and the wheel axis were used. The wheel axis

consisted oftwo points (wl, w2), one at either end ofthe wheel. The bushings

centerline consisted ofthree points, one at the center ofthe face that faces the

base (bI), one at the center ofthe face that faces the wheel (b2), and one at the

end ofthe bushing (b3). The two point move consisted ofmoving b2 to wI

(coincident), and moving b3 collinear with w2. The assembled left bushing is

shown in Figure 11 and the right bushing in Figure 12. 27

Figure 11. Assembly ofthe Left Bushing to the Wheel.

~ 3_Dcs - (SEP9:2:Graph] Pir.:l ~

Figure 12. Assembly ofRight Bushing to the Wheel. 28

5. The wheel subassembly was moved to the shaft assuming a gravity fit with a

two-point move. A gravity fit is used in conjunction with a clearance move

where the wheel is dropped onto the shaft the amount the clearance allows

after the two point move has been completed. The first object and target

points ofthe two point move are the center ofthe left bushing outer face (b1)

to the center ofthe shaft left face where the journal begins. The second object

and target points are the center ofthe right bushing outer face to the center of

the shaft right face. Note that this can cause interference on the left-hand side.

However, because the product characteristic is a sum clearance, the

interference will simply be subtracted from the clearance on the other side.

This subtraction represents the freedom the wheel has to move along the shaft

and is thus acceptable. The assembled wheel subassembly to the shaft is

shown in Figure 13. 29

Figure 13. Assembly ofthe Wheel Sub-Assembly to the Shaft

6. The final assembly step is to move the hanger to both the base and the right

side ofthe shaft using a three-two-one point move. This was very difficult

because there were several ways this part could be located. For example, the

part could be completely located by the two base holes the base. But, then the

shaft position would not affect the hanger position at all. Hence, it was

decided to use the shaft as a secondary locating feature. The primary locating

features were the centers ofthe two hanger holes and the bottom center ofthe

hanger front face. This was aligned with were the centers ofthe two base

front holes and the center ofthe base step. The secondary locating surface 30

was defined by the center points the shaft journal. These were aligned with

the two center points ofthe axle support hole. The tertiary feature was simply

the center ofthe hanger front face, which was aligned with the bottom center

ofthe hanger front face. The three points position the hanger on the base in

the z-axis, the two points position the hanger to the shaft in the y-axis, and the

one point positions the hanger to the shaft in the x-axis. The completed

assembly is shown in Figure 14.

Figure 14. Assembly ofthe Hanger to the Base and the Shaft.

Tolerance specification

The next step is to specify the tolerances for each dimension. This involves stating the upper and lower specification limit, selecting a distribution, and stating how 31 many sigma units the tolerance range represents. All tolerances in the study were assumed to be normal with a +/- 3 sigma spread between the upper and lower tolerance limit. An example tolerance specification window is shown in Figure 15. This information is primarily used for the software's Monte Carlo simulation, where the resulting product performance distribution can be determined. Since this thesis did not use the Monte Carlo simulation it will not be explained here. However, the tolerances were still entered and selectively turned on and offaccording to the various experimental designs.

Two tolerance levels were used in the experiments: a loose and a tight level. The loose levels were used in the standard and double fractional factorial designs. The loose and the tight levels were used in the Taguchi design. The specific tolerances and their usage are explained in more detail in Chapter 4.

Figure 15. Example Tolerance Specification Window. Lower and Upper Tolerances specified for each Independent Variable 32

CHAPTER 4

METHOD

This chapter describes the method used to evaluate the different experimental designs. To conduct the experimental investigation and construct the various idler wheels, a 3Dcs software simulation program was used to create and assemble the components and measure the sum clearance between the left & right bushings and base

and hanger axle supports respectively. The specific prototypes constructed were

determined according to the different experimental designs. The results ofthe

experiments were then compared with a HLM report provided by the simulation software

3Dcs.

Usually complete randomization should be used during an experiment to prevent

the influence ofunknown and uncontrolled variables (Mead, 1988; Baker, 1990).

However, since the experiment was conducted using computer simulation, all variables

were known and controlled, and thus, randomization was not necessary

This chapter will first explain the HLM analysis, followed be the 3 experimental

designs, and finally the evaluation criteria by which suitability ofthe designs to tolerance

analysis were determined. 33

HLM Analysis

In 3Dcs, HLM (High-Low-Median) analysis, also known as a sensitivity analysis, is used to determine the percentage ofvariation the tolerances cause in the assembly characteristics being measured. HLM simulations vary each dimension to its upper tolerance value (high), lower tolerance value (low), and median value (the 0.9987,

0.0013, and 0.50 percentile points ofthe input distribution respectively). This is done for each dimension, one at a time, while holding all other dimensions at their nominal values.

As the ith tolerance is varied through it high, low, and median value, the maximum

(Max.) and minimum (Min.) values ofthe assembly characteristic are recorded. The range, R, is then computed and the variance, Sj2, estimated by:

~~D. = Max.1 - Min.1 [2]

[3]

The variance contribution ofeach tolerance in the assembly are then summed to determine the total variance in the assembly characteristic Y, S;:

k si = IS} [4] ;=]

where k is the number oftolerances in the assembly. The percentage, Pi' that the ith tolerance contributes to the overall variation ofthe assembly characteristic:

[5] 34

The HLM report presents these percentages in decreasing order along with their tolerances. The report was assumed to be the correct analysis ofthe case study and used to compare the results ofthe experimental design results.

The HLM analysis assumes that all dimensions are normally distributed, which they were in the simulation. It also assumes that only main effects are significant and interaction effects are negligible. A main effect is the effect ofa variable all by itself on the output measurement results. An interaction effect is when the result oftwo or more variables combined is greater than the sum ofeach oftheir main effects. The fractional factorial designs also assume that interaction effects are negligible relative to main effects. rrhe Experimental Designs

Three experimental designs were evaluated as to their applicability to allocate tolerances using the specific case study ofan idler wheel. The three designs are:

1. Standard fractional factorial.

2. Double fractional factorial consisting ofa

a) Positive fractional factorial.

b) Negative fractional factorial.

3. Taguchi's inner and outer arrays.

The experimental designs attempt to determine how much each tolerance contributes to the variation in the dependent variable. Those that have a large 35 contribution will be held to their tight values, and those that contribute little will be held to their loose tolerance value. Each experimental design will be explained using the case study.

To conduct the experimental investigation and simulate the various idler wheels,

Monte Carlo simulation in the 3Dcs software was used to measure the sum clearance between the left & right bushings and base and hanger axle supports respectively. HLM simulation was used to generate the independent variables percentage contribution lists

Usually complete randomization should be used during an experiment to prevent the influence ofunknown and uncontrolled variables (Mead, 1988; Baker, 1990).

However, since the experiment was conducted using computer simulation, all variables were known and controlled, and thus, randomization was not necessary

Standard Fractional Factorial

In any experimental design, there are four classes ofvariables, dependent, independent, constant, and others. Since simulation is used in obtaining the data for the four experimental designs, there are only the dependent and independent variables.

Constant variables are those held constant by the experimenter to reduce experimental runs. The remaining variables, called "rest", are those not controlled during the experiment including "lurking" variables that are unknown to the experimenter and may effect the results. However, a computer model has no "others" or "lurking" variables.

Any variable that does not change by design, i.e. is not an independent variable, is by definition constant. 36

The dependent variable is the sum clearance between the left & right bushings and base & hanger axle supports respectively (Figure 3). The variation in the sum clearance is affected by fourteen independent variables (Figure 4).

A single replicate or a 214-7 resolution III fractional factorial design with 16 runs was used instead ofa full factorial design to reduce the number ofexperimental runs required. This was done because in reality one would wish to reduce the cost of producing prototype components and the time ofconducting the experiments. The design is shown in Table 3.

Resolution III designs are designs in which no main effects are aliased with any other main effect, but main effects are aliased with two-factor interactions and two factors interactions may be aliased with each other (Montgomery, 1984). Since there are

14 factors, an L16 table with 15 degrees offreedom is required. The L16 table in standard order is shown in Table 3. The plus and minus ones in the table represent the high and low levels ofthe independent variables. Thus, each row represents a particular assembly for which the sum clearance can be determined. For example the first row is a shaft built to the lower tolerance (A) assembled to a base support at its lower perpendicularity tolerance (B) and lower hole true position tolerance (C) etc.

Table 2 summarizes the independent variables and there high and low levels, which are the upper and lower specification limits ofthe loose tolerances. The loose, as opposed to the tight tolerances were selected to increase the precision and applicability of the results. 37

14 7 A single replicate or a 2 - resolution III fractional factorial design with 16 runs was used instead ofa full factorial design to reduce the number ofexperimental runs required. This was done because in reality one would wish to reduce the cost of producing prototype components and the time ofconducting the experiments. The design is shown in Table 3.

Table 2. Independent variables and their values (Loose tolerance)

LEVEL

Independent Variables High Nominal Low

A : Shaft length 3.505 3.5 3.495

B : Base Axle Support (perpendicular) 1.6325 1.63 1.6275

C : Base Axle Support hole true position 2.89 2.88 2.87

D : Base holes diameter 0.5025 0.5 0.4975

E : Base right hole true position 6.2 6.19 6.18

F : Base left hole true position 6.2 6.19 6.18

G : Hanger Axle Support (perpendicular) 0.255 0.25 0.245

H : Hanger right hole true position 1.32 1.31 1.3

I : Hanger Axle Support hole true position 2.51 2.5 2.49

J : Hanger holes diameter 0.5025 0.5 0.4975

K : Hanger left hole true position 1.32 1.31 1.3

L : Wheel length 3.26 3.25 3.24

M : Right Bushing head thickness 0.13 0.125 0.12

N : Left Bushing head thickness 0.13 0.125 0.12 39

Table 3. Standard fractional factorial L16 design. ii~luns A B C D EFG H I J K L MN 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1

2 1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 -1

J -1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 1

4 1 1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 1

5 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1

6 1 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1

7 -1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 -1

8 1 1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1

9 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 1 1

10 1 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1

11 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1

12 1 1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 -1

13 -1 -1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1

14 1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 -1

15 -1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1

16 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Double fractional factorial

The standard fractional factorial suffers from a major disadvantage. Since the

standard only uses two levels, it can only capture relationships that are approximately 40 linear (see Figure 16a). Ifthe relationship is not linear, for example parabolic (see Figure

16b), the standard will not correctly identify the variance contribution ofthe tolerance.

However, using two fractional factorial designs that vary from the nominal to the upper tolerance limit (positive design) and one that varies from the nominal to the lower tolerance limit (negative design) may alleviate the problem and still not require as many runs a full factorial or a Taguchi design.

Standard fractional factorial y y Double fractional factorial

Tolerance Tolerance (a) (b)

Figure 16. Comparison ofStandard Fractional Factorial and Double Fractional Factorial

for the Approximately Linear (a) and Non-Linear (b) Cases.

Double fractional factorial follows the same procedure as a standard fractional factorial except for performing two designs, negative and positive, each design with 16 runs. The negative design tests the variation between the low tolerance (-1) and nominal values (0) for each variable (Table 4). While, the positive design tests the variation between the nominal (0) and upper tolerance values (+1) for each variable (Table 5). 41

Table 4. L16 Array for Positive Design

Runs A B C D E F G H I J K L M N 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0

2 1 0 0 0 0 0 0 1 1 1 1 1 1 0

3 0 1 0 0 0 1 1 0 0 1 1 1 0 1

4 1 1 0 0 1 0 0 0 0 1 0 0 1 1

5 0 0 1 0 1 0 1 0 1 0 1 0 1 1

6 1 0 1 0 0 1 0 0 1 0 0 1 0 1

7 0 1 1 0 0 0 1 1 0 0 0 1 1 0

8 1 1 1 0 1 1 0 1 0 0 1 0 0 0

9 0 0 0 1 1 1 0 1 0 0 0 1 1 1

10 1 0 0 1 0 0 1 1 0 0 1 0 0 1

11 0 1 0 1 0 1 0 0 1 0 1 0 1 0

12 1 1 0 1 1 0 1 0 1 0 0 1 0 0

13 0 0 1 1 1 0 0 0 0 1 1 1 0 0

14 1 0 1 1 0 1 1 0 0 1 0 0 1 0

15 0 1 1 1 0 0 0 1 1 1 0 0 0 1

16 1 1 1 1 1 1 1 1 1 1 1 1 1 1 42

Table 5. L16 Array for Negative Design

Runs A B C D EF G H I J K L M N 1 -1 -1 -1 -1 0 0 0 0 0 0 -1 -1 -1 -1

2 0 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 -1

3 -1 0 -1 -1 -1 0 0 -1 -1 0 0 0 -1 0

4 0 0 -1 -1 0 -1 -1 -1 -1 0 -1 -1 0 0

5 -1 -1 0 -1 0 -1 0 -1 0 -1 0 -1 0 0

6 0 -1 0 -1 -1 0 -1 -1 0 -1 -1 0 -1 0

7 -1 0 0 -1 -1 -1 0 0 -1 -1 -1 0 0 -1

8 0 0 0 -1 0 0 -1 0 -1 -1 0 -1 -1 -1

9 -1 -1 -1 0 0 0 -1 0 -1 -1 -1 0 0 0

10 0 -1 -1 0 -1 -1 0 0 -1 -1 0 -1 -1 0

11 -1 0 -1 0 -1 0 -1 -1 0 -1 0 -1 0 -1

12 0 0 -1 0 0 -1 0 -1 0 -1 -1 0 -1 -1

13 -1 -1 0 0 0 -1 -1 -1 -1 0 0 0 -1 -1

14 0 -1 0 0 -1 0 0 -1 -1 0 -1 -1 0 -1

15 -1 0 0 0 -1 -1 -1 0 0 0 -1 -1 -1 0

16 0 0 0 0 0 0 0 0 0 0 0 0 0 0

The purpose ofrunning both a standard fractional factorial and a double fractional

factorial design is to determine whether the assembly characteristic response is linear or

non-linear. In the case where the response is linear, both designs should generate the 43 same output. However in case ofnon-linear response, for example a parabolic curve, the results generated from both experiments would differ. In a non-linear response the positive and negative designs should show variables that are significant that did not show when running a standard fractional factorial design.

Taguchi's Inner and Outer Arrays

Taguchi's method ofparameter design using inner and outer arrays was presented in the literature review. This section explains how the method is applied to the case study ofthe idler wheel.

Taguchi's technique differs from the other experiments in that the levels are not the upper and lower specification ofthe independent variables, but the tolerance levels themselves, i.e., the loose or tight tolerance being considered. Since the major decision is the tolerance level, the inner array consists ofan L16 orthogonal array (OA). The independent variables and their levels are shown in Table 6.

The inner array specifies the combination oftolerances (loose==l, tight==2) used for each control variable (component feature). The outer array specifies the level ofnoise combinations, i.e., the direction (upper tolerance == +, lower tolerance = -) that the tolerances deviate from their nominal direction. Together, they define a unique product, which component features are at the extremes oftheir tolerances. The structure ofthe inner and outer orthogonal array is shown in Figure 17. 44

Table 6. Tight and Loose Tolerance Levels for Taguchi's Inner-Outer Array.

Loose tolerance Tight tolerance

A Shaft length 0.05 0.025

B Base Axle Support (perpendicular) 0.025 0.0125

C Base Axle Support hole true position 0.01 0.005

D Base holes diameter 0.025 0.0125

E Base right hole true position 0.01 0.005

F Base left hole true position 0.01 0.005

G Hanger Axle Support (perpendicular) 0.005 0.0025

H Hanger right hole true position 0.01 0.005

I Hanger Axle Support hole true position 0.01 0.005

J Hanger holes diameter 0.025 0.0125

K Hanger left hole true position 0.01 0.005

L Wheel length 0.01 0.005

M Right Bushing head thickness 0.005 0.0025

N Left Bushing head thickness 0.005 0.0025 45

Outer Array

Run 1 2 3 ... 16

A -1 +1 -1 ... +1

B -1 -1 +1 ... +1

C +1 -1 -1 ... +1

......

Inner array N -1 -1 +1 ... +1

Run A B C ... N

1 -1 -1 +1 -1 Y1,1 Y 1,2 Y 1,3 ... Y 1,16

2 +1 -1 -1 -1 Y 2,1 Y 2,2 Y 2,3 ... Y 2,16

3 -1 +1 -1 +1 Y 3,1 Y 3,2 Y 3,3 ... Y 3,16

......

16 +1 +1 +1 +1 Y 16,1 Y 16,2 Y 16,3 ... Y 16,16

Figure 17. Taguchi's Inner and Outer Array

Using the case study ofthe idler wheel and referring to Figure 17, the box marked

Y1,1 in the output array represents the run which is a combination ofthe first row ofthe

inner array with the first column ofthe outer array. The second box marked Y1,2 represents the run, which is a combination ofthe first row ofthe inner array with the second column ofthe outer array, and so on until all 16 runs ofthe first row are conducted. Then the process is repeated for the second row ofthe inner array and so on. 46

As the inner array consists of 16 rows and the outer array consists of 16 columns the total number ofrequired runs is 16 x 16 = 256 runs.

In general, each outer OA noise combination is treated as a replication ofthe inner

OA. Thus, ifan outer OA requires Lo runs for each run ofan inner OA that require L, runs, the size ofthe experiment is L, * L, runs. This usually results in an excessive number ofruns, which makes the Taguchi technique unsuitable for tolerance design in practice. It is, however, included here as another comparison for the standard and double fractional factorials.

Since each column is a replicate ofthe inner array, one can determine which factors affect the mean and which affect the variance ofthe product characteristic. Hence two ANOVAs are computed in Taguchi designs compared to the standard fractional factorial.

Evaluation Measures

Clearly, the primary measure ofapplicability is how closely the various designs identify the variance contributions ofthe various tolerances compared to the HLM analysis. This is measured in two ways: 1) determining which factors are shown to be significant, and 2) determining the variance contributions ofthe significant variables.

The variance contributions from the experiments were computed as the ratio ofthe sum of squares ofthe various tolerances. Close to the HLM analysis was arbitrarily selected as within 5% ofthe HLM results. 47

Two secondary measures are the number ofcomponents built and the number of tests required. In this regard, all the above designs will be evaluated as well as Bisgaard's pre and post fractionation. In Bisgaard's original work, a positive design was used, i. e., the component dimensions varied between the nominal dimension and upper tolerance limits. Thus, although the exact design is somewhat different resulting in a different number ofrequired components and assemblies, the variance contributions are the same as in the positive design. Hence, Bisgaard's design was evaluated for the primary measure, but was evaluated for the secondary measure, the number ofcomponents and test required.

Although the secondary measure could be evaluated at this time, i. e., prior to running the simulations, they will be discussed in the results chapter. 48

CHAPTER 5

RESULTS

The three designs were evaluated according to the following primary measures:

1. The significant variables identified by ANOVA for the three designs.

2. The number ofsignificant independent variables correctly identified (within ±5%)

compared to the 3Dcs HLM analysis report. and as secondary measures:

3. Number ofexperiments required for each method (number oftests/ assemblies

required).

4. Number ofparts to be manufactured to run the experiments for each method.

ANOVA Results

The following analysis where conducted:

1. ANOVA for standard fractional factorial (Table 7).

2. ANOVA for positive fractional factorial (Table 8).

3. ANOVA for negative fractional factorial (Table 9).

4. ANOVA for Taguchi's mean response (Table 10).

5. ANOVA for Taguchi's variance response (Table 11). 49

The analyses were conducted with JMP (SAS, 1994). The significant effects were determined from a normal probability plot, and the non-significant effects were pooled into the error term. The pooled ANOVA tables are presented here.

Table 7. Pooled ANOVA for Standard Fractional Factorial.

Source DOF SS MS F p

B 1 1.01E-04 1.01E-04 3.11E+02 4.6E-07

E 1 4.49E-05 4.49E-05 1.38E+02 7.2E-06

F 1 1.01E-04 1.01E-04 3.11E+02 4.6E-07

G 1 1.0IE-04 1.01E-04 3.11E+02 4.6E-07

K 1 9.27E-05 9.27E-05 2.86E+02 6.2E-07

L 1 3.98E-04 3.98E-04 1.23E+03 4.0E-09

M 1 9.93E-05 9.93E-05 3.06£+02 4.9£-07

N 1 1.01£-04 1.0IE-04 3.11£+02 4.6£-07

Residual 7 2.27E-06 3.24E-07

Total 15 1.04E-03 50

Table 8. Pooled ANOVA for Positive Fractional Factorial.

Source DOF SS MS F P E 1 2.20E-05 2.20E-05 8.31 0.018

F 1 1.89E-05 1.89E-05 7.14 0.026

H 1 1.83E-05 1.83E-05 6.91 0.027

L 1 8.69E-05 8.69E-05 32.83 0.000

M 1 2.01E-05 2.01E-05 7.59 0.022

N 1 1.46E-05 1.46E-05 5.52 0.043

Residual 9 2.38E-05 2.65E-06

Total 15 2.05E-04

Table 9. Pooled ANOVA for Negative Fractional Factorial.

Source DOF SS MS F p

E 1 4.40E-05 4.40E-05 23.66 1.25E-03

F 1 1.85E-05 1.85E-05 9.95 1.35E-02

G 1 5.34E-05 5.34E-05 28.71 6.79E-04

H 1 1.58E-05 1.58E-05 8.49 1.95E-02

L 1.26E-04 1.26E-04 67.74 3.56E-05

M 1 4.84E-05 4.84E-05 26.02 9.29E-04

N 1 1.62E-05 1.62E-05 8.71 1.84E-02

Residual 8 1.49E-05 1.86E-06

Total 15 3.37E-04 51

Table 10. Pooled ANOVA for Taguchi Mean Response.

Source DOF SS MS F p

A 1 3.7E-07 3.70E-07 15.26 0.0036

C 1 1.3E-07 1.30E-07 5.36 0.0458

D 1 2E-07 2.00E-07 8.25 0.0184

E 1 3E-07 3.00E-07 12.37 0.0065

G 1 3.7E-07 3.70E-07 15.26 0.0036

I 1 2.8E-07 2.80E-07 11.55 0.0079

Residual 9 2.18E-07 2.43E-08

Total 15 1.87E-06

Table 11. Pooled ANOVA for Taguchi Variance Response.

Source DOF SS MS F p

B 1 1.30E-I0 1.30E-I0 12.61 6.21E-03

G 1 1.36E-I0 1.36E-I0 13.19 5.47E-03

I 1 1.22E-I0 1.22E-I0 11.83 7.39E-03

L 1 1.77E-09 1.77E-09 171.69 3.63E-07

M 1 2.42E-I0 2.42E-I0 23.47 9.15E-04

N 1 1.38E-I0 1.38E-I0 13.39 5.25E-03

Residual 9 9.28E-l1 1.03E-ll

Total 15 2.63E-09 52

Number ofsignificant independent variables:

From the ANOVA results one can clearly see:

1. The standard design showed variables B, E, F, G, K, L, M, and N are

significant.

2. The positive design showed variables E, F, H, L, M, and N are

significant.

3. The negative design showed variables E, F, G, H, L, M, and N are

significant.

4. The Taguchi's mean response showed variables A, C, D, E, G, and I

are significant.

5. The Taguchi's variance response showed variables B, G, I, L, M, and

N are significant.

6. The 3Dcs showed variables B, E, F, G, K, L, M, and N are

significant. Since there is no variation in an HLM analysis,

significance was arbitrarily selected as all dimensions that

contributed more than 5% ofthe performance characteristic

variation. ~ ~ sr~ (t) ...... tv Standard Positive Negative TaguchiMean TaguchiVariance HLM o: ~ Tolerance SS % SS % SS % SS % SS % % 3 3 A 1.00E-08 0.00% 3.10E-07 0.15% 3.29E-06 0.98% 3.70£-07 19.80% 8.IDE-13 0.03% 0.00% ~ ~ B 1.01£-04 9.70% 2.00E-06 0.98% 2.66E-06 0.79% 6.97E-09 0.37% 1.30E-tO 4.940/0 8.20% ~ ~xr C 5.80£-07 0.06% 1.16£-06 0.57% 1.OOE-070.03% 1.30E-07 6.96% 1.32E-12 0.05% 0.00% (t) 0 tJ Ho) D 4.00E-07 0.04% 1.80E-07 0.09% 4.28E-06 1.27% 2.00E-07 10.71% 9.00E-14 0.00% 0.00% (t) r:/l CI'J ~ 1-06' (JQ 3 E 4.49£-05 4.31ok 2.20E-05 10.75% 4.40E-05 13.08% 3.00E-07 16.06% 2.07E-l1 0.79% 8.20% =::s CI'J CI'J 0 § Ho) 9.70% 9.240/0 1.85E-05 9.31E-09 0.500/0 1.51% 8.200~ tr: F 1.01£-04 1.89E-05 5.50% 3.97E-l1 0.. ~ f""'+- ~ =:r G 1.0lE-04 9.70% 7.51E-06 3.67% 5.34E-05 15.88% 3.70E-07 19.80% 1.36E-IO 5.17% 8.20% (t) ~ (t) ::r: CI'J H 0.00£+00 0.00% 1.83E-05 8.94% 1.58E-05 4.70ok 2.00E-08 1.070/0 1.76£-11 0.67% 0.00% ~ § ~ 0.. I 7.00E-07 0.07% 4.88£-06 2.380/0 1.39£-06 0.41% 2.80E-07 14.990/01.22E-I0 4.64% 8.200~ ~ > (t) =::s ~ ()1-1 J 5.60E-07 0.05% 4.60E-07 0.22% 4.00E-07 0.12% 1.00E-08 0.54% 6.25E-12 0.24% 0.00% ~ (t) CI'J =::s 1-06- f""'+- 9.27E-05 ok 5.50E-06 2.69% 1.55E-06 0.46% 3.00E-08 1.61% 6.25E-14 0.00% 8.20% ~ o K 8.90 0

1-1a L 3.98E-04 38.23% 8.69E-05 42.470/01.26E-0437.47% 1.98E-09 0.11% 1.77E-09 67.280/0 34.40% 1-06- sr ~ % 0 f"""+- 9.93£-05 9.54 2.01E-05 9.82% 4.84E-05 14.39% 4.00E-08 2.14% 2.42E-I0 9.20°,4 8.20k 1-06. M 0 t:S CI'J 9.700~ 4.82% 5.25% 8.20% N 1.01E-04 1.46E-05 7.140/0 1.62£-05 8.00E-08 4.28% 1.38E-I0 0 Ho) f"""+- O.OOE+OO 0.890/0 0.09% 2.00E-08 1.07% 0.24% O.OOE+OO =:r Residual 0.00% 1.82E-06 3.10E-07 6.25E-12 (t) tJ Sum l.04E-03 1000/0 2.05£-04 1000/0 3.36E-04 100% 1.87E-06 100% 2.63E-09 100% 1000/0 ~. ~ (t) 1-1 (t) t:S Vl f"""+- W 54

These results are more readily interpreted with Table 12. The body ofthe table shows the percent contribution ofeach tolerance to the product characteristic. The percentage, P is computed by:

P = ( SSi / L SSi) * 100 where SSi = Sum ofsquares ofeach variable.

The significant variables are in bold and underlined.

One can readily see that the standard design most closely matches the HLM analysis both in terms ofthe significant tolerances as well as the variance contribution.

The only tolerance that was significant for the HLM and not for the standard fraction was

I. Both, the positive and negative fraction performed similarly to the standard fraction and HLM analysis. The discrepancies are B, I, and K which neither the positive nor the negative show as significant; G, which the positive did not identify as significant, H, which the HLM did not identify as significant. The Taguchi mean analysis did not perform well at all. It identified A, and D as significant, which the HLM did not, and did not identify B, F, K, L, M, and N which the HLM did. In particular, L, which accounts for the greatest amount ofvariation and was identified by all other designs was not found significant by the Taguchi mean analysis. The Taguchi variance analysis was closer to the HLM and the other designs than the mean analysis. It did not identify E, F, and K.

Otherwise it identified all the variables correctly. The percent contributions, however, are very different, with L being twice as large as the HLM analysis. 55

An interesting point concerns the residual. One can see that there is not residual for the HLM analysis. This is because the residual in the other designs represents the contribution due to interaction effects. Indeed, since the HLM as a one-factor-at-a-time experiment, it cannot capture interaction effects, even when they are strong. The experiments, however, do capture the interaction effects, which in this case are all small: around 1% or less. But, it does point out that the HLM analysis is not exact.

Number ofExperiments & Parts Required

Standard Design

One can deduce the number ofparts required from the experimental design (see

"fable 3). The base had 5 variables, say A, B, C, D, and E. One can see that there are 8 unique combinations ofA, B, C, E, and F which are in rows 1 through 8. These rows are repeated in rows 9 through 16. It is similar for the hanger. Ifvariables D, H, I, J, and N represent the 5 dimensions that are varied on the hanger, then there are only 8 hangers necessary. One can see in Table 3, that the every two rows are identical in the factor settings for D, H, I, H, and N. Since there is only 1 dimension for all the other parts, two parts are always required. Hence the total number ofparts required for the standard fractional factorial is 24 (see Table 13). 56

Table 13. Number ofParts Required in the Standard Fractional Factorial.

Part No. ofDimensions No. ofParts Total Parts

Base 5 25-2 8

Hanger 5 25-2 8

Shaft 1 21 2

Left Bushing 1 21 2

Right Bushing 1 21 2

Wheel 1 21 2

Total 14 24 parts

The number ofexperimental runs were the 16 trials specified by the design.

Double Design (positive & negative designs)

One might think that since the design for the positive and negative fraction are exactly the same as for the standard, the double fractional factorial requires twice as many parts and twice as many trials. Specifically, there are no parts that overlap between the two designs, even though they both utilize the nominal dimension as a common level, because there is always some dimension that is at the upper or lower tolerance limit. This is only true for the hanger and the base. For the remaining components, the nominal is in common and can be used in both designs. Hence the total number ofrequired parts for the double fractional factorial is 44 (see Table 14). 57

Table 14. Number ofParts Required in the Double Fractional Factorial.

Part No. ofDimensions No. ofParts Total Parts

5 2 Base 5 2*2 - 16

Hanger 5 2*2 5-2 16

Shaft 1 3 3

Left Bushing 1 3 3

Right Bushing 1 3 3

Wheel 1 3 3

Total 14 44 parts

14 10 The number ofexperimental runs required was 2 - = 16 for each both the positive and negative designs, which gives a total of32 runs.

Taguchi Inner and Outer Arrays Design

It is much more difficult to compute the number ofparts required in the inner outer array schema. For the base, there are 8 combinations oftight and loose toleranced dimensions manufactured at 8 distinct upper and lower specification limits resulting in 64 unique bases. The same rational applies to the hanger resulting in 64 hangers. The other components are built at 2 tolerance levels, each to their upper and lower specification limit resulting in 4 components each. Hence the total number ofparts required is 144

(see Table 15). 58

Table 15. Number ofParts Required by the Taguchi Design.

Part # ofVariables # ofParts Total Parts

Base 5 25-2*25-2 64

Hanger 5 25-2*25-2 64

Shaft 1 2*2 4

Left Bushing 1 2*2 4

Right Bushing 1 2*2 4

Wheel 1 2*2 4

Total 14 144 parts

14 10 The number ofexperimental runs required = 2 - = 16 runs/array. Taguchi's method is a 16*16 matrix = 256 runs.

Bisgaard's pre and post-fractionation

Bisgaard's prefractionation requires 24 parts, the same as the standard fractional factorial (see Table 16). All possible combinations ofthe 24 parts would require

8*8*2*2*2*2 = 1024 assemblies. Utilizing post fractionation, the number ofassemblies could be reduced to16 in this case. 59

Table 16. Number ofParts Required by Bisgaard's Prefractionation.

Part No. ofDimensions No. ofParts Total Parts

Base 5 25-2 8

Hanger 5 25-2 8

Shaft 1 21 2

Left Bushing 1 21 2

Right Bushing 1 21 2

Wheel 1 21 2

Total 14 24 parts

Comparing the number ofexperiments and number ofparts required to run each design ofexperiment, the standard design and Bisgaard's method have the lowest number ofparts (24) and runs (16 runs). 60

CHAPTER 6

DISCUSSIONS AND CONCLUSION

This thesis compared three experimental designs: the standard fractional factorial, the double fractional factorial, and Taguchi designs using the same simulation model, tolerances, and product characteristic. From the experiments and the data analysis results, the following observations can be made:

1. The standard and double fractional factorial performed the best in the case

study. They are also very similar. This leads on to conclude that the actual

response function is approximately linear (see Figure 18). In the standard

fractional factorial, the independent variables were varied between low and

high tolerance levels, resulting in a linear approximation ofthe actual response

function. While, in double fractional factorial, the independent variables were

varied in the negative fractional factorial between the lower specification limit

and the nominal value, then varied in positive fractional factorial between the

nominal value and the upper specification limit. Clearly, depending on the

strength ofthe actual response function, the slopes ofthe various lines would

more or less steep. This difference in slope indicates the different effect ofthe

various variables and explains why certain variables show up significant,

while others do not. It also explains the slight differences in the contribution

percentages. 61

Actual function curve

Standard fractional factoria

Negative fractional factorial

Figure 18. Standard and Double Fractional Factorial.

2. The large difference in Taguchi results are a little more complicated to

explain. In the standard fractional factorial, the independent variables were

varied from upper and lower specification limit as defined by the loose

tolerance range. In Taguchi's design, however, the independent variables were

varied randomly from upper and lower specification limit (outer array) for

both the loose and tight (inner array) tolerance ranges (see Figure 19). Thus,

the mean analysis shows what happens when the tolerances are varied from

the loose to the tight level. In graphical terms this means which variables

cause the difference in slope between the two lines. This is in contrast to

which variables create the line, which what the standard fractional factorial

determines.

3. The difference between Taguchi's mean responses and variance response

could be explained as follows. The mean response reflects the variation in the

independent variables as they change from the loose to the tight tolerance, i.e., 62

a change in the slope. While, the variance response reflects the variation in the

independent variables from the upper specification limit or the lower

specification (outer array), i. e., what causes the slope. Therefore, the variance

response is akin to the standard fractional factorial and showed the same

variables as significant. It did not, however, develop the same variance

contributions, because the tolerance levels were being changed

simultaneously.

Actual function curve

Loose tolerance slope

Tight tolerance slope

Figure 19. Comparison ofStandard Fractional Factorial Array and Taguchi's Inner-Outer

Array Design.

4. The HLM analysis does not consider interaction effects. Ifinteractions are

present, then sequential DOE studies can be used to identify them. This

implies that the DOE studies may be superior to traditional HLM analyses.

C:onclusions

Three experimental design techniques were investigated. The results clearly show the standard fractional factorial to be superior to the other designs both in terms of 63 determining the significant variables as well as doing so with fewer parts and runs. The double fractional factorial did not prove to be particularly useful, and Taguchi's required too many parts and runs and did not provide superior results. Thus, it is not recommended for tolerance design.

However, it is also clear that the particular case study had a linear response.

Hence the designs were not tested for a non-linear response where the difference between the various designs would be much more pronounced. Any further research in this area would require that the case study have a non-linear response. In this case, two level designs would clearly be inadequate. Alternative designs, such as the central composite design should be considered.

Lastly, an alternative to HLM analysis as the base line must be established. HLM analysis inability to capture interaction effects is a serious problem when using it as the measure ofcorrectness. A pure analytical study using Taylor series approximation, where the interactions can be explicitly modeled would be preferable.

Designed experimental methods show much promise in the area oftolerance analysis. They are the only quantitative method available when the stackup function is unknown. A major limitation ofthe method is that it requires custom made components and assemblies, which may limit the number oftolerances that can be investigated in a stack. 64

References

1. American National Standard Institute. "Dimensions and Tolerancing- ANSIY 14.5 M -1994~~, New York, New York: The American Society ofMechanical Engineers, 1995.

2. Araj, S. and D.S. Ermer. "Integrated Simultaneous Engineering Tolerancing", In Quality Improvement Techniques for Manufacturing Products and Services, editor A.H., Abdelmonem. 97-114. Dearborn, Michigan, USA: American Society of Mechanical Engineers, 1989.

3. Bisgaard, S., "Designing Experiments for Tolerancing Assembled Products", Center for Quality and Productivity Improvement, University ofWisconsin, May 1993.

4. Bjorke, Oyvind. "Computer-Aided Tolerancing", 2nd edition. New York, New York: ASME Press, 1989.

5. Byrne, K.W. and Taguchi, Shin. "The Taguchi Approach to Parameter Design", 40th Annual Quality Congress Transactions, 1987.

6. Chase, K. W., "Least Cost Tolerance Allocation for Mechanical Assemblies with Automated Process Selection", Manufacturing Review 3, 1 (March, 1990): 49-59.

7. 3-Dcs, Dimensional Control Systems Inc., Troy, Michigan, 1996.

8. Evans, D. H. "Statistical Tolerancing: The State ofthe Art, Part I : Background", Journal ofQuality Technology, 188-195, October, 1974.

9. Evans, D. H. "Statistical Tolerancing: The State ofthe Art, Part II: Methods for Estimating Moments", Journal ofQuality Technology, 1-12, January, 1975. 65

10. Gerth, R. l., "Demonstration ofa Process Control Methodology using Multiple Regression and Tolerance Analysis", Ph.D. Dissertation, University ofMichigan, 1992.

11. Gerth, R. l., "A spreadsheet approach to minimum cost tolerancing for rocket engines", Computers and Industrial Engineering. 549-552, Vol. 27, No. 1-4, 1994.

12. Gerth, R. J., "Engineering Tolerancing: A Review ofTolerance Analysis and Allocation Methods", Engineering Design and Automation 2(1),3-22,1996.

13. Gerth, R. J., "Advanced Tolerancing Techniques", New York, New York, USA: John Wiley and Sons, 1997.

14. Islam, Z., "A Design ofExperiment Approach to Tolerance Allocation", M Sc., Industrial and Manufacturing Systems Engineering, June 95.

15. lMP IN Software. SAS Institute, Inc. 1994.

16. Lee, Woo-long and Tony C. Woo. "Tolerance Synthesis for Nonlinear Systems Based on Nonlinear Programming", lIE Transactions, Volume 25, Number 1, January 1993.

17. Liou, Y. H. A., "Tolerance Specification ofRobot Kinematics Parameters using an Experimental Design Technique - The Taguchi Method", Robotics & Computer Integrated Manufacturing, 199-206, June, 1993.

18. Microsoft Office 97 for Windows, Microsoft Corporation, Seattle, WA, 1996.

19. Montgomery, D.C. "Design and Analysis ofExperiments", New York, New York, USA: John Wiley and Sons, 1984.

20. Sayed, S.E.Y., Kheir, N. A. " An Efficient Techniques for Minimum Cost Tolerance Assignment", Simulations, 189-195, April, 1985. 66

21. Spotts, M.F. " Allocation ofTolerances to Minimize Cost ofAssembly", Transaction ofthe ASME, 762-764, August, 1993.

22. Taguchi, G., Wu, Y. "Introduction to off-line Quality Control", Nagaya, Japan: Central Japan Quality Control Organization, 1979. 67

Appendices

APPENDIX A. STANDARD FRACTIONAL FACTORIAL DESIGN 68

APPENDIX B. POSITIVE FRACTIONAL FACTORIAL DESIGN 69

APPENDIX C. NEGATIVE FRACTIONAL FACTORIAL DESIGN 70

APPENDIX D. TAGUCHI'S INNER-OUTER ARRAY WITH DATA 71

APPENDIX E. IDLER WHEEL DIMENSIONS 72 68

Appendix A. Standard Fractional Factorial Design.

I +

I 69

Appendix B. Positive Fractional Factorial Design.

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Appendix C. Negative Fractional Factorial Design. -.)

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