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NON LINEAR TOLERANCE ANALYSIS BY RESPONSE SURFACE METHODOLOGY
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 of Science Th by Misako Hata June, 2001
OHIO UNIVERSITY LIBRARY 111
TABLE OF CONTENTS
TABLE OF CONTENTS 111
LIST OF TABLES VI
LIST OF FIGURES IX
DEFINITIONS OF SYMBOLS AND ABBREVIATIONS XI
1 INTRODUCTION 1
1.1 BACKGROUND 2
1.2 PROBLEM 8TATEMENT 6
1.3 SCOPE 7
1.4 STRUCTURE 7
2 LITERATURE REVIEW 9
2.1 TOLERANCE ANALYSIS 9
2.2 PROBLEMS IN DOE APPROACH 12
2.1 TOLERANCE ANALYSIS USING DOE WHEN STACKUP EQUATION IS KNOWN 13
2.2 TOLERANCE ANALYSIS WHEN STACK UP EQUATION IS UNKNOWN 14
3 RESEARCH OBJECTIVE 21
3.1 RESEARCH OBJECTIVE 22
4 METHOD 23
4.1 RESEARCH STRUCTURE 23
4.2 DESIGN METRICS 25
4.3 SIMULATION METHOD 25
4.4 EXPERIMENTAL DESIGNS 28
5 PHASE I. MATHEMATICAL MODEL 34 IV
5.1 PHASE I PROCEDURE · ·····.··························· 34
5.2 STACKUP EQUATIONS 34
5.3 PHASE I DESIGN METRICS 37
5.4 TOLERANCE ANALYSIS OF KNOWN EQUATIONS 39
5.5 EXPERIMENTAL DESIGN 42
5.6 RESULTS 49
6 PHASE II. A SIMPLE-THREE-BRACKET ASSEMBLY 71
6.1 PHASE II PROCEDURE 71
6.2 CASE STUDY PRODUCT 72
6.3 MC SIMULATION 74
6.4 HLM ANALYSIS 78
6.5 PHASE II DESIGN METRICS 80
6.6 EXPERIMENTAL DESIGN 82
6.7 DOE ANALYSIS METHOD 93
6.8 RESULTS 94
7 DISCUSSIONS AND CONCLUSION 123
7.1 PHASE I CONCLUSION 123
7.2 PHASE II CONCLUSION 126
7.3 CONCLUSIONS 129
REFERENCES ...... •...... •...... 124
APPENDICES 126
ApPENDIX A MOMENT GENERATING FUNCTIONS (MGF) 127
ApPENDIX B PHASE I TRUE SIMULA TION(MC) 130
ApPENDIX C PHASE I TOLERANCE ANALYSIS (TSE) 133
ApPENDIX D REGRESSION ANALYSIS 137 v
ApPENDIX E PHASE I MOMENTS BY MC 145
ApPENDIX F PHASE I MOMENTS BY MOFs 153
ApPENDIX G PHASE II GD &T BRACKET DRAWING 156
ApPENDIX H PHASE II VSL CODE AND RESULT 160
ApPENDIX I PHASE II VSL CODE FORDOE SIMULATION AND ITS RESULT 169
ApPENDIX J PHASE II SIMULATION ON DOE ESTIMATED STACKUP 184
ApPENDIX K METRIC MANPULATrONS 187
ApPENDIX L NORMAL PROBABILITY PLOTS 189
ApPENDIX M MGF ON FeCD MODEL 194
ApPENDIX N DOE SIMULATIONS FOR VALIDATION 196 VI
LIST OF TABLES
Table 5-1 The distribution characteristics ofxl and x2 for equation (6) and (7) 35
Table 5-2 Moments computed from MGFs for 1st order TSE 41
Table 5-3 Moments computed from MGFs for 2nd order TSE 42
Table 5-4 Full factorial design and data for equation (6) 43
Table 5-5 Full factorial design and data for equation (7) 44
Table 5-6 Double factorial design and data for equation (6) 45
Table 5-7 Double factorial design and data for equation (7) 45
Table 5-8 Face centered cube design for equation (6) 46
Table 5-9 Face centered cube designs for equation (7) 47
Table 5-10 Central composite design for equation (6) 48
Table 5-11 Central composite design for equation (7) 49
2 Table 5-12 Coordinate ofthe maximum APM and E distance for equation (6) 58
2 Table 5-13 Coordinate ofthe maximum APM and E distance for equation (7) 58
Table 5-14 Equation (6) moment comparison oftrue vs. TSE and DOE equation 59
Table 5-15 Equation (7) moment comparison oftrue vs. TSE and DOE equation 60
Table 5-16 Equation (6) moment comparison ofDOE, TSE and the true equation 61
Table 5-17 Equation (7) moment comparison ofDOE, TSE and the true equation 61
Table 5-18 Number ofexperimental runs 63
Table 5-19 Number ofexperimental run required given number ofvariables 63
Table 5-20 APM comparison ofestimated equation (6) on inner design points 66
Table 5-21 APM comparison ofestimated equation (6) on four outer design points 67 VII
Table 5-22 APM comparison of estimated equation (7) on inner design points 68
Table 5-23 APM comparison ofestimated equation (7) on four outer design points 69
Table 6-1 VSL file types 75
Table 6-2 VSA programming structure 75
Table 6-3 List ofindependent variables for Phase II DOE 83
Table 6-4 Design Generators for FFD, and DFFD 85
Table 6-5 Design generators for FCCD 87
Table 6-6 Fractional Factorial Design 88
Table 6-7 (Part 1 of2) Double Fractional Factorial Design (lower design) 89
Table 6-8 (Part 2 of2) Double Fractional Factorial Design continued (upper design) .. 90
Table 6-9 (Part 1 of 1) Face Centered Cube Design (fractional factorial part) 91
Table 6-10 (Part 2 of2) Face Centered Cube Design continued (axial points and center
points) 92
Table 6-11 ANOVA table for NPP selected factors by FFD 94
Table 6-12 ANOVA table for NPP selected factors by DFFD (lower design) 95
Table 6-13 ANOVA table for NPP selected factors by DFFD (upper design) 96
Table 6-14 Significant factors and their coefficients in lower DFFD design 96
Table 6-15 Significant factors and their coefficients in upper DFFD design 97
Table 6-16 Significant effects from upper and lower design combined 98
Table 6-17 ANOVA table for estimated stackup by DFFD (95%) 99
Table 6-18 ANOVA table for FCCD regression model. 100
Table 6-19 The number of significant independent variables identified 101 VIII
Table 6-20 ANOVA for reduced DFFD 102
Table 6-21 The number ofsignificant independent variables identified 102
Table 6-22 The number ofsignificant independent variables identified 103
Table 6-23 Comparison ofAPM distribution moments 104
Table 6-24. FCeD equation moment comparison 105
Table 6-25 Confidence Interval ofmean and standard deviation from True moments
(mean, and standard deviation) 105
Table 6-26 Metric 3: Number ofexperimental runs required 106
Table 6-27. Validation design points for FFD and estimated stackup results 107
Table 6-28. Validation design points for DFFD and estimated stackup results 107
Table 6-29. Validation design points for FCCD and estimated stackup results 108
Table 6-30 Comparison ofestimated APM to MC simulation 109
Table 7-1 Summary results for Phase I, equation (6) 124
Table 7-2 Summary results for Phase I, equation (7) 124
Table 7-3 Summary results for Phase II 127 IX
LIST OF FIGURES
Figure 1-1 Structure of a product. 1
Figure 1-2 A straight line response 4
Figure 1-3 A parabolic response 5
Figure 1-4 A trigonometric response 5
Figure 2-1 Taguchi's Inner and Outer Array Design (adapted from [6J) 16
Figure 2-2 Taguchi's parameter design. (adapted from [5]) 17
Figure 2-3 A parabolic response 18
Figure 2-4 the Simple Triangular Bracket (adapted from [10J) 20
Figure 4-1 A change in accumulative average mean 27
Figure 5-1 Two graphs showing equation (6) from different angles 36
Figure 5-2 Two graphs showing equation (7) from different angles 36
Figure 5-3 The design points for FFD for equation (6) and equation (7) 43
Figure 5-4 The design points for DFD for equation (6) and equation (7) 44
Figure 5-5 The design points for FCCD for dquation (6) and equation (7) 46
Figure 5-6 The design points for CCD for Equation (6) and Equation (7) 48
Figure 5-7 2-D response surface comparison (true vs. FFD) 50
Figure 5-8 2-D response surface comparison (true vs. FFD) 50
Figure 5-9 2-D response surface comparison (true vs. DFD) 51
Figure 5-10 2-D response surface comparison (true vs. DFD) 51
Figure 5-11 response surface comparison (true vs. 2nd order FCeD) 52
Figure 5-12 2-D response surface comparison (true vs. full term FCeD) 52 x
Figure 5-13 2-D response surface comparison (true vs. 2nd order FCCD) 53
Figure 5-14 2-D response surface comparison (true vs. full term FCCD) 53
Figure 5-15 2-D response surface comparison (true vs. 2nd order CCD) 54
Figure 5-16 2-D response surface comparison (true vs. full term CCD) 54
Figure 5-17 2-D response surface comparison (true vs. 2nd order CCD) 55
Figure 5-18 2-D response surface comparison (true vs. full term CCD) 55
Figure 5-19 2-D response surface comparison (true vs. 2nd order TSE) 56
Figure 5-20 2-D response surface comparison (true vs. 2nd order TSE) 56
Figure 5-21 Experimental runs vs. number ofvariables for DFFD and RSM designs. 64
Figure 6-1 Bracket Assembly 73
Figure 6-2 The true position combined levels 84 Xl
DEFINITIONS OF SYlVIBOLS AND ABBREVIATIONS
.ym CHAPTERl APM Assembly Performance Measure Ts APM tolerance tel, Component tolerances k number ofcomponent feature dimensions I counter ofcomponent feature dimensions Ys theAPM the ith component dimension f function ofAPM LSL Lower specification Limit USL Upper specification Limit ~ DOE .gn ofF,x It': CHAPTER 2 Nominal value ofthe APM Nominal value ofthe ith component variables the nominal value index Jlki k th moment ofthe ith component variables CTSA Cost tolerance sensitivity analysis CT Cost tolerance LTL Lower Tolerance Limit UTL Upper Tolerance Limit Standard deviation ofcomponent tolerance i Standard deviation ofAPM tolerance partial derivatives ofthe stackup coefficient ofith component tolerance FFD Fractional Factorial design
PBD Plackett-Bunnan oeSl9110" CHAPTER 3 RSM Response Surface Method FCCD Face Centered Cube Design CCD Central Composite Design
MGF Moment G r1 ~ Function CHAPTER 4 MC Monte slm1l1~ti{)n FFD Full Factorial design (Phase 1)/ Fractional Factorial design (P II) XII
symbols BBD Box-Behnken design P the fraction ofthe full factorial design F number of factorial points (runs) number ofcenter points (runs) a axial runs N scaled prediction variance Varl'y(x) (J2 N number ofexperiments, which makes quantity to reflect variance on a per observation basis (J2 variance ofy, scale free factor Y·J(. .\ vanance 'pl~1..1 ~,...l vaI ue CHAPTERS TSE Taylor Series Expansion dij Second order partial derivative with respect to i and j d., Second order partial derivative with respect to i TRUE Data based on given stackup equations (Equation (1), and Equation (2)) radial distance nd higher order (> 2 ) stackups estimated by RSM designs the skewness coefficient the kurtosis coefficient number of L~rIC cal runs CHAPTER 6 ASME The American Society ofMechanical Engineers GD&T Geometric Dimensioning and Tolerancing HLM High-Low-Median analysis MMC Maximum Material Condition Ri Variation range ofcomponent tolerance i NPP Normal Probability Plot ANOVA Analysis ofvariance 2 R ratio ofthe model portion and total v ~ ~1 lity 1 INTRODUCTION
Tolerancing is an engineering process which attempts to assure a quality product, specifically to ensure the interchangeability ofmodem mass production parts
[1]. The definition of a product, in this context, is an assembly consists of components.
Figure 1-1 shows a simple structure ofa product assembly. Shown in Figure 1-1, depending on a complexity ofa product a product may consist ofsubassemblies.
Components consist offeatures which have dimensions, and dimensions are specified by a tolerance range around their intended nominal dimensions. Therefore, components become interchangeable ifeach component feature's explicit or implicit nominal dimensions are within its associated bilateral or unilateral tolerance. That is, when components are exchanged in a product the product still performs as intended [2].
is an assembly / may consists of sub assemblies
consists of components
has features and each feature has dimension. feature 1 feature 1 feature 1 feature 1 dimension +- tol dimension+- to1 dimension +-tol dimension +- tol feature 2 feature 2 feature 2 dimension +- tol dimension+- to1 dimension +- to1 feature 3 dimension +- tol
Figure 1-1 Structure of a product. 2
In the context ofthis thesis, there are two separate product functions. The first is that the components will fit together into the assembled product, i. e., the product can be assembled. The second is the product or assembly will perform its intended function. These functions are typically measured by product assembly performance measures (APM), which are a function ofthe component feature dimensions and component properties. The functional relationship between a product function (APM) and the component feature tolerances is called the stackup function. Knowing this relationship will allow one to determine whether the designed component feature dimensions and its tolerances will result in a product that functions within acceptable assembly tolerances.
This research is aimed at empirically estimating the stackup function.
1.1 BACKGROUND
There are two main areas oftolerancing theory: tolerance analysis and tolerance allocation [2], [3]. Tolerance analysis determines the resulting APM tolerance Ts ' when the individual feature dimension tolerances, tol, are given. Conversely, tolerance allocation determines the tolerances on the feature dimension tol., when the APM tolerance, Ts is given. Tolerance analysis and tolerance allocation both attempt to rationally determine the tolerances required on the individual component features to ensure that the components can be assembled correctly. This thesis will be concerned with the concept oftolerance analysis. 3
Traditionally, designers have used one or more ofthe following tolerancing
methods: tabulated values, past designs, rules ofthumb, blanket tolerances, and CAD
system default settings [2]. These methods are relatively easy and quick. However,
they often do not work, i. e., result in products that cannot be assembled or have a poor production yield, or component tolerances are assigned overly conservatively, which results in higher production costs [2]. Also, with the exception ofpast designs and
simple pin-hole features in handbooks, these methods do not explicitly consider
assembly variation, e.g., variation from clearances between mating parts, variation in tooling fixtures, etc. In order to overcome the limitations ofthese conventional tolerancing methods, tolerance analysis methods, such as worst case (WC) tolerancing, statistical tolerancing, Constant Factor method, Estimated Mean Shift, and Monte Carlo
(MC) methods have been developed [2], [3]. Ofthese, WC and MC have found broad acceptance.
Central to tolerance analysis methods is the stackup equation. The stackup equation is a mathematically defined relationship between the assembly performance measure (APM) and the k component feature dimensions which affect it (see equation
(1)). All tolerance analysis methods require the stackup equation between component dimensions and APM be known.
(1) where
Ys == the APM, and
Xi == the ith component dimension. 4
The function.j', may be linear or non-linear. Ifthe function is non-linear it is common to use a first order Taylor series expansion to linearize the system. From the
stackup equation, the first order partial derivatives; aj/aXi ' can be interpreted as the rate ofchange ofAPM per unit change in Xi. The first order Taylor series fits the 2 dimensional equivalent ofa straight line to the non-linear function.j', as shown in dotted line of Figure 1-2. The general argument used to justify this simplification is that even ifthe function.j, is generally highly non-linear, it will behave approximately linearly over the narrow tolerance range ofinterest [4].
Linear (1st order) Response f(xl)
.. .- .. LSL USL
xi
Figure 1-2 A straight line response.
Ifthe functional relationship is highly non-linear over the range ofinterest such as the nominal design is a maximum or a minimum, a first order Taylor series may be inadequate. Figure 1-3 shows that ifthe true stackup equation is second (or higher) order in nature, a linear Taylor series expansion will not correctly capture the effect of the dimensional variation in x; 5
Non-linear (2nd order) Response f(xi) First order Talyor ser es captures no rate of change in Y(x2) ...... )......
LSL xi USL
Figure 1-3 A parabolic response.
Ifa conventional first order tolerance analysis were conducted on such a highly non-liner response, the component response would indicate no rate ofchange, and the
APM tolerance would be seriously miscalculated.
The required order ofthe estimated response is related to the tolerance range of interest. For example, assume the true APM response is sin x (see Figure 1-4). Ifthe tolerance range is from LSL1 to USL1, a first order expansion is sufficient to estimate the response, since it is relatively linear within the LSL1 and USLI range. However, if tolerance range ofinterest is from L8L2 to USL2, a first order expansion is insufficient, and a third order expansion is required.
f(xi) =SIN (xi) + 0.3
I I I I . -~ ~_1_.-,_-
I I I I I I f(xi) I I LSL2 LSL1 xi USL 1 USL2
Figure 1-4 A trigonometric response. 6
1.2 PROBLEM STATElVIENT
All tolerance analysis methods to date assume the stackup equation is known.
This assumption has been justified, since most tolerance work has been conducted on dimensional APMs for mechanical parts, where the stackup equation is usually defined by the geometric relationships between the individual component features. However, there are many systems where the stackup equation is not known, but is still a function ofcomponent feature dimensions or defined by measurable characteristics. Examples include systems where the APM is typically not a dimensional characteristic, but rather some other characteristic, such as noise or friction force [2J [4J. Other examples include where the components are not rigid, but rather compliant parts. Hence, their dimensions change based on assembly forces. In these cases, traditional tolerance analysis can not be applied. This problem is further complicated when the stackup function is highly non-linear. In such cases, where 1) the APM is non-dimensional and
2) the response is not monotonically increasing or decreasing, empirical estimation of the stackup equation may be useful.
One empirical estimation method is to build prototype parts and perform designed experiments. Through experiments, investigators can estimate the unknown true functional relationship, determine the empirical stackup model, and utilize tolerance analysis and allocation techniques. Therefore this research proposes a method for estimating the stackup function by design ofexperiment (DOE) for conducting tolerance analysis. 7
1.3 SCOPE
The scope ofthis research is concerned with the occasion when a design engineer assigns tolerance limits for assembled products. The assumption is that the stackup equation is not known for an assembly performance measure (APM), and that experiments will be conducted by assembling prototype components and measuring the
APM. However, it is not within the scope of this research to make prototypes and actually run experiments. Therefore, computer simulations will be used to simulate the manufacturing and assembly processes. It also is assumed that the probability distribution ofeach component feature dimension is known. Two additional assumptions are that all component feature distributions are normally and independently distributed.
1.4 STRUCTURE
This thesis has seven chapters including the Introduction. This is followed by the Literature Review, which describes past work on using DOE methods for tolerance analysis and its related area. Since this thesis is an extension ofwork completed by
Islam [5J [6] and by Eloseily [4J the literature review critically reviews their work and describes the problems they encountered. The research objective is presented in
Chapter 3. The research was conducted in two phases: each phase studied a different case under slightly different assumptions. Therefore the remaining chapters are related to the various phases ofthe research. Chapter 4, Method, presents methods common to both phases. Chapter 5, Phase I, and Chapter 6, Phase II, describe the details and results 8 of each phase including an illustration ofthe case study used to demonstrate the utility ofthe proposed DOE methods. Finally in Chapter 7, Discussions and Conclusions, the results ofthe two phases are integrated and their impact on tolerance analysis discussed. 9
2 LITERATURE REVIEW
2.1 TOLERANCE ANALYSIS
In this research, DOE is used to estimate the stackup equation. The estimated stackup equation can then be used to conduct a tolerance analysis and obtain the distribution characteristics ofthe APM.
The basic idea in tolerance analysis is to linearize the stackup function [7J. A multivariable Taylor series expansion is used to convert a non-linear functional relationship as a series of linear expressions as shown in equation (2). The expanded
Taylor series can then be used to estimate the moments ofthe dependent variable distribution. The distribution ofthe dependent variable can be approximated from the
Peason family or other distribution families by matching the first four moments [7J.
nd The general formula for a 2 order Taylor series expansion off(x1, X2, A Xk) is given by:
2 ~k af 0 0 k 1 (JOa f ° 2 Ys = Yso + (- ) (Xi - Xi )+ L- -2 (Xi - Xi ) 1=1 OXi 1=1 2 oX i o 2 L~k-l k a f ° ° + ( (Xi -Xi XXi -Xi) (2) 1= ]>1 OXiOX j J + third and higher order terms
The notation is based on [7]. The analysis makes 3 assumptions:
1. The component feature dimensions are uncorrelated, i.e., independent, 10
2. Third and higher order derivatives are 0 (2nd order Taylor series expansion
only), and
3. Fifth and higher order moments ofthe'component variables are negligible,
l.e.,
fLki=O 'tj k e S
where fLki = kth moment ofthe ith component variable.
The partial derivatives are evaluated at the mean or nominal values ofthe
0 component variables, Xi °and X j • The nominal response, Ys°,is obtained by substituting the mean or nominal values ofcomponent variables into the original
,x~ stackup equation, i.e., Yso = f(x 10 ,K ,xZ ). Typically for tolerance analysis, the first- order Taylor series is used (linear tolerance analysis). This is justified by the rational that most stackup functions behave approximately linearly over the narrow tolerance range ofinterest. However, there are instances in practice where this is not true, and a second order analysis is required.
The approximation for Ys is determined by the order ofthe term. Highly non- linear functional equations require higher order terms. The terms not included in the equation (2) define the magnitude ofthe error. For example, the first order Taylor series expansion is given by:
k Ys = Yso + L [Of- J(Xi - x iO ) + error term (3) i=l oX i 0 11
2 2 k 1 If 0 f ] 2 k-l k 0 f ] X) Notice the term L 2" --2 (Xi - X iO) + ~ L axax (Xi - X;o X; - X jO has been 1=1 \ oX i 0 1- ]>1 1 J 0
omitted and is now included in the error. Therefore, ifthe true functional relationship is
highly non-linear over the tolerance range a second or higher order tolerance analysis
should be applied.
From equation (2), which was truncated at the desired order, the first four
moments of Ys can be approximated based on the Pearson distribution family [7] [13].
This moment generating method is referred to as statistical error propagation or the
delta method [12]. Several references contain the equations for determining the moments ofan output variable from a second order Taylor series [7] [12] [13], and these equations are listed in appendix A.
Unfortunately, the equations are not consistent with each other. For the first and second moments, it was readily determined which reference was in error. For the third and fourth moment, however, it is not clear which reference is correct. Hence, the results from all three references are compared to the Monte Carlo simulation results in
Phase I.
In this research, tolerance analysis was conducted on the stackup equation r'(x.
X2, i1Xk) estimated from the DOE. The pitfalls ofthe DOE estimation method are explained in next session followed by several articles related to this research. 12
2.2 PROBLEMS IN DOE APPROACH
As explained in Chapter 1, this research aims to establish the tolerance analysis procedure when the stackup function is unknown. In order to determine the unknown stackup function, experiments are conducted, and empirical data are collected to statistically establish the functional relationship. Since experiments are estimates of a true functional relationship, the estimated stackup function will depend on the choice of experimental design. Moreover, there are certain risks ofmisidentifying the variables in the stackup equation ifa poor experimental design is selected.
In any experiment one should run the minimum number ofexperiments necessary to build the model ofthe desired fidelity. The number of experiments required is a function ofthe number of factors in the experiment, the number oflevels for each factor, and the functional form ofthe equation one wishes to estimate. The most basic experimental design is a two-level factorial. In any 2-level experiment, only linear effects can be estimated. For example, ifthere are three component dimensions named A, B, and C (n == 3), a full factorial design would require 23 == 8 experimental runs. The two-level full factorial design provides information on all main effects and interaction effects.
However, ifall experimenter can only conduct four experimental runs, the
3 1 experimenter could use a fractional factorial design, in our example a 2 - resolution III design. The trade offin reducing the number ofexperiments is that the main effects are aliased with interaction and higher order effects, i.e., one loses information, There is a rule ofthumb, which states that main effects are generally stronger and much more 13 significant than interaction effects [11]. Hence, the tradeoff between reducing the number ofexperiments and losing information on interaction effects is generally considered a good one. However, if a two level fractional factorial design is chosen, the experimenter must be sure that the linear approximation is appropriate and that interaction effects are unlikely.
The choice of experimental design should be selected with care since the true stackup function may include second or higher order effects, which cannot be detected by two level factorial designs. As explained in a later section, each design has advantages and disadvantages based on the type offunctions that can be estimated and the numberofexperimental runs required to estimate the function.
2.1 TOLERANCE: ANALYSIS USING DOE WHEN STACKUP EQUATION IS
KNOWN
Few articles h.ave been published on using DOE for tolerance analysis. Gerth
[8J applied the DOE method called CTSA (Cost Tolerance Sensitivity Analysis) to measure the trade-offbetween CT (Cost Tolerance) curves and the assembly performance measure (APM). The primary purpose ofusing DOE for this method was not to estimate the functional relationship, but rather to determine which component tolerances are sensitive to errors in the cost estimates. The functional relationship was assumed to be known.
Similarly, Kusiak and Feng [9J applied DOE to select between process alternatives in a tolerance allocation problem. In their example, the cost was the APM.
They allocated the component tolerances based on the cost. Since production costs and 14
component tolerances have monotonically decreasing relationships (the tighter the
tolerance, the lower cost), a two level factorial design was used. Again, the stackup
function was assumecl to be known.
These two articles assumed the stackup function was already known. DOE was
used in both cases to help select between alternative processing methods, thereby
determining individual component feature tolerances. However, because they assumed
the stackup function was known, their work is not directly applicable to this research.
2.2 TOLERANCE; ANALYSIS WHEN STACK UP EQUATION IS UNKNOWN
Four articles have been found in the area oftolerancing using DOE when a
stackup equation is unknown.
2.2.1 Bisgaard's pre and post fractional Factorial Method
Bisgaard [1] presents a unique approach for determining the tolerance limits for
mating components ofan assembled product. It is assumed the APM is linear and can
be detected by two level fractional factorial designs. A further assumption is that the
product can be disassembled and reassembled without significant change in output
performance due to assembly variation. Hence, components can be reused.
The primary purpose ofthis article was to reduce the cost ofexperiments by reducing the number ofexperimental runs. The number ofruns is an important factor
for DOE, since tolerancing problems that utilize DOE require prototype parts, which are
often very costly and time consuming to produce. Bisgaard reduces the number of
required experiments by introducing the concepts ofpre- and post-fractionation ofthe 15
two-level fractional designs. Pre-fractionation reduces the number ofprototype
components that must be fabricated, and post-fractionation reduces number of
assemblies
that must be created from those components. Ifit is known that the response is linear,
then this method is an excellent way ofconducting linear tolerance analysis. However,
ifthe response is higher in order, then alternative designs must be used.
2.2.2 Islam's Taguchi method for Tolerance Allocation
Islam [5] and Gerth alld Islam [6] present a case study estimating the stackup function
of a non-fixing bench vice. Islam applied Taguchi's parameter design to the tolerance
allocation problem.
Taguchi's parameter design is a way to identify the controllable factor levels
that are least sensitive to noise by deliberately introducing uncontrollable factors (noise) into the experiment. For example, assume a stack up involves seven feature dimensions
(A-G), and the designer is considering two different tolerances (loose and tight) for each
dimension. The inner control array contains the combinations ofloose and tight tolerances, and the outer array contains the direction the dimension deviates from the nominal dimension for each particular combination of the control array (see Figure 2-1). 16
Outer / Noise Arrav G - + + - + - - + F + + - - - + + E + - + - - + - + D - - - - + + + + C + - - + + - - + B - - + + - - + + Inner / control Arrav A - + - + - + - + RUN A B C D E F G SIN X S 1 L1 L1 L1 L1 L1 L1 L1 2 L2 L1 l.2 L1 L2 L1 L2 3 L1 L2 l.2 L1 L1 L2 L2 4 L2 L2 L1 L1 L2 L2 L1 5 L1 L1 l.1 L2 L2 L2 L2 6 L2 L1 l.2 L2 L1 L2 L1 7 L1 L2 L2 L2 L2 L1 L1 8 L2 L2 L1 L2 L1 L1 L2
Figure 2-1 Taguchi's Inner and Outer Array Design (adapted from [6]).
The crossing point ofinner and outer array is one experiment. Each experiment requires creating a prototype assembly where the dimensions are manufactured to the tolerances defined by the inner array, and to the upper or lower limit ofthat tolerance as defined by the outer array. Since the inner and outer arrays must be ofthe same dimensionality, the method requires a large number ofexperiments (8 * 8 == 64) compared to the regular fractional factorial (8 runs).
In essence, Taguchi's parameter design for tolerance allocation creates a five level factorial design (LTL with large tolerance, LTL with tight tolerance, nominal,
UTL with tight tolerance, UTL with loose tolerance), as shown in Figure 2-2. In the
Figure 2-2, the curved line shows a true response. Dotted lines show the change in response between tight and loose tolerances. The inner-outer array structure makes the method sensitive to differences between going from the tight tolerance to the loose tolerance. This works well for both linear and highly non-linear cases (see Figure 2-2), but ultimately requires too many experiments to be practical. 17
Non-linear Response estimated by Taguchi method
tighttolerance
loose tolerance
Figure 2-2 Taguchi's parameter design. (adapted from [5J). t
2.2.3 Eloseily's DOE approach for Tolerance Allocation
Eloseily utilized a combination oftwo fractional factorial designs (Double
Fractional Factorial) to detect quadratic effects. In the first FFD each component
feature is varied from its LTL to its nominal dimension, and in the second FFD each
feature is varied from its nominal dimension to its UTL (see Figure 2-3). A double fractional factorial design detects the rate ofchange in the component feature from its
nominal. For a case with 7 component features, a standard fractional factorial design would require 27-4 = 8 experimental runs. The double fractional factorial would thus require a total of 16 runs. This is vastly less than the 64 runs required by the Taguchi method. 18
Non-linear Response estimated by OFFmethod f(xi)
firs t orde estimate Double fractional factorial detects deviation from the nominal
LTL xi UTL
Figure 2-3 A parabolic response.
Eloseily's dou.ble fractional factorial design case study, an idler wheel, was
supposed to detect the pure second order response shown in the Figure 1-3. However, the case study response was approximately linear (see Figure 1-2) and, therefore, it did not exhibit the response he wished to prove. While the dual fractional factorial may detects the significance effects for the non-linear response, it sacrificed experimental runs, which could have been used to detect interaction effects.
2.2.4 Bisgaard's Functional Tolerancing With DOE and CAD
Bisgaard, Graves, and Shin [10] took a similar approach to what this research tries to accomplish for the Tolerance Analysis method by using DOE. Their emphasis is on the statistical tolerance analysis when the stackup (functional relationship) equation is unknown. Fractional Factorial design and Plackett-Bunnan (PB) designs are used to estimate the functional relationship ofthe simple triangular bracket in Figure
2-4. The estimated functional relationship is used for estimating the variance ofoutput 19 by first order error transmission formula (4) which is what this research refers to as the second moments ofstackup equation truncated at the first order (5).
2 n af 2 (4) cry = L - JCJ"i i=l [ aX i X=Jl
k /lY2 = d;2 /liZ (linear case) L (5) i=l
Bisgaard's approach is to estimate the assembly tolerance by the linear tolerance analysis method. The linear assumption does not require an estimate ofthe third and fourth moments since equation (4) yields the resulting distribution to be normal from the Central Limit Theorem. Ifthe linear assumption is inappropriate or ifthe resulting distributions are non-normal, then the assembly tolerance range may not be acceptable and there may be an excessive number ofunacceptable APM values.
Bisgaard clairns that higher order terms are assumed to be negligible over the narrow range oftolerance intervals, and hence a linear model is used. This is the traditional linear stackup argument. Therefore, only experimental designs which estimate linear effects, such as Fractional Factorial (FF) and Plackett-Burman (PB) designs are used. This research argues that the linear assumption is not adequate for some applications even in the narrow range oftolerance intervals. 20
Top bar (XB)
Point deviation range due to the deviation from factors
-:': ~.~.~.~.~.~.~.~.~.: ·::·0
Base (XA) y Support bar (x c )
Wall x
Figure 2-4 the Simple Triangular Bracket (adapted from [10]) 21
3 RESEARCH OBJECTIVE
In the Islam and Eloseily's case studies, the research was conducted without knowing the true functional relationship. Therefore, they could not clearly prove the advantages oftheir proposed designs.
This research used Response Surface Methods (RSM) for estimating the stack up function. RSM is a DOE methodology used to estimate the response ofhighly non linear systems near the response maximum/minimum [19]. RSM method has been successfully applied to estimate second order responses with relatively few experimental runs. Two RSM designs were investigated: the face centered cube design
(FFD) and the central composite design (CCD).
The research was conducted in two phases:
1. In Phase I different experimental designs were compared on their ability to
estimate a known stackup function. In this phase the amount ofimprovement
gained from a first to a second order analysis was determined and the various
moment generating equations were compared. Lastly, the estimated functions
from the DOEs were compared to a second order Taylor series expansion.
2. In Phase II different experimental designs were evaluated on their ability to
estimate an unknown stackup function. The applicability and efficacy ofeach
design was compared to MC simulation results. 22
3.1 RESEARCH OBJECTIVE
The objective ofthis research is to determine the applicability and efficacy of
RSM as a stackup equation estimation method for tolerance analysis.
The hypotheses were:
1. The distribution characteristics (first four moments) ofAPM from the stackup
generated by RSM design will be closer to the first four moments ofthe true
response than the first four moments ofAPM from the stackups generated by tIle
other designs.
2. The estimated stackup coefficients from RSM will be closer to the coefficients
obtained from the 2nd order Taylor series expansion ofthe true stackup
relationship than the estimated coefficients from the other experimental designs.
3. Estimated stackup by RSM design will be so close the true response that
moment generating functions (MGFs) can be utilized for mathematical
application oftolerance analysis.
4. RSM design will be efficiently estimate the response compared to the other
designs meaning that RSM design provides the more accurate stackup
estimation with fewer number ofexperimental runs. 23
4 METHOD
4.1 RESEARCH STRUCTURE
The objective ofthis research is to determine the efficacy ofRSM as a stackup
equation estimation method. As explained in Chapter 3, Research Objective, this research was conducted in two phases. This chapter explains the aspects common to both phase I and phase II. The first section explains the basic structure ofeach phase followed by 4.2 Design Metrics, 4.3 Simulation Method and 4.4 Experimental Designs.
The details, procedures, and results for each phase are explained in the following chapters.
4.1.1 Phase I
The first phase evaluates the different experimental designs relative to a mathematically created and known functional relationship. This phase appraised the applicability and flexibility ofthe different designs in the case ofa highly non-linear function. In order to avoid the difficulties encountered by Islam and Eloseily two stackup functions were created with the following properties:
1. The functions increase monotonically over the range ofinterest, ensuring that
the function is highly non-linear and that a linear tolerance analysis will be
inadequate. This allows estimating the error from assuming that a linear
tolerance analysis is adequate. 24
2. The number ofindependent variables are limited to two to facilitate graphical
interpretation ofthe results.
3. The apex ofthe functions are at nominal for case 1: equation (6) and off
nominal for case 2: equation (7).
Phase I involves three major comparisons. The first is a comparison of a first to a second order analysis on the stackup functions. It evaluates how a second order analysis gains accuracy over the linear assumption. The second is a comparison ofthe first four moments computed from moment generating functions from different references with the results from a 50,000 run Monte Carlo simulation. The experimental design, which empirical stackup function results in moments that are closest to the MC simulation results, will be deemed best for generating stackup equations suitable for tolerance analysis. The third is a comparison ofthe coefficients of the estimated functions from the different experimental designs. This is a measurement for the flexibility ofthe empirical models to capture second order effects, particularly with the apex offthe nominal design point.
4.1.2 Phase II
The second phase also evaluates the same set ofexperimental designs based on a simple mechanical assembled part. Phase II is designed to see how the different experimental designs will work on more realistic applications, i. e. a simple three bracket assembly. The bracket was used as the case study because:
1. from the nature ofthe product, the APM is known to be highly non-linear;
2. the APM is a geometric measure, and hence easily simulated on a computer; and 25
3. it is relatively" simple, so the results can be readily interpreted.
4.2 DESIGN METRICS
In both phases, the different experimental designs are compared by a predetermined set ofdesign metrics. There are three types ofmetrics.
1. The accuracy metric is the measurement ofhow closely each design estimated
the known stackup (Phase I) or the MC simulation result (Phase I and Phase II).
Accuracy metrics include a comparison ofthe variables included in the stackup
equations, a comparison ofthe APM moments computed from the stackup
equation to the true moments.
2. The efficiency metric is the measurement how efficient the design is. The
objective is to accurately estimate the stackup function with as few experimental
runs as possible.
3. The validation metric is the measurement ofhow closely each stackup model
estimated the known APM given in the each phase. Validation metric is a gauge
how useful stackup models are for tolerance analysis application.
Details ofeach design metric are specified in each phase. The same sets of experimental designs are compared for Phase I and Phase II unless some experimental designs are not applicable to the specific case study ofthe phases.
4.3 SIlVIULATION METHOD
There are two simulations involved in this research, Monte Carlo simulation and
DOE simulation. The former is done for obtaining the first four moments ofthe APM 26 distribution. The latter is done to conduct the experiments by simulating making prototypes components. Both simulations are done by a VSA program [14] [18].
4.3.1 Monte Carlo Simulation
The distribution characteristics ofthe stackup equations can be obtained by
Monte Carlo simulation (Me simulation). If the functional relationship is known or geometrically defined" then this is the simplest and the most popular method for nonlinear tolerance analysis [10]. During a Monte Carlo simulation a computer program generates random values for specified variables based on their density functions. The variables typically represent component feature dimensions and the density functions in the design phase ofa product are based on the assumed manufacturing process distribution and specification nominal dimensions and tolerances. The program then computes the APM from the known mathematical relationship and saves the result. This process is repeated a large number oftimes.
From the stored APM: values the program computes the distribution characteristics of the APM. Monte Carlo simulation was conducted using the VSA Monte Carlo simulation package [14]. The VSL program written for MC simulation is provided in
Appendix B for Phase I, and .Appendix H for Phase II.
In both phases, the MC simulation based on the given stackup equations or functional relationship are considered to be the "true" result since the simulation does not rely on any approximatio:ns in the form ofthe stackup function or the computation ofthe moments. The moments from the MC simulation results are compared with the 27 first and second order tolerance analysis ofthe estimated stackup function to estimate the error involved in the DOE estimation processes.
The advantage ofMonte Carlo simulation is the large number ofsamples that can be obtained by simulation. The number ofsimulation runs was selected based on when the moments stabilize. (see Figure 4-1). In this case, accumulative average mean with increment of2000 runs are observed since the first moment is common to all
higher moments, it follows that the higher moments stabilize shortly after mean. Fifty
thousand simulation runs were found to be sufficient.
Average Mean (increment of 2000 simulations)
:: 3.497 +----J-l,.,------I=-...... ------J- ~ ~ E 3.496 Qj ';f 3.495 l. Qj <3.494 ~---l--\-l------.....------~ 3.493-1---l------
3.492 -t---,.---r---r-.---,-----,-J---,---,-----,----,----,----r---..,.---....------l O.OE+O l.OE+04 2.0E+O 3.0E+O 4.0E+O 5.0E+O 6.0E+O 7.0£+0 8.0E+O 9.0E+O l.OE+05 1.1£+05 1.2E+05 1.3£+05 1.4£+05 o 444 4 4 4 4 4 runs
Figure 4-1 A change in accumulative average mean.
4.3.2 DOE simulation
Computer simulation is used to simulate the manufacturing and assembly processes as ifDOE were conducted on a real part. As the levels ofthe component feature dimensions are set at their ranges as given by the DOE, VSA calculates the
APM according to the programmed functional relationship (stackup equation). In Phase
I, the stackup equation is a known mathematical function. In Phase II, the stackup equation is based on the assembly relationships between the features ofa simple three- 28 link bracket. VSA [18] is used to produce the variation in the assembled mechanical parts given the component feature dimension tolerance range and distribution shape. It
then assembles the various co:mponents and measures the APM.
The programming code for DOE simulations is given in Appendix J for Phase II.
The experimental designs for Phase I and Phase II are shown in Chapter 5 and Chapter
6 respectively.
4.4 EXPERIlVIENTAL DESIGNS
4.4.1 Design Selections
A large variety ofexperimental designs are available, and their selection
depends on the constraints ofthe experiments in the Phase I and Phase II case studies.
This research is aimed at finding the most versatile and efficient experimental design
for tolerance analysis when the functional relationship between component feature
dimensions is unknown. Therefore, any design which would not have fit the constraints
ofeither case study was not considered.
Four experimental designs were evaluated as to their applicability to estimate
the stackup equation. The four designs were:
1. Standard full (fractional) factorial design (FFD)
2. Double full (fractional) factorial design (DFFD)
3. Face centered cube design (FCeD)
4. Central composite design (CCD) 29
Taguchi's parameter design was not considered, since it requires too many experimental runs to be considered as a useful experimental design for estimating a
stackup equation, especially for phase II where there are over twenty component tolerances. The face centered cube design (FFCD) was chosen from the RSM
experimental designs since its design space is limited to the feature dimension tolerance
range. The central composite design (CCD) was used, as an alternative to the FeCD
given there is no strict design space limitation.
Box-Behnken designs (BBD) could also be an alternative, however there has to
be certain number ofvariables in order to have fewer number ofexperiments than the
FCCD or CeDe For example, ifk = 4, BBD contains 12 + center runs, whereas CCD
with a full factorial contains 14 + center runs. In this case the BBD is more efficient.
However, as k increases the BBD is not as efficient as the CCD, which can include a
fractional factorial component. Also, with the BBD one cannot predict the response at the extremes ofthe design space. For tolerance analysis, being able to predict the response at the extremes ofthe design space, i.e. where the dimensions are at their maximum tolerance limits, is an important factor. Therefore the BBD was not considered. Details ofthe CC:D and BBD designs are provided in [23] for the interested readers.
The advantages and disadvantages for each experimental designs used in the research are explained in following sections. 30
4.4.2 Variables
There are four types ofDOE variables for this study; independent, dependent, constant, and others [11]. The independent variables are the component feature dimensions. The number ofindependent variables, k, are two for Phase I and twenty two for Phase II. The dependent variables (APMs) are:
Phase I: functional value ofthe mathematical equation.
Phase II: the distance between the nominal and deviated measurement hole.
Typically, an experimenter will hold certain variables that are not the focus ofthe investigation, but might impact the outcome ofthe investigation, constant so as to reduce the amount ofnoise in the experiment. Other, or lurking, variables are the remaining variables the experimenter mayor may not know about, but chooses not to control in any way. In simulation, neither ofthese variables are a concern, because there are no lurking variables, and any variable that is not controlled will automatically be held constant.
4.4.3 Fractional Factorial Design
A two level full factorial design with k factors requires 2 x 2 ... x2 == 2k observations and results in a model that can estimate all main effects and all interaction effects up to the kth order. By conducting only a fraction ofthe full factorial experiment, a fractional factorial design (FFD) can estimate the important main and low
k p order interaction effects. Therefore, it requires 2 - observations, p is a function ofthe fraction ofthe full factorial design that is actually run. Since each factor only has two 31 levels, it is assumed that the response is approximately linear, in other words monotonically increasing or decreasing.
Assuming the sparsity of effects principle [11] applies, which states, "when there are several variables, the system or process is likely to be driven primarily by some ofthe main effects and low-order interactions." then only a fraction ofthe experimental runs are required. Hence, the number ofexperimental runs can be reduced by sacrificing information on higher order interaction effects. This method is useful when many factors are investigated for significance. The disadvantage is that the FFD is unable to detect higher order effects. Therefore ifthe unknown functional relationship is non-linear, this design is not recommended.
4.4.4 Double Fractional Factorial Design
This is a variation ofthe FFD. The DFFD is two symmetric FFDs: one with variable levels that vary from. the nominal to the upper specification limit and one with variables that vary from the nominal to the lower specification limit. Therefore, the design actually covers 3 factor levels. As the name suggests, the FFD is doubled, and hence, the number ofexperiments is doubled. The advantage is the design detects the
2nd order curvature, especially whe~ the peak point ofthe response is close to the nominal point.
4.4.5 Face Centered Cube Design
The face centered cube design (FCCD) is a type ofresponse surface method designs, and is widely used for fitting second-order response surfaces. Unlike the 32 central composite design (CCD), which is used to search for the optimal response over relatively large possible response space, FeCD is designed to study the system response only within the strictly defined range ofindependent variables [23]. This design is used since our assumption is that a designer has already defined a set ofcomponent feature tolerance ranges, and the product will not function beyond these tolerance ranges.
The FCCD consists ofthe F factorial points, 2*k axial points, and nc center points. The full factorial or resolution V FFD estimates the linear terms and two factor interactions. Axial points and center points are used to detect curvature (quadratic terms) in the system. The axial distance, a, is set at one (coded level) to stay within the region ofindependent variables. The center points are also used to estimate the pure
error term. The number ofcenter runs, nc ' has an impact on the scale ofthe prediction
N· Var y(x) vanance ') . In this research, however, the number ofcenter runs is 1, a- because there is no run to run variation due to the exact nature ofsimulation.
4.4.6 Central Composite Design
The major difference between the FCCD and the CCD is the axial distance, a.
As stated above, in a FCeD a == 1. In three dimensions, the design would form a cube.
However, in a CCD a = 1.41. In three dimensions the design would form a sphere (see
Figure 5-6). Because the CCD sphere is symmetric in all directions, the design is said to be rotatable. Rotatable designs provide a uniform prediction variance value 33 throughout the design region. Hence, the prediction confidence interval is the same over the entire design space. This is not true ofthe non-rotatable FCCD design. 34
5 PHASE I. MATHEMATICAL MODEL
5.1 PHASE I PROCEDURE
The procedure followed for Phase I was:
1. Develop two stackup equations (see Equation (6), and (7)) and determine the
tolerance range for the independent variables.
2. Determine the design metrics.
3. Conduct the Monte Carlo simulation on the two stackup equations.
4. Conduct a first and second order tolerance analysis ofthe two stackup equations.
5. Execute the various experimental designs for each stackup equation.
6. Compare the DOE results with a traditional TSE based tolerance analysis and
with the true equations based on various accuracy and efficiency metrics.
7. Draw conclusions on the best experimental design and applicability ofDOE to
tolerance analysis.
5.2 STACKUP EQUATIONS
The two equations created for this phase are:
) ) ) • )] y = COS(x1 + COS(2x2 + 2 .[COS(x1 COS(1.2x 2 (6)
- ) • )] y = COS(2.5x1 +1.75) + COS(x2 0.8) +[2.8· COS(xl COS(x2 (7)
The APM for equations (6) and (7), y, is to be maximized. These equations are considered to reflect "true" functional relationship ofsome mechanical part. For the 35
Phase I study, only two independent variables were used so that the stackup equation
could be visualized by plotting its surface response. The independent variables, x, and
X2, are comparable to component feature dimensions. They have an associated nominal
value and tolerance. The distribution characteristics ofX 1 and X2 are listed in Table 5-1.
Table 5-1 The distribution characteristics ofxl and x2 for equation (6) and (7).
variables mean sigma Jl3i Jl4i distribution
Xl 0 1/3 o 3 o 3/81 normal X2 0 1/3 o 3 o 3/81 normal
where:
J.11i = first moment (mean) of Xi
J.1 2i = 0'2 = second moment (variance) of Xi
f.1 3i =.JP: X f.1 2i 1.5 = third moment of Xi 2 J.14i = /32 X J.12i = fourth moment of Xi
.JP:(the skewness coefficient) = f.13Y_ l.~ f12Y
Jl4y /32 (the kurtosis coefficient) = 2 J.12Y
Both equations have one desirable APM extreme point (in this case, a maximum) within the region ofinterest (between UTL and LTL). In equation (6), the apex ofthe APM is at the nominal values ofx(=O and ..Y2==O (see Figure 5-1). In equation (7), the apex is offthis nominal vector at x, ==-0.25 and x2==-0.5 (see Figure
5-2). 36
Max APM at the nominal wrt x1 Max APM at the nominal wrt x1
4 3.5 3.5 3 3 2.5 2.5 APM 2 2 APM 1.5 1.5 1
0.5 -1 0.5 0 x2 o x1
x1 x2
Figure 5-1 Two graphs showing equation (6) from different angles.
Max PPM off the nominal wrtx1 Max PPM off the nominal wrtx2
4.5 4.5 4 4 3.5 3.5 3 3 2.5 2 APM 2.5 2 APM 1.5 1.5 1 0.5 1 o -1 0.5 -0.5 o x2 -0.5
~ I x1 x1 x2
Figure 5-2 Two graphs showing equation (7) from different angles.
The specifications for x J and X2 are 0 ±1. Hence, the region ofinterest and area ofthe APM response is limited to this tolerance range. This range ensures that the polynomial and trigonometric stackup functions have only one maximum in the region ofinterest. 37
5.3 PHASE I DESIGN METRICS
Accuracy Metrics
1. Response Surface Comparison: Comparison ofthe graphical representation of
the true and estimated stackup equations obtained from the different
experimental designs.
2-D contour graphs are plotted to see where the design apex is and how the
response is skewed. To mathematically compare the peak point coordinate
distances between the estimated stackup and true response surface, the radial
2) distance (E for each experimental design is calculated. The experimental
design with the smallest f2 value is considered the best design.
2. MC Moment Comparison: Comparison ofthe APM distribution moments with
the moments obtained from an MC simulation ofthe original (true) equation, the
estimated DOE stackup equations and a TSE equation.
In this comparison the moments were computed by running the original, DOE,
and TSE equations in the Me simulator. TSE equations are original equations
expanded to 2nd order Taylor series. When the stackup relationship is known,
Monte Carlo simulation is the favored tolerance analysis method due to the
complex formulae ofthe TSE and moment generating functions [21]. The
experimental design, which yielded the moments closest to the moments ofthe
true equations, is considered to be the best design. 38
3. Moment Generating Function Comparison: Comparison ofthe APM distribution
moments computed from the moment generating functions ofthe true equation,
the estimated DOE stackup equation, and the TSE equation.
In this comparison, the APM moments were obtained from MC simulation, but
the moments from the DOE estimated equations were computed using moment
generating functions (MGFs). The experimental design, which yielded the
moments closest to the moments ofthe true equation is considered to be the best
design.
4. APM point estimates: The prediction error between the true equation and the
various estimated stackup models at 4 points at the extreme ofthe design space
and 4 arbitrary points which were not in any ofthe experimental designs.
The difference between the true model and the estimated models was computed
at four arbitrarily chosen points within the tolerance limits, which were not used
in any ofthe DOEs, and for four points which are at the extreme ofthe design
space. The former is used to validate the models' interpolation accuracy where
design points are not covered, and the latter is used to validate the models
usefulness at extreme ofthe design space where most engineers have the
greatest concern for product performance. The stackup equations with the
smallest prediction error are considered to be the best.
Efficiency Metrics
5. Number ofExperimental Runs: The number ofexperimental runs required for
each design. 39
The design the fewest number ofexperimental runs, is considered to be the best
design, given it is also relatively accurate as determined from the accuracy
metrics.
5.4 TOLERANCE ANALYSIS OF KNOWN EQUATIONS
A linear and 2nd order TSE tolerance analysis were conducted on stackup
equations (6) and (7). For phase I, the TSE tolerance analysis and DOE were conducted on the true functional relationship so that:
1. the equation coefficients from the DOE could be compared with TSE ofthe
original stackup equation, and
2. the moments computed from the tolerance analysis equations and the DOE
equations could be compared to the moments obtained from the MC simulation
(true moments).
3. the higher order polynomial and trigonometric stackup functions could be
applied to the moment generating functions (MGF).
The first four moments were estimated by the moment equations in Appendix A.
As briefly mentioned in section 2.1 Tolerance Analysis, three sets ofequations from three different sources, Cox [7], Hahn and Shapiro [12], and Evans [13] are listed. All three references had contradictory second order moment generating functions. Cox's [7] equations were eliminated due to the incompleteness ofthe equations (the full moment was missing.) and an obvious error in the 1st moment equation. Also, Hahn and
Shapiro's equations were eliminated due to the dissimilar sets ofsecond order equations relative to other two sources. Therefore, Evans' [13] moment generating functions were 40 used for tolerance analysis in this research. Details ofthe moment calculations are shown in Appendix F. The tolerance analysis results and the moments calculated from
Evans' moment generating equations are shown in the following sub sections.
5.4.1 First and Second-Order TSE of Equation (1) and Equation (2)
First-Order TSE for equation (1) is Y==4 (8) and for equation (2) is
(9) Y== 3.3185 - 2.460 x] + 0.7174x2
The second-order TSE for equation (1) is
(10) and for equation (2) is
Y = 3.3185 - 2.460 x, + 0.7174 X2 - 0.8405 x/- 1.749 xl (11)
5.4.2 First-Order Moment Generating Function
The MGFs for the 1st order TSE are:
(12)
(13)
(14)
k k-l k Jl4Y = Id/Jl4i +6I Id/dJJl2iJl2) (15) i=l i=l j=i+l 41
Symbols and computations are shown in Appendix C.
5.4.3 Second-Order Moment Generating Function
The MGFs for the 2nd order TSE as referenced in [13] are:
I/IY = 1/", (J.14· - J.1;.)~ ~ ~ d: 1/ , 11" . (17) rr: ""(d?L...J I r:i. + d.d··J13'1 II I +-.!.d,24 II I ...1 + L...J L...J lJr2lr_J i i=l )=£+1
-3J..l2/)-8d/diiJ..l3iJ..l2J+12~ J..l4Y = I [d/(J..l4i Id/djdijJ..l3iJ..l2J +3J..l;y (19) i=l i=l )=£+1
Symbols and computations are shown in Appendix C.
5.4.4 Computed Moments from Equation (6) and Equation (7)
The resulting moments for the 1st order TSE are shown in Table 5-2. The calculation ofthe partial derivatives is shown in Appendix C.
Table 5-2 Moments computed from MGFs for 1st order TSE.
Jl2Y Equation (6) 4 o o o Equation (7) 3.3185 0.7295 o -1.1256
The resulting moments for the 2nd order TSE are shown in Table 5-3.
Calculations for differentials are shown in Appendix F. 42
Table 5-3 Moments computed from MGFs for 2nd order TSE.
1l2Y 1l4Y Equation (6) 3.4511 0.3477 o 0.3628 Equation (7) 3.0307 0.6976 -0.4434 1.4700
5.5 EXPERIMENTAL DESIGN
Four experimental designs (DOEs) were evaluated in Phase I; a full factorial
(FFD) design, a double factorial design (DFD), a face centered cube design (FCCD),
and a central composite design (CCD). The DOEs specify the levels ofthe independent
variables, Xl and X2. Each experimental run was calculated by Microsoft Excel [25J (see
Table 5-6 through Table 5-11, and Statgraphics [23] was used to construct the regression models (see Appendix D). The response, y, was tabulated based on the given stackup equations (6) and (7). It should be mentioned that in a non-computer simulation based DOE, experimental error would appear in each experimental run.
However, this case study is based on known stackup equations, so random error is not present and simple tabulation by Microsoft Excel is possible. The experimental designs and estimated stackup equations are described in the following subsections.
5.5.1 Full Factorial Design
A 2 2 full factorial design without replicates was used. Since stackup equations
(6), and (7) have only two variables a fractional factorial design cannot be conducted.
The Figure 5-3 shows the design points for the full factorial design. 44
Table 5-5 Full factorial design and data for equation (7).
Run # xl x2 y 1 -1 -1 1.3219 2 1 -1 0.1441 3 -1 1 2.5291 4 1 1 1.3514
The stackup equation obtained from the Table 5-5 is:
y =1.3366 - 0.5889x1 + 0.6036x2 (21)
5.5.2 Double Factorial Design
The double factorial design (DFD) consisted oftwo 2 2 full factorials resulting in
8 runs. The difference between the regular full factorial and the DFD is the DFD divides the experimental design region into two symmetric spaces that range from the nominal to upper tolerance limit and from the nominal to lower tolerance limit. The design points ofDFD are shown in Figure 5-4. As one can see the two sets ofFFD are combined in one design space.
o xl ------.------
-1 ~------1 0 : I x2:I
Figure 5-4 The design points for DFD for equation (6) and equation (7). 46
5.5.3 Face Centered Cube Design
The FCC design consists of4 factorial runs, 4 axial points, and 1 center point.
The axial points are selected at the UTL and LTL respectively. The design points of
DFD are shown in Figure 5-5.
...... •...... -
__---...-----tII. -1 -1 0 : I I xl :
Figure 5-5 The design points for FCCD for dquation (6) and equation (7).
The data for the FCC design applied to equation (6) and equation (7) are shown in Table
5-8 and Table 5-9 respectively.
Table 5-8 Face centered cube design for equation (6).
Run # xl x2 y 1 -1 -1 0.5157 2 1 -1 0.5157 3 -1 1 0.5157 4 1 1 0.5157 5 -1 0 2.6209 6 1 0 2.6209 7 0 -1 1.3086 8 a 1 1.3086 9 0 0 4.0000
In general, the FCCD and CCD are used to develop a second order regression equation with two independent variables as given by: 47
(24)
However, it determined that the RSM designs can result in models with higher order terms due to their additional degrees of freedom. Hence, the FCCD and CCD could also generate higher order equations ofthe form:
2 Y = f30 + f31xl + fJ2 x2 + f311 X1 + fJ22.:'(2 2 + fJ12Xlx2 (25) 2 2 + f31fJ22 Xl x2 2 + fJllfJ2 X1 X 2 + f3llfJ22 X1 Xl2 + &
Hence, both a second and a higher order response surface model were constructed. The higher order model is called the full-term model. The second order (26) and full term
(27) stackup equations obtained from the Table 5-8 data are:
2 - 2 Y =3.7395 - O.9883x1 2.3006X2 (26)
2 2 2 2 y=4.0-1.3791X1 -2.6915x2 +O.5862x"1 x 2 (27)
Table 5-9 Face centered cube designs for equation (7).
Run # xl x2 y 1 -1 -1 1.3219 2 1 -1 0.1441 3 -1 1 2.5291 4 1 1 1.3514 5 -1 0 2.9412 6 1 0 1.7635 7 0 -1 1.1074 8 0 1 2.3147 9 0 0 3.3185
The second (28) and fourth (29) order stackup equations obtained from the Table 5-9 are:
.,v =3.0555-0.5889x +O.6036x -O.5717x 2 -1.2130x 2 I 2 I 2 (28) 48
5.5.4 Central Composite Design
This design consists of4 factorial runs, 4 axial points, and 1 center point. As
shown in Figure 5-6, axial point were selected to be at 1.41 * ~ the tolerance range to
ensure a rotatable design (24).
·I -~ 1.41 .. • .. 1
.....•...... •...... •....I 0 xl
I ·I ·I 0: ... ·1 I -1.41 -1 1 ~ - ·1.41 x2 :
Figure 5-6 The design points for CCD for Equation (6) and Equation (7).
The data for the CCD design applied to equation (6) and equation (7) are shown in
Table 5-10 and Table 5-11 respectively.
Table 5-10 Central composite design for equation (6).
Run # xl x2 y 1 -1 -1 0.5157 2 1 -1 0.5157 3 -1 1 0.5157 4 1 1 0.5157 5 -1.41 0 1.4803 6 1.41 0 1.4803 7 0 -1.41 -0.1905 8 0 1.41 -0.1905 9 0 0 4
The second order (30) and full term (31) stackup equations obtained from the Table
5-10 are: 49
2 ~v - 2 = 4.0013 -1.2955x1 2.1359x2 (30)
2 2 2 2 Y = 4.0 -1.2674x] - 2.1 078x 2 - 0.1091x] x 2 (31)
Table 5-11 Central composite design for equation (7).
Run # xl x2 y 1 -1 -1 1.3219 2 1 -1 0.1441 3 -1 1 2.5291 4 1 1 1.3514 5 -1.41 0 0.9422 6 1.41 0 1.6784 7 0 -1.41 -0.3265 8 0 1.41 1.0897 9 0 0 3.3185
The second order (32) and full term (33) stackup equations obtained from the
Table 5-11 are:
2 2 y=3.3125-0.1652x] +0.5531x2 -O.8800X1 -1.3471x2 (32)
y=3.3185-0.2611.x] +0.5022.;t -1.0101x 2 -1.4772x 2 2 1 2 (33) 2 2 2 2 - + 0.1014x1 x 2 0.8499.x]x2 + 0.5055x1 x 2
5.6 RESULTS
5.6.1 lVIetric 1. Response Surface Comparison
Two-dimensional graphical representations ofthe response surfaces ofthe original and estimated stackup equations are shown in Figure 5-7 through Figure 5-17 50
Response Surface Response Surface Estimated Equation (6) by FFD Equation(6) 1
,---.j.--+-+--..... 0.75 0.75
-'---to---+- 0.5 0.5 0.25 0.25 o x2 o x2 -0.25 -0.25
4---1---4--0.5 -0.5
'--.j.-...,C-4--..... -0.75 -0.75 APM ~+--+--+--+--+-f--+--~-1 APM , -1 ,- LO LO LO 0 LO LO LO ,- ,- LO LO LO 0 L() LO LO , I I'- • N N· I'- I to--: 0 N N· I'- 9 9 9 x1 0 0 0 9 I 9 x1 0 0 0 1_0-1 01-2 02-3 .3-41 III 0-0.2 110.2-0.4 ..0.4-0.61
Figure 5-7 2-D response surface comparison (true vs. FFD).
Response Surface Response Surface Estimated equation (7) by FFD equation (7) 1 0.75 0.5 0.25 o x2 -0.25 t--~Irl--+--+--+----+---+~ -0.5 t--+--f--J'M---+---+----+----+---+ -0.75 APM !--+---+-~--+---+---+--+-+ -1 ~ L() o L() ~ L() I 9 o I 9 x1 APM x1 0-1 01-2 02-3 1 [J -1-0 .0-1 01-2 02-3 .3-4 lLl 4-5 1 I_ 1
Figure 5-8 2-D response surface comparison (true vs. FFD). 51
Response Surface Response Surface Estimated Equation (6) by DFD Equation (6) __..,.--r---,.---r----r-or---r-'"1III 1 ..-..,.-r-,-~---r-~-r---.__ 1
-~4- 0.75 ~-t----+ 0.75 0.5 0.5 0.25 0.25 o x2 o x2 -0.25 -0.25
-0.5 -r-""l"--.... -0.5 ~~4--"'" -0.75 -0.75 APM -1 -1 ~ ~ ~ ~ 0 ~ ~ ~ ~ t.n ~ ~ 0 L() LO I I'- . C\I C\I' I'- I'- . N N' 9 '( 9 x1 0 0 0 9 '( 9 x1 0 0
1 11II 0-1 0 1-2 02-3 .3-4 [EJ-1-0 .0-1 01-202-3 .3-4 1 1
Figure 5-9 2-D response surface comparison (true vs. DFD).
Response Surface Response Surface Estimated equation (7) by DFD Equation (7)
-+--+--+---+-+---+---+0.75 0.75 ~----11-7'-+---+-~~---+- 0.5 0.5 ~--?-'~-+---+---+---+~ 0.25 0.25 o x2 0 x2 -0.25 -0.25 -0.5 -0.5 -0.75 -0.75 -1 ~ L() L() L() 0 L() ~ ~ ~ APM t---+--+---+----I-+---+-__¥_--+-1 ~ L() L() L() L() L() ~ I ": 0 LO 0 "! "! 0 ": I ,...... • N C\I' I'- APM 9 I 9 x1 0 0 9 9 9 x1 0 0 0
I EJ -1-0 .0-1 01-2 02-3 .3-4 UJ 4-5 2 1 i 1 1_- - rnH-Q _ 0-1 01-2 02-3 - 3-4
Figure 5-10 2-D response surface comparison (true vs. DFD). 52
Response Surface Response Surface Estimated Equation (6) by FCCD (2nd) Equation (6) __.....---,..--,.----.-----r-~_,._1!!IIIIr1 -.,..---,...,.----.----,-~~ro--T"---1
~+----T+--.... 0.75 ----f----+f--4-O.75 0.5 0.5 0.25 0.25 a x2 a x2 -0.25 -0.25 -0.5 -0.5 ~I---+--l--"" -0.75 ----+-~--&- -0.75 APM ~~-+-~-+--~---l-':-..-j-~-1 APM -1 ~ L() ~ ~ ~ ~ ~ ~ ~ ~ o L{) 0 I I I' . C\I C\I' I' o 9 9 9 x1 0 0 0 9 x1 1_0-1 01-2 02-3 .3-41 111I0-1 01-2 02-3 .3-41
Figure 5-11 response surface comparison (true vs. 2nd order FCCD).
Response Surfa ce Response Surface Esimtaed Equation (6) by FCCD (Full) Equation (6) __....--__-,.---,-~__---r 1 -~r-o------...-----1
--+--T-+---1t- 0.75 -+---Y---4-0.75 -+---+---+0.5 0.5 0.25 0.25 a x2 o x2 -0.25 -0.25 -0.5 -0.5 ----+--+-+-----0.75 ~--.'I--4--0. 75
,....-!--4---4---+-~~-+~-1 ,-.-~t..+---+---+--4---4-'~~-1 ~ ~ ~ ~ L.C') I ~ x1° I x1 0 9 o <;> o APM I_ 0-1 01-2 02-3 .3-41 APM 1_0-1 01-2 02-3 .3-4 1
Figure 5-12 2-D response surface comparison (true vs. full term FCCD). 53
Response Surface Response Surface Estimated equation (7) by FCCD (2nd) Equation (7) r-r----r--r----r-r---r----r 1 ___,....--...,.-....,.-~--T---,-~1
0.....+--+---+---+-+---+--+ 0.75 -t--t-"7't---+--+-t--~-+ 0.75 -+---+~-t--~-+---+ 0.5 ___+--ff----+--+---+-----1I--+--4-0.5
~~--+---+----if---+--~-+ 0.25 I---t-t--t--t---+--+-t--+--+0.25 o x2 o x2 -0.25 -0.25 -0.5 -0.5 -0.75 -0.75 -1 ~ ~ ~ ~ 0 ~ ~ ~ ~ AP M ~ LO L() LO 0 u? i.O L() ~-1 I I'-- . N N' I'-- I ": a"!"! a l'-: APM 9 x1 0 9 9 a a 9 I 9 x1 0 0 I 0 -1-0 • 0-1 0 1-2 0 2-3 • 3-4 [] 4-51 I_ 0-1 0 1-2 0 2-3 • 3-4 I
Figure 5-13 2-D response surface comparison (true vs. 2nd order FCCD).
Response Surface Response Surface Esitmated Equation (7) by FCCD (full) Equation (7)
~-+--+--+-t--+---+ 0.75 +--0""'+-"7""+---+---+-+--~-+ 0.75
--+~~+--+-~~--+ 0.5 __+--oft--+---+---+-+---+-~0.5 ~--f,o'--+-t---+--+---t-~ 0.25 0.25 o x2 o x2 -0.25 -0.25 -0.5 -0.5 -0.75 -0.75 -1 ~ LO LO ~ 6 LO ~ io ~-1 ~ ~ LO LO 0 L() LO ~ ~ I I'-- . N N' f"- I ": 0 "! "! 0 ": er> 9 er> x1 ci 0 0 APM 9 I 9 x1 0 0 1ILl -1-0 .0-1 01-2 02-3 .3-4 UJ 4-5 APMi- 0-1 01-2 02-3 .3-41 1
Figure 5-14 2-D response surface comparison (true vs. full term FeeD). 54
Response Surlace Response Surlace Estimated Equation (6) by ceo (2nd) Equation (6)
-~~----o. 75 :--+--oft---+0.75
-+---tl-----t- 0.5 0.5
-++---+0.25 0.25 o x2 o x2 -++---+-0.25 -0.25
'-+----f----t--0.5 -0.5
----+----,f-+-__._ -0.75 '--I--'f'----+-0.75
,--..-t---+---+-+----+----+--t---,..-1 P---+-~----t--+----+--+""--+--""-1 ~ ~ ~ ~ a ~ ~ ~ ~ I ~ 0 '"'! '"'! 0 ~ 9 I 9 x1 a a APM APM1-.-0--1- -... 4--5- I_ 0-1 01-2 02-3 .3-41 0-1--2-0-2--3-.-3-4-LJ 1
Figure 5-15 2-D response surface comparison (true vs. 2nd order CCD).
Response Surface Response Surface Esimtated Equation (6) by CCO (full) Equation (6) 1 1 0.75 0.75 0.5 0.5 0.25 0.25 o x2 o x2 -0.25 -0.25 -0.5 -0.5 -0.75 -0.75 -1 ~ ~ -1 io io io a eo co io ~ I ,...... ,...... ~ ~ L{) LO io io ~ I ,...... a N N 0 N N ,...... 9 0 9 0 9 9 x1 0 9 9 x1 0 0 APM APM I_0-1 01-2 02-3 _ 3-4 1 I_ 0-1 0 1-2 02-3 _ 3-4 1
Figure 5-16 2-D response surface comparison (true vs. full term CCD). 55
Response Surface Response Surface Esimtated Equation (7) by CCO (2nd) Equation (7) r-r---r----r----r--r---.-....,.. 1 _".---,--,.",..-r----,--.,.-,---o--...,.- 1
-i--t---t----t--r--;---;-0.75 ___+--'7"'t"---t---t---t-t--""f---'lo&0.75
---+---If?"-T---t---r->'~---t- 0.5 :-+---t---t-0.5 -+----¥--+---+----1--t---~:_t_ 0.25 0.25 o x2 o x2 -0.25 -0.25 -0.5 -0.5 -0.75 I---++-+---+----+-~~-+-+--+ -0.75
-1 ".-.-+--..,----+---+---+-+--+--'-+ -1 ~ ~ ~ ~ 0 ~ ~ ~ ~ ~ ~ ~ ~ 0 ~ ~ ~ ~ I I'- . N N' I'- I I'- . N N' I'- APM 9 9 9 x1 0 0 0 o 9 0 x1 0 0 0 APM ~ 1 WI -1-0 110-1 01-2 02-3 .3-4 0 4-5 'I_ 0-1 1-2 02-3 _ 3-41 1
Figure 5-17 2-D response surface comparison (true vs. 2nd order CCD).
Response Surface Response surface Esimtated Equation (7) by CCO (full) Equation (7)
-+---+---+---f-t---t---t-0.75 ----t-~-t--_+___+_""~___+ 0.75
~-~+--_+_-+-"~---t- 0.5 1IIIIIaC-+----+'--+---+--I-+--~"M- 0.5 4---j,.o'---+---+---ir--+--~:-t- 0.25 0.25 o x2 o x2 -0.25 -0.25 -0.5 -0.5 -0.75 -0.75
~ ~ ~ ~ 6 ~ io L.O ~-1 ~ ~ ~ io 0 io 1.0 io ~-1 I I'- . N N' I'- I ~ 0 "! "! 0 ~ APM 9 9 9 x1 0 0 0 9 I 9 x1 0 0 B1-1-Q _ 0-1 01-2 02-3 _ 3-41 1 m-1-0 .0-1 01-2 02-3 .3-4 [E]4-5 APj 1
Figure 5-18 2-D response surface comparison (true vs. full term CCD). 56
Response Surface Response Surface Esimtated Equation (6) by TSE Equation (6) 1 0.75 0.75
0.5 0.5
0.25 0.25 0 x2 x2
-0.25 -0.25
-0.5 -0.5
-0.75 -0.75
-1 -1 ~ l.C) io io 0 io L{') io ~ I C'\l C'\l ~ co L{') io 0 io io L() ~ "" 9 x1 0 "" I C'\l C'\l 0 r-, APM 9 9 0 0 APM 9"" 9 9 x1 0 ci I_ 0-1 01-2 02-3 .3-41 1[[1 -1-0 11I0-1 01-2 02-3 .3-41
Figure 5-19 2-D response surface comparison (true vs. 2nd order TSE).
Response Surface Response Surface Esimtated equation (7) by TSE Equation (7) .-.----.---.---...-..--,.,...-...... 1 1 -+----4----4-----+--+---+--+0.75 0.75 ---+-----+.,..A.-~4__~-+--+ 0.5 ~.f---+--"""-""---f---+-~ 0.5 ~~--+-___+____1f---+---~:-+ 0.25 -+,.L--+---+----+--II--~:_+ 0.25 Ox2 o x2 -0.25 -0.25 -0.5 -0.5 -0.75 -0.75 -1 ~ L() l{) l{) 0 L() L{') L{') ~ -1 I ,...... • C'\l C'\l' "" ~ APM 9 9 9 x1 0 0 0 APM "7 C'\l C'\l.,...... 9 9 x1 0 0 0 IlEI -1-0 • 0-1 0 1-2 0 2-3 • 3-4 lEI 4-5 1 1_-2- 1 E:]-1-D _ 0-1 01-2 02-3 _ 3-4 1iJ4-sl
Figure 5-20 2-D response surface comparison (true vs. 2nd order TSE). 57
The obvious result is that the FFD did not detect the curvature ofequation (6) and (7) at all (see Figure 5-7 and Figure 5-8). This is understandable, since the FFD is only capable ofmodeling planes. The other designs, DFD, FCCD, and CCD all estimated equation (6) well, which has its maximum point at the nominal design point.
With regards to equation (7) the FFD could again not detect the curvature. The
FFD model is an inclined plane. Again, these results are understandable, since the FFD can only model planes. However, the other designs did detect the curvature, and the fact that the maximum is not at the nominal design point. But, they could not detect the maximum peak boundary (the closest contour line surrounding the peak point), because none ofthe experimental design points fell on equation (7) peak point: xl = 0.25, and x2 = -0.5. In order to make closer estimates, experimental designs with more than two levels, or follow-up experiments centered about the hypothesized peak point would be required.
In order to determine how close the response surface ofeach experimental design estimated the true response, their peak points, and radial distances were compared. The peak point comparison compares the maximum APM value predicted by each DOE model with the true maximum APM value. The radial distance error, e; is a measure ofhow far the estimated APM point is from the true APM maximum. The error, 8, is obtained from:
2 (2 2) (2 2) & = Xl + X 2 - Xlnominal + X2nominal (34)
Hence, the best experimental design is the one with the smallest difference in peak point value and smallest radial distance. 58
Table 5-12 and Table 5-13 show the coordinates ofthe APM peak, the radial distance, and the peak APM value.
2 Table 5-12 Coordinate ofthe maximum APM and E distance for equation (6).
FFD DFD FCCD FFCD CCD CCD TSE TRUE (full) (full) Xl 0 0 0 0 0 0 0 X2 0 0 0 0 0 0 0 PeakAPM 0.5157 4 3.7395 4 4.0010 4 4 4 2 E 0 0 0 0 0 0
2 Table 5-13 Coordinate ofthe maximum APM and E distance for equation (7).
FFD DFD FCCD FCCD CCD CCD TSE TRUE (full) (full) Xl -1 0.25 0.25 0.25 0.25 0.25 0.25 0.25 X2 -1 -0.25 -0.25 -0.25 0 -0.50 -1 -0.50 PeakAPM 2.5591 3.4188 3.2821 3.4581 3.3665 3.555 3.3185 4.11 E2 1.69 0.19 0.19 0.19 0.25 0 0.75
All experimental designs except the FFD estimated the maximum point at the nominal exactly. This is because the maximum point was one ofthe experimental design points.
The E ofentries except the peak APM for the FFD are blank since the estimated response is a plane with a constant APM of 0.5157 throughout the design boundary (see
Figure 5-7). The DFD, and full FCCD and CCD models estimated the same APM as the true function at the nominal point.
Upon examining the results for equation (7), the full CCD model had the
2 minimum E = 0 and the closest APM estimate (86.5% ofthe TRUE maximum APM).
This means the CCD correctly estimated the exact peak point ofEquation (7), even though the peak point coordinates were not on a design point. Among the other designs the DFD and FFCD had the closest radial distance to the true maximum. The second 59
order CCD equation did not estimate the true off-nominal equation well. Hence for
these metrics, the full term CCD is considered to be the best.
5.6.2 Metric 2. Me Moments Comparison
The APM distribution moments obtained from the MC simulations ofthe DOE
estimated equations and the true equation are compared in Table 5-14 and Table 5-15.
The Me simulation results are shown in Appendix E. Me simulation result based on
original equation is labeled as "MC". For equation (6), the estimated mean is within
5% ofthe true result, and the standard deviation and skewness are within 20% and 35%
ofthe true result respectively. However, the estimated kurtosis is twice as much as the
kurtosis oftrue equation. In other words, the higher the moment the less accurate the
DOE equations can accurately estimate the moment. The equation (7) results are even
worse. Though the mean is estimated closely, the estimated standard deviation is half
the deviation ofthe true equation. Skewness and Kurtosis are twice to 30 times as large
as the true moments. TSE equations in general yield more errors than DOE estimated equations.
Table 5-14 Equation (6) moment comparison oftrue vs. TSE and DOE equation.
Mean %Diff. Std %Diff. Skewness %Diff. Kurtosis %Diff. Dev FFD 0.5157 -85.25% ------DFD 3.5511 1.57% 0.4766 -4.0% -2.3792 29.5% 9.3911 105.4% FCCD 3.3764 -3.43% 0.3927 -20.9% -2.4801 35.0% 10.3967 127.4% FFCD (4th) 3.5576 1.75% 0.4604 -7.2% -2.2836 24.3% 8.5395 86.8% CCD 3.6222 3.60% 0.3913 -21.1% -2.2744 23.8% 9.1942 101.1% CCD (4th) 3.6258 3.70% 0.3880 -21.80/0 -2.3108 25.8% 9.7491 113.30/0 2nd order 3.3257 -4.88% 0.8519 71.7% -0.0108 -99.4% -0.0367 -100.80/0 TSE MC(true) 3.4963 100.0% 0.4962 100.00/0 -1.8368 100.00/0 4.5711 100.0% 60
Table 5-15 Equation (7) moment comparison oftrue vs. TSE and DOE equation.
Mean %Diff. Std %Diff. Skewness %Diff. Kurtosis %Diff. Dev FFD 1.3378 -56.1 % 0.2792 -60.1 % 0.0097 -101.8% -0.0308 -87.6% DFD 3.0359 -0.30/0 0.4074 -41.8% -2.0485 285.3°~ 6.1677 -2588.0% FCCD 2.8596 -6.10/0 0.3485 -50.2% -1.5718 195.7% 3.7459 -1611.1% FCCD (4th) 3.0425 -0.1% 0.3942 -43.7% -1.7443 228.1% 4.3692 -1862.50/0 CCD 3.0662 0.6% 0.3155 -55.0% -2.1144 297.7% 7.2416 -3021.20/0 CCD (4th) 3.0505 0.1% 0.3386 -51.70/0 -2.1170 298.2% 6.8979 -2882.50/0 2nd order 3.3098 8.6% 0.9028 28.9°~ -0.6669 25.50/0 0.5832 -335.3% TSE MC(true) 3.0464 100.0% 0.7005 100.00~ -0.5316 100.0% -0.2479 100.0%
5.6.3 Metric 3. Moments Generating Function Comparison
In contrast to using MC simulation to compute the moments, the moments were also computed based on the moment generating functions (MGFs) in Appendix A. This comparison was done to determine the applicability ofthe MGFs on the estimated DOE
nd equations. The MGFs were computed from both a standard 1st order (linear) and a 2 order tolerance analysis. The moment computation for the DOE equations are shown in
Appendix F. The APM distribution moments ofthe DOE equations and the TSE are compared to the moments ofthe true equation (Refer equation (20) through (32)). The standard deviation, skewness and kurtosis ofthe MC simulation were converted to the
2nd, 3rd, and 4th moments (/liY) for this comparison (see Appendix B). 61
Table 5-16 Equation (6) moment comparison ofDOE, TSE and the true equation.
1st order MGF 2nd order MGF illY 1l2Y /l3Y /l4Y illY /l2Y 1l3Y /l4Y FFD 0.5157 0 a 0 0.5157 0 0 0 DFD 4.0000 0 0 0 3.5477 0.2300 0 0.1587 FCCD 3.7395 0 0 0 3.3740 0.1547 0 0.0718 CCD 4.0010 0 0 0 3.6197 0.1541 0 0.0712 TSE 4 0 0 0 3.4511 0.3477 0 0.3628 TRUE 3.4963 0.2462 -.2244 0.0952 3.4963 0.2462 -.2244 0.0952
Table 5-17 Equation (7) moment comparison ofDOE, TSE and the true equation.
1st order Tolerance Analysis 2nd order Tolerance Analysis IllY 1l2Y /l3Y /l4Y /llY 1l2Y Jl3Y 1l4Y FFD 1.3366 0.0790 0 0.0187 1.3366 0.0790 0 0.0187 DFD 3.3185 0.0790 0 0.0187 3.0325 0.1702 -0.0838 0.0869 FCCD 3.0555 0.0790 0 0.0187 2.8725 0.1234 -0.4742 0.0457 CCD 3.3124 0.037 0 0.0041 3.0500 0.1010 -0.0323 0.0306 TSE 3.3185 0.7295 0 -1.125 3.0307 0.6976 -0.4434 1.4600 TRUE 3.0464 0.4907 -0.1827 -0.7820 3.0464 0.4907 -0.1827 -0.7820
J.11i = first moment (mean) of Xi
J.12i = (j2 = second moment (variance) of Xi f.1 3i =.fii: X f.1 2i J.5 = third moment of Xi 2 J.14i = /32 X J.12i = fourth moment of Xi
There are several conclusions that can be drawn from the tables. First, as was already shown in the response surface graphs, the 1st order DOE designs (FFD) do not work well since there are no design points to estimate curvature effects regardless ofthe true equation or the MGF order.
Second, a 1st order MGF also estimates the moments poorly compared to a 2nd order MGF. As can be seen from Table 5-16 and Table 5-17 the 2nd and higher order moments, 1l2Y, /l3Y, and /l4Y from the first order MGF are all 0 or close to 0 for both 62
equation (6) and equation (7). Hence, given a non-linear response, a first order tolerance
analysis produces large errors and is not recommended.
Third, as with the MC moments, the moments for Equation (6) were closer to
the true moments than Equation (7). This is again because the peak was off-nominal,
and none ofthe DOE design points were on the peak point ofthe true response
(equation (7)).
Fourth, the moments computed from the Me simulation were generally closer to the true moments than those computed by MGFs. This is due to the limited order ofthe
MGFs. For example, ifthe MGFs were ofhigher order then they may have resulted in closer estimates ofthe moments. However, due to the complexity ofthe MGFs and the difficulty ofobtaining the higher order moments on the independent variables that would be required to estimate the parameters ofa higher order MGF, obtaining the moments through MC simulation is recommended.
5.6.4 Metric 4. Number of Experiments
The number ofexperimental runs are listed in Table 5-18. In Phase I, since there were only two variables, the number ofexperimental runs did not seem to indicate a practically significant advantage for one experimental design over another. The
DFFD design has one fewer experimental run than the other RSM designs (FCCD and
CCD). However, in metric 2 the full term FeeD and CeD resulted in better estimates than the DFD or other models which analyses were limited to 2nd order. Table 5-19 shows a comparison ofthe number ofexperimental runs for the DF and RSNI designs with respect to the number of independent variables. The table assumes the same 63 fractionation ofthe factorial portion ofthe designs. The table compares which design is more efficient given the design estimates closer to the true response. The table clearly shows that even as the number of variables increases, the DFFD is not that more efficient over the RSM designs. The effect is even more evident in Figure 5-21, which shows the number ofexperiments required for DFFD and RSM designs assuming only
1 center point.
Table 5-18 Number ofexperimental runs.
Number ofexperimental runs required FFD 4 DFD 8 FCCD 9 CCD 9
Table 5-19 Number ofexperimental run required given number ofvariables.
number of DFFD FCCD/CCD variables (double fractional factorial) (wi minimum fractional factorial)
5 2 5 2 2 - + 10 axial runs +center runs 5 2 - * 2 = 16 = 18 + c
10 6 10 6 2 - + 20axia . 1runs +center runs 10 2 - * 2 = 32 = 36 + c
10 6 10 6 2 - + 30axia . 1runs +center runs 15 2 - * 2 = 32 =46 + c
10 15 20 15 2 - + 40axia . 1runs +center runs 20 2 - * 2 = 64 = 72 + c 64
350
300
e 250 tIJ" c: :J ... 200 cu -+-DFFD C Q) E ---RSM 'i: 150 Q) a. >< W 100
50
0 0 20 40 60 80 100 120 Variables, k
Figure 5-21 Experimental runs vs. number ofvariables for DFFD and RSM designs.
5.6.5 Metric 5. Validation
The previous sections evaluated which experimental design generated stackup models closer to the true response. In this section, all estimated stackup equations are evaluated on how well they predict the APM response at points there were both part of and not part ofthe original design. This form ofvalidation tests the usefulness ofthe models as to their prediction ability.
The predictability ofthe various equations was evaluated by comparing the computed APM for various combinations ofx I and X2 design points with the true APM value. There were two sets ofvalidation points. The first set consisted ofpoints that were not in any ofthe original experimental designs, but were within the experimental design space: (xl, x2) = (-0.5, -0.5), (-0.5,0.5), (0.5, -0.5), and (0.5,0.5). These points 65
test the interpolation ability ofthe estimated models, and the results are shown in Table
5-20 and Table 5-22 for equations (6) and (7) respectively. The second set ofthe
validation points were at the outer edges ofthe experimental designs, which also
correspond to the feature dimension specification limits: (x 1,x2) == (-1, -1), (-1, 1), (1, -
1), and (1, 1). These points test the pure prediction ability ofthe models and are also
their results are the points where most engineers are concerned about product
performance.
The results are shown in Table 5-21 and Table 5-23 for equations (6) and (7)
respectively. The difference was said to be large ifit exceeded ± 1.6450", which
corresponds to a 95% CI range ofthe MC simulation. The DOE cell, which is within
this range, is marked with one star (*). Ifthe difference is within the range of± 1c, the
APM estimate is said to be close to the true value, and the cell is marked with two stars
(**). This is one ofthe measures ofhow well each equation estimates the true
functional relationship. R2 prediction is the other measure ofhow much variability in
the stackup model may have [24]. This is calculated as follows:
4 ~)Yi _yJ2 2 1 i=l R prediction = - -4---- (35) L(Yi _)7)2 i=l
In Table 5-20 and Table 5-21, R2 values are all 0, since validation points are all on the same value ofy and denominator becomes O. Therefore, in this special case R2
values are not used for the evaluation. 66
Table 5-20 APM comparison ofestimated equation (6) on inner design points.
xl x2 True Estimated % f Actual tR2 APM APM difference difference predicted <1.65a? Examination ofTable 5-20 shows that all models except the FFD model interpolate well. The difference between DOE point estimates and the true APM were within 1.0 sigma. Among these designs, only the full CCD estimated the true response 67 exactly. All CCD design of any order had close estimates with uniform errors since the CCD used here are rotatable. Table 5-21 APM comparison ofestimated equation (6) on four outer design points. 2 xl x2 True Estimated % tActual tR APM APM difference difference predicted Examination ofTable 5-21 shows that all models except the DFFD and 2nd order TSE point estimates were within the sigma range. Among the DOEs, the full CCD, full 68 FCCD, and the FFD estimated the true response exactly. However, with the FFD this is understandable since the outer design points are exactly the points measured in the FFD. Hence the validation points are the same points used to compute the .FFD model, which is a plain without any 2nd order effects (see Figure 5-7). Table 5-22 APM comparison ofestimated equation (7) on inner design points. xl x2 True Estimated % Actual R2 APM APM difference difference predicted <1.65a? Examination ofTable 5-22 shows that the 2nd order TSE, full CCD and full FCCD resulted in the best models, followed by the FCCD. The DFFD and CCD did not fare as well. This clearly shows that the higher order models are more capable of interpolating the true effects when the response is offnominal. Table 5-23 APM comparison ofestimated equation (7) on four outer design points. xl x2 True Estimated % Actual R2 APM APM difference difference predicted <1.65a? Examination ofTable 5-23 shows that the RSM designs performed well, with the FCeD slightly outperforming the CCD. One should note that the full FCCD had zero error at all four design points. The TSE had more difficulty, as the extreme design points represent a greater extrapolation from the point about which the series is expanded. Hence, the larger error and poor performance ofthe TSE. 71 6 PHASE II. A SIMPLE-THREE-BRACKET ASSEMBLY 6.1 PHASE II PROCEDURE The research procedure for Phase II was as follows: 1. Determine the product for the case study. 2. Draw the parts according to the AMSE Y14.5M 1994 Geometric Dimensioning and Tolerancing (GD&T) standard [16]. 3. Set the tolerance ranges for the component features. 4. Determine the design metrics. 5. Code simulation program based on the GD &T part drawing. 6. Conduct Monte Carlo (MC) simulation and HLM analysis. 7. Code simulation program for experimental designs. 8. Conduct DOE simulation. 9. Compare the significant component features between MC and DOE simulations. 10. Conduct a simulation based on the estimated DOE stackup equation, and compare APM distribution characteristics with the MC simulation result. 11. Draw conclusions on the best experimental design and applicability ofDOE for stackup estimation method. 72 6.2 CASE STUDY PRODUCT The product used in Phase II was a simple three-link bracket assembly (see Figure 6-1). It consists ofthree (iron) parts: base, top bar, and support bar. The base is affixed to a wall and the top bar is attached to holes in the upper side ofthe base. Nuts and bolts are used to attach the bars through the holes on each bracket end. Similarly the support bar is attached to holes in the lower side ofthe base. After the top and support bars are attached to the base, the top bar slides into the end slot ofthe support bar and is affixed. The actual size and location ofthe bracket is affected by part feature variations and clearances. The APM ofthe assembly is the two-dimensional distance (x-y coordinate in Figure 6-1 between the nominal center ofthe measurement hole (hole M), and the actual center (displaced by parts deviation) ofhole M. The independent variables are the size and position ofthe various holes in the bar and the bolt size. The APM has a minimum at nominal and deviates in a positive direction as the assembly deviates from nominal. Hence, the i\PM is a 2nd order response. Also, the assembly is sufficiently complex that the stackup function is difficult, albeit not impossible, to determine directly. Finally, there are large number (23) ofcomponent feature dimensions that can affect the APM. Hence, the case study exhibits all the necessary characteristics to evaluate the differences between the DOEs. 73 Top bar o o Hole M .:: ~~~~~~~:~: ...... y x Base (X,y,z) = (0,0,0) Support bar o Wall Figure 6-1 Bracket Assembly. The part drawings for the base, top and support bar are shown in Appendix G and follow the ASME Y14.5M 1994 GD&T standard [16]. The names ofthe component features and tolerances shown in the drawings correspond to the variable names used in the simulation code. The tolerances ofall features were selected from the default values for material and manufacturing processes in the Product Design Handbook [15]. Bolt dimensions were based on a manufacturer's default values [17] and are also listed in Appendix G. The assumptions ofthis case study are: 1. Each component feature dimension is manufactured independently ofany other feature. 2. Component feature distributions are normally distributed. 74 3. The datums are based on the function ofthe assembled part (functional datuming) and not based on other considerations, such as manufacturing setup or ease ofinspection. 6.3 MC SIMULATION Unlike Phase I, the Phase II stackup equation is not defined directly for the bracket assembly. Instead, it is defined by a series ofassembly statements that indirectly provide the mathematical relationship between the component features. Therefore, Me simulation can be used to provide the resulting APM distribution characteristics. The first four moments determined from the MC simulation results are considered to be the actual behavior ofthe assembly. Any reference to the results from these simulation runs are labeled "MC" in this thesis. 6.3.1 Coding The assembly simulation is coded in Variation Systems Language (VSL) [18], and the simulation program is composed of five file types as shown in Table 6-1. The program requires components (base, top bar, support bar, and pins), variation on each component feature dimension (size and true position), component assembly statements, and output measurements (distance between the nominal and deviated hole M). The simple three-bracket assembly VSL programming structure is shown in Table 6-2. The Me simulation code and its results are provided in Appendix H. 75 Table 6-1 VSL file types. Name Extensions Purpose ...... M.~~~ PE~.~.~~ ~..~.~.! ~..~!.!.~ ~.!.! ~.!~.~ E ~.!.~.~ . ...... Declaration file .inc General declarations ...... I.~}.~E.~~~.~ §.!.~ :.!.~} ~.~~~.~!.~.~~.~ ~~.~ ~.~~P.~.~~~.! g.~~.~~.!!1 ~.!.~!.~!P.~~.!.~ ...... ~.~..~.~~~.!.y g.!~ ~.~ .~!P: ~.~.~.~.~p.!x i.~~.!~~!.~.~.~~ .. Measurement .mes Output measurements file Table 6-2 VSA programming structure. Phase II file File Name File Description dec hata.inc global variable declarations. base hata.tol Top_hata.tol Spt_hata.tol variation and component geometry statements. pinAhata.tol pinBhata.tol pinChata. to1 positions the base against the wall in global bkt hata. vsl t asmhata.asm coordinates. (Main attaches the top bar to the base. program) attaches the support bar to the base assembly. includes moves the support bar to the top bar. moves the top bar to the support bar. s asmhata.asm repeats the move until difference between the top attachment hole and support bar hole is less than 0.001 mm. measures the distance between the nominal position and mated position mes hata.mes (DOE simulations do not include the measurement files). In the declaration file, global variables are declared. In the tolerance (.tol) files, the point coordinates ofeach component are defined, and variations are applied to the points. The point numbers in the code correspond to the numbers in the part drawing in Appendix G. A bonus tolerance is also modeled for all position tolerances with an 76 MMC (maximum material condition) modifier. The bonus tolerance is the difference between the actual feature size and the MMC feature size and is added to the geometric positional tolerance [20]. In the MC simulation, the bonus tolerance is multiplied by a probability factor since, in reality, only a portion ofthe total available positional tolerance would be used. The assembly (.asm) files contain the move statements that move one component to the next. For clarity, a separate assembly file is created for each component that is moved. A move statement is a VSL code that moves the 3D objects to 3D targets. The most common move uses 6 points on the object that are to be mapped to 6 target points. The move itselfhandles all translation and orientation effects based on the vectors that are computed from the specified object and target points. Part biasing due to gravity and clearances between the holes and the bolts are also modeled in the simulation program. The gravity effect means that bolt and hole edges are in positive contact with each other. The origin (x,y,z) == (0,0,0), ofeach component (base, top bar and support bar) is shown on each part drawing (see Appendix G). In the t_asmhata.asm file, the simulation positions the base bracket in the assembly coordinate system, which is the center ofthe base datum B hole that attaches to the wall (Refer to Figure 6-1). After the base bracket is in its assembled position, the top bar is attached to the base. Point tHOLE_dtmB on the top bar (see Appendix G) is moved to the point Hole_Eon the base. Then pin A is inserted through both holes to attach the top bar to the base. Similarly, sHOLE_B on the support bar is moved to point Hole_D on the 77 base, and pin B is inserted to hold the support bar to the base. Pinhole clearances are taken into the account and parts are biased down due to gravity. The last assembly move is to attach sHaLE_Con the support bar to tHOLE_dtmC on the top bar. Both holes should line up considering pin-hole clearances and force bias. Since VSA can only move one part at a time, an iteration loop is required to first move one part into position, measure the distance between the holes, then move the other part into position, and again measure the distance between the holes. This back and forth movement is continued until the distance between the holes is very small « 0.001 mm). Then Pin C is inserted through the matched hole to fix all the brackets. After all assembly routines have been executed, the specified APM is measured in the .mes file. The measurement is the difference between the nominal and the varied position ofHole M. 6.3.2 Simulations There are two types of simulations in Phase II: Me simulation and DOE simulation. The MC simulation code is shown in Appendix H and DOE simulation code is shown in Appendix 1. The MC simulation is designed to simulate the production of50,000 brackets by a mass manufacturing facility, i.e., parts and part features are created randomly according to their assumed distributions that are a function ofthe part feature specifications (nominal and tolerance). The DOE simulation is designed to simulate prototyped parts and part features that were constructed to 78 precise dimensions according to the DOE matrix. The basic code structure is the same for both simulation types. There are slight differences between the MC and DOE simulations, due to the difference in their purpose. In MC simulation, the purpose is to estimate the APM distribution based on random input variables. Hence, the bonus tolerance is multiplied by a probability factor that represents the percent ofthe bonus tolerance that is experienced in the assembly. The DOE simulation is designed to estimate a stackup equation based on specific cases as defined by the experimental design. Hence, the bonus tolerance is fixed at its maximum value and changed in direction since all the factors are set at their extreme values. 6.4 HLM ANALYSIS MC simulation generates a HLM (High-Low-Median) analysis, which is used to determine the percent contribution ofeach component tolerance to the variation in the APM [14]. This analysis applies only to the MC bracket simulation since during the DOE simulation the variables are already fixed at their extreme values. In a HLM simulation the ith component variable (i == 0 to k) is varied to its high, median, and low value ofits distribution. High, low and median levels are the -3cr, +3cr, and J.1 ofthe component distribution [14]. The simulation is conducted for each component dimension, one at a time, while all other component dimensions are held at their nominal values. 79 The resulting maximum and minimum value of Ys (APM) is used to compute the range, R: (see equation (37)[14J), which quantify the amount ofspread in the APM due to variation in the /th component feature dimension, assuming the feature is normally distributed. R. = max.-min. l l l (36) and the variance, S, [14J: (37) which quantify the amount of spread in the APM due to variation in the ith component feature dimension, assuming the feature is normally distributed. The total variance ofthe APM, Sy2, is the sum ofall component variances: (38) 2 Pi is the percent variance contribution ofSi on Sy2: P. = S;2 (39) l S 2 Y Since the HLM analysis is based on a one-factor-a-at-time experiment, it is only accurate when the response consists only ofmain effects, i.e., when interaction effects are not present [14]. To compensate for this problem, it is possible to group two variables together. Grouped variables are varied according to a full factorial design (all combinations) ofhigh, low, and median values. An example ofgrouping is the true position of a hole. The true position of a hole can be modeled as the convolution of 80 normally distributed radial distribution with a uniformly distributed angular distribution. In order to capture the effects ofthese two variables, the must be grouped so that all possible combinations ofhigh and low radii at all angles can be captured in a HLM simulation (see Figure 6-2). Since the contribution ofeach variable can no longer be separated due to possible interaction effects, the contribution ofthe whole group is presented in the HLM report. In this study, HLM analysis (see Appendix H) is used as one ofthe metrics for how well the DOE estimated stackup equation captures the significant component variables. However, ifinteraction effects are present, some factors may show up as significant in the DOE, which do not show up as high contributors in the HLM report. 6.5 PHASE II DESIGN METRICS There are four metrics for this phase: Metrics 1 and 2 compare the accuracy of the experimental designs. Since the true functional relationship is unknown, the DOE estimates can only be compared to the Monte Carlo simulation. Accuracy Metrics 1. Significant variables: The statistically significant independent variables determined from a normal probability plot (NPP) and their percent contribution compared to the HLM analysis results. This metric compares significant factors identified from the NPP with the significant factors identified in the HLM analysis. 81 2. MC moments comparison: Comparison ofthe APM distribution moments obtained from the MC simulation ofthe true bracket model with another Me simulation based on the DOE estimated equations. This metric compares the first four moments obtained from the "true" Me simulation analysis with the first four moments obtained from a VSA simulation analysis ofthe DOE equation. Ifthe simulated moments based on the DOE equation are close to the true moments it will be concluded that the experimental design successfully estimated the distribution characteristics ofAPM. Efficiency Metric 3. Number ofExperiments: The number ofexperimental runs required for each design. The design, which has the smallest number ofexperimental runs, is considered to be the best design, given it is also relatively accurate as determined from the accuracy metrics. Validation Metric 4. APM point estimates: Stackup model prediction for APM on points which were not experimental design points to test the prediction power ofthe various DOE models. Four points were arbitrarily selected. These points were within the tolerance limits, not used in any ofthe experimental designs, and at the limits ofthe design space. The stackup equations, which most closely estimate the predicted APM points are considered to be the best. 82 6.6 EXPERIMENTAL DESIGN Each experimental run was simulated by VSA [14]. Three designs, Fractional Factorial (FFD), Double Fractional Factorial (DFFD) and Face Centered Cube (FCCD) were compared. The Central Composite Design was not considered in Phase II, since it requires variable values that are beyond the tolerance ranges stated on the drawing. 6.6.1 Variables for DOE simulation The APM ofthis case study is the two-dimensional (XY) distance between the nominal center ofhole M and the deviated position ofhole M. Component variable features which vary along the z-axis are not controlled and allowed to vary randomly in the DOE simulations. The component feature variables included in the DOE are listed in Table 6-3. Each component name is shown on the part drawings in Appendix G. 83 Table 6-3 List ofindependent variables for Phase II DOE. Variable Name Nominal Range (±) 1 HOLE sizeC 10.000 0.062 2 HOLE sizeD 8.200 0.062 3 HOLE sizeE 8.200 0.062 4 HOLE radC 0.000 0.150 5 HOLE radD 0.000 0.150 6 HOLE radE 0.000 0.150 7 HOLE_angC 0°,90° 8 HOLE_angD 0°,90° 9 HOLE_angE 0°,90° 10 tHOLE sizeB 8.200 0.062 11 tHOLE sizeC 8.200 0.062 12 tHOLE sizeM 8.200 0.062 13 tHOLE radC 0.000 0.150 14 tHOLE radmes 0.000 0.150 15 tHOLE_angC 0°,90° 16 tHOLE_angmes 0°,90° 17 sHOLE sizeB 8.200 0.062 18 sHOLE sizeC 8.200 0.062 19 sHaLE radC 0.000 0.150 20 sHOLE_angC 0°,90° 21 pinA_size 7.9375 0.035 22 pinE_size 7.9375 0.035 23 pinC size 7.9375 0.035 Levels ofthe independent variables Xl to X23 are set at their upper specification limit (USL), lower specification limit (LSL), or nominal design point as specified by the experimental design. Each variable was set on a coded scale: -1 = LSL, +1 = USL or 0 = nominal. Since each variable is set to a specific value according to the experimental design, there are no variation statements in the .tol files except for component features, which are not included in the independent variable list. These features were not deemed to have a significant effect on the APM as their variation was along the axis (z-axis) perpendicular to the plane in which the measurement is taken. 84 Hole_rad is the true position deviation ofthe corresponding component hole. Hole_ang is the angular deviation ofthe true position. Combinations ofbilateral true position and angle levels of0, and 90 degrees cover the deviation from the nominal towards four directions (at 0°,90°, 180°, and 270°). Hole_rad and Hole_ang are grouped together to specify the deviation from the nominal position at the -1 and +1 level. Figure 6-2 shows all possible combinations ofRole_rad and Hole_ang. For example, the 7th row in Table 6-3 specifies the angle levels (0 and 90 degrees) of HOLE_angC, which covers only two angular deviations ofthe hole center position. However, combined with HOLE_rade, which specifies the radial deviation from the nominal position at -1, and +1 level, there are four point deviations from the nominal. The Figure 6-2 shows the all the possible combination for creating positional deviation from the nominal. (HOLE_angC, HOLE_fade) =~90 deg, -1) (0 deg , 0) ...... ~ . . o x2 . (0 deg, -1) (0 deg , 1) -1 -1 0: 1 I :(-90 deg , 1) xl : I Figure 6-2 The true position combined levels. 85 6.6.2 Fractional Factorial Design The Fractional Factorial Design (FFD), a 2(23-18) resolution III design with 23 factors, requires 32 runs. The design and data are shown in Table 6-6. The independent variables were set at the extremes oftheir tolerance range, i.e., -1 = LSL and +1 = USL. The design generators for the FFD design are shown in Table 6-4. Table 6-4 Design Generators for FFD, and DFFD. variable treatment number combination 1 A 2 B 3 C 4 D 5 E 6 BD 7 AE 8 CDE 9 BDE 10 BCE 11 BCD 12 ADE 13 ACE 14 ACD 15 ABE 16 ABD 17 ABC 18 BCDE 19 ACDE 20 ABDE 21 ABCE 22 ABCD 23 ABCDE 86 6.6.3 Double Fractional Factorial Design The Double Fractional Factorial Design (DFFD) consists oftwo 2 (23- 18) fractional factorials. The variable levels for the first 32 runs (lower design) were set at the lower specification limit (-1), and the nominal design point (0). The second 32 runs (upper design) were exactly like the first, except the variables were set at the nominal point (0), and the upper specification limit (1). The design and data are shown in Table 6-7 and Table 6-8. Both upper and lower design design generators are the same as those for the FFD (see Table 6-4). This resulted in a total of64 runs. 6.6.4 Face Centered Cube Design The Face Centered Cube Design (FCCD) required a total of79 experimental runs. It essentially adds 23 * 2 == 46 axial points and 1 center point to the 32 run FFD. The design generators are shown in Table 6-5. The design and data are shown in Table 6-9 and Table 6-10. 87 Table 6-5 Design generators for FCCD. variable treatment treatment number combination combination 1 A A2 2 2 B B 2 3 C C 2 4 D D 2 5· E E 6 BD (BD)2 7 AE (AE)2 8 CDE (CDE)2 9 BDE (BDE)2 10 BCE (BCE)2 11 BCD (BCD)2 12 ADE (ADE)2 13 ACE (ACE)2 14 ACD (ACD)2 15 ABE (ABE)2 16 ABD (ABD)2 17 ABC (ABC)2 18 BCDE (BCDE)2 19 ACDE (ACDE)2 20 ABDE (ABDE)2 21 ABCE (ABCE)2 22 ABCD (ABCD)2 23 ABCDE (ABCDE)2 Table 6-6 Fractional Factorial Design. 0/······ r\~E I-\n~OF n A B C E·······HI1 AE shE BeE ··BCD ADE< ACF A~O ARF ARO ARC .RCOF ACOE ARDE ·ABCE.ffl~C[' V 1 -1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 -1 0.5849 ? 1 -1 -1 -1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 0 ?123 3 -1 1 -1 -1 -1 -1 1 -1 1 1 1 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 03108 4 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 1 -1 0.3175 5 -1 -1 1 -1 -1 1 1 1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 -1 1 0.3854 6 1 -1 1 -1 -1 1 -1 1 -1 1 1 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 0.434 7 -1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 1 1 1 -1 1 -1 -1 1 1 -1 0.253 8 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 1 0.4242 9 -1 -1 -1 1 -1 -1 1 1 1 -1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 -1 1 0.4558 10 1 -1 -1 1 -1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 0.4827 11 -1 1 -1 1 -1 1 1 1 -1 1 -1 1 -1 1 1 -1 1 1 -1 1 -1 1 -1 0.4051 12 1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 1 00627 13 -1 -1 1 1 -1 -1 1 -1 1 1 -1 1 1 -1 -1 1 1 1 1 -1 -1 1 -1 07991 14 1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 1 0.2273 15 -1 1 1 1 -1 1 1 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 1 1 -1 1 0.5941 16 1 1 1 1 -1 1 -1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 -1 -1 -1 1 -1 0.8772 17 -1 -1 -1 -1 1 1 -1 1 1 1 -1 1 1 -1 1 -1 -1 -1 -1 -1 -1 1 1 04824 18 1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 1 -1 1 1 -1 1 1 1 -1 -1 0 1601 19 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 1 1 -1 -1 1 1 1 -1 1 1 -1 -1 0.0921 20 1 1 -1 -1 1 -1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 0 ;W47 21 -1 -1 1 -1 1 1 -1 -1 1 -1 1 1 -1 1 1 -1 1 1 1 -1 1 -1 -1 0.5088 22 1 -1 1 -1 1 1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 0.5066 ?3 -1 1 1 -1 1 -1 -1 -1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 1 -1 1 1 0.2648 24 1 1 1 -1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 -1 1 -1 -1 -1 1 -1 -1 0.164 25 -1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 1 1 1 -1 1 1 1 -1 -1 -1 0.2151 26 1 -1 -1 1 1 -1 1 -1 -1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 0.2446 27 -1 1 -1 1 1 1 -1 -1 1 -1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 1 1 1 1083 28 1 1 -1 1 1 1 1 -1 1 -1 -1 1 -1 -1 1 1 -1 -1 -1 1 -1 -1 -1 0 7635 29 -1 -1 1 1 1 -1 -1 1 -1 -1 -1 -1 -1 -1 1 1 1 -1 -1 1 1 1 1 0.2284 30 1 -1 1 1 1 -1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 0 3874 31 -1 1 1 1 1 1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 -1 -1 -1 -1 -1 0.2001 32 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 03692 00 00 Table 6-7 (Part 1 of 2) Double Fractional Factorial Design (lower design) ...... '·i.~·,",,,,,· n A is C D '·i.e BDE Rht: .Rr.·n····Ahr:: ACE ACD ABE ABD ABC BCDe ACDE ABDE ABCE ABCD ABCDE v 1 -1 -1 -1 -1 -1 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 0 -1 0.4551 2 0 -1 -1 -1 -1 0 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0.1487 3 -1 0 -1 -1 -1 -1 0 -1 0 0 0 -1 -1 -1 0 0 0 -1 0 -1 -1 -1 0 0.3263 4 0 0 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 -1 -1 -1 -1 -1 0 0 0 -1 0.4879 5 -1 -1 0 -1 -1 0 0 0 -1 0 0 -1 0 0 -1 -1 0 -1 -1 0 -1 -1 0 0.4304 6 0 -1 0 -1 -1 0 -1 0 -1 0 0 0 -1 -1 0 0 -1 -1 0 -1 0 0 -1 0.518 7 -1 0 0 -1 -1 -1 0 0 0 -1 -1 -1 0 0 0 0 -1 0 -1 -1 0 0 -1 0.2173 8 0 0 0 -1 -1 -1 -1 0 0 -1 -1 0 -1 -1 -1 -1 0 0 0 0 -1 -1 0 0.3056 9 -1 -1 -1 0 -1 -1 0 0 0 -1 0 0 -1 0 -1 0 -1 -1 -1 -1 0 -1 0 0.5669 10 0 -1 -1 0 -1 -1 -1 0 0 -1 0 -1 0 -1 0 -1 0 -1 0 0 -1 0 -1 0.2403 11 -1 0 -1 0 -1 0 0 0 -1 0 -1 0 -1 0 0 -1 0 0 -1 0 -1 0 -1 0.4121 12 0 0 -1 0 -1 0 -1 0 -1 0 -1 -1 0 -1 -1 0 -1 0 0 -1 0 -1 0 0.2419 13 -1 -1 0 0 -1 -1 0 -1 0 0 -1 0 0 -1 -1 0 0 0 0 -1 -1 0 -1 0.2694 14 0 -1 0 0 -1 -1 -1 -1 0 0 -1 -1 -1 0 0 -1 -1 0 -1 0 0 -1 0 0.2875 15 -1 0 0 0 -1 0 0 -1 -1 -1 0 0 0 -1 0 -1 -1 -1 0 0 0 -1 0 0.2622 16 0 0 0 0 -1 0 -1 -1 -1 -1 0 -1 -1 0 -1 0 0 -1 -1 -1 -1 0 -1 0.7038 17 -1 -1 -1 -1 0 0 -1 0 0 0 -1 0 0 -1 0 -1 -1 -1 -1 -1 -1 0 0 0.3169 18 0 -1 -1 -1 0 0 0 0 0 0 -1 -1 -1 0 -1 0 0 -1 0 0 0 -1 -1 0.1003 19 -1 0 -1 -1 0 -1 -1 0 -1 -1 0 0 0 -1 -1 0 0 0 -1 0 0 -1 -1 0.1969 20 0 0 -1 -1 0 -1 0 0 -1 -1 0 -1 -1 0 0 -1 -1 0 0 -1 -1 0 0 0.1629 21 -1 -1 0 -1 0 0 -1 -1 0 -1 0 0 -1 0 0 -1 0 0 0 -1 0 -1 -1 0.2187 22 0 -1 0 -1 0 0 0 -1 0 -1 0 -1 0 -1 -1 0 -1 0 -1 0 -1 0 0 0.1731 23 -1 0 0 -1 0 -1 -1 -1 -1 0 -1 0 -1 0 -1 0 -1 -1 0 0 -1 0 0 0.057 24 0 0 0 -1 0 -1 0 -1 -1 0 -1 -1 0 -1 0 -1 0 -1 -1 -1 0 -1 -1 0.1825 25 -1 -1 -1 0 0 -1 -1 -1 -1 0 0 -1 0 0 0 0 -.1 0 0 0 -1 -1 -1 0.03 26 0 -1 -1 0 0 -1 0 -1 -1 0 0 0 -1 -1 -1 -1 0 0 -1 -1 0 0 0 0.2424 27 -1 0 -1 0 0 0 -1 -1 0 -1 -1 -1 0 0 -1 -1 0 -1 0 -1 0 0 0 0.1092 28 0 0 -1 0 0 0 0 -1 0 -1 -1 0 -1 -1 0 0 -1 -1 -1 0 -1 -1 -1 0.4037 29 -1 -1 0 0 0 -1 -1 0 -1 -1 -1 -1 -1 -1 0 0 0 -1 -1 0 0 0 0 0.3265 30 0 -1 0 0 0 -1 0 0 -1 -1 -1 0 0 0 -1 -1 -1 -1 0 -1 -1 -1 -1 0.0656 31 -1 0 0 0 0 0 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 0 -1 -1 -1 -1 -1 0.2861 32 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 \0 Table 6-8 (Part 2 of 2) Double Fractional Factorial Design continued (upper design) 33 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0.1465 34 1 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 0.3768 35 0 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 0 1 0 0 0 1 0.0795 36 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 1 0 0.272 37 0 0 1 0 0 1 1 1 0 1 1 0 1 1 0 0 1 0 0 1 0 0 1 0.2443 38 1 0 1 0 0 1 0 1 0 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0.1816 39 0 1 1 0 0 0 1 1 1 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0.4035 40 1 1 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 1 0 0 1 0.0794 41 0 0 0 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 0 0 1 0 1 0.0139 42 1 0 0 1 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 0.3403 43 0 1 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 1 0 1 0 1 0 0.2304 44 1 1 0 1 0 1 0 1 0 1 0 0 1 0 0 1 0 1 1 0 1 0 1 0.1966 45 0 0 1 1 0 0 1 0 1 1 0 1 1 0 0 1 1 1 1 0 0 1 0 0.2544 46 1 0 1 1 0 0 0 0 1 1 0 0 0 1 1 0 0 1 0 1 1 0 1 0.1811 47 0 1 1 1 0 1 1 0 0 0 1 1 1 0 1 0 0 0 1 1 1 0 1 0.3276 48 1 1 1 1 0 1 0 0 0 0 1 0 0 1 0 1 1 0 0 0 0 1 0 0.0978 49 0 0 0 0 1 1 0 1 1 1 0 1 1 0 1 0 0 0 0 0 0 1 1 0.3176 50 1 0 0 0 1 1 1 1 1 1 0 0 0 1 0 1 1 0 1 1 1 0 0 0.1866 51 0 1 0 0 1 0 0 1 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 0.0134 52 1 1 0 0 1 0 1 1 0 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0.6527 53 0 0 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0.4287 54 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 1 0.6264 55 0 1 1 0 1 0 0 0 0 1 0 1 0 1 0 1 0 0 1 1 0 1 1 0.3098 56 1 1 1 0 1 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 1 0 0 0.116 57 0 0 0 1 1 0 0 0 0 1 1 0 1 1 1 1 0 1 1 1 0 0 0 0.1419 58 1 0 0 1 1 0 1 0 0 1 1 1 0 0 0 0 1 1 0 0 1 1 1 0.0959 59 0 1 0 1 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 1 1 1 0.8449 60 1 1 0 1 1 1 1 0 1 0 0 1 0 0 1 1 0 0 0 1 0 0 0 0.5462 61 0 0 1 1 1 0 0 1 0 0 0 0 0 0 1 1 1 0 0 1 1 1 1 0.1421 62 1 0 1 1 1 0 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 0.2127 63 0 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0.2737 64 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.3738 o\.0 Table 6-9 (Part 1 of 1) Face Centered Cube Design (fractional factorial part). ...... '., i n A" B C' <,>0 .•...ro' en 1\ P ,a;;;...... - AB'E > ABO ABC BCDE ACDE ABDE ABCE ABCD ABCDE V 1 -1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 -1 0.5849 2 1 -1 -1 -1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 0.2123 3 -1 1 -1 -1 -1 -1 1 -1 1 1 1 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 0.3108 4 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 1 -1 0.3175 5 -1 -1 1 -1 -1 1 1 1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 -1 1 0.3854 6 1 -1 1 -1 -1 1 -1 1 -1 1 1 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 0.434 7 -1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 1 1 1 -1 1 -1 -1 1 1 -1 0.253 8 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 1 0.4242 9 -1 -1 -1 1 -1 -1 1 1 1 -1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 -1 1 0.4558 10 1 -1 -1 1 -1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 0.4827 11 -1 1 -1 1 -1 1 1 1 -1 1 -1 1 -1 1 1 -1 1 1 -1 1 -1 1 -1 0.4051 12 1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 1 0.0627 13 -1 -1 1 1 -1 -1 1 -1 1 1 -1 1 1 -1 -1 1 1 1 1 -1 -1 1 -1 0.7991 14 1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 1 0.2273 15 -1 1 1 1 -1 1 1 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 1 1 -1 1 0.5941 16 1 1 1 1 -1 1 -1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 -1 -1 -1 1 -1 0.8772 17 -1 -1 -1 -1 1 1 -1 1 1 1 -1 1 1 -1 1 -1 -1 -1 -1 -1 -1 1 1 0.4824 18 1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 1 -1 1 1 -1 1 1 1 -1 -1 0.1601 19 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 1 1 -1 -1 1 1 1 -1 1 1 -1 -1 0.0921 20 1 1 -1 -1 1 -1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 0.3447 21 -1 -1 1 -1 1 1 -1 -1 1 -1 1 1 -1 1 1 -1 1 1 1 -1 1 -1 -1 0.5088 22 1 -1 1 -1 1 1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 0.5066 23 -1 1 1 -1 1 -1 -1 -1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 1 -1 1 1 0.2648 24 1 1 1 -1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 -1 1 -1 -1 -1 1 -1 -1 0.164 25 -1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 1 1 1 -1 1 1 1 -1 -1 -1 0.2151 26 1 -1 -1 1 1 -1 1 -1 -1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 0.2446 27 -1 1 -1 1 1 1 -1 -1 1 -1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 1 1 1.1083 28 1 1 -1 1 1 1 1 -1 1 -1 -1 1 -1 -1 1 1 -1 -1 -1 1 -1 -1 -1 0.7635 29 -1 -1 1 1 1 -1 -1 1 -1 -1 -1 -1 -1 -1 1 1 1 -1 -1 1 1 1 1 0.2284 30 1 -1 1 1 1 -1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 0.3874 31 -1 1 1 1 1 1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 -1 -1 -1 -1 -1 0.2001 32 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.3692 ~\0 Table 6-10 (Part 2 of 2) Face Centered Cube Design continued (axial points and center points). 33 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 34 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 35 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0425 36 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0435 37 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0117 38 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0117 39 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 40 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 41 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.2498 42 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.2481 43 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0671 44 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0688 45 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 46 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 47 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 48 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 49 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 50 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 51 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0117 52 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0117 53 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0.0435 54 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0.0425 55 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 56 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 57 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0.1877 58 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0.1858 59 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0.181 60 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0.181 61 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 62 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 63 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 65 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 00425 66 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0.0435 67 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0.0435 68 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0.1499 69 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 70 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 71 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 -~ 72 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 73 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0.0132 74 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 00132 75 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0.049 76 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0.04Ji 77 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0.048 78 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0.049 79 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \0 N 93 6.7 DOE ANALYSIS METHOD There are twenty-three independent variables in the case study (see Table 6-3). The data were analyzed in Excel using ANOVA (Yate's algorithm) and regression analysis (matrix manipulation). The FFD was analzyed via ANOVA. The DFFD was analyzed via a combination ofANOVA and regression. The FCCD were analzyed via regression (see Appendix K). Yate's algorithm and regression analysis provided the effects for all variables. Since it is nonsensical to run replicates in a computer-simulated experiment, there are no degrees offreedom for the error. Hence, normal probability plots (NPPs) were used to identify the significant variables. The underling concept is that negligible effects are normally distributed with mean zero and variance (52. Hence, insignificant effects will tend to fall along a straight line, and significant effects will fall away from the straight line [11]. The significant effects in the case study are labeled on the plots and are shown in Appendix L. The resulting significant factors and stackup equations for each experimental design are shown in the Results section. Since NPP is the graphical way to choose significant factor effects, the chosen effects were statistically checked by an ANOVA table. An ANOV A table summarizes the variance contributions ofthe significant variables. The sums ofsquares for the individual regression variables was obtained by Yates algorithm. The R2 value is a measure ofthe proportion ofthe total variability explained by the stackup equation. If the R2 value is small, there are three possibilities: 94 1. There may be lurking variables, i. e. variables that have a significant effect, but were not included in the DOE. 2. There may be additional higher order interaction effects that have not been estimated. 3. There may be other interaction effects that are aliased with the factors included or excluded from the model that have a significant effect. As this is a computer simulation, option 1 is impossible. The other two options are possible, but at this stage, the R2 value cannot be improved without running additional experiments. 6.8 RESULTS 6.8.1 Fractional Factorial Design In order to verify the stackup equation an ANOVA table was constructed containing only those variables identified on the NPP. Table 6-11 ANOVA table for NPP selected factors by FFD. ANOYA SS DOF MS FO F (0.05, 1,24)= 4.26 x10 0.1209 1 0.1209 11.0442 Significant x8 0.1001 1 0.1001 9.1469 Significant x18 0.0604 1 0.0604 5.5197 Significant x9 0.0548 1 0.0548 5.0046 Significant x4 0.0609 1 0.0609 5.5669 Significant x6 0.0933 1 0.0933 8.5192 Significant x22 0.1007 1 0.1007 9.1990 Significant Error 0.2628 24 0.0109 Total 0.8540 31 R2 = 0.6923 The estimated stackup equation from the FFD is: 95 y = 0.402069 + 0.061719 * x4 + 0.076350 * x6 - 0.079113 * x8 (40) + 0.058519 * x9 - 0.086931 * x l 0 - 0.061456 * x 18 + 0.079338 * x22 The R2 value is low (== 0.69), since the FFD cannot estimate a quadratic te1TI1. 6.8.2 Double Fractional Factorial Design As the DFFD is a non-standard design, the analysis method used requires some explanation. The DFFD consists oftwo fractional factorial designs, an upper (nominal to USL) and a lower design (nominal to LSL). The designs were first analyzed separately, and then combined to create a single model. First an ANOVA was conducted on the lower (see Table 6-12) and upper design (see Table 6-13) separately. Table 6-12 ANOVA table for NPP selected factors by DFFD (lower design). ANOVA SS DOF MS FO F (0.05, 1, 20)== 4.3513 ab == x18*x19 0.0333 1 0.0333 21.0633 significant ac 0.0048 1 0.0048 3.0339 x13 0.0625 1 0.0625 39.5929 significant x5 0.1408 1 0.1408 89.1255 significant x6 0.0104 1 0.0104 6.5739 significant xl l 0.0140 1 0.0140 8.8640 significant x22 0.0064 1 0.0064 4.0342 significant x19 0.0638 1 0.0638 40.3724 significant x18 0.0328 1 0.0328 20.7928 significant x23 0.0108 1 0.0108 6.8181 significant Error 0.0316 20 0.0016 Total 0.4111 31 R2 == 0.9231 The letter expressions, ab and ac, are interaction effects defined by the alias structure. According to the sparsity ofeffects principle [11] (Refer section 4.4.3.) the ab interaction effect is an interaction ofsignificant main effects. Examination ofthe alias structure ofthe main effects shows that the ab interaction effects is aliased with the -,1:18 (BCDE) and X19 (ACDE) interaction (CDE cancels out). 96 Table 6-13 ANOVA table for NPP selected factors by DFFD (upper design). ANOVA SS DOF MS FO F (0.05, 1, 20)== 4.3513 x10 0.0505 I 0.0505 8.5064 significant be == x10*x5 0.0354 I 0.0354 5.9690 significant ac == x6*x22 0.0304 1 0.0304 5.1171 significant xl 0.0021 1 0.0021 0.3482 x19 0.0101 1 0.0101 1.7095 x2 0.0134 1 0.0134 2.2594 x15 0.0147 1 0.0147 2.4843 x23 0.0162 1 0.0162 2.7208 be 0.0166 1 0.0166 2.7962 x14 0.0238 1 0.0238 4.0062 xI3 0.0313 1 0.0313 5.2799 significant x9 0.0471 1 0.0471 7.9316 significant x5 0.0539 1 0.0539 9.0739 significant x22 0.0547 1 0.0547 9.2172 significant x6 0.0683 1 0.0683 11.5074 significant Error 0.0950 16 0.0059 Total 0.5635 31 R2 == 0.8143 Similarly to the lower design, the be interaction effect is aliased with the XIO (BeE) and Xs (E) interaction (E cancels out) effect, and the ac interaction effect is aliased with the X6 (BD) and X22 (ABeD) interaction effect (BD cancels out). The significant effects and their coefficient values for upper and lower DFFD are listed in Table 6-14 and Table 6-15 respectively. Table 6-14 Significant factors and their coefficients in lower DFFD design. significant effects coefficients x, -0.1876 Xl9 -0.1263 Xl3 -0.1250 X18* X19 0.0912 XI8 -0.0906 XII 0.0592 X23 -0.0519 X6 0.0510 X22 0.0399 97 Table 6-15 Significant factors and their coefficients in upper DFFD design. significant effects coefficients X6 0.1307 X22 0.1170 Xs 0.1160 XIO -0.1124 X9 0.1085 X10* x, -0.0941 Xl3 0.0885 X6* X22 -0.0871 The upper and lower designs have identified different sets of significant variables. In order to integrate the two disparate results into a single model, a regression model was constructed as follows: 1. Ifa variable is significant in both equations, and ifboth equations are ofthe same sign (both positive or both negative), then response is monotonically increasing with respect to the factor. Hence, the regression model is likely to contain a strong linear, and possibly a weaker quadratic term in that variable. 2. If a variable is significant in both equations and the coefficients are of opposite sign, the regression equation is likely to contain a pure quadratic term for that variable. 3. Ifa variable only shows up in one and not the other equation, then the regression equation may have a linear and a quadratic term for the variable. 4. Ifthe variable does not show up in either equation, then it is dropped from further consideration. 98 The above cases assume that the functional form ofthe regression model is second order. According to these cases from a. through d. The significant variables from Table 6-14 and Table 6-15 were selected according to the above criteria. The results are shown in Table 6-16. Ifthe term is likely to be linear, quadratic, or an interaction, then it is marked with an "X" in the appropriate column. Ifthe term might be linear or quadratic, then it is marked with a"?". Table 6-16 Significant effects from upper and lower design combined. case variables linear quadratic interaction term term term a X6 X ? X22 X ? b Xs ? X XI3 ? X C X6*X22 X XI9 X ? XIS X ? XIS*XI9 X XIO X ? XIO*XS X X9 X ? Xll X ? X23 X ? All terms marked with an "X" or "?" were included in building the regression model using Stat-Graphics [22]. The result is shown in Table 6-17. 99 Table 6-17 ANOYA table for estimated stackup by DFFD (95%). ANOYA SS DOF MS FO p-value Significance x5 0.028206 1 0.028206 2.22 0.1445 x6 0.131873 1 0.131873 10.36 0.0026 X x9 0.03729 1 0.03729 2.93 0.0948 xl0 0.091053 1 0.091053 7.15 0.0108 X xlI 0.003168 1 0.003168 0.25 0.6206 x13 0.008921 1 0.008921 0.7 0.4076 x18 0.03103 1 0.03103 2.44 0.1264 x19 0.031345 1 0.031345 2.46 0.1245 x22 0.096846 1 0.096846 7.61 0.0087 X x23 4.36E-05 1 4.36E-05 0 0.9536 x6*x6 0.034987 1 0.034987 2.75 0.1052 x22*x22 0.033579 1 0.033579 2.64 0.1123 x5*x5 0.369714 1 0.369714 29.03 0 X x13*x13 0.182425 1 0.182425 14.33 0.0005 X x19*x19 0.032106 1 0.032106 2.52 0.1202 x18*x18 0.006233 1 0.006233 0.49 0.4882 x10*xl0 0.002729 1 0.002729 0.21 0.6459 x9*x9 0.054213 1 0.054213 4.26 0.0456 X xl1*x11 0.027544 1 0.027544 2.16 0.1492 x23*x23 0.053298 1 0.053298 4.19 0.0474 X x6*x22 0.011038 1 0.011038 0.87 0.3574 x10*x5 0.063813 1 0.063813 5.01 0.0308 X x18*x19 0.009973 1 0.009973 0.78 0.3815 Residual 0.509355 40 0.012734 Total 1.94924 63 R2 =0.73869 The estimated stackup equation from the DFFD is: y =-0.0134844 + 0.0865608 * x6 - 0.0719267 * x10 - 0.0741795 * x22 + 0.214975 * x5 * x5 - 0.0225142 * x13 * x13 + 0.0582094 * x9 * x9 (41) + 0.0577156 * x23 * x23 - 0.126306 * x5 * xl0 The R2 value is 0.74. Ifonly the significant factors are included in the model, the R2 value drops to 0.54. Since, the DFFD is a non-symmetric, nonstandard design, it is likely have missed identifying some significant factors. 100 6.8.3 Face Centered Cube Design For a Face Centered Cube Design, the significant factors as determined from the NPP were used to build a regression model (see Appendix L). Table 6-18 ANOVA table for FCCD regression model. ANOVA SS DOF MS FO F (0.05, 12,66)= 4.26 Model 5.3357 12 0.7622 48.5488 Significant Error 1.0362 66 0.0157 Total 6.3719 78 == 0.8374 The resulting FCCD regression model is given by: y = 0.028497 - 0.074459 * x8 - 0.081818 *xl0 - 0.054712 * x18 - 0.036959 * x21 + 0.074641 * x22 + 0.219805 * x5 * x5 (42) - 0.029145 * x9 * x9 + 0.157605 *x13 * x13 + 0.151855 * x14 * x14 - 0.029145 * x15 * xIS - 0.029145 * x16 * x16 - 0.029145 * x19 * x19 6.8.4 Metric 1. Significant Variables The significant independent variables identified from the HLM analysis and from the NPPs were compared. The full HLM analysis is shown in Appendix H. 6.8.4.1 FFD Table 6-19 lists the significant variables ofthe FFD model compared to the HLM results in order ofincreasing HLM significance. The significant FFD variables that are also significant HLM variables are denoted by a *. 101 Table 6-19 The number ofsignificant independent variables identified. Order of HLM %of FFD %of Significance effect effect 1 X19, X20 24.88% 14.16% XIO 2 X6, X9 17.36% X22 11.790/0 3 Xs, Xs 15.94% xs* 11.72% 4 Xl3, XIS 9.00% X6* 10.930/0 5 X14,X16 8.36% X4 7.13% 6 XIS 8.36% XlS* 7.07% 7 X2 4.30% X9* 6.42% 8 XII 4.21% 9 X3 3.55% 10 X12 1.43% total 97.39% total 69.22% Number offactors -- 4 correctly identified Only four out of 10 effects were identified by this method. 6.8.4.2 DFFD Table 6-21 lists the significant variables ofthe FFD model compared to the HLM results in order ofincreasing HLM significance. The % ofeffects were obtained from the ANOVA with reduced variables (Table 6-20), and is the ratio ofthe sum of squares for that variable to the total sum ofsquares. 102 Table 6-20 ANOVA for reduced DFFD. ANOVA SS DOF MS FO p-value x6 0.090498 1 0.090498 5.32 0.0249 x10 0.185928 1 0.185928 10.93 0.0017 x22 0.059361 1 0.059361 3.49 0.0671 x5*x5 0.451512 1 0.451512 26.53 0 x13*x13 0.182425 1 0.182425 10.72 0.0018 x9*x9 0.054213 1 0.054213 3.19 0.0798 x23*x23 0.053298 1 0.053298 3.13 0.0823 x10*x5 0.093031 1 0.093031 5.47 0.023 Residual 0.936009 55 0.017018 Total 1.94924 63 R2 = 0.5339 Table 6-21 The number ofsignificant independent variables identified. Order of HLM %of DFFD 0/0 of Significance effect effect 1 X19, X20 24.88% *X52* 21.44% 2 2 X6, X9 17.360/0 *XI3 * 8.66% 3 xs, Xs 15.94% *X6* 4.30% 4 X13, XIS 9.00% X22 2.82% 5 X14, Xl6 8.36% XIO 8.83 % 6 XIS 8.36% *XIO*XS* 4.420/0 2* 7 X2 4.30% *X9 2.57% 2 8 XII 4.21% X23 2.53% 9 X3 3.55% 10 XI2 1.43% total 97.39% total 55.56% Number offactors -- 5 correctly identified Only 5 out of 10 variables were identified, and the total % effect they accounted for was less than the % effect accounted for by the FFD model. 6.8.4.3 FCCD Table 6-22 lists the significant variables ofthe FCCD model compared to the HLM results in order ofincreasing HLM significance. 103 Table 6-22 The number ofsignificant independent variables identified. Order of HLM %of FCCD %of Significance effect effect 1 X19, X20 24.88% XIO - 2 X6, X9 17.36% xs* - 3 Xs, Xs 15.94% XlS* - 2 4 X13, XIS 9.00% xs * - 5 X14,X16 8.360/0 X2I - 2* 6 XIS 8.360/0 X9 - 2* 7 X2 4.30% X16 - 2 8 XII 4.21% XI3 * - 2* 9 X3 3.55% XI9 - 2* 10 Xl2 1.43% XIS - 2* 11 X14 - total 97.39% total - Number of factors -- 9 correctly identified The FCeD model correctly identified 9 ofthe leading 11 factors, more than any other design. 6.8.4.4 Metric 1. Analysis The FFD performed worst ofthe three designs. Its model has a low R2 value, and it identified only four effects. The model does not contain any quadratic terms due to the reduced number ofexperiments. The FFD model is clearly not realistic in this case as quadratic effects are clearly present. Both the DFFD and FCCD detected the effect ofthe second order quadratic terms and the very nature ofthe APM is non-linear, since any variation will result in a positive deviation from its the nominal value (see Figure 6-1). The DFFD did not fare much better than the FFD. Although the R2 value ofeach design is high (upper: 810;0; lower: 92%), the combined result missed many significant 104 variables. The R2 value ofthe combined model was very low (530/0), and only five significant variables are common to the HL~1 analysis. It is interesting to note that the FFD identified fewer significant variables (4), but has a higher R2 value. The FCCD detected the quadratic terms and yielded the best results with an R2 value of 830/0. Nine variables are correctly identified compared with HLM analysis. 6.8.5 Metric 2. Moments Comparison APM distribution moments obtained from the true simulation (50,000 runs) and the DOE estimated equations (40), (41) and (42) were compared. The moments for the DOE estimated equations were computed by running a MC simulation with the stackup equations (see Table 6-23). The simulation program for this analysis is shown in Appendix J. Table 6-23 Comparison ofAPM distribution moments. Mean %diff Std Dev %diff FFD 0.4022 228.6% 0.0641 -18.3% DFFD 0.0209 -82.9% 0.0596 -24.1% FCCD 0.0748 -38.9% 0.0706 -10.1% TRUE 0.1224 0.0785 Skewness %diff Kurtosis %diff FFD -0.0012 -100.10/0 0.0060 -99.50/0 DFFD 0.7140 -32.0% 2.0072 62.8% FeeD 0.6150 -41.5% 1.1593 -6.0% TRUE 1.0506 1.2330 Overall, the FCCD generated the closest moments among the three experimental designs. The moments computed from the FFD model, and DFFD model were, as expected, not close to the true values since both models had low R2 values. 105 Table 6-24. FCCD equation moment comparison. Mean %diff Std Dev %diff Skewness %diff Kurtosis %diff FCCD by 0.0748 -38.90/0 0.0706 -10.1 % 0.6150 -41.50/0 1.1593 -6.0% MC FCCD by , 0.0459 -62.50/0 0.0518 -34.00/0 0.0052 -99.5% 0.0081 -99.3% MGF True 0.1224 0.0% 0.0785 0.00/0 1.0506 0.00/0 1.2330 0.0% Since the FCCD performed so well, a comparison ofthe MC generated moments and moments computed from moment generating functions (MGFs) was conducted. The moments for each independent variable and the calculation ofthe APM moments by MGFs are shown in Appendix M. The results are shown in Table 6-24. As can be seen the moments do not compare too favorably. Both the MC and MGF estimated moments are significantly different than the true MC moments. As Table 6-25 shows the differences between the FCCD moments computed by MC are not within the 90% confidence interval ofthe true MC simulation. Indeed, the percent difference for the skewness and kurtosis ofthe MGF estimated moments is nearly 100%. Hence, although neither is particularly good, estimating the moments by MC simulation is preferable to MGFs. Table 6-25 Confidence Interval ofmean and standard deviation from True moments (mean, and standard deviation). Estimated Upper CI Lower CI MC Within Moments (90%) (900/0) Moments CI? FCCD Mean 0.0459 0.1218 0.1230 0.1224 no (MGF) Std Dev 0.0518 0.0781 0.0789 0.0785 no None ofthe TSE estimated moments fall within the 90% CI ofthe true moments. 106 6.8.6 Metric 3. Number of experimental runs Table 6-26 Metric 3: Number ofexperimental runs required. DOE No. ofex . runs FFD 32 DFD 64 FCCD 79 FFD has the smallest number ofexperimental runs. However only the main effects could be estimated, and it obviously resulted in a poor model. The difficulty of this type of analysis is that it is not certain whether there are second order effects present, since a true functional relationship is presumed to be unknown. Ifthere were second order interaction effects, more experimental runs would be needed. The factorial part ofFCCD was a resolution III design. Hence certain second order interaction effects were aliased with main effects. Ifinteraction effects are likely to be significant the fractional part is expandable to a resolution V design which would separate all 2nd order interaction effects from main effects and from each other so one could identify the contributions ofthe interaction effects. Again, the significance of each variable could not be expressed in FCCD. 6.8.7 Metric 4. Validation Similar to Phase I, in this section, all estimated stackup equations were evaluated as to how well they predict at design points and / or points that were not in the original design. The predictability ofthe various stackup equations was evaluated by comparing the computed stackup APM for various combinations ofdesign points to the 107 simulated APM value ofthe same design points. The points were chosen arbitrarily. All non-significant variables in the stackup equations were set to O. For the validating simulations, significant variables were set at levels ofthe validation points, and non- significant variables were varied randomly as programmed in the MC simulation (see section 6.3). Validating simulations for each experimental designs are shown in Appendix N. The set ofvalidation points were arbitrary chosen so that the points are not on any original experimental design within the experimental design space, that is, variables are set at either 0.5, or -0.5 levels. Another set ofvalidation points are variables set at- 1 or 1, which are at the edge ofthe design space. Combinations ofpoints for significant variables of FFD, DFFD and FCeD and the results in stackup calculations at each levels are shown in Table 5-20, Table 6-28, and Table 6-29 respectively. Table 6-27. Validation design points for FFD and estimated stackup results. x4 x6 x8 x9 xl0 x18 x22 y 1 -1 -1 -1 -1 1 1 1 0.215545 2 1 1 1 1 -1 -1 -1 0.588593 3 -0.5 0.5 -0.5 0.5 -0.5 0.5 -0.5 0.451269 4 0.5 -0.5 0.5 -0.5 0.5 -0.5 0.5 0.352869 5 0 0 0 0 0 0 0 0.402069 Table 6-28. Validation design points for DFFD and estimated stackup results. x5 x6 x9 xl0 xI3 x22 x23 y 1 -1 -1 -1 -1 1 1 1 0.065416 2 1 1 1 1 -1 -1 -1 0.124961 3 -0.5 0.5 -0.5 0.5 -0.5 0.5 -0.5 0.361662 4 0.5 -0.5 0.5 -0.5 0.5 -0.5 0.5 0.480753 5 0 0 0 0 0 0 0 -0.01348 108 Table 6-29. Validation design points for FCCD and estimated stackup results. x5 x8 x9 xIO xI3 x14 xI5 xI6 x18 x19 x21 y 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 0.376576 2 1 -1 1 -1 1 -1 1 -1 1 -1 1 0.505788 3 -0.5 0.5 -0.5 0.5 -0.5 0.5 -0.5 0.5 -0.5 0.5 -0.5 0.136686 4 0.5 -1 0.5 -0.5 0.5 -0.5 0.5 -0.5 0.5 -0.5 0.5 0.126651 5 0 0 0 0 0 0 0 0 0 0 0 0.028497 The difference was said to be large ifit exceeded ± 1.645a, which corresponds to a 950/0 CI range ofthe Me simulation (see Appendix N for a). The DOE cell, which is within this range, is marked with one star (*). Ifthe difference is within the range of ± l c, the APM estimate is said to be close to the true value, and the cell is marked with two stars (**). This is a measure ofhow well each equation estimates the true functional relationship. Another measurement is R2 prediction (see section 5.6.5 of equation (35)). Comparison results are shown in Table 6-30. 109 Table 6-30 Comparison ofestimated APM to MC simulation. Point True Estimated 0/0 Actual R2 APM APM difference difference predicted <1.65a? Examination ofTable 6-30 shows that DFFD model did not interpolate well. In terms of% difference FCCD had the minimum difference ranging 500/0 to 120%. All points estimated by FFD were within the certain confidence range, however c differes from MC simulation due to the less number ofsignificant variables. Overall, FCeD design had close point estimates. 123 7 DISCUSSIONS AND CONCLUSION This thesis was conducted in two phases, comparing different experimental designs for stackup estimation purposes. The primary purpose ofthis research wasto verify that second order response surface designs (CCD, or FCCD) are the best designs for estimating a non-linear (second order) response. The secondary purpose ofthis research was the model estimated would be applied for moment generating equations of tolerance analysis method. In order to verify efficacy and effectiveness ofthe response surface designs for tolerance analysis method, comparisons were made based on accuracy and efficiency metrics. Based on the results, the conclusions are as follows. 7.1 PHASE I CONCLUSION Phase I conducted DOE on two known trigonometric equations, which response were approximately 2nd order within the range ofthe component tolerances. Equation (6) had its APM maximum at the design nominal, and Equation (7) had its maximum APM point offthe nominal design point. Table 7-1 and Table 7-2 summarize the metrics in a subjective manner. 124 Table 7-1 Summary results for Phase I, equation (6). Design £-? Validation Moments MC Moments Efficiency (value) (avg. % (difference) 2nd order MGF (No. of 8t difference) (1 and (diff/1 st and 2nd exp.) 2ndonly) only) FFD N/A Poor Poor Poor Excellent ...... J.?9.~2 (:.~.?!:~2. J.~.?? : =..l ..9.Q.~~) (~2. . DFFD Excellent Good Excellent Excellent Good ...... (Q..:.Q) (!..:.?~) {:~ : ~.~?2 (:.? ~ ~.~.2. (~.2 .. FCCD Excellent Excellent Good Good Poor ...... ,(9..:.Q) ..<.~.~?) (:.~ =~..!..~) {:.~ =:}..7.~) (~2. .. FCCD Excellent Good Excellent N/A Poor ...... f~.!} (.9..:..Q) (.?~?) ,(.:.7 : ? ~.) (.~) . CCD Excellent Good Good Good Poor ...... (9..:.92 <..! ..9.~) .<~ : ~J..~.) (:} =~.9.~) .<.?.2.. .. CCD Excellent Excellent Good N/A Poor ...... [~.!.L ,,(9..:..Q2 (Q.~) (=~ =~.~.~) .<.~2. .. TSE Excellent Poor Good (0.0) (-5%-72%) (-4-21%) Table 7-2 Summary results for Phase I, equation (7). ., Design e: Validation Moments Me Moments Efficiency (value) (avg. % difference) (difference) 2ndorderMGF (No.ofexp.) (1st and 2nd onl) (diff! 1st and 2nd oni ) FFD Poor Poor Poor Poor Excellent ,...... (~.:.~~) .(~~.~>. (?~:~.g.0!) (~.?~ :.. ~~1~J (~) .. DFFD Excellent Excellent Good Poor Good ...... (Q:.~.~) (~.0!) (Q.~.~~~) (9 ..: ..:.~. ? !? 2 (~J .. FCCD Excellent Excellent Good Poor Poor 0...... (Q:.~.~) (~.~) (~.~.~.2~) (~~..:..~.??!?) {~) .. FCCD full Excellent Excellent Good N/A Poor ...... (Q:.~.~) (~.Q~?) J~~~ ..:..Q.~) {~) .. CCD Good Good Good Poor Poor ...... (q.:~~) .(~.~~) (~?~ ..:..q.~) (Q..:..?~.!?2 (~) .. CCD full Excellent Excellent Good N/A Poor ...... (Q:.QQ) (~.~J (~?~ ..:..q.~). (?J .. TSE Good Good Good (0.75) (7-29%) (0-42%) [Note: For Moments comparison only 1st and 2nd order moments were compared since 3ed and 4th order moments involved too much error (> 100%). The % shows the range ofdifference for 1st and 2nd moments. 4th order FCCD and CCD MGF moments were not be able to be obtained since 4th order MGF equation was not available. 125 1. Among the four experimental designs compared (FFD, DFD, FCCD, and CCD), FFD did not estimate the second order response, since the FFD is a design meant for linear response estimation. The other designs (DFFD, FCCD, and CCD) detected the curvature of the true response and in general did a good to excellent detection ofthe nominal point for both the on and offdesign center maximum. 2. The validation analysis shows similar results, with FFD being quite poor and the other designs rating good to excellent. It is interesting to note the differences between the 2nd order CCD and the full CCD models. The 2nd order always only ranks a good versus the full CCD, which always outperforms all other designs. Another unexpected result is the FCCD full underperforms the FCCD 2nd order model. The reason for this is unknown. 3. For equation (7), the moment generation based on MC simulation was not favorable. With the exception ofthe full CCD model, none ofthe experimental designs correctly identified the maximum APM coordinates for the offnominal case. Since the remaining higher order moments depend on the maximum point coordinates, the error was propagated. 4. Computing the APM distribution moments by Me simulation was always superior to using the MGFs. This is probably because the 2nd order MGFs are themselves approximations. It is believed that the approximations are too inaccuarate to correctly compute higher order moments for a 2nd order function. In addition, it is believed that for the 3rd and higher order moments, input distribution moments above the 4th moment playa stronger role and fare 126 required. The fact that none of the moments estimated by MC simulation were statistically the same as the true MC simulation moments is less ofa concern, because the confidence intervals on the moments were based on the MC simulation. With 50,000 runs the confidence interval is quite tight. Indeed, as the confidence interval is a function ofthe number ofsimulation runs, it can be made to an arbitrary width. Hence, it is always recommended to use MC simulation on the estimated equations to predict the APM distribution. S. With regards to the number ofexperimental runs, clearly FFD is superior. Unfortunately, it completely misses the curvature in the function. The other designs have about the same number ofexperiments. However, the DFFD will have significantly fewer runs than the FCCD or the CCD when the number of variables becomes large. 6. Considering all the metrics, CCD and FCCD were the best designs ofall the experimental designs compared in terms ofthe closer estimate ofthe true functional relationship (effectiveness), and number ofexperiments (efficiency). It also found that FCCD had closer point estimates and CCD had closer distribution estimates compared to the true values due to their design spaces. The DFFD may prove to be superior, however, as the number ofvariables increases. It also performed quite well in both the on and offnominal cases. 7.2 PHASE II CONCLUSION In Phase II, stackup equations were estimated by three different experimental designs based on the simple bracket assembly. There were 23 component feature 127 dimensions which affect the distance between the varied and nominal position ofa point (hole M). The three different experimental designs were FFD, DFFD, and FCCD. In this phase, the true results are the MC simulation and HLM results ofthe varied bracket. Table 7-3 Summary results for Phase II. Design Number of MomentsMC Moments Efficiency Validation significant (difference) 2nd order MGF (No. of (avg. % variables (1st and (diff/ 1st and 2nd exp.) difference) defined 2ndonly) only) FFD Poor Excellent Good 5 N/A (-18 - 230%) (32) (1100/0) DFFD Good Good Poor 4 N/A (-83 - -24%) (64) (220%) FCCD Excellent Poor Poor Excellent 9 (-10 - -39%) (-60 - -35%) (79) (85.6%) 1. First, among the three experimental designs compared (FFD, DFD, and FCCD), FFD did not estimate the second order response, since the FFD is a design meant for linear response estimation. The other designs (DFFD, and FCCD) detected the curvature ofthe true response. Comparing those two designs which detected the curvature ofthe true response, FCCD identified most ofthe main effects defined by HLM simulations. FCCD model was found to be the best design by ANOVA. 2. Similar to the Phase I, the moments generated by MC, based on estimated stackups, were compared. FCCD had the best stackup model, which moments are close to the true MC simulation. FFD and DFFD had large difference even in first and the second moments. These models were not used for MGF, which accumulates errors, by approximation. 128 3. The moment generating method was also used on the FCCD stackup. As expected the result were not close to the true MC simulation. Same as Phase I, computing the APM distribution moments by MC simulation was superior to using the MGFs since MGF is the approximation on DOE estimated models. Error propagates by MGF as well as order ofmoments. Errors in 3ed and 4th MGF moments were the largest, even 3ed and 4th estimated moments by MC were not close to be useful. 4. Since the DFFD could not directly obtain the quadratic term due to the nature of the factorial design, and lack ofstatistical software which deals with large number ofindependent variables, DFFD was separated to upper and lower design in order to analyze significant effects. The each separated designs, (upper and lower) gave good R-squared values, however the combined regression analysis based on significant variable from both design did not generate a desirable result. That may be the reason for DFFD design did not yield the favorable result in this phase. S. FCCD required 79 experiments, and 46 degrees offreedoms were used. Since coefficients were obtained manually, it was too tedious to define interaction terms. Statistical software could have fully utilized the all the experimental runs. DFFD required 64 experimental runs, yet the design did not generate a close estimate ofa stackup. 6. Validation analysis could not be applied straightforward. The problem lay in the Me simulation method for defining the true position. The two true position 129 factors, the radius distance and angle, were interacted each other and considered as one grouped variable in MC simulation, whereas in DOE analysis those two factors were separated in order to be defined by design levels. Therefore, DOE models have significance ofeither variable was modified in the MC simulation code. The FCCD had the best overall point estimates within the confidence interval range. 7.3 CONCLUSIONS This research clearly showed the inadequacy ofthe linear stackup assumption when faced with a non-linear tolerancing problem. Response surface designs are a superior method ofestimating the stackup equations when the stackup equation is unknown. It was clear that response surface designs were clearly capable ofdetecting the non-linear responses, and estimate the complex curvature with regression models with higher terms ofcoefficients. However the question is whether the estimated functions are actually dependable for tolerance analysis applications. The Phase I research showed that estimation ofthe APM distribution is highly dependent on accurately locating the APM maximum. Ifthe experimental design points hit on the peak point, the moments will be estimated well. However, ifthe coordinates ofthe peak point are not on an experimental design point, the APM distribution mean and higher order moments estimated as well. Therefore, it is imperative to ensure that one attempt to identify the APM maximum prior to designing the experiments so that the experiments can be centered about this point. In many applications, this will not be a significant problem, as the maximum is often the design nominal. 130 The research also showed that current published 2nd order statistical tolerancing models are also inconsistent and inadequate. MGFs were shown to be an undesirable way for moments estimation method due to the complexity ofthe process, uncertainty ofthe formula, and error in the results. The altenative is to use Me simulation with the estimated DOE stackup model. It provides a much closer estimate ofthe APM distribution than can be obtained from MGFs. Additional research would be necessary to derive the more complete MGF which included higher order moments. One must question the practicality ofthis approach, however, given that the 5th and higher order moments ofthe input distributions are rarely available. In phase II, response surface method was proven to be the better design then linear models in more complex system than phase I. Although, phase II has shown that the DOE method can be applied to the Tolerance Analysis ofthe complex system, implementation ofthe method has to be considered. First, there is a constraint for number ofvariables handled in statistical software. Manual manipulation methods are too tedious to be practical for implementation oftolerance analysis. Secondly, the simulation method for true position has to be analyzed. Since some ofthe mechanical parts do not have bi-directional tolerance and DOE designs for these cases have to be considered and left for further research. 124 REFERENCES [1] Bisgaard, S., 1993, "Design ofExperiments for Tolerancing Assembled Products", Center for Quality and Productivity Improvement, Report No. 99 University of Wisconsin. [2] Gerth, R.l., 1996, "Engineering Tolerancing: A Review ofTolerance Analysis and Allocation Methods", Engineering Design and Automation, Vol. 2, No.1, Spring pp. 3-22. [3] Zhang, H.C., and Huq, M. E., 1992 "Tolerancing Techniques: the State-of-the-Art", International Journal ofProduction Research, 30 (9), 2111-2135. [4] Eloseily, A. M., 1998, "A Comparison ofThree Experimental Designs For Tolerance Allocation", M Sc., Industrial Systems and Manufacturing Engineering, Ohio University. [5] Islam, Z., 1995, "A Design ofExperiment Approach to Tolerance Allocation", M Sc., Industrial Systems and Manufacturing Engineering, Ohio University. [6] Gerth, R. J., and Islam, Z., 1996, "Towards A Designed Experiments Approach to Tolerance Design" 5th International Seminar on Computer Aided Tolerancing, [7] Cox, D. N., 1986 "How to Perform Statistical Tolerance Analysis", The ASQC basic references in quality control: statistical techniques; Vol. 11, pp. 4-11, Milwaukee, Wisconsin. [8] Gerth, R. J., and Pfeiffer, T., 1999 "Minimum Cost Tolerancing Under Uncertain Cost Estimates", lEE transactions. [9] Kusiak, A., and Feng, C. X., 1996 "Robust Tolerance Design for Quality", Journal ofEngineering/or Industry, 118, February, pp. 166-169. [10] Bisgaard, S., 2000, "Tolerancing Mechanical Assemblies With CAD and DOE", Journal ofQuality Technology, Vol. 32, No.3, pp. 231-240. [11] Montgomery, D.C., 1997, Design and Analysis ofExperiments, 4th ed., John Wiley & Sons. [12] Hahn J. G, and Shapiro S. S., 1967, Statistical Models in Engineering, John Wiley & Sons. 125 [13] Evans H. D., 1975, "Statistical Tolerancing: The State ofthe Art, Part II. Methods for Estimating Moments" , Journal ofQuality Technology, Vol. 7, No. 1 pp. 1-11. [14] Applied computer Solutions, VSA SIM-3D User's Manual, 1998 Applied Computer Solutions, St. Clair Shores, Michigan. [15] Bralla, J. G., 1986, Handbook ofProduct design for Manufacturing, McGraw Hill [16] Dimensioning and Tolerancing, ISME Y14.5M-1994, 1995, The American Society ofMechanical Engineers, New York, New York. [17] http://www.nutsandbolts.net/specifications/hexcapscrews.cfm [18] VSL User's Reference, 1997, Variation Systems Analysis, Inc, St. Clair Shores, Michigan. [19] Myers, R. H., 1999, "Response Surface Methodology Current Status and Future Directions", Journal ofQuality Technology, Vol. 31, No.1, pp. 30-43. [20] McCuistion, P. J., 1999, "Geometric Dimensioning", Course supplement, Ohio University [21] Nigam D. S., Turner U. J., 1995, "Review ofstatistical approaches to tolerance analysis", Computer-Aided Design, Vol. 27, No.1, pp. 6-14. [22] STATGRAPICS Plus for windows 4.0,1999, Manugistics, Inc., Rockville, Maryland. [23] Myers H. R., Montgomery C.D, 1995, Response Surface Methodology, John Wiley & Sons. [24] Montgomery, D.C., Elizabeth, A.P., 1992, Introduction to Linear Regression Analysis, 2nd ed., John Wiley & Sons. [25] MicrosofusrExcel. 1997,2000, Microsoft Corp., Redmond, WA. 126 APPENDICES 127 APPENDIX A MOMENT GENERATING FUNCTIONS (MGF) The first and second-order tolerance analysis formulas from Cox [7], Hahn and Shapiro [12], and Evans [13] are listed below. When applied to a first order TSE, each reference had the same formula. When applied to a second order TSE, however, each reference had a slightly different formula. It was difficult to determine which were correct and which were not. In some cases it was not possible to make a clear determination. All equations are shown as the direct moments. The notation follows Cox's notation [7] and is given in the Definitions of Symbols and Abbreviations. Assumptions: 1. The component variables are uncorrelated, i.e., independent, 2. Third and higher order derivatives are 0 (2nd order Taylor series expansion only), and 3. Fifth and higher order moments ofthe component feature distributions are negligible, i.e., JLki = 0 V k ~ 5 Moment Generating Functions for a First-Order TSE (A - 1) (A - 2) (A - 3) 128 k k-l k 2d]J.12iJ.12j J.14Y = Id/J.14i + 6I Idi (A - 4) i=l i=l j=i+1 Moment Generating Functions for a Second-Order TSE From Evans's reference [13]: (A - 5) 1 ~ k-1 k II?y := ~ d: f-L? + d.d··f-L3· +-d?(11 · - II;.) + " ~ d?f-L,)·f-L2· (A - 6) r: _ L-J ( I _1 1 II 1 4 II r: 41 r: _1 L-J L-J lj _I J i i=l j=i+1 ~(3 3 3 J.1 3Y = ~ d, J.13i +2,di2d; (J.14i - J.12i2) -2,didii2J.13iJ.12i ) + k-l k (A - 7) ~ ~(6.d.d.d"f-L2·11,. +3.(d.d.~ +d .. d .. d.)"/ L-J L-J I J lJ 1 r -J 1 lJ II lJ J r 3·f-L2·)I J i=l j=i+l From Hahn and Shapiro's reference [12]: (A - 9) 2 J.12Y = I (di J.12i + did iiJ.13i ) (A -10) i (A -11) k k-1 k J.14Y = Id/J.14i +6I Id/d]J.12iJ.12j (A -12) i=l i=l j=i+l From Cox's equation [7] (corrected for f-LIY) (A - 13) 129 (A - 14) k-l k /I ~ ~ ~ I ~ ,/ I", ,) r3Y = L...J d l r" 3,l + "L...J "(dL...J tir"//3'ir://3}. + 6 . d.dl }.dlJr!.lr-l.. / + i i=1 )=i+l (A - 15) n-2 11-1 n ~3d.d~//3,1I2' ~ ~ ~6·d L...J l lJr tr: } + L...J L...J L...J lJ.. d·kd·k",.,·I,.,·II"'k 1 } r ... t r: ... Jr_ i=) i=1 )=i+lk=)+1 k k-l k 4 2 II Y = '" d. /1 , + 6~ ~ d. d : //"./1",. (I' ) (A - 16) r4 L...J l r 4 1 L...J L...J l } r: zir: -.1 Inear case i=1 i=1 )=i+l Cox's equations for direct central moments were a function (recursive relationship) ofthe raw moment equations, which were generated by a computer program, SOERP. And yet, his published second-order equation for the 1st raw moment k (mean) had an error (see equation A-13)). The Id jd.1 2i tenn is missing the coefficient i=1 1 Since the raw moments were computer-generated equations, it is impossible to be 2 assured that the other raw moment equations do not have errors as well. For the fourth direct central moment, the equation is too complex to be calculated by hand. Hahn and Shapiro's [12] second order moment generating function is dissimilar to the Evan's and Cox's. Hence, it is believed that their equation may be incorrect as the other two references agree. This leaves the Evan's reference as the only one without a reason for doubt. Although there remains some uncertainty as to which moment equations are correct, ifany, Evans [13] second order moment generating function was used in this research. 130 APPENDIXB PHASE I TRUE SIMULATION(MC) True APM Monte Carlo Simulation Code /****************************************************************/ /* Filenarre: Phasel .vsl */ /* */ /* Written by: Misako Rata */ /* Date: 04/16/2000 */ /* */ /* RJRroSE: Phase I sirrclation */ /* */ /* */ /* Assurrptions: */ 1* */ /* Revisions: */ /* Date fvbdeler Rev Nature and Purpose of Chanae */ /* */ /****************************************************************/ constant pi=3.l4l59265359; value xl, x2, a, b, C; variation vl, no:r:rral = 0 +- 1: v2 no:r:rral = 0 +- 1 ; xl = vI * l80/pi; //IConvert radians to degrees x2 = v2 * ISO/pi; //IConvert radians to degrees a = 2.5; b = 1.75 * ISO/pi; C = 0.8 * ISO/pi; output yl = cos(xl)+cos(2*x2)+2*(oos(xl)*oos(1.2*x2)); // IPhase I ec;ruation A 131 True APM Monte Carlo Simulation Result (50,000 runs) Session: 20000721-112948 y1 -> 50000 samples Sample Est. of (Variance)A O. 5 90% Conf. Interval Statistics Pop. in Est. of Pop. in Est. of Pop. Paramo ------Nominal 4.0000 Mean 3.4963 3.4963 0.0022 3.4927 3.5000 Std Dev 0.4962 0.4962 0.0027 0.4937 0.4988 Cp N/A N/A N/A N/A N/A Cpk N/A N/A N/A N/A N/A Skew -1.8368 LDL/UDL N/A / N/A Kurt 4.5711 Distribution Pearson-Beta Sample Est. Sample ** Est. % < Low Limit N/A N/A Low Val -0.9526 0.8800 % > High Limit N/A N/A High Val 4.0000 3.9866 % Out of Spec N/A N/A Range 4.9525 3.1066 ** Estimated range of 99.7300% Session: 20000721-112948 y2 -> 50000 samples Sample Est. of (Variance)A O. 5 90% Conf. Interval Statistics Pop. in Est. of Pop. in Est. of Pop. Paramo Nominal 3.3185 Mean 3.0464 3.0464 0.0031 3.0413 3.0516 Std Dev 0.7005 0.7004 0.0038 0.6968 0.7041 Cp N/A N/A N/A N/A N/A Cpk N/A N/A N/A N/A N/A Skew -0.5316 LDL/UDL N/A / N/A Kurt -0.2479 Distribution Pearson-Beta Sample Est. Sample ** Est. % < Low Limit N/A N/A Low Val -1.5647 0.8473 % > High Limit N/A N/A High Val 4.1128 4.2881 % Out of Spec N/A N/A Range 5.6775 3.4407 ** Estimated range of 99.7300% 132 The simulation results from equation (6) are adjusted for MGF moments comparison. JLl y = 3.4963 2 rII"..y = 0'2 = 0.4962 = 0.2462 J13y =fii: X J12i1.5 =-1.8368 *0.24621.5 =-0.2244 2 * J14y = fJ2 x J12/ =(4.5711- 3) * 0.2462 = 0.0952 The simulation results from equation (7) are adjusted for MGF moments comparison. J'ly =3.0464 2 JL"...y = 0'2 = 0.7005 = 0.4907 J13y =fii: X J12i 1.5 =-0.5316 * 0.49071.5 = -0.1827 2 * J14y = fJ2 x J12/ =(-0.2479 - 3) * 0.4907 =-0.7820 *Note: VSA defaults represents kurtosis as {32+3 in simulation results, therefore (32 value has to be adjusted by subtracting 3 in Jl4y calculations 133 APPENDIXC PHASE I TOLERANCE ANALYSIS (TSE) Conducting a first and second order tolerance analysis involves: 1. Computing the first four moments according to the equations in Appendix A, and 2. Computing the first and second order TSE. To compute the first four moments the partial derivatives and distribution characteristics ofthe independent variables for equation A and B are shown here. Notation: d i =( of)O = first order partial derivative of x i at its mean oX i 2 d .. = (a 1)° = second order partial derivative of Xl' at its mean II 2 aX'i 2 Xi X j dii =( a f )0 = second order partial derivatives of and at their means DxDxl }. J.11i = first moment (mean) of Xi J.12i = (J2 =second moment (variance) of Xi Jl3i = -J7i: X Jl2i 1.5 = third moment of Xi 2 J.14i = /32 XJ.12i =fourth moment of Xi -J7i:(the skewness coefficient) = Jl3~.5 J.12Y J.1 4Y /32 (the kurtosis coefficient) = 2 J.12Y 134 The skewness and kurtosis coefficients .J7f:, and /32 are dimensionless characteristics ofthe distribution. Coefficient .J7f: measures how symmetric the distribution is, and /32 measures how tapered the distribution is [7]. For the normal distribution, the skewness coefficient is 0, meaning the distribution is symmetrical, and the kurtosis coefficient is 3. When the skewness coefficient is negative the density has a longer tail to the left, and when the skewness coefficient is positive the density has a longer tail to the right. When kurtosis coefficient is greater than 3, the distribution is more peaked than the normal distribution. The independent variables, xl and x2, are N(O, 1/9) distributed for both Phase I equation C-1 and C-2. Their distribution characteristics ofare shown in Table C-2. Table C - 1 Characteristic ofthe independent variables. variables Jlu Jl2i ~3i 1l4i J7f: [32 distribution Xl 0 1/9 0 3/81 0 3 normal X2 0 1/9 0 3/81 0 3 normal Equation (6) has a maximum at the nominal value (Xj=O and X2=O). y = COS(Xl ) + cOS(2x2 ) + 2· COS(Xl ). cOS(1.2x2 ) (C - 1) The partial derivatives for equation C-l are shown in Table C - 1. Substitution oftheir values into equation (2) and (1) (Refer to chapter 2. Literature Review) yields the first and the second order TSE ofequation (C-l) respectively: Y=4 (C - 2) 2 2 (C - 3) Y = 4 - 1.5 x j - 3.44 x2 135 Table C - 2 Partial Derivatives for Equation (C-1). APM partial Partial Derivatives derivative at Xl=O and X2=0 constant 4 ~ )· ) = -SIN(x]) - 2· SIN(xj COS(l.2x2 o oX l ) - ) 8y = -2 . SIN(2x2 2.4 · COS(x]) ·SIN(1.2x 2 o OX 2 2 ~8 = -COS(x]) - 2· COS(x])· COS(1.2x2 ) -3 OX - 1 a2 y -;=-4 . COS(2x2 ) - 2.88 . COS(.;"l) . COS(1.2x2 ) -6.88 - oX 2 8 2 y --2= 2.4· SIN(xl ) • SIN(1.2x2 ) o OXl X 2 Equation (6) has maximum point offthe nominal at the nominal value (xj=-O.25 and - )· )] y = COS(2.5x1 +1.75) + COS(x2 0.8) + [2.8· COS(xl COS(x2 (C - 4) The partial derivatives for equation C-4 are shown in Table. Similarly to the equation (6), substitution oftheir values into equation (2) and (1) (Refer to chapter 2. Literature Review) yields the first and the second order TSE ofequation (C-4) respectively: (C - 5) Y = 3.3185 - 2.460 Xj + 0.7174 X2 (C - 6) y= 3.3185 - 2.460 x, + 0.7174 X2 - 0.8405 Xj2_ 1.749.'1722 136 Table C - 3 Partial Derivatives for Equation (C - 4). partial derivative at Partial Derivatives xl==-0.25 and x2==-O.5 constant 3.3185 By COS(~'(2) - = -2.5 . SIN(2.5w'(1 + 1.75)- 2.8 . SIN(x1 )· - 2.460 aX t ) • By = -SIN(x? - 0.8) - 2.8 . COS(xl SIN(x?) 0.7174 - - aX 2 8 2 y -2= -6.25· COS(2.5x} + 1.75) - 2.8· COS(x COS(x ) -1.681 ax} 1)· 2 a'J = -COS(x? - 0.8) - 2.8· COS(x ) • COS(x ) -22 l 2 - 3.498 ax ... 2 82 ) • ) --Y-2 = 2.8 · SIN(x1 SIN(x2 o OXt X 2 137 APPENDIXD REGRESSION ANALYSIS FFDl (Equation (6» General Linear Models Number of dependent variables: 1 Number of categorical factors: 0 Number of quantitative factors: 2 Cannot perform analysis. Data values are all equal. FFD2 (Equation (7» ANOVA Analysis of Variance for y Source Sum of Squares Df Mean Square F-Ratio P-Value Model 2.84465 2 1.42233 Residual 0.0 1 0.0 Total (Corr.) 2.84465 3 Type III Sums of Squares Source Sum of Squares Df Mean Square F-Ratio P-Value xl 1.38716 1 1.38716 x2 1.4575 1 1.4575 Residual 0.0 1 0.0 Total (corrected) 2.84465 3 All F-ratios are based on the residual mean square error. R-Squared = 100.0 percent R-Squared (adjusted for d.f.) 100.0 percent Standard Error of Est. = 0.0 95.0% confidence intervals for coefficient estimates (y) Standard Parameter Estimate Error Lower Limit Upper Limit V.I.F. CONSTANT 1.33663 0.0 1.33663 1.33663 xl -0.588888 0.0 -0.588888 -0.588888 1.0 x2 0.603634 0.0 0.603634 0.603634 1.0 138 DFFDl (Equation (6» ANOVA Source Sum of Squares Df Mean Square F-Ratio P-Value Model 14.0343 3 4.67809 Residual 0.0 4 0.0 Total (Corr.) 14.0343 7 Type III Sums of Squares Source Sum of Squares Df Mean Square F-Ratio P-Value xl*x1 1.9019 1 1.9019 x2*x2 7.2438 1 7.2438 x1*x2 0.171842 1 0.171842 Residual 0.0 4 0.0 Total (corrected) 14.0343 7 All F-ratios are based on the residual mean square error. R-Squared = 100.0 percent R-Squared (adjusted for d.f.) 100.0 percent Standard Error of Est. = 0.0 95.0% confidence intervals for coefficient estimates (y) Standard Parameter Estimate Error Lower Limit Upper Limit V.I.F. CONSTANT 4.0 0.0 4.0 4.0 x1*x1 -1.37909 0.0 -1.37909 -1.37909 2.0 x2*x2 -2.69143 0.0 -2.69143 -2.69143 2.0 x1*x2 0.586245 0.0 0.586245 0.586245 3.0 DFFD2 (Equation (7» ANOVA Source Sum of Squares Df Mean Square F-Ratio P-Value Model 5.93677 5 1.18735 Residual 0.0 2 0.0 Total (Corr.) 5.93677 7 Type III Sums of Squares Source Sum of Squares Df Mean Square F-Ratio P-Value xl 1.04037 1 1.04037 x2 1.09312 1 1.09312 x1*x2 0.175055 1 0.175055 x1*x1 0.933362 1 0.933362 x2*x2 2.58382 1 2.58382 Residual 0.0 2 0.0 Total (corrected) 5.93677 7 All F-ratios are based on the residual mean square error. R-Squared = 100.0 percent R-Squared (adjusted for d.f.) 100.0 percent Standard Error of Est. = 0.0 139 95.0% confidence intervals for coefficient estimates (y) ------Standard Parameter Estimate Error Lower Limit Upper Limit V.I.F. ------CONSTANT 3.31846 0.0 3.31846 3.31846 xl -0.588888 0.0 -0.588888 -0.588888 1.33333 x2 0.603634 0.0 0.603634 0.603634 1.33333 x1*x2 0.591702 0.0 0.591702 0.591702 3.0 x1*x1 -0.966107 0.0 -0.966107 -0.966107 2.0 x2*x2 -1.60743 0.0 -1.60743 -1.60743 2.0 ------ FeeDl (Equation (6» 2nd order ANOVA Analysis of Variance for y Source Sum of Squares Df Mean Square F-Ratio P-Value Model 12.5389 5 2.50777 49.25 0.0044 Residual 0.152748 3 0.0509161 Total (Corr.) 12.6916 8 Type III Sums of Squares Source Sum of Squares Df Mean Square F-Ratio P-Value xl 0.0 1 0.0 0.00 1.0000 x2 0.0 1 0.0 0.00 1.0000 x1*x2 0.0 1 0.0 0.00 1.0000 x1*x1 1.95333 1 1.95333 38.36 0.0085 x2*x2 10.5855 1 10.5855 207.90 0.0007 Residual 0.152748 3 0.0509161 Total (corrected) 12.6916 8 All F-ratios are based on the residual mean square error. R-Squared = 98.7965 percent R-Squared (adjusted for d.f.) = 96.7906 percent Standard Error of Est. = 0.225646 95.0% confidence intervals for coefficient estimates (y) Standard Parameter Estimate Error Lower Limit Upper Limit V.I.F. CONSTANT 3.73945 0.168187 3.2042 4.27469 xl 0.0 0.0921196 -0.293166 0.293166 1.0 x2 0.0 0.0921196 -0.293166 0.293166 1.0 x1*x2 0.0 0.112823 -0.359053 0.359053 1.0 x1*x1 -0.988263 0.159556 -1.49604 -0.480485 1.0 x2*x2 -2.3006 0.159556 -2.80838 -1.79282 1.0 4th order ANOVA with pooled error Analysis of Variance for y Source Sum of Squares Df Mean Square F-Ratio P-Value Model 12.6916 3 4.23054 Residual 0.0 5 0.0 Total (Corr.) 12.6916 8 140 Type III Sums of Squares Source Sum of Squares Df Mean Square F-Ratio P-Value x1*x1 1.26793 1 1.26793 x2*x2 4.8292 1 4.8292 x1*x1*x2*x2 0.152748 1 0.152748 Residual 0.0 5 0.0 Total (corrected) 12.6916 8 All F-ratios are based on the residual mean square error. R-Squared = 100.0 percent R-Squared (adjusted for d.f.) 100.0 percent Standard Error of Est. = 0.0 95.0% confidence intervals for coefficient estimates (y) Standard Parameter Estimate Error Lower Limit Upper Limit V.I.F. CONSTANT 4.0 0.0 4.0 4.0 x1*x1 -1.37909 0.0 -1.37909 -1.37909 3.0 x2*x2 -2.69143 0.0 -2.69143 -2.69143 3.0 x1*x1*x2*x2 0.586245 0.0 0.586245 0.586245 5.0 FCCD2 (Equation (7)) 2nd order ANOVA Analysis of Variance for y Source Sum of Squares Df Mean Square F-Ratio P-Value Model 8.01868 5 1.60374 Residual 0.0 3 0.0 Total (Carr.) 8.01868 8 Type III Sums of Squares Source Sum of Squares Df Mean Square F-Ratio P-Value xl 2.08074 1 2.08074 x2 2.18625 1 2.18625 x1*x1 0.622242 1 0.622242 x2*x2 1.72255 1 1.72255 x1*x1*x2*x2 0.155605 1 0.155605 Residual 0.0 3 0.0 Total (corrected) 8.01868 8 All F-ratios are based on the residual mean square error. R-Squared = 100.0 percent R-Squared (adjusted for d.f.) 100.0 percent Standard Error of Est. = 0.0 Mean absolute error = 0.0 Durbin-Watson statistic = 141 95.0% confidence intervals for coefficient estimates (y) ------Standard Parameter Estimate Error Lower Limit Upper Limit V.I.F. ------CONSTANT 3.31846 0.0 3.31846 3.31846 xl -0.588888 0.0 -0.588888 -0.588888 1.0 x2 0.603634 0.0 0.603634 0.603634 1.0 x1*x1 -0.966107 0.0 -0.966107 -0.966107 3.0 x2*x2 -1.60743 0.0 -1.60743 -1.60743 3.0 x1*x1*x2*x2 0.591702 0.0 0.591702 0.591702 5.0 95.0% confidence intervals for coefficient estimates (y) Standard Parameter Estimate Error Lower Limit Upper Limit V.I.F. CONSTANT 3.05548 0.169752 2.51526 3.59571 xl -0.588888 0.0929769 -0.884783 -0.292994 1.0 x2 0.603634 0.0929769 0.30774 0.899529 1.0 xl*x2 0.0 0.113873 -0.362395 0.362395 1.0 x1*x1 -0.571639 0.161041 -1.08414 -0.0591351 1.0 x2*x2 -1.21296 0.161041 -1.72546 -0.700456 1.0 4th order ANOVA with pooled error Analysis of Variance for y Source Sum of Squares Df Mean Square F-Ratio P-Value Model 8.01868 5 1.60374 Residual 0.0 3 0.0 Total (Corr.) 8.01868 8 Type III Sums of Squares Source Sum of Squares Df Mean Square F-Ratio P-Value xl 2.08074 1 2.08074 x2 2.18625 1 2.18625 x1*x1 0.622242 1 0.622242 x2*x2 1.72255 1 1.72255 x1*x1*x2*x2 0.155605 1 0.155605 Residual 0.0 3 0.0 Total (corrected) 8.01868 8 All F-ratios are based on the residual mean square error. R-Squared = 100.0 percent R-Squared (adjusted for d.f.) 100.0 percent Standard Error of Est. = 0.0 Mean absolute error = 0.0 Durbin-Watson statistic = 142 95.0% confidence intervals for coefficient estimates (y) Standard Parameter Estimate Error Lower Limit Upper Limit V.I.F. CONSTANT 3.31846 0.0 3.31846 3.31846 xl -0.588888 0.0 -0.588888 -0.588888 1.0 x2 0.603634 0.0 0.603634 0.603634 1.0 x1*x1 -0.966107 0.0 -0.966107 -0.966107 3.0 x2*x2 -1.60743 0.0 -1.60743 -1.60743 3.0 x1*x1*x2*x2 0.591702 0.0 0.591702 0.591702 5.0 CCDl (Equation (6)) 2nd order ANOVA Analysis of Variance for y Source Sum of Squares Df Mean Square F-Ratio P-Value Model 13.1965 5 2.63929 334.75 0.0003 Residual 0.0236529 3 0.0078843 Total (Corr.) 13.2201 8 Type III Sums of Squares Source Sum of Squares Df Mean Square F-Ratio P-Value xl 0.0 1 0.0 0.00 1.0000 x2 0.0 1 0.0 0.00 1.0000 x1*x2 0.0 1 0.0 0.00 1.0000 x1*x1 4.84817 1 4.84817 614.91 0.0001 x2*x2 13.1791 1 13.1791 1671.56 0.0000 Residual 0.0236529 3 0.0078843 Total (corrected) 13.2201 8 All F-ratios are based on the residual mean square error. R-Squared = 99.8211 percent R-Squared (adjusted for d.f.) = 99.5229 percent Standard Error of Est. = 0.0887936 se the smallest data value was less than or equal to 0.0. 95.0% confidence intervals for coefficient estimates (y) Standard Parameter Estimate Error Lower Limit Upper Limit V.I.F. CONSTANT 4.0013 0.0887904 3.71873 4.28387 xl 0.0 0.0314401 -0.100056 0.100056 1.0 x2 0.0 0.0314401 -0.100056 0.100056 1.0 x1*x2 0.0 0.0443968 -0.141291 0.141291 1.0 x1*x1 -1.29547 0.0522419 -1.46173 -1.12921 1.6741 x2*x2 -2.1359 0.0522419 -2.30216 -1.96964 1.6741 143 4th order ANOVA with pooled error Analysis of variance for y Source Sum of Squares Df Mean Square F-Ratio P-Value Model 13.2201 3 4.4067 Residual 0.0 5 0.0 Total (Corr.) 13.2201 8 Type III Sums of Squares Source Sum of Squares Df Mean Square F-Ratio P-Value x1*x1 4.23255 1 4.23255 x2*x2 11.7071 1 11.7071 x1*x1*x2*x2 0.0236529 1 0.0236529 Residual 0.0 5 0.0 Total (corrected) 13.2201 8 All F-ratios are based on the residual mean square error. R-Squared = 100.0 percent R-Squared (adjusted for d.f.) = 100.0 percent Standard Error of Est. = 0.0 95.0% confidence intervals for coefficient estimates (y) Standard Parameter Estimate Error Lower Limit Upper Limit V.I.F. CONSTANT 4.0 0.0 4.0 4.0 x1*x1 -1.26738 0.0 -1.26738 -1.26738 1.83536 x2*x2 -2.10782 0.0 -2.10782 -2.10782 1.83536 x1*x1*x2*x2 -0.109079 0.0 -0.109079 -0.109079 1.11786 CCD2 (Equation (7» 2nd order ANOVA Analysis of Variance for y Source Sum of Squares Df Mean Square F-Ratio P-Value Model 7.90269 5 1.58054 2.41 0.2501 Residual 1.96896 3 0.656319 Total (Carr.) 9.87165 8 Type III Sums of Squares Source Sum of Squares Df Mean Square F-Ratio P-Value xl 0.217634 1 0.217634 0.33 0.6051 x2 2.4398 1 2.4398 3.72 0.1495 x1*x1 2.23686 1 2.23686 3.41 0.1620 x2*x2 5.2422 1 5.2422 7.99 0.0664 x1*x2 0.0 1 0.0 0.00 1.0000 Residual 1.96896 3 0.656319 Total (corrected) 9.87165 8 All F-ratios are based on the residual mean square error. R-Squared = 80.0544 percent R-Squared (adjusted for d.f.) = 46.8118 percent Standard Error of Est. = 0.810135 144 95.0% confidence intervals for coefficient estimates (y) Standard Parameter Estimate Error Lower Limit Upper Limit V.I.F. CONSTANT 3.31245 0.810106 0.734325 5.89057 xl -0.165183 0.286853 -1.07808 0.747712 1.0 x2 0.553069 0.286853 -0.359826 1.46596 1.0 x1*x1 -0.879948 0.476645 -2.39685 0.63695 1.6741 x2*x2 -1.34708 0.476645 -2.86398 0.169815 1.6741 x1*x2 0.0 0.405067 -1.28911 1.28911 1.0 4th order ANOVA with pooled error Analysis of Variance for y Source Sum of Squares Df Mean Square F-Ratio P-Value Model 9.87165 7 1.41024 Residual 0.0 1 0.0 Total (Corr.) 9.87165 8 Type III Sums of Squares Source Sum of Squares Df Mean Square F-Ratio P-Value xl 0.270983 1 0.270983 x2 1.00282 1 1.00282 x1*x1 2.68846 1 2.68846 x2*x2 5.75013 1 5.75013 x1*x1*x2 0.0205164 1 0.0205164 x1*x2*x2 1.44051 1 1.44051 x1*x1*x2*x2 0.507933 1 0.507933 Residual 0.0 1 0.0 Total (corrected) 9.87165 8 All F-ratios are based on the residual mean square error. R-Squared = 100.0 percent R-Squared (adjusted for d.f.) 100.0 percent Standard Error of Est. = 0.0 95.0% confidence intervals for coefficient estimates (y) Standard Parameter Estimate Error Lower Limit Upper Limit V.I.F. CONSTANT 3.31846 0.0 3.31846 3.31846 xl 0.261058 0.0 0.261058 0.261058 2.00599 x2 0.5022 0.0 0.5022 0.5022 2.00599 x1*x1 -1.01009 0.0 -1.01009 -1.01009 1.83536 x2*x2 -1.47722 0.0 -1.47722 -1.47722 1.83536 x1*x1*x2 0.101434 0.0 0.101434 0.101434 2.00599 x1*x2*x2 -0.849946 0.0 -0.849946 -0.849946 2.00599 x1*x1*x2*x2 0.505479 0.0 0.505479 0.505479 1.11786 145 APPENDIXE PHASE I MOMENTS BY Me Session: 20010203-131056 CCD1(Equation (6))-> 50000 samples Sample Est. of (Variance) "'0.5 90% Conf. Interval Statistics Pop. in Est. of Pop. in Est. of Pop. Paramo Nominal 4.0013 Mean 3.6222 3.6222 0.0018 3.6193 3.6251 Std Dev 0.3913 0.3913 0.0021 0.3893 0.3934 Cp N/A N/A N/A N/A N/A Cpk N/A N/A N/A N/A N/A Skew -2.2744 LDL/UDL N/A / N/A Kurt 9.1942 Distribution Pearson-VI Sample Est. Sample ** Est. % < Low Limit N/A N/A Low Val -2.8658 1.2448 % > High Limit N/A N/A High Val 4.0013 4.0327 % Out of Spec N/A N/A Range 6.8671 2.7879 Session: 20010203-131056 CCD1FULL(Equation (6)) -> 50000 samples Sample Est. of (Variance) "'0.5 90% Conf. Interval Statistics Pop. in Est. of Pop. in.Est. of Pop. Paramo Nominal 4.0000 Mean 3.6258 3.6258 0.0017 3.6230 3.6287 Std Dev 0.3880 0.3880 0.0021 0.3860 0.3900 Cp N/A N/A N/A N/A N/A Cpk N/A N/A N/A N/A N/A Skew -2.3108 LDL/UDL N/A / N/A Kurt 9.7491 Distribution Pearson-VI Sample Est. Sample ** Est. % < Low Limit N/A N/A Low Val -3.1787 1.2479 % > High Limit N/A N/A High Val 4.0000 4.0382 % Out of Spec N/A N/A Range 7.1787 2.7903 ** Estimated range of 99.7300% 146 Session: 20010203-131056 CCD2(Equation (6))-> 50000 samples Sample Est. of (Variance) A O. 5 90% Conf. Interval Statistics Pop. in Est. of Pop. in Est. of Pop. Paramo Nominal 3.3125 Mean 3.0662 3.0662 0.0014 3.0639 3.0686 Std Dev 0.3155 0.3155 0.0017 0.3139 0.3172 Cp N/A N/A N/A N/A N/A Cpk N/A N/A N/A N/A N/A Skew -2.1144 LDL/UDL N/A / N/A Kurt 7.2418 Distribution Pearson-VI Sample Est. Sample ** Est. % < Low Limit N/A N/A Low Val -1.5375 1.2282 % > High Limit N/A N/A High Val 3.3770 3.3886 % Out of Spec N/A N/A Range 4.9145 2.1604 Session: 20010203-131056 CCD2FULL(Equation (7)) -> 50000 samples Sample Est. of (Variance) A O. 5 90% Conf. Interval Statistics Pop. in Est. of Pop. in Est. of Pop. Paramo Nominal 3.3185 Mean 3.0505 3.0505 0.0015 3.0480 3.0530 Std Dev 0.3386 0.3386 0.0019 0.3369 0.3404 Cp N/A N/A N/A N/A N/A Cpk N/A N/A N/A N/A N/A Skew -2.1170 LDL/UDL N/A / N/A Kurt 6.8979 Distribution Pearson-VI Sample Est. Sample ** Est. % < Low Limit N/A N/A Low Val -0.4766 1.0943 % > High Limit N/A N/A High Val 3.3825 3.3531 % Out of Spec N/A N/A Range 3.8591 2.2588 147 Session: 20010203-131056 DFFD2(Equation (7» -> 50000 samples Sample Est. of (Variance) A O. 5 90% Conf. Interval Statistics Pop. in Est. of Pop. in Est. of Pop. Paramo Nominal 3.3185 Mean 3.0359 3.0359 0.0018 3.0329 3.0389 Std Dev 0.4074 0.4074 0.0022 0.4053 0.4096 Cp N/A N/A N/A N/A N/A Cpk N/A N/A N/A N/A N/A Skew -2.0485 LDL/UDL N/A / N/A Kurt 6.1677 Distribution Pearson-Beta Sample Est. Sample ** Est. % < Low Limit N/A N/A Low Val -0.7901 0.7376 % > High Limit N/A N/A High Val 3.4378 3.4254 % Out of Spec N/A N/A Range 4.2279 2.6878 ** Estimated range of 99.7300% Session: 20010203-131056 FCCD1(Equation (6» -> 50000 samples Sample Est. of (Variance) A O. 5 90% Conf. Interval Statistics Pop. in Est. of Pop. in Est. of Pop. Paramo Nominal 3.7395 Mean 3.3764 3.3764 0.0018 3.3735 3.3793 Std Dev 0.3927 0.3927 0.0022 0.3907 0.3947 Cp N/A N/A N/A N/A N/A Cpk N/A N/A N/A N/A N/A Skew -2.4801 LDL/UDL N/A / N/A Kurt 10.3967 Distribution Pearson-VI Sample Est. Sample ** Est. % < Low Limit N/A N/A Low Val -2.3200 0.9148 % > High Limit N/A N/A High Val 3.7395 3.7284 % Out of Spec N/A N/A Range 6.0595 2.8135 148 Session: 20010203-131056 FCCD1FULL(Equation (6» -> 50000 samples Sample Est. of (Variance) A O. 5 90% Conf. Interval Statistics Pop. in Est. of Pop. in Est. of Pop. Paramo Nominal 4.0000 Mean 3.5576 3.5576 0.0021 3.5542 3.5610 Std Dev 0.4604 0.4604 0.0025 0.4580 0.4628 Cp N/A N/A N/A N/A N/A Cpk N/A N/A N/A N/A N/A Skew -2.2836 LDL/UDL N/A / N/A Kurt 8.5395 Distribution Pearson-VI Sample Est. Sample ** Est. % < Low Limit N/A N/A Low Val -2.1059 0.7818 % > High Limit N/A N/A High Val 4.0000 3.9984 % Out of Spec N/A N/A Range 6.1059 3.2166 ** Estimated range of 99.7300% Session: 20010203-131056 FCCD2(Equation (7»-> 50000 samples Sample Est. of (Variance) A O. 5 90% Conf. Interval Statistics Pop. in Est. of Pop. in Est. of Pop. Paramo Nominal 3.0555 Mean 2.8596 2.8596 0.0016 2.8571 2.8622 Std Dev 0.3485 0.3485 0.0019 0.3467 0.3503 Cp N/A N/A N/A N/A N/A Cpk N/A N/A N/A N/A N/A Skew -1.5718 LDL/UDL N/A / N/A Kurt 3.7459 Distribution Pearson-VI Sample Est. Sample ** Est. % < Low Limit N/A N/A Low Val -0.1173 1.0855 % > High Limit N/A N/A High Val 3.2822 3.3012 % Out of Spec N/A N/A Range 3.3995 2.2157 149 Session: 20010203-131056 FCCD2FULL(Equation (7»-> 50000 samples Sample Est. of (Variance) A O. 5 90% Conf. Interval Statistics Pop. in Est. of Pop. in Est. of Pop. Paramo Nominal 3.3185 Mean 3.0425 3.0425 0.0018 3.0396 3.0454 Std Dev 0.3942 0.3942 0.0022 0.3922 0.3963 Cp N/A N/A N/A N/A N/A Cpk N/A N/A N/A N/A N/A Skew -1.7443 LDL/UDL N/A / N/A Kurt 4.3692 Distribution Pearson-Beta Sample Est. Sample ** Est. % < Low Limit N/A N/A Low Val -0.5188 0.9762 % > High Limit N/A N/A High Val 3.4670 3.4744 % Out of Spec N/A 'N/A Range 3.9858 2.4982 ** Estimated range of 99.7300% Session: 20010203-131056 FFD1(Equation (6» -> 50000 samples Variance and/or Standard Deviation less than current value of Precision ( 1.000000E-010 ). Data cannot be analyzed. Session: 20010203-131056 FFD2(Equation (7» -> 50000 samples Sample Est. of (Variance) A O. 5 90% Conf. Interval Statistics Pop. in Est. of Pop. in Est. of Pop. Paramo Nominal 1.3366 Mean 1.3378 1.3378 0.0012 1.3357 1.3398 Std Dev 0.2792 0.2792 0.0015 0.2778 0.2807 Cp N/A N/A N/A N/A N/A Cpk N/A N/A N/A N/A N/A Skew 0.0097 LDL/UDL N/A / N/A Kurt -0.0308 Distribution Tested Normal Sample Est. Sample ** Est. % < Low Limit N/A N/A Low Val 0.2533 0.5001 % > High Limit N/A N/A High Val 2.6097 2.1754 % Out of Spec N/A N/A Range 2.3563 1.6752 150 VSA CODE FOR OBTAINING MOlVIENT OF TSE EQUATION /*********************************************************************/ /* Filename: TSE.vsl */ /* */ /* Written by: Misako Rata */ /* Date: 03/31/2001 */ /* */ /* PURPOSE: Phase I simulation Me MGF */ /* */ /* */ /* Assumptions: */ /* */ /* Revisions: */ /* Date Modeler Rev Nature and Purpose of Change */ /* */ /*********************************************************************/ value xl, X2i variation vI normal 0 +- 1, v2 normal 0 +- li xl v I, x2 v2i output TSEl yl 4i // Iphase I equation (1) output TSEl y2 3.3185 - 2.460*x1 + 0.7174*x2i // Iphase I equation (2) output TSE2 yl 4 - 1.5*xl*xl - 3.44*x2*x2i // Iphase I equation (1) output TSE2 y2 3.3185 - 2.460*x1 + O.7l74*x2 - 0.8405*xl*xl - 1.749*x2*x2i // Iphase I equation (2) /*********************************************************************/ /********************* END OF FILE**********************************/ /*********************************************************************/ 151 MOMENT BY Me ON TSE EQUATIONS First Order TSE Session: 20010331-090300 TSE1_y1(Equation (6)) -> 50000 samples variance and/or Standard Deviation less than current value of Precision ( 1.000000E-010 ). Data cannot be analyzed. Report bypassed for this variable. session: 20010331-090300 TSE1_y2(Equation (7)) -> 50000 samples Sample Est. of (Variance) A O. 5 90% Conf. Interval Statistics Pop. in Est. of Pop. in Est. of Pop. Paramo Nominal 3.3185 Mean 3.3257 3.3257 0.0038 3.3195 3.3320 Std Dev 0.8519 0.8518 0.0047 0.8475 0.8563 Cp N/A N/A N/A N/A N/A Cpk N/A N/A N/A N/A N/A Skew -0.0108 LDL/UDL N/A / N/A Kurt -0.0367 Distribution Tested Normal Sample Est. Sample ** Est. % < Low Limit N/A N/A Low Val -0.0424 0.7702 % > High Limit N/A N/A High Val 6.9834 5.8813 % Out of Spec N/A N/A Range 7.0258 5.1111 152 Second order TSE Session: 20010331-090300 TSE2_y1(Equation (6)) -> 50000 samples Sample Est. of (Variance) A O. 5 90% Conf. Interval Statistics Pop. in Est. of Pop. in Est. of Pop. Paramo Nominal 4.0000 Mean 3.4546 3.4546 0.0026 3.4503 3.4590 Std Dev 0.5885 0.5885 0.0032 0.5855 0.5916 Cp N/A N/A N/A N/A N/A Cpk N/A N/A N/A N/A N/A Skew -2.4714 LDL/UDL N/A / N/A Kurt 10.3440 Distribution Pearson-VI Sample Est. Sample ** Est. % < Low Limit N/A N/A Low Val -5.1336 -0.2298 % > High Limit N/A N/A High Val 4.0000 3.9853 % Out of Spec N/A N/A Range 9.1336 4.2151 Session: 20010331-090300 TSE2_y2(Equation (7)) -> 50000 samples Sample Est. of (Variance) A O. 5 90% Conf. Interval Statistics Pop. in Est. of Pop. in Est. of Pop. Paramo Nominal 3.3185 Mean 3.0398 3.0398 0.0040 3.0332 3.0465 Std Dev 0.9028 0.9028 0.0049 0.8981 0.9075 Cp N/A N/A N/A N/A N/A Cpk N/A N/A N/A N/A N/A Skew -0.6669 LDL/UDL N/A / N/A Kurt 0.5832 Distribution Pearson-Beta Sample Est. Sample ** Est. % < Low Limit N/A N/A Low Val -1.9535 -0.4313 % > High Limit N/A N/A High Val 5.1189 4.9253 % Out of Spec N/A N/A Range 7.0724 5.3566 ** Estimated range of 99.7300% 153 APPENDIXF PHASE I MOMENTS BY MGFS. The DOE estimated stackups are used for MGF in Appendix A. In order to plug in MGF, partial derivatives ofstackup coefficients are calculated in Table. The second order TSE is used in this section to compare with the other DOE stackups. Equation (6) Table F - 1 Coefficients ofEstimated Stackups. y= constant xl x2 x1x2 x1x1 x2x2 FF 0.5157 0 0 0 0 0 DF 4 0 0 0.5862 -1.3791 -2.6914 FCC 3.7395 0 0 0 -0.9883 -2.3001 CCD 4.001 0 0 0 -1.2955 -2.1359 TSE 4 0 0 0 -1.5000 -3.4400 Table F - 2 Partial derivatives ofEstimated Stackups. FF DF FCC CCD TSE 8y +0.5862x2 0 -1.9766xI -2.591xl -3 Xl ax} -2.7582xI ay +0.5862xl 0 -4.6002x2 -4.2718x2 -6.88 X2 aX 2 -5.3828x2 a2y 2 0 -2.7582 -1.9766 -2.591 -3 aX1 a2y -5.3828 -4.6002 2 0 -4.2718 -6.88 ax 2 a2y 2 0 0.5862 0 0 0 aX1X2 154 Table F - 3 Partial Derivatives at Nominal Value. d. d.. d2 d22 d12 FF 0 0 0 0 0 DF 0 -2.7582 0 -5.3828 0.5862 FCC 0 -1.9766 0 -4.6002 0 CCD 0 -2.591 0 -4.2718 0 TSE 0 -3 0 -6.88 0 NOTE: d, = first order partial derivative at mean value ofith component tolerance. dii = second order partial derivative around at mean value ofith component tolerance. dij = partial derivatives with respect ofith and jth component tolerance at their mean. Table F - 4 Estimated moments ofStackups. 1st order Tolerance Analysis 2nd order Tolerance Analysis III Y 1l2Y Jl3Y !l4Y J-Ll Y 1l2Y !l3Y !l4Y FF 0.5157 0 0 0 0.5157 0 0 0 DF 4 0 0 0 3.5477 0.2300 0 0.1587 FCC 3.7395 0 0 0 3.3740 0.1547 0 0.0718 CCD 4.001 0 0 0 3.6197 0.1541 0 0.0712 TSE 4 0 0 0 3.4511 0.3477 0 0.3628 Equation (7) Table F - 5 Coefficients ofEstimated Stackups. y= constant xl x2 xlx2 x1x1 x2x2 FF 1.3366 -0.5889 0.6036 0 0 0 DF 3.3185 -0.5889 0.6036 0.5917 -0.9661 -1.6074 FCC 3.0555 -0.5889 0.6036 0 -0.5717 -1.2130 CCD 3.3124 -0.1651 0.5530 0 -0.8800 -1.3471 TSE 3.3185 -2.460 0.7174 0 -0.8405 -1.7490 155 Table F - 6 Partial derivatives ofEstimated Stackups. FF DF FCC CCD TSE -0.58889 8y -0.5889 -0.1651 -2.460 -0.5889 -1.93222xl -1.1434xl -1.76 x. -1.681 x. aX l +0.597102x2 0.6036 ay 0.60365 0.5530 0.7174 0.6036 -3.2148x2 -2.426x2 -2.6942x2 -3.498x2 aX 2 +0.5917xl a2y 2 0 -1.93222 -1.1434 -1.76 -1.681 aX l a2 y -3.2148 2 0 -2.426 -2.6942 -3.498 aX 2 a2 y 2 0 0.5917 0 0 0 aXlX2 Table F- 7 Partial Derivatives at Nominal Value. d l d11 d2 d22 d12 FF -0.5889 0 0.6036 0 0 DF -0.5889 -1.9322 0.6036 -3.2148 0.5917 FCC -0.5889 -1.1434 0.6036 -2.426 0 CCD -0.1651 -1.76 0.5530 -2.6942 0 TSE -2.460 -1.681 0.7174 -3.498 0 Table F - 8 Estimated moments ofStackups. 1st order Tolerance Analysis 2nd order Tolerance Analysis ~lY ~2Y ~3Y ~4Y ~lY ~2Y ~3Y ~4Y FF 1.3366 0.0790 0 0.0187 1.3366 0.0790 0 0.0187 DF 3.3185 0.0790 0 0.0187 3.0325 0.1702 -0.0838 0.0869 FCC 3.0555 0.0790 0 0.0187 2.8725 0.1234 -0.4742 0.0457 CCD 3.3124 0.037 0 0.0041 3.0500 0.1010 -0.0323 0.0306 TSE 3.3185 0.7295 0 -1.1256 3.0307 0.6976 -0.4434 1.4700 156 0z ~ ~ ~ ~ ~ c ~ ~ ~ u ~ z ~ ~ ~ ~ ~ ~ < ~ c~ ~ ~ ~ rJJ < ~== ---J ~ Vl 2 ... 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B.R~PPlIES - - - , .. ~J\' APPFO,.lED_S'" CHECt:::ED_BY 14.5 DFAIf ~.oT3C'~LE -- Ut~LESS U:I'i ~FERTC4.. •. ..... DO ~ CI-P. ~N6LU.RTa.. 424 CHECn:O [rox' --- U" USED - HIT. - OR a: 182UED H:REI::4-I :::07.5 - DOO".J "FCfllJ~l'l:t~ E810~ E~ITIS nJS ... 3HO",IN dtmB PEf)J REProe-JCED ~1l_1 W.ll_Z .~l'Ero.L 1..,6 Li DOCl es Hf3lOlliE a: .r(,lHII-6 FE • RIG lHI3 ~.~lHlt-16 "'FJTTE." ~ ~.l(lHIH6. __ WOTTO ----- 18 -B I1jto3;g.IJoI tHOLE !-38.2±O.062 HERECH. PERSOO2 - ~NO 0 - TO PR::aPRE"nl.R'I' Z 2 \-..-- '\'ITHOJTTHE ~~""'-'M\Cl1JRE .. \ tJ 12.5±O.7 ct.~E \ - TO on::E \ .. "-1 ~ DE CL~ 1 f\ , CO.aFDEHCE ,. P~-':=ES3_Z PA:CE3S_1 PFrJ::ESS . --~~ -- lB 113±O.15 view view Front Bottom Top Bracket \0 Ul ~ l W ~ ([ 0 ~ .. 1."1 C~I ..... I I ~36±O"15 point ~sthickr,e'3s2) ~;ize 1:1 O.O~32 0110 ~ float OLE Et-~S. ,¥7 sHOLE_C ~ j§r0);}:}j>I.tJ.16(R -.(i~~- ej8.2± TH .1 A I - / r+lo.H~IAIElcll~ (slotvvidth) 13~O.13 / (sH >--1 I:.=..:J J~::-:]. 1 L\ c:»: 114111) BRACKET "l:JT .• ' •.. . j p,J:,M-":; ANYTl-IING !1NIIlIIIG_ .~ -"\·'i. 11'1.& 6 4-\ / '\I ~2 ~ : -:a1 )( ~ ~ 1IlU~ r..:.:e ~-.: D:i."R ..e:CFIED ~ LHIllft JtIft,D ~ :t#' '!t:.1 Q1"I.n'Nlsr:: • • TA arJQ1. HLJ.. UWU;U .... I)()Y<'JI"~ ~ QM ~1A1I"n.'J... I;I~ID IaWIII ~ EO IU IA: [I a~ OR [I Cf= E211E HEFE 12 [I0::U~HlT. t~FORW.llC"" In.:Ju 10m 3HQ"a,'U EF)J REPRaJOUC~ P ftI.E TOlliE 1lA1'!JIfdl U a: DO::U~E.IT HT& RECELII~611-11Z RIG lHI2 ~U(lHt~6 ~..,nHIU6. ~.fo.l(THIU6 _"I/FrTlE .~OTTO OG2 E O. HEREa~. PER~on: ~H[I ± 2 TO PF>:JPRE"n\RI" lrrHOJTTHE .. __ 8 ~~~UF-)'C'Tl~E 7 (~,8. sHOLE_B ~ GB.~.:31)7{II.6.1 TO ~crn:;E [lECLceE[I CLAI~J CCHFI:IE.~CE ~ \----- 1) \\ ------512.6 \ \ 5 S e F12.5±Oo i :.t .. l., ..C I.I· l 2 t- t s .. ~\ I t·.· It·/· ~ (F:HS). .. . 1.~ t IB~:~lc~1 er ------~.\------j/---,l-- ~5±O.15 2{LHS)) 14+lo21,t1, Ifloat» lTI 113±O. -r- bracket Fro,at Top Rear Bottom Support 160 APPENDIXH PHASE II VSL CODE AND RESULT MC simulation code for true functional relationship 1*********************************************************************1 1* Filename: bracket.vsl *1 1* *1 1* Written by: Misako Hata *1 1* Date: 06/14/2000 *1 1* *1 1* PURPOSE: Phase II programming */ /* "TRUE" simulation */ /* *1 1* Assumptions: Components variations are set as if brackets were */ 1* manufactured and assembled for production, which is considered */ I * as true APM response. */ /* *1 /* Revisions: *1 1* Date Modeler Rev Nature and Purpose of Change */ 1* *1 /*********************************************************************/ /1 II #inelude Idee hata.ine' II declarations #inelude 'base hata.tol' II base part description #inelude 'top_hata.tol l // top bar part description #inelude 'spt_hata.tol' // support bar part description #include 'pinAhata.tol' // pin A description #include 'pinBhata.tol' /1 pin B description #include IpinChata. tal , II pin B description #include 't asmhata.asm' 1/ assembly operation for top bar #include 's asmhata.asm' 1/ assembly operation for top bar #include 'mes_hata2.mes' 1/ measurement file /***************************************************** ****************1 1********************* END OF FILE *******************************1 /*********************************************************************/ 1/IDeclarations file constant x=l, y=2, z=3; /1 !coordinate system constant xy = 1,yz = 2,zx 3'I /1 Icoordinate system /11 .tol variables value bonus_tal, MMC; II Iassembly variables value clr asmi 1/1 clear~nce (w/prob) between top or support bar and baseslot. gravity effect of pins on top or support bar side holes value dist; value i; point topfloat, sptfloat; point transstop, rotstop; point floatl; floatl = 512.5,0,0; II Ifloat point for support bar point float2, float3; I/Ifloat point for base slots 161 point tarl, tar2, tar3; tarl 0,0,0; tar2 1,0,0; tar3 0,250,0; III Tolerance File for the base object base(8); context base; variation Ilivariation for the base HOLE_sizeC normal 10 +- 0.062, Illdia 10 +- 0.062 hole C size HOLE sizeD normal 8.2 +- 0.062, Illdia 8 +- 0.062 hole D size HOLE=sizeE normal 8.2 +- 0.062, Illdia 8 +- 0.062 hole E size bonus_probl normal 0 +- I, bonus_prob2 normal 0 +- I, bonus_prob3 normal 0 +- 1, bonus_prob4 normal 0 +- 1, bonus_probS normal 0 +- 1, float2var 0 +- 0.1, Illtrue position 0.2 MMC (RHS) float3var 0 +- 0.1, Illtrue position 0.2 MMC (LHS) bslotwidth1 normal 15 +- 0.13, III size of the base slot along y axis (RHS) bslotwidth2 normal 15 +- 0.13, III size of the base slot along y axis (LHS) HOLE_radC normal 0 +- 0.15, II I [POS Idia 0.3_MMCIAIB_MMCIC] hole C HOLE radD normal 0 +- 0.15, II I [POS Idia 0.3_MMCIAIB_MMClc_MMCjdtmD] hole D HOLE radE normal 0 +- 0.15, II I [POS Idia 0.3_MMcjAID_MMcIC_MMC] hole E HOLE_angC uniform 0 +- 180, II I [POS Idia 0.3_MMCIAIB_MMCIC] hole C HOLE_angD uniform 0 +- 180, II I [POS Idia 0.3_MMcIAIB_MMclc_MMCldtmD] hole D HOLE_angE uniform 0 +- 180; III [POSldia 0.3_MMCIAID_MMCIC_MMC] hole E group HOLE_C is level 2 HOLE_radC level 5 HOLE angC • [POSldia O.lS_MMCIAIB_MMC/dtmC]hole C' ; group HOLE_D is level 2 HOLE_radD level 5 HOLE angD , [POSIdia 0.15_MMCIAIB_MMCIC_MMCldtmD] hole D'; group HOLE_E is level 2 HOLE_radE level 5 HOLE angE , [POSldia O.lS_MMCIAID_MMCIC_MMCJ hole E'; group MMC_bC is level 3 HOLE- sizeC, bonus_prob1 'MMC on hole C' ; group MMC bD is level 3 HOLE sizeD, bonus_prob2 'MMC on hole D' ; group MMC b56 is level 3 float2var, bonus_prob3 'MMC on base pointS, 6' ; group MMC bE is level 3 HOLE sizeE, bonus_prob4 'MMC on hole E' . group MMC b78 is level 3 float3var, bonus_probS 'MMC on base point7, 8' ; #01 0,0,0; /1 Ibottom of datum B (DRF) #02 0,0,30; /iltop of the datum B hole #03 -250,0,0; Illbottom of datum C #04 -250,0,30; I/Itop of the datum C hole #05 80,O-(bslotwidthl/2) ,15; Ilipoint inside the slot #06 80, 0+ (bslotwidthl/2) ,15; /Ilpoint inside the slot #07 #06.x-410,O-(bslotwidth2/2) ,15; Illpoint inside the slot #08 #07.x,O+(bslotwidth2/2) ,IS; /Ilpoint inside the slot bonus_tol = HOLE_sizec/2 - (9.938/2); MMC = bonus_tol * bonus_probl; #03.x = -250 + HOLE_radC + MMC; 162 II Ivariation along x axis for datum C hole #04.x = -250 + HOLE radC + MMC; II Ivariation along ~ axis for datum D hole bonus tal = HOLE sizeD/2 - ((8.2-0.062)/2); MMC =-bonus_tol ~ bonus_prob2; base.05 = point varied from point base.OS by HOLE_radD+MMC radially at angle HOLE_angD in standard csystem zx; base.06 point varied from point base.06 by HOLE_radD+MMC radially at angle HOLE_angD in standard csystem zx; bonus tol = 14.87 - (-#OS.y + #06.y); MMC =-bonus tol * bonus prob3; base.OS.y -base.oS.y +-float2var + MMC; Illpoint #05 with TP and MMC base.06.y = base.06.y + float2var + MMC; Illpoint #06 with TP and MMC bonus tol = HOLE sizeE/2 - ((8.2-0.062)/2); MMC =-bonus tol ~ bonus prob4; base.07 = p;int varied from point base.07 by HOLE_radE+MMC radially at angle HOLE_angE in standard csystem zx; base.08 point varied from point base.08 by HOLE_radE+MMC radially at angle HOLE_angE in standard csystem zx; bonus tol = 14.87 - (-#07.y + #08.y); /llbo;us tal = Actual size (radius) - MMC condition MMC = bonus_tol * bonus_probS; base. 07.y base.07.y + float3var + MMC; //Ipoint #07 with TP and MMC base. 08 . Y base.08.y + float3var + MMC; /Ilpoint #08 with TP and MMC III Tolerance File for the top object top(6); context top; variation Illvariation for the top bar tHOLE- sizeB normal 8.2 +- 0.062, Illdia 8.2 +- 0.062 top hole B size 8.2 0.062, 8.2 tHOLE- sizeC normal +- Illdia +- 0.062 top hole C size tHOLE- sizeM normal 8.2 +- 0.062, Illdia 8.2 +- 0.062 top measurement hole bonus_prob6 normal o +- 1, bonus_prob7 normal o +- 1, tthickness normal = 13 +- 0.15, 11113 +- 0.15 bar thickness tHOLE_radC normal = 0 +- 0.15, III [POSldia 0.3_MMCIA!B_MMC\dtmC] top hole C tHOLE_radmes normal 0 +- 0.15, III [POSldia 0.3_MMCIAIB_MMClc_MMCl mes hole tHOLE_angC uniform = o +- 180, II I [POS I dia O. 3_MMC IA I B_MMC 1dtmC] top hole C tHOLE_angmes uniform o +- 180; III [POSldia 0.3_MMCIAIB_MMCIC_MMC] mes hole group tHOLE C is level 2 tHOLE_radC level 5 tHOLE_angC 163 I [POSldia 0.3_MMcIA_MMcIBldtmC] top hole C'; group tHOLE_mes is level 2 tHOLE_radmes levelS tHOLE_angmes '[Posjdia 0.3_MMCIB_MMClc_MMCJmes hole'; group MMC_tC is level 3 tHOLE_sizeC, bonus_prob6 'MMC on top hole C' ; group MMC_tM is level 3 tHOLE_sizeM, bonus_prob7 'MMC on top mes hole' ; #01 0,0,-tthickness/2; II Ibottom of the datum B hole #02 o,O,tthickness/2; Illtop of the datum B hole #03 #01.x+307.5,0,-tthickness/2; II bottom of the datum C hole #04 #02.x+307.5,O,tthickness/2; II top of the datum C hole #05 #01.x+424,O,-tthickness/2; II bottom of the measurement hole #06 #02.x+424,O,tthickness/2; II top of the measurement hole bonus tal = tHOLE sizeC/2 - (8.2-0.062)/2; MMC =-bonus tol *-bonus prob6; top.03.x top.03.x + (tHOLE_radC + MMC); top.04.x = top.04.x + (tHOLE_radC + MMC); bonus_tol = tHOLE_sizeM/2 - (8.2-0.062)/2; MMC = bonus_tal * bonus_prob7; top.OS = point varied from point top.OS by tHOLE_radmes + MMC radially at angle tHOLE_angmes in standard csystem xy; top.06 point varied from point top.06 by tHOLE_radmes + MMC radially at angle tHOLE_angmes in standard csystem xy; III Tolerance File for the support bar object support (6) ; context support; variation Ilvariation for the support bar sHOLE sizeB normal 8.2 +- 0.062, Ilidia 8.2 +- 0.062 top hole B size SHOLE_sizeC normal 8.2 +- 0.062, Illdia 8.2 +- 0.062 top hole C size bonus_prob8 normal 0 +- 1, bonus_prob9 normal 0 +- 1, bonus_prob10 normal = 0 +- I, floatvar1 0 +- 0.05, Illtrue position 0.1 MMC floatvar2 0 +- 0.05, II!true position 0.1 MMC sthickness1 normal = 13 +- 0.15, III size of the bar along z axis sthickness2 normal = 35 +- 0.15, III size of the support part along z axis slotwidth normal = 13 +- 0.15, III size of the slot along z axis sHOLE radC normal = 0 +- 0.15, III [POSldia 0.3_MMCIAIB_MMCjdtmC] support hole C sHOLE angC uniform = 0 +- 180; 11\ lPOS\dia O.3_MMCIAIB_MMCldtmC] support hole C group sHOLE C is level 2 sHOLE radC level 5 sHOLE_angC I [POSldia O.3_MMCIA_MMC\B!dtmC] support hole C' ; sHOLE sizeC, bonus_prob8 group MMC- sC is level 3 'MMC on top hole C' ; 1 group MMC_s34 is level 3 floatvarl, bonus_prob9 'MMC on spt pt 3,4 ; group MMC- s56 is level 3 floatvar2, bonus_prob10 'MMC on spt pt 5,6' ; 164 #01 0,0,-sthicknessl/2; // bottom of the datum B hole #02 0,0,sthicknessl/2; II top of the datum B hole #03 512.5,0,float1.z - sthickness2/2; II point #03 wrt support part size #04 512.5,0,floatl.z + sthickness2/2; 1/ point #04 wrt support part size #05 512.5,O,floatl.z - slotwidth/2; //Ipoint #05 wrt slot size #06 512.5,0,float1.z + slotwidth/2; I/Ipoint #06 wrt slot size bonus tol = sHOLE sizeC/2 - (8.2-0.062)/2; /llbo~us toleranc~ for dtmC hole MMC = bonus_tol * bonus_prob8; support.03.x = support.03.x + (sHOLE_radC+MMC); I/Ibonus tol on C wrt x-axis support.04.x = support.04.x + (sHOLE_radC+MMC); //Ibonus talon C wrt x-axis support.05.x = support.05.x + (sHOLE_radC+MMC); //lbonus tol on C wrt x-axis support.06.x = support.06.x + (sHOLE_radC+MMC); I/Ibonus tol on C wrt x-axis bonus tol = 35.15 - (-#03.z + #04.z); //Ibo~us tol for support part MMC = bonus_tol * bonus_prob9; support.03.z = support.03.z + floatvarl + MMC; Illpoint #03 with TP and MMC support.04.z = support.04.z + floatvarl + MMC; I/Ipoint #04 with TP and MMC bonus tal = (-#05.z + #06.z) - 12.87; I/Ibo~us tol for slot width MMC = bonus_tol * bonus_problO; support.05.z = support.05.z + floatvar2 + MMC; I/Ipoint #05 with TP and MMC support.06.z = support.06.z + floatvar2 + MMC; Illpoint #06 with TP and MMC III Tolerance File for three pins (PinA,PinB,PinC) object pinA(2) ; context pinA; variation pinA_size normal 7.9375 +- 0.035; /iidia 7.9375 + 0.035 pin A size #01 0,0,0; Iljbottom of pin A #02 0,0,35 ; Illtop of pin A object pinB(2); context pinB; variation pinB_size normal 7.9375 +- 0.035; //Idia 7.9375 + 0.035 pin B size #01 0, 0, 0; /1 Ibottom of pin B #02 0, a135; Illtop of pin B object pinC(2); context pinC; variation pinC_size normal 7.9375 +- 0.035; //Idia 7.9375 + 0.035 pin C size #01 0, a10; //Ibottom of pin C #02 o10 135; /iitop of pin C 165 III Assembly file to attach top to the base variation probuniforml uniform = 0 +- 1; group b_t_asm is level 3 bslotwidth2, tthickness, probuniforml 'clearance bwn base and top' ; II\flip the base against the wall move base positioning axis base.Ol, base.02 on axis tarl, tar2 translation stop point base.Ol with offset 0 on tarl with offset 0 rotation stop point base.03 on point tar3; IIIAdd gravity and clearance effect clr_asm = (bslotwidth2 - tthickness)/2 * probuniforml; //Iprob_asmclr for top float3 = point at distance (bslotwidth2/2) + clr_asm along line base.07, base.OB; clr asm = -(HOLE sizeE/2) + pinA size - (tHOLE_sizeB/2); /I!gravityeffect floatl = base.07; floatl.y = floatl.y + clr_asm; float2 = base.08; float2.y = float2.y + clr_asm; Illmove the top bar to the base transstop tarl; 1110,0,0 rotstop = 500 + base.07.x, base.07.y, base.07.z; I/Irotational stop (base.07 + positive x) move top positioning axis top. aI, top.02 on axis floatl, float2 translation stop point transstop with offset 0 on point float3 with offset a rotation stop point top.03 on point rotstop; III Assembly file to attach support to the base and attach support to the top III until distance between the two gets less than .Olmm. variation probuniform2 uniform = a +- 1; group b_s_asm is level 3 bslotwidthl, sthicknessl, probuniform2 'clearance bwn spt and base' ; IIlAdd gravity and clearance effect clr_asm = (bslotwidthl - sthicknessl)/2 * probuniform2; /Ilprob_asmclr for spt float3 = point at distance bslotwidthl/2 + clr_asm along line base.OS, base.06; clr asm= -(HOLE_sizeD/2) + pinB size - (sHOLE_sizeB/2); 166 //jgravityeffect floatl = base. OS; floatl.y = floatl.y + clr_asm; float2 = base.06; float2.y = float2.y + clr_asm; rotstop = top.03; clr_asm = -(tHOLE_sizeC/2) + pine size - (SHOLE_sizeC/2); rotstop.y rotstop.y + clr_asm; transstop tarl; move support positioning axis support.OI, support.02 on axis floatl, float2 translation stop point transstop with offset 0 on point float3 with offset 0 rotation stop point support.OS on point rotstop; floatl = support.OS.x, support.OS.y, 0; rotstop.z = 0; dist = distance between floatl and rotstop; if (dist > 0.001) { i = 0; repeat { i = i+l; if (i>10) { print 'no conversions in s_asmhata.asm', totsim; reject; } rotstop support. 0S ; rotstop.y = rotstop.y - clr_asm; move top positioning axis top.OI, top.02 on axis top.OI, top.02 translation stop point top.01 with offset 0 on point top.OI with offset 0 rotation stop point top.03 on point rotstop; floatl = top.03.x, top.03.y, 0; rotstop.z = 0; dist = distance between floatl and rotstop; if (dist>O.OOI) { rotstop = top.03; rotstop.y = rotstop.y + clr_asm; 167 move support positioning axis support.OI, support.02 on axis support.OI, support.02 translation stop point support.Ol with offset 0 on point support.Ol with offset 0 rotation stop point support.OS on point rotstop; floatl = support.OS.x, support.OS.y, o· rotstop.z = 0; dist = distance between floatl and rotstop; } } until (dist < 0.001); print 'rptdist2 ',dist; } III Measurement file to measure the APM distance point nompt; float1 = point at distance (top.OS.z - top.06.z)/2 along line top.OS, top.06; if (nomflag) { nompt = floatl; } print nompt; output dist_mes = distance between point nompt and floatl; 1113-D distance between nominal point to the deviated point. output dist mes2 = (nornpt.x - float1.x)**2 + (nompt.y - floatl.y)**2; dist-mes2 = sqrt(dist mes2); print dist mes2i - 1112-D distance between nominal point to the deviated point 168 Me Simulation result from 50,000 runs Session: 20000926-224041 dist_mes2 (true response)-> 50000 samples Sample Est. of (Variance) A O. 5 90% Conf. Interval Statistics Pop. in Est. of Pop. in Est. of Pop. Paramo Nominal 0.0000 Mean : 0.1224 0.1224 0.0004 0.1218 0.1230 Std Dev 0.0785 0.0785 0.0004 0.0781 0.0789 Cp : N/A N/A N/A N/A N/A Cpk : N/A N/A N/A N/A N/A Skew 1.0506 LDLjUDL N/A / N/A Kurt 1.2330 Distribution Pearson-Beta Sample Est. Sample ** Est. % < Low Limit N/A N/A Low Val 0.0004 0.0053 % > High Limit N/A N/A High Val 0.7031 0.4500 % Out of Spec N/A N/A Range 0.7027 0.4447 Session: 20000926-224041 dist_mes2 (true response HLM contributors) Nominal at Median: 0.0000 High-Law-Median Study HLM Variance: 0.0075 Tolerance Effect sHaLE C- (X19 / X20) -> [POSldia 0.3_MMCIA_MMCIBldtmC] support hole C 24.88% HOLE E (X6, / X9) -> [POS!dia 0.15_MMcIAID_MMclc_MMCJ hole E 17.36% HOLE D (x, / xa) -> [POSldia 0.15_MMCIAIB_MMClc_MMCldtmD] hole D 15.94% tHOLE C (xu j XIS) -> [POSldia 0.3_MMcIA_MMCIBldtmCJ top hole C 9.00% tHOLE mes (X14 / X16) -> [POSldia 0.3_MMCIB_MMCIC_MMCJmes hole 8.36% MMC_sC (XIS) -> MMC on top hole C 8.36% MMC_bD (X2) -> MMC on hole D 4.30% MMC_tC (xu) -> MMC on top hole C 4.21% MMC_bE (X3) -> MMC on hole E 3.55% MMC_tM (Xl2) -> MMC on top mes hole 1.43% ------97.40% 169 APPENDIX I PHASE II VSL CODE FOR DOE SIMULATION AND ITS RESULT FFD simulation code (declaration file only) II \Declarations constant x=l, y=2, z=3; II Icoordinate system constant xy = l,yz = 2,zx 3 ; II lcoordinate system III .tol variables variation Illvariation for the base bonus_probl normal 0 +- I, bonus_prob2 normal = 0 +- I, float2var o +- 0.1, Illtrue position 0.2 MMC (RHS) float3var a +- 0.1, Illtrue position 0.2 MMC (LHS) bslotwidthl normal 15 +- 0.13/ III size of the base slot along y axis (RHS) bslotwidth2 normal 15 +- 0.13, III size of the base slot along y axis (LHS) Illvariation for the top bar tthickness normal = 13 +- 0.15, 11113 +- 0.15 bar thickness Ilvariation for the support bar bonus_prob9 normal = 0 +- 1/ bonus_problO normal = 0 +- 1, floatvar1 a +- 0.05, Ilitrue position 0.1 MMC floatvar2 a +- 0.05/ Illtrue position 0.1 MMC sthickness1 normal = 13 +- 0.15/ III size of the bar along z axis sthickness2 normal = 35 +- 0.15, III size of the support part along z axis slotwidth normal = 13 +- 0.15, III size of the slot along z axis Ilvariation for t asm probuniforml unif;rm = a +- 1/ Ilvariation for s_asm probuniform2 uniform = 0 +- 1; Illgrouping for the base group MMC_b56 is level 3 float2var, bonus_probl 'MMC on base pointS,6'; group MMC_b78 is level 3 float3var, bonus_prob2 'MMC on base point7,8'; Illgrouping for the support bar group MMC s34 is level 3 floatvarl, bonus_prob9 'MMC on spt pt 3,4'; group MMC=S56 is level 3 floatvar2, bonus_problO 'MMC on spt pt 5,6'; I/]grouping for the t_asmhata group b_t_asm is level 3 bslotwidth2, tthickness, probuniforml 'clearance bwn base and top' ; I/lgrouping for the s_asmhata group b_s_asm is level 3 bslotwidth1, sthickness1, probuniform2 'clearance bwn spt and base' ; 170 value bonus_tal, MMC; II \assembly variables value clr asm; III clear~nce (w/prob) between top or support bar and baseslot. gravity effect of pins on top or support bar side holes value j; III loop for nominal point value dist; point topfloat, sptfloat; point transstop, rotstop; point floatl; float1 = 512.5,0,0; II\float point for support bar point float2, float3; III float point for base slots point tar1, tar2, tar3; tar1 0,0,0; tar2 1,0,0; tar3 0,250,0; value k; IIIDOE loop counter value i· array DOE(33,23); IllFractional Factorial design (array) repeat for i = 1 to 31 by 2 II Ifactor A { DOE(i,l) = -1; DOE(i+1,1) = 1; repeat for i = 1 to 29 by 4 II Ifactor B { DOE(i, 2) = -1; DOE(i+1,2) -1; DOE(i+2,2) 1; DOE(i +3 , 2) 1 ; repeat for i = 1 to 31 by 8 III factor C { DOE(i,3) = -1; DOE(i+1,3) -1; DOE(i+2,3) -1; DOE(i+3,3) -1; DOE(i+4,3) 1; DOE(i+5,3) 1; DOE(i+6,3) 1; DOE(i+7,3) 1; repeat for i = 1 to 32 by 1 II\factor D { if (( i>=l&i <=8) I (i>=l7 &i<=24) ) { DOE( i , 4 ) = - 1 ; } else { DOE(i,4) 1 ; } repeat for i 1 to 16 by 1 III factor E 171 { DOE (i, 5) = -1; } repeat for i = 17 to 32 by 1 II\factor E { DOE (i , 5) = 1; } repeat for i = 1 to 32 by 1 II\factor F bd { DOE(i,6) = DOE(i,2)*DOE(i,4); } repeat for i = 1 to 32 by 1 II\factor G ae { DOE(i,7) = DOE(i,l)*DOE(i,5); } repeat for i = 1 to 32 by 1 III factor H ede { DOE(i,8) = DOE(i,3)*DOE(i,4)*DOE(i,S); } repeat for i = 1 to 32 by 1 III factor I bde { DOE(i,9) = DOE(i/2)*DOE(i,4)*DOE(i,S); } repeat for i = 1 to 32 by 1 III factor J bee { DOE (i, 10) DOE(i,2)*DOE(i,3)*DOE(i,S) ; } repeat for i = 1 to 32 by 1 II \factor K bed { DOE(i,11) DOE(i,2)*DOE(i,3)*DOE(i,4) ; } repeat for i = 1 to 32 by 1 II\faetor L ade { DOE(i,12) DOE (i, 1) *DOE (i, 4) *DOE (i , S) ; } repeat for i = 1 to 32 by 1 III factor M ace { DOE(i,13) DOE(i,1)*DOE(i,3)*DOE(i,S) ; } repeat for i = 1 to 32 by 1 III factor N acd { DOE(i,14) DOE (i, 1) *DOE (i, 3) *DOE (i,4) ; } repeat for i = 1 to 32 by 1 II! factor 0 abe { DOE (i,lS) DOE(i,l)*DOE(i,2)*DOE(i,S) ; } repeat for i = 1 to 32 by 1 II\factor P abd { DOE (i, 16) DOE (i,l) *DOE (i, 2) *DOE (i, 4) ; } repeat for i = I to 32 by 1 III factor Q abc { DOE(i,17) = DOE(i,1)*DOE(i,2)*DOE(i,3); } 172 repeat for i = 1 to 32 by 1 III factor R = bcde { DOE(i,18) = DOE(i,2)*DOE(i,3)*DOE(i,4)*DOE(i,5); } repeat for i = 1 to 32 by 1 III factor S = acde { DOE(i,19) = DOE(i,1)*DOE(i,3)*DOE(i,4)*DOE(i,5); } repeat for i = 1 to 32 by 1 III factor T = abde { DOE(i,20) = DOE(i,1)*DOE(i,2)*DOE(i,4)*DOE(i,5); } repeat for i = 1 to 32 by 1 II/factor U = abce { DOE(i,21) = DOE(i,1)*DOE(i,2)*DOE(i,3)*DOE(i,5); } repeat for i = 1 to 32 by 1 III factor V = abcd { DOE(i,22) = DOE(i,1)*DOE(i,2)*DOE(i,3)*DOE(i,4); } repeat for i = 1 to 32 by 1 II/factor W = abcde { DOE(i,23) = DOE(i,1)*DOE(i,2)*DOE(i,3)*DOE(i,4)*DOE(i,5); } III variable for measurement value dist_mes, dist_mes2; FFD simulation result 50,000 simulation result (distribution of Y) with estimated functions Session: 20000926-160501 FF -> 50000 samples Sample Est. of (Variance) "0.5 90% Conf. Interval Statistics Pop. in Est. of Pop. in Est. of Pop. Paramo Nominal 0.4232 Mean : 0.4231 0.4231 0.0004 0.4224 0.4238 Std Dev 0.0938 0.0938 0.0005 0.0933 0.0943 Cp : N/A N/A N/A N/A N/A Cpk : N/A N/A N/A N/A N/A Skew 0.0082 LDL/uDL N/A I N/A Kurt -0.1817 Distribution Pearson-II Sample Est. Sample ** Est. % < Low Limit N/A N/A Low Val 0.0792 0.1554 % > High Limit N/A N/A High Val 0.7879 0.6908 % Out of Spec N/A N/A Range 0.7087 0.5354 173 DFFD simulation code (declaration file only) II IDeclarations constant x=l, y=2, z=3; II Icoordinate system constant xy = 1,yz = 2,zx 3 ; II Icoordinate system III .tol variables variation Illvariation for the base bonus_prob1 normal 0 +- 1, bonus_prob2 normal = 0 +- 1, float2var o +- 0.1, Illtrue position 0.2 MMC (RHS) float3var o +- 0.1, Illtrue position 0.2 MMC (LHS) bslotwidth1 normal 15 +- 0.13, III size of the base slot along y axis (RHS) bslotwidth2 normal 15 +- 0.13, III size of the base slot along y axis (LHS) Illvariation for the top bar tthickness normal = 13 +- 0.15, 11113 +- 0.15 bar thickness Ilvariation for the support bar bonus_prob9 normal = a +- 1, bonus_problO normal = a +- 1, floatvarl o +- 0.05, Illtrue position 0.1 MMC floatvar2 o +- 0.05, Illtrue position 0.1 MMC sthickness1 normal = 13 +- 0.15, III size of the bar along z axis sthickness2 normal = 35 +- 0.15, III size of the support part along z axis slotwidth normal = 13 +- 0.15/ III size of the slot along z axis Ilvariation for t_asm probuniform1 uniform = a +- 1, Ilvariation for s_asm probuniform2 uniform = a +- 1; Illgrouping for the base group MMC_bS6 is level 3 float2var, bonus_probl 'MMC on base pointS,6'; group MMC_b78 is level 3 float3var, bonus_prob2 'MMC on base point7,8'; Illgrouping for the support bar group MMC s34 is level 3 floatvar1, bonus_prob9 'MMC on spt pt 3,4'; group MMC=SS6 is level 3 floatvar2, bonus_prob10 'MMC on spt pt 5,6'; II Igrouping for the t_asrnhata group b_t_asm is level 3 bslotwidth2, tthickness, probuniforrnl 'clearance bwn base and top' ; Illgrouping for the s asrnhata group b_s_asm is level 3 bslotwidth1, sthickness1, probuniform2 'clearance bwn spt and base' ; value bonus_tol, MMC; II Iassembly variables value clr asm; III clear~nce (w/prob) between top or support bar and baseslot. gravity effect of pins on top or support bar side holes value j, k; 174 III loop for nominal point value dist; point topfloat, sptfloat; point transstop, rotstop; point float1; float1 = 512.5,0,0; III float point for support bar point float2, float3; III float point for base slots point tar1, tar2, tar3; tar1 0,0,0; tar2 1,0,0; tar3 0,250,0; value i, 1; array DOE(64,23); II\Fractional Factorial design (array) repeat for i = 1 to 31 by 2 III factor A { DOE(i, 1) = -1; DOE(i+1,l) = 1; repeat for i = 1 to 29 by 4 III factor B { DOE(i,2) = -1; DOE (i+1, 2) -1; DOE(i+2,2) 1; DOE(i+3,2) 1; repeat for i = 1 to 31 by 8 I/Ifactor C { DOE(i,3) = -1; DOE(i+1,3) -1; DOE(i+2,3) -1; DOE(i+3,3) -1; DOE(i+4,3) 1 ; DOE(i+S,3) l' DOE(i+6,3) 1; DOE(i+7,3) l' repeat for i = 1 to 32 by 1 /Ilfactor D { if ((i>=l&i <=8) I (i>=17&i<=24)) { DOE(i,4) = -1; } else { DOE(i,4) 1· } repeat for i = 1 to 16 by 1 III factor E { DOE(i, 5) = -1; } repeat for i = 17 to 32 by 1 /I\factor E { DOE(i,5) = 1; } repeat for i = 1 to 32 by 1 III factor F bd { DOE(i,6) = DOE(i,2)*DOE(i,4); 175 repeat for i = 1 to 32 by 1 Illfaetor G ae { DOE(i,7) = DOE(i,I)*DOE(i,S); } repeat for i = 1 to 32 by 1 Illfaetor H ede { DOE(i,8) = DOE(i,3)*DOE(i,4)*DOE(i,S); } repeat for i = 1 to 32 by 1 I/Ifaetor I bde { DOE(i,9) = DOE(i,2)*DOE(i,4)*DOE(i,S); } repeat for i = 1 to 32 by 1 II\faetor J bee { DOE(i,10) = DOE(i,2)*DOE(i,3)*DOE(i,S); } repeat for i = 1 to 32 by 1 I/Ifaetor K = bed { DOE(i,11) = DOE(i,2)*DOE(i,3)*DOE(i,4); } repeat for i = 1 to 32 by 1 Illfaetor L ade { DOE(i, 12) = DOE(i, 1) *DOE (i, 4) *DOE (i, 5) ; } repeat for i = 1 to 32 by 1 Illfaetor M ace { DOE(i,13) = DOE(i,1)*DOE(i,3)*DOE(i,S); } repeat for i = 1 to 32 by 1 Illfaetor N aed { DOE(i,14) DOE(i,1)*DOE(i,3)*DOE(i,4) ; } repeat for i = 1 to 32 by 1 Illfaetor 0 abe { DOE(i,lS) = DOE(i,1)*DOE(i,2)*DOE(i,S); } repeat for i = 1 to 32 by 1 Illfaetor P abd { DOE(i,16) DOE(i,I)*DOE(i,2)*DOE(i,4) ; } repeat for i = 1 to 32 by 1 II\faetor Q abc { DOE(i, 17) DOE(i,1)*DOE(i,2)*DOE(i,3) ; } repeat for i = 1 to 32 by 1 /Ilfactor R = bede { DOE(i, 18) DOE(i,2)*DOE(i,3)*DOE(i,4)*DOE(i,S) ; } repeat for i = 1 to 32 by 1 III factor S = aede { DOE(i, 19) DOE(i,I)*DOE(i,3)*DOE(i,4)*DOE(i,S) ; } repeat for i 1 to 32 by 1 /Ilfaetor T abde 176 DOE(i,20) DOE(i,I)*DOE(i,2)*DOE(i,4)*DOE(i,5) ; repeat for i = 1 to 32 by 1 III factor U = abce { DOE(i, 21) DOE(i,I)*DOE(i,2)*DOE(i,3)*DOE(i,S) ; } repeat for i = 1 to 32 by 1 III factor V = abed { DOE(i,22) = DOE(i,I)*DOE(i,2)*DOE(i,3)*DOE(i,4); } repeat for i = 1 to 32 by 1 III factor W = abcde { DOE(i,23) = DOE(i,I)*DOE(i,2)*DOE(i,3)*DOE(i,4)*DOE(i,5); } repeat for 1 = 1 to 23 by 1 { repeat for i = 1 to 32 by 1 II -1 to 0 conversion { if (DOE(i,l)== 1) DOE(i,l) = O· III Convert second half -Is to Os repeat for i = 33 to 63 by 2 III factor AA { DOE(i,l) = -1; DOE(i+l,l) = 1· repeat for i = 33 to 61 by 4 III factor BB { DOE(i,2) = -1; DOE(i+l,2) -1; DOE(i+2,2) 1; DOE(i+3,2) 1; repeat for i = 33 to 57 by 8 III factor CC { DOE(i,3) = -1; DOE(i+l,3) -1; DOE(i+2,3) -1; DOE(i+3,3) -1; DOE(i+4,3) 1; DOE(i+S,3) 1; DOE(i+6,3) 1· DOE(i+7,3) 1; repeat for i = 33 to 64 by 1 II Ifaetor DD { if ((i>=33&i<=40) I (i>=49&i<=S6)) { DOE(i,4) = -1; } else { DOE(i,4) 1 ; } 177 repeat for i = 33 to 48 by 1 III factor EE { DOE(i,S) = -1; } repeat for i = 49 to 64 by 1 III factor EE { DOE(i,S) = 1; } repeat for i = 33 to 64 by 1 II!faetor F bd { DOE(i,6) = DOE(i,2)*DOE(i,4); } repeat for i = 33 to 64 by 1 III factor G ae { DOE(i,7) = DOE(i,I)*DOE(i,S); } repeat for i = 33 to 64 by 1 III factor H ede { DOE(i,8) = DOE(i,3)*DOE(i,4)*DOE(i,S); } repeat for i = 33 to 64 by 1 III factor I bde { DOE(i,9) = DOE(i,2)*DOE(i,4)*DOE(i,5); } repeat for i = 33 to 64 by 1 II\faetor J bee { DOE(i,10) = DOE(i,2)*DOE(i,3)*DOE(i,S); } repeat for i = 33 to 64 by 1 III factor K = bed { DOE(i,ll) = DOE(i/2)*DOE(i,3)*DOE(i,4); } repeat for i = 33 to 64 by 1 II Ifactor L ade { DOE(i,12) = DOE(i,I)*DOE(i,4)*DOE(i,S); } repeat for i = 33 to 64 by 1 III factor M ace { DOE(i,13) = DOE(i,I)*DOE(i,3)*DOE(i,S); } repeat for i = 33 to 64 by 1 II ! factor N aed { DOE(i,14) DOE(i,I)*DOE(i,3)*DOE(i,4) ; } repeat for i = 33 to 64 by 1 II Ifactor 0 abe { DOE(i,IS) = DOE(i,I)*DOE(i,2)*DOE(i,S); } repeat for i = 33 to 64 by 1 Illfactor P abd { DOE(i,16) DOE(i,l)*DOE(i,2)*DOE(i,4) ; } repeat for i = 33 to 64 by 1 II Ifactor Q abc { DOE(i/17) = DOE(i,1)*DOE(i,2)*DOE(i,3); 178 repeat for i = 33 to 64 by 1 /I\faetor R = bcde { DOE(i,18) = DOE(i,2)*DOE(i,3)*DOE(i,4)*DOE(i,5); } repeat for i = 33 to 64 by 1 /Ilfaetor S = acde { DOE(i,19) = DOE(i,I)*DOE(i,3)*DOE(i,4)*DOE(i,5); } repeat for i = 33 to 64 by 1 //\faetor T = abde { DOE(i,20) = DOE(i,I)*DOE(i,2)*DOE(i,4)*DOE(i,5); } repeat for i = 33 to 64 by 1 /Ilfaetor U = abee { } repeat for i = 33 to 64 by 1 I/Ifaetor V = abed { DOE(i,22) = DOE(i,I)*DOE(i,2)*DOE(i,3)*DOE(i,4); } repeat for i = 33 to 64 by 1 1/1 factor W = abcde { DOE(i,23) = DOE(i,I)*DOE(i,2)*DOE(i,3)*DOE(i,4)*DOE(i,S); } III Convert second half of -Is to Os repeat for 1 = 1 to 23 by 1 { repeat for i = 33 to 64 by 1 II -1 to 0 conversion { if (DOE(i,l)== -1) DOE(i , 1) = 0; III variable for measurement value dist_mes, dist_mes2; DFFD simulation result Session: 20000926-160501 DFFD -> 50000 samples Sample Est. of (Variance) A O. 5 90% Conf. Interval Statistics Pop. in Est. of Pop. in Est. of Pop. Paramo Nominal 0.3547 Mean : 0.3546 0.3546 0.0003 0.3541 0.3552 Std Dev 0.0756 0.0756 0.0004 0.0752 0.0760 Cp : N/A N/A N/A N/A N/A Cpk : N/A N/A N/A N/A N/A 179 Skew -0.0001 LDL/UDL N/A / N/A Kurt -0.0492 Distribution Pearson-II Sample Est. Sample ** Est. % < Low Limit N/A N/A Low Val 0.0355 0.1307 % > High Limit N/A N/A High Val 0.6688 0.5785 % Out of Spec N/A N/A Range 0.6333 0.4479 180 FCCD simulation code (declaration only) II!Declarations constant x=l, y=2, z=3; II Icoordinate system constant xy = 1,yz = 2,zx 3 ; II \coordinate system III .tol variables variation jj)variation for the base bonus prob1 normal a +- 1, bonus=prob2 normal = a +- 1, float2var a +- 0.1, Illtrue position 0.2 MMC (RHS) float3var a +- 0.1, Illtrue position 0.2 MMC (LHS) bslotwidth1 normal 15 +- 0.13, III size of the base slot along y axis (RHS) bslotwidth2 normal 15 +- 0.13, Illsize of the base slot along y axis (LHS) II Ivariation for the top bar tthickness normal = 13 +- 0.15, 11113 +- 0.15 bar thickness Ilvariation for the support bar bonus_prob9 normal = a +- 1, bonus_prob10 normal = a +- I, floatvar1 a +- 0.05, II Itrue position 0.1 MMC floatvar2 o +- 0.05, Illtrue position 0.1 MMC sthickness1 normal = 13 +- 0.15, III size of the bar along z axis sthickness2 normal = 35 +- 0.15, III size of the support part along z axis slotwidth normal = 13 +- 0.15, III size of the slot along z axis Ilvariation for t asm probuniforml unif~rm = 0 +- 1, Ilvariation for s_asm probuniform2 uniform = a +- 1; Illgrouping for the base group MMC b56 is level 3 float2var, bonus_prob1 'MMC on base point5,6'; group MMC=b78 is level 3 float3var, bonus_prob2 'MMC on base point7,8'; Illgrouping for the support bar group MMC s34 is level 3 floatvar1, bonus_prob9 'MMC on spt pt 3,4'; group MMC=S56 is level 3 floatvar2, bonus_prob10 'MMC on spt pt 5,6' i Iligrouping for the t_asmhata group b_t_asm is level 3 bslotwidth2, tthickness, probuniform1 'clearance bwn base and t op t : Illgrouping for the s asmhata group b_s_asm is level 3 bslotwidth1, sthickness1, probuniform2 'clearance bwn spt and base' ; value bonus_tal, MMC; II Iassembly variables value clr asm; III clear~nce (w/prob) between top or support bar and baseslot. gravity effect of pins on top or support bar side holes value i, j, k; 181 III loop for nominal point value dist; point topfloat, sptfloat; point transstop, rotstop; point floatl; floatl = 512.S,O,O; III float point for support bar point float2, float3; III float point for base slots point tarl, tar2, tar3; tar1 0,0,0; tar2 1,0,0; tar3 0,2S0,0; array DOE(79,23); IllCentral Composite design (array) repeat for i = 1 to 31 by 2 III factor A { DOE(i,l) = -1; DOE(i+l,l) = 1; repeat for i = 1 to 29 by 4 /Ilfactor B { DOE(i, 2) = -1; DOE(i+1,2) -1; DOE(i+2,2) l' DOE(i +3 , 2) 1 ; repeat for i = 1 to 31 by 8 /Ilfactor C { DOE(i,3) = -1; DOE(i+l,3) -1; DOE(i+2,3) -1; DOE(i+3,3) -1; DOE(i+4,3) 1; DOE(i+5,3) 1 ; DOE(i+6,3) 1 ; DOE(i+7,3) 1; repeat for i = 1 to 32 by 1 /Ilfactor D { if ((i>=I&i <=8) I (i>=17&i<=24)) { DOE(i,4) = -1; } else { DOE(i,4) 1; } repeat for i = 1 to 16 by 1 .Ill factor E { DOE(i,S) = -1; } repeat for i = 17 to 32 by 1 III factor E { DOE(i,S) = 1 ; } repeat for i = 1 to 32 by 1 Illfactor F bd { DOE(i,6) = DOE(i,2)*DOE(i,4); 182 repeat for i = 1 to 32 by 1 III factor G ae { DOE(i,7) = DOE(i,l)*DOE(i,S); } repeat for i = 1 to 32 by 1 III factor H cde { DOE(i,8) = DOE(i,3)*DOE(i,4)*DOE(i,S); } repeat for i = 1 to 32 by 1 III factor I bde { DOE(i,9) = DOE(i,2)*DOE(i,4)*DOE(i,S); } repeat for i = 1 to 32 by 1 III factor J bce { DOE(i,lO) = DOE(i,2)*DOE(i,3)*DOE(i,S); } repeat for i = 1 to 32 by 1 III factor K bcd { DOE(i,ll) = DOE(i,2)*DOE(i,3)*DOE(i,4); } repeat for i = 1 to 32 by 1 III factor L ade { DOE (i, 12) DOE(i,1)*DOE(i,4)*DOE(i,S) ; } repeat for i = 1 to 32 by 1 I/Ifactor M ace { DOE(i,13) = DOE(i,1)*DOE(i,3)*DOE(i,S); } repeat for i = 1 to 32 by 1 I/Ifactor N acd { DOE(i,14) = DOE(i,1)*DOE(i,3)*DOE(i,4); } repeat for i = 1 to 32 by 1 I/Ifactor 0 abe { DOE(i,lS) DOE(i,I)*DOE(i,2)*DOE(i,S) ; } repeat for i = 1 to 32 by 1 I/Ifactor P abd { DOE(i,16) = DOE(i,1)*DOE(i,2)*DOE(i,4); } repeat for i = 1 to 32 by 1 1/\ factor Q abc { DOE (i, 17) DOE(i,l) *DOE(i,2) *DOE(i,3); } repeat for i = 1 to 32 by 1 1/1 factor R = bcde { DOE(i, 18) DOE(i,2)*DOE(i,3)*DOE(i,4)*DOE(i,S) ; } repeat for i = 1 to 32 by 1 I/Ifactor S = acde { DOE(i,19) DOE(i,1)*DOE(i,3)*DOE(i,4)*DOE(i,S) ; } repeat for i 1 to 32 by 1 II!factor T abde 183 DOE (i/ 20) DOE(i/l)*DOE(i/2)*DOE(i/4)*DOE(i/5) ; repeat for i = 1 to 32 by 1 //Ifactor U = abce { DOE(i/21) = DOE(i/l)*DOE(i/2)*DOE(i/3)*DOE(i/5); } repeat for i = 1 to 32 by 1 //\factor V = abed { DOE(i/22) = DOE(i,1)*DOE(i/2)*DOE(i/3)*DOE(i/4); } repeat for i = 1 to 32 by 1 //1 factor W = abcde { DOE(i/23) = DOE(i/l)*DOE(i/2)*DOE(i/3)*DOE(i/4)*DOE(i/5); } repeat for j = 1 to 23 by 1 { repeat for i = 33 to 79 by 1 { DOE( i / j) = 0; } i = 33; repeat for 1 to 23 by 1 { DOE(i / j) = -1; DOE(i+1/j) = 1; i = i + 2; //1 variable for measurement value dist_mes/ dist_mes2; FCCD simulation result Session: 20000926-160501 FFCD-> 50000 samples Sample Est. of (Variance) "'0.5 90% Conf. Interval Statistics Pop. in Est. of Pop. in Est. of Pop. Paramo Nominal 0.2615 Mean : 0.2615 0.2615 0.0004 0.2609 0.2622 Std Dev 0.0895 0.0895 0.0005 0.0891 0.0900 Cp : N/A N/A N/A N/A N/A Cpk : N/A N/A N/A N/A N/A Skew 0.0046 LDL/UDL N/A / N/A Kurt -0.1767 Distribution Pearson-II Sample Est. Sample ** Est. % < Low Limit N/A N/A Low Val -0.0858 0.0056 % :> High Limit N/A N/A High Val 0.5808 0.5175 % Out of Spec N/A N/A Range 0.6666 0.5119 184 APPENDIXJ PHASE II SIMULATION ON DOE ESTIMATED STACKUP Simulation for estimated stackups /*********************************************************************/ /* Filename: bracket.vsl */ /* */ /* Written by: Misako Hata */ /* Date: 06/14/2000 */ /* */ /* PURPOSE: Phase II programming */ /* estimated DOE simulation */ /* */ /* Assumptions: Components variations are set as if brackets were*/ /* manufactured and assembled for production, */ /* which is considered as true APM response. */ /* */ /* Revisions: */ /* Date Modeler Rev Nature and Purpose of Change */ /* --- */ /*********************************************************************/ variation xl normal 0 +- I, x2 normal 0 +- I, x3 normal 0 +- I, x4 normal 0 +- I, xS normal 0 +- I, x6 normal 0 +- I, x7 normal 0 +- I, x8 normal 0 +- I, x9 normal 0 +- I, x10 normal 0 +- I, xlI normal 0 +- I, x12 normal 0 +- I, x13 normal 0 +- I, x14 normal 0 +- I, xIS normal 0 +- I, x16 normal 0 +- I, x17 normal 0 +- I, x18 normal 0 +- I, x19 normal 0 +- I, x20 normal 0 +- I, x21 normal 0 +- I, x22 normal 0 +- I, x23 normal 0 +- 1 ; output FFDy = 0.402069 + 0.061719*x4 + 0.076350*x6 - 0.079113*x8 + 0.058519*x9 0.086931~xl0 - 0.061456*x18 + 0.079338*x22; output 185 DFDy = -0.0134844 + 0.0865608 * x6 -0.0719267 * x10 -0.0741795 * x22 + 0.214975*xS*xS -0.0225142*x13*x13 + 0.0582094*x9*x9 + 0.0577156*x23*x23 - 0.126306*x5*xl0; output FCCDy = 0.028497 - 0.074459* x8 - 0.081818* x10 - 0.054712* x18 - 0.036959*x21 + 0.074641* x22 + 0.219805* x5* x5 - 0.029145*x9* x9 + 0.157605*x13*x13 + 0.151855* x14*x14 - 0.029145* x15* x15 - 0.029145* x16* x16 - 0.029145* x19*x19; Output of simulation for estimated stackups Session: 20010405-112949 FFDy -> 50000 samples Sample Est. of (Variance) A O. 5 90% Conf. Interval Statistics Pop. in Est. of Pop. in Est. of Pop. Paramo Nominal 0.4021 Mean 0.4022 0.4022 0.0003 0.4018 0.4027 Std Dev 0.0641 0.0641 0.0004 0.0637 0.0644 Cp N/A N/A. N/A N/A N/A Cpk N/A N/A. N/A N/A N/A Skew -0.0012 LDL/UDL N/A / N/A Kurt -0.0060 Distribution Tested Normal Sample Est. Sample ** Est. % < Low Limit N/A N/A Low Val 0.1396 0.2100 % > High Limit N/A N/A High Val 0.6607 0.5944 % Out of Spec N/A N/A Range 0.5212 0.3844 Session: 20010405-112949 DFDy -> 50000 samples Sample Est. of (Variance) A O. 5 90% Conf. Interval Statistics Pop. in Est. of Pop. in Est. of Pop. Paramo Nominal -0.0135 Mean 0.0209 0.0209 0.0003 0.0205 0.0214 Std Dev 0.0596 0.0596 0.0003 0.0593 0.0599 Cp N/A N/A N/A N/A N/A Cpk N/A N/A N/A N/A N/A Skew 0.7140 LDL/UDL N/A / N/A Kurt 2.0072 Distribution Pearson-IV Sample Est. Sample ** Est. % < Low Limit N/A N/A Low Val -0.1903 -0.1412 % > High Limit N/A N/A High Val 0.5971 0.2847 % Out of Spec N/A N/A Range 0.7874 0.4259 186 Session: 20010405-112949 FCCDy -> 50000 samples Sample Est. of (Variance) "0.5 90% Conf. Interval Statistics Pop. in Est. of Pop. in Est. of Pop. Paramo Nominal 0.0285 Mean 0.0748 0.0748 0.0003 0.0743 0.0753 Std Dev 0.0706 0.0706 0.0004 0.0702 0.0709 Cp N/A N/A N/A N/A N/A Cpk N/A N/A N/A N/A N/A Skew 0.6150 LDL/UDL N/A / N/A Kurt 1.1593 Distribution Pearson-IV Sample Est. Sample ** Est. % < Low Limit N/A N/A Low Val -0.1572 -0.1060 % > High Limit N/A N/A High Val 0.5539 0.3652 % Out of Spec N/A N/A Range 0.7111 0.4713 187 APPENDIXK METRIC MANPULATIONS MATRIX MANIPULATION FOR FULL TERM STACKUP EQUATION Matrix manipulation was conductecl on each experimental design. The data structure of multiple linear regression for all the experimental design are shown Table K - 1. Table K - 1. Data structure ofmultiple linear regression Y Xl X2 Xk YI XII X2I XIk Y2 X2I X22 Yn Xln Xn2 Xnk The model equations is then specified as, k Y; =/30 + 'LPjXij +&p i=1,2, ...,n (K - 1) j=1 The equation is can be written in matrix notation as (K - 2) where X YI 1 XII l 2 A X 1K Y2 1 X 21 X 22 A X 3= (n x 1) vector, J(= 2k (n x p) matrix M M M M M Y n 1 x nl x n2 A x nk /30 8} /31 8 2 $= (p xI) vector, and 8= (n x l) vector M M /3k 8 n 188 The least square estimator of ~ is (K - 3) Therefore, full term fitted regression model becomes (K - 4) Based on equations from (K - 2) through (K - 4), full term stackup coefficients are obtained in Table K - 2. Table K - 2. All the coefficients estimated by matrix manipulation. DFFD FeeD variables FFD Lower Upper FCCD squared FCCD term constant 0.4021 -0.0339 0.0310 0.0285 - - 2 xl -0.0284 -0.0136 0.0227 -0.0268 x1 -0.0291 2 x2 0.0074 -0.0022 0.0579 0.0070 x2 0.0139 x3 0.0119 -0.0086 -0.0126 0.0112 x32 -0.0174 x4 0.0617 0.0094 -0.0101 0.0581 x42 -0.0291 x5 -0.0246 -0.1876 0.1160 -0.0232 x52 0.2198 x6 0.0764 0.0510 0.1307 0.0719 x62 0.0388 x7 0.0185 -0.0128 0.0195 0.0174 x72 -0.0291 x8 -0.0791 0.0019 -0.0614 -0.0745 x82 -0.0291 x9 0.0585 -0.0079 0.1085 0.0551 x92 -0.0291 x10 -0.0869 -0.0230 -0.1124 -0.0818 xl02 -0.0174 xlI -0.0059 0.0592 -0.0238 -0.0056 x11 2 0.0139 x12 0.0201 0.0124 -0.0400 0.0189 x12 2 -0.0291 x13 -0.0001 -0.1250 0.0885 -0.0001 x13 2 0.1576 x14 0.0037 -0.0468 0.0771 0.0035 x142 0.1519 xIS -0.0274 -0.0399 0.0607 -0.0257 xlS2 -0.0291 x16 -0.0268 -0.0116 -0.0512 -0.0252 x16 2 -0.0291 x17 0.0212 -0.0199 -0.0562 0.0200 x17 2 0.0139 x18 -0.0615 -0.0906 0.0151 -0.0547 x18 2 0.0676 x19 0.0386 -0.1263 0.0504 0.0363 x19 2 -0.0291 x20 -0.0258 -0.0255 -0.0240 -0.0242 x20 2 -0.0291 x21 -0.0393 0.0051 -0.0537 -0.0370 x21 2 -0.0159 x22 0.0793 0.0399 0.1170 0.0746 x22 2 0.0194 x23 -0.0132 -0.0519 0.0635 -0.0124 x23 2 0.0194 189 APPENDIXL NORMAI.J PROBABILITY PLOTS FFD data and NPP Table L - 1. Data sheet for NNP FFD. j y(i) NORMSINV (j-(3/8))/(32+0.25) significant var 1 bce -0.0869 -2.0667 0.0194 x10 2 cde -0.0791 -1 .6411 0.0504 x8 3 bcde -0.0615 -1 .3957 0.0814 x18 4 ce -0.0608 -1 .2138 0.1124 5 abce -0.0393 -1.0651 0.1434 6 a -0.0284 -0.9368 0.1744 7 be -0.0280 -0.8224 0.2054 8 abe -0.0274 -0.7178 0.2364 9 abd -0.0268 -0.6206 0.2674 10 abde -0.0258 -0.5289 0.2984 11 e -0.0246 -0.4414 0.3295 12 cd -0.0153 -0.3572 . 0.360 5 13 abcde -0.0132 -0.2755 0.391 5 14 ad -0.0085 -0 .1956 0.4225 15 bcd -0.0059 -0.1169 0.4535 16 ace -0.0001 -0.0389 0.4845 17 de 0.0 0 04 0.0389 0.5155 18 acd 0.0037 0.1169 0.5465 19 b 0.0074 0.1956 0.5775 20 c 0.0119 0.2755 0.6085 21 ae 0.0185 0.3572 0.6395 22 ade 0.0201 0.4414 0.6705 23 abc 0.0212 0.5289 0.7016 24 be 0.0284 0.620 6 0.7326 25 ab 0.0344 0.7178 0.7636 26 ac 0.0382 0.8224 0.7946 27 acde 0.0386 0.9368 0.8256 28 bde 0.0585 1 .0651 0.8566 x9 29 d 0.0617 1 .21 38 0.8876 x4 30 bd 0.0764 1.3957 0.9186 x6 31 abed 0.0793 1.6411 0.9496 x22 Normal Probability Plot by FFD 2 >. ~4 := ~ :g x9 .cC'G ...o Q. -0.1 ce-0.05 0.05 0.1 Ci ...E x~ t zo • xI8 -2 xlO -3 y(i) Figure L - 1 Normal Probability Plot ofFFD. 190 DFFD data and NPP Table L - 2. Data sheet for NNP DFFD lower design. y(i) NORMSINV (j-(3/8 ))/(32+0 .25) sig nificant var 1 e -0.0938 -2.0667 0.0194 x5 2 acde -0.0631 -1.6411 0.0504 x19 3 ace -0.0625 -1.3957 0.0814 x13 4 bcde -0.0453 -1.2138 0.1124 x18 5 abede -0.0259 -1.0651 0.1434 6 aed -0.0234 -0.9368 0.1744 7 abe -0.0199 -0.8224 0.2054 8 be -0.0161 -0.7178 0.2364 9 abde -0.0128 -0.6206 0.2674 10 bee -0.0115 -0.5289 0.2984 11 ee -0.0115 -0.4414 0.3295 12 abc -0.0100 -0.3572 0.3605 13 a -0.0068 -0.2755 0.3915 14 ae -0.0064 -0.1956 0.4225 15 abd -0.0058 -0.1169 0.4535 16 c -0.0043 -0.0389 0.4845 17 bde -0.0040 0.0389 0.5155 18 be -0.0036 0.1169 0.5465 19 de -0.0012 0.1956 0.5775 20 b -0.0011 0.2755 0.6085 21 ede 0.0009 0.3572 0.6395 22 cd 0.0015 0.4414 0.6705 23 ad 0.0020 0.5289 0.7016 24 abee 0.0025 0.6206 0.7326 25 d 0.0047 0.7178 0.7636 26 ade 0.0062 0.8224 0.7946 27 ac 0.0173 0.9368 0.8256 ae 28 abed 0.0200 1.0651 0.8566 x22 29 bd 0.0255 1.2138 0.8876 x6 30 bed 0.0296 1.3957 0.9186 x11 31 ab 0.0456 1.6411 0.9496 ab NPP-DFFD(lower) NPP significant effect by DFFD (Lower design) ab • xlf • ~ ~ a c••• x6 x22 zeu .c o ~ Q. -0.15 -0.1 -0.05 0.05 0.1 (ij E x13. •-1 ~ o • x23 -1.5 z x19 • x18 x5 • -2 -2.5 y(i) Figure L - 2 Normal Probability Plot ofDFFD (lower design). 191 Table L - 3. Data sheet for NNP DFFD upper design. j y(i) NORMSINV U-(3/8))/(32+0.25) significant var 1 bce -0.0562 -2.0667 0.0194 x10 2 bc -0.0471 -1.6411 0.0504 be 3 ac -0.0436 -1.3957 0.0814 ac 4 cde -0.0307 -1.2138 0.1124 x8 5 abc -0.0281 -1.0651 0.1434 6 cd -0.0279 -0.9368 0.1744 7 abce -0.0269 -0.8224 0.2054 8 abd -0.0256 -0.7178 0.2364 9 ad -0.0229 -0.6206 0.2674 10 ab -0.0206 -0.5289 0.2984 11 ade -0.0200 -0.4414 0.3295 12 ce -0.0134 -0.3572 0.3605 13 abde -0.0120 -0.2755 0.3915 14 bed -0.0119 -0.1956 0.4225 15 c -0.0063 -0.1169 0.4535 16 d -0.0050 -0.0389 0.4845 17 de 0.0038 0.0389 0.5155 18 bcde 0.0076 0.1169 0.5465 19 ae 0.0098 0.1956 0.5775 20 a 0.0114 0.2755 0.6085 x1 21 acde 0.0252 0.3572 0.6395 x19 22 b 0.0290 0.4414 0.6705 x2 23 abe 0.0304 0.5289 0.7016 x15 24 abcde 0.0318 0.6206 0.7326 x23 25 be 0.0322 0.7178 0.7636 be 26 acd 0.0386 0.8224 0.7946 x14 27 ace 0.0443 0.9368 0.8256 x13 28 bde 0.0542 1.0651 0.8566 x9 29 e 0.0580 1.2138 0.8876 x5 30 abcd 0.0585 1.3957 0.9186 x22 31 bd 0.0653 1.6411 0.9496 x6 NPP-DFFD(upper) NPP significant effect by DFFD (Upper design) x~ .6 ~ x/~ •• x22 ~ •• x9 :aeu .4' x14 .Q o x19,x2,x15,x23,be L. ~ -0.08 -0.06 -0.04 0.02 0.04 0.06 0.08 E oL. Z y(i) Figure L - 3 Normal Probability Plot ofDFFD (upper design). 192 FCCD data and NPP Table L - 4. Data sheet for NNP FCCD. y(i) NORMSINV (j-(3/8))/(47+0.25) significant? 1 x10 -0.0818 -2.2195 0.0132 x10 2 x8 -0.0745 -1.8198 0.0344 x8 3 x18 -0.0547 -1.5932 0.0556 x18 4 x21 -0.0370 -1.4275 0.0767 5 xx9 -0.0291 -1.2937 0.0979 6 xx16 -0.0291 -1.1798 0.1190 7 xx19 -0.0291 -1.0794 0.1402 8 xx15 -0.0291 -0.9888 0.1614 9 xx20 -0.0291 -0.9057 0.1825 10 xx4 -0.0291 -0.8285 0.2037 11 xx8 -0.0291 -0.7559 0.2249 12 xx12 -0.0291 -0.6870 0.2460 13 xx7 -0.0291 -0.6213 0.2672 14 xx1 -0.0291 -0.5582 0.2884 15 x1 -0.0268 -0.4972 0.3095 16 x15 -0.0257 -0.4380 0.3307 17 x16 -0.0252 -0.3803 0.3519 18 x20 -0.0242 -0.3239 0.3730 19 x5 -0.0232 -0.2684 0.3942 20 xx3 -0.0174 -0.2138 0.4153 21 xx10 -0.01 74 -0. 1598 0.4365 22 xx21 -0.0159 -0.1063 0.4577 23 x23 -0.0124 -0.0531 0.4788 24 x11 -0.0056 0.0000 0.5000 25 x13 -0.0001 0.0531 0.5212 26 x14 0.0035 0.1063 0.5423 27 x2 0.0070 0.1598 0.5635 28 x3 0.0112 0.2138 0.5847 29 xx17 0.0139 0.2684 0.6058 30 xx2 0.0139 0.3239 0.6270 31 xx11 0.0139 0.3803 0.6481 32 x7 0.0174 0.4380 0.6693 33 x12 0.0189 0.4972 0.6905 34 xx23 0.0194 0.5582 0.7116 35 xx22 0.0194 0.6213 0.7328 36 x17 0.0200 0.6870 0.7540 37 canst 0.0285 0.7559 0.7751 38 x19 0.0363 0.8285 0.7963 39 xx6 0.0388 0.9057 0.8175 40 x9 0.0551 0.9888 0.8386 41 x4 0.0581 1.0794 0.8598 42 xx18 0.0676 1.1798 0.8810 43 x6 0.0719 1.2937 0.9021 44 x22 0.0746 1.4275 0.9233 x22 45 xx14 0.1519 1.5932 0.9444 xx14 46 xx13 0.1576 1.8198 0.9656 xx13 47 xx5 0.2198 2.2195 0.9868 xx5 193 NPP-FCCD Normal Probability Plot by FCCD x10 2.5 x8 x21 • x91\2 • ~ t18 ~, x16/\2 'n; ..,; x191\2 .c= o x151\2 c.~ (ij -0.2 -0.1 -. 0 E 0.1 0.2 0.3 ~ x22 1 - o z x141\2. .5 x131\2· -2 x51\2 • -2.5 y(i) Figure L - 4 Normal Probability Plot ofFCCD. 194 APPENDIXM MGlf ON FCCD MODEL Estimated stackup by FeCD The Equation (1) has max point at the nominal value y = 0.028497 - 0.074459 *x8 - 0.081818 * xl0 - 0.054712 * x18 - 0.036959 * x21 + 0.074641 * x22 + 0.219805 * x5 * x5 (M - 1) - 0.029145 * x9 * x9 + 0.157605 * x13 * x13 + 0.151855 * x14* x14 - 0.029145 * x15 * xIS - 0.029145 * x16 * x16 - 0.029145 * x19 * x19 Partial derivatives: Table M - 1. Partial Derivatives ofEquation (M - 1). 2 oy a y Variables Coefficients 2 oxi OX i x8 -0.0745 -0.0745 0.0000 x10 -0.0818 -0.0818 0.0000 x18 -0.0547 -0.0547 0.0000 x21 -0.0370 -0.0370 0.0000 x22 0.0746 0.0746 0.0000 xx5 0.2198 0.4396 0.4396 xx9 -0.0291 -0.0583 -0.0583 xx13 0.1576 0.3152 0.3152 xx14 0.1519 0.3037 0.3037 xx15 -0.0291 -0.0583 -0.0583 xx16 -0.0291 -0.0583 -0.0583 xx19 -0.0291 -0.0583 -0.0583 Table M - 2. Characteristic of all the independent variables for Equation (M - 1). variable mean sigma [j3; /32 ~31 ~4i distribution s all 0 1/3 0 3 0 3/81 normal Where: 195 J.11i = 0 2 u 2i = a 2 = ( ~) ~ ~3i 2 = fi{ X flu1.5 = 0 X (i) = 0 ~4i ~2/ =P2 X =3 X (i)2 :1 196 APPENDIXN DOE SIMULATIONS FOR VALIDATION FFD validation program code (declaration only) 1*********************************************************************1 1* Filename: bracket.vsl *1 1* *1 1* Written by: Misako Hata *1 1* Date: 06/14/2000 *1 1* *1 1* PURPOSE: Phase II programming *1 1* "DOE" simulation *1 1* *1 1* Assumptions: Components variations are set as if brackets were *1 1* manufactured and assembled for DOE prototype, which is considered*1 1* as tested APM response. */ /* *1 1* Revisions: *1 1* Date Modeler Rev Nature and Purpose of Change *1 1* *1 1*********************************************************************1 II I #include 'dec hata.inc' II Ideclarations repeat for k=l to 5 by 1 { #include 'base hata.tol' II Ibase part description #include 'top_hata.tol' II Itop part description #include 'spt_hata.tol' II I support part description #include 'pinAhata.tol' II I support part description #include 't asmhata.asm' II lassembly operation for top bar #include 's asmhata.asm' 1/ lassembly operation for top bar } 1/ I 1/ I 1/ I 1*********************************************************************1 1********************* END OF FILE: *******************************1 1*********************************************************************/ II I Declarations constant x=l, y=2, z=3; II Icoordinate system constant xy l,yz = 2,zx 3; II Icoordinate system variation Illvariation for the base HOLE sizeC normal 10 +- 0.062, Illdia 10 + 0.062 hole C size HOLE sizeD normal 8.2 +- 0.062, Illdia 8 + 0.062 hole 0 size HOLE sizeE normal 8.2 +- 0.062, Illdia 8 + 0.062 hole E size bonus_probl normal 0 +- I, bonus_prob2 normal 0 +- I, bonus_prob3 normal 0 +- I, bonus_prob4 normal 0 +- 1, bonus_probS normal 0 +- 1, float2var 0 +- 0.1, II I true position 0.2 MMC (RHS) float3var 0 +- 0.1, Illtrue position 0.2 MMC (LHS) bslotwidthl normal 15 +- 0.13, III size of the base slot along yaxis (RHS) bslotwidth2 normal 15 +- 0.13, Illsize of the base slot along y axis (LHS) I*HOLE_radC:::: DOE(k,l) normal = 0 +-- 0.15, III [POSldia O.3_MMCIAIB__MMCldtmC] hole C *1 197 HOLE radO normal = 0 +- 0.15, III [POSldia 0.3 MMCIAIB MMCIC MMC\ dtmO] hole 0 I*HOLE_radE = DOE(k,2) normal = 0 +- 0.15, 11\[POSldia 0.3_MMCIAID_MMCIC_MMCJ hole E*/ HOLE angC uniform = 0 +- 180, III [POSldia 0.3 MMCIAIB MMCldtmC] hole C I*HOLE_angO DOE(k,3) uniform o +- 180, II I [POS I dia- O. 3_MMCI A I B_MMC I C_MMC! dtmDl hole D*I I*HOLE_angE DOE(k,4) uniform o +- 180; III [POSldia 0.3_MMCIAID_MMCIC_MMC] hole E *1 Illvariation for the top bar l*tHOLE_sizeB = DOE(k,5) normal = 8.2 +- 0.062, Illdia 8.2 +- 0.062 top hole B size*1 tHOLE sizeC normal 8.2 +- 0.062, Illdia 8.2 +- 0.062 top hole C size tHOLE sizeM normal 8.2 +- 0.062, Illdia 8.2 +- 0.062 top measurement hole bonus_prob6 normal 0 +- 1, bonus_prob7 normal 0 +- 1, tthickness normal 13 +- 0.15, 11113 +- 0.15 bar thickness +- 0.15, III [POSldia O.3_MMCIAIB_MMCldtmC] top hole tHOLE- radC normal 0 C tHOLE radmes normal 0 +- 0.15, III [POSldia 0.3_MMCIAIB_MMCIC_MMC] mes hole tHOLE_angC uniform = 0 +- 180, III [POSldia 0.3 MMCIAIB MMCldtmC] top hole C tHOLE_angmes uniform = 0 +- 180, III [POSIC1ia 0.3_lMMCIAIB_MMCIC_MMCJ mes hole I*group tHOLE C is level 2 tHOLE radC levelS tHOLE angC 1 [POSldia 0.3_MMCIA_MMCIBldtmC] top hole C '; group tHOLE mes is level 2 tHOLE racooes levelS tHOLE angmes 1 [POSldia 0.3_MMCIB_MMCIC_MMC]mes hole'; group MMC tC is level 3 tHOLE_sizeC, bonus_prob6 'MMC on top hole C'; group MMC tM is level 3 tHOLE_sizeM, bonus_prob7 'MMC on top mes hole';*1 Ilvariation for the support bar sHOLE sizeB normal = 8.2 +- 0.062, Illdia 8.2 +- 0.062 top hole B size l*sHOLE sizeC normal 8.2 +- 0.062, Illdia 8.2 +- 0.062 top hole C size*1 bonus_prob8 normal = 0 +- 1, bonus_prob9 normal = 0 +- 1, bonus_prob10 normal = 0 +- 1, floatvar1 o +- 0.05, Illtrue position 0.1 MMC floatvar2 o +- 0.05, II/true position 0.1 MMC sthicknessl normal = 13 +- 0.15, III size of the bar along z axis sthickness2 normal = 35 +- 0.15, III size of the support part along z aX1S slotwidth normal = 13 +- 0.15, III size of the slot along z axis sHOLE radC normal = 0 +- 0.15, II I [POS Idia 0.3 MMCIAIB MMCI dtmC] support hole C sHOLE_angC uniform 0 +- 180, II I [POS Idia 0.3=MMCIAIB=MMCldtmC] support hole C I*group sHOLE C is level 2 sHOLE radC levelS sHOLE_angC 1 [POSldia 0.3_MMCIA_MMCIBldtmC] support hole C'; group MMC sC is level 3 SHOLE_sizeC, bonus_prob8 'MMC on top hole C '; group MMC s34 is level 3 floatvarl, bonus_prob9 'MMC on spt pt 3,4 '; group MMC s56 is level 3 floatvar2, bonus_problO 'MMC on spt pt 5,6';*1 III variation for pin sizes pinA size normal = 7.9375 +- 0.035, Illdia 7.9375 +- 0.035 pin A size l*pinB_size normal = 7.9375 +- 0.035, Illdia 7.9375 +- 0.035 pin B size*/ pinC_size normal = 7.9375 +- 0.035, Illdia 7.9375 +- 0.035 pin C size III variation for t and s asmhata.asm probuniforml uniform = 0 +- 1, 198 probuniform2 uniform = 0 +- 1; I*group b_s_asm is level 3 bslotwidthl, sthicknessl, probuniform2 'clearance bwn spt and base' ; group b t asm is level 3 bslotwidth2, tthickness, probuniforml 'clearance bwn base and top';*I- - III .tol variables value bonus tol; value MMC; II Iassembly variables value clr asm; /11 clearance (w/prob) between top or support bar and baseslot. gravity effect of pins on top or support bar side holes value dist mes2; value dist; value i; point topfloat, sptfloat; point transstop, rotstop; point floatl; floatl = 512.5,0,0; III float point for support bar point float2, float3; III float point for base slots point tar1, tar2, tar3; tarl 0,0,0; tar2 1,0,0; tar3 0,250,0; value k; array DOE(5,7); k = 1; repeat for k = 1 to 5 by I{ if (k == 1) { DOE(I,I) -1; DOE(I,2) 1; DOE(I,3) -1; DOE(I,4) 1; DOE(I,5) -1; DOE(1, 6) 1; DOE(I,7) -1; } if (k == 2) { DOE(2,I) 1; DOE(2,2) -1; DOE(2,3) 1; DOE(2,4) -1; DOE(2,5) 1; DOE(2,6) -1; DOE(2,7) 1; } if (k == 3) { DOE(3,I) -0.5; DOE(3,2) 0.5; DOE(3,3) -0.5; DOE(3,4) 0.5; DOE(3,5) -0.5; DOE(3,6) 0.5; DOE(3,7) -0.5; } if (k == 4) { DOE(4,1) 0.5; DOE(4,2) = -0.5; 199 DOE(4,3) 0.5; DOE(4,4) -0.5; DOE(4,S) 0.5; DOE (4,6) -0.5; DOE(4,7) 0.5; } if (k == 5) { DOE(S,l) 0; DOE(S,2) 0; DOE(S,3) 0; DOE(S,4) 0; DOE(S,S) 0; DOE(S,6) 0; DOE(S,7) 0; } } print DOE; object base(8); context base; value HOLE radC, HOLE_radE; HOLE radC 0.15; HOLE-radE = 0.15; #01 0,0,0; //Ibottom of datum B (DRF) #02 0,0,30; //Itop of the datum B hole #03 -250,0,0; //Ibottom of datum C #04 -250,0,30; //Itop of the datum C hole #05 80,0-(bslotwidthl/2),15; //Ipoint inside the slot #06 80,0+(bslotwidthl/2),lS; I/Ipoint inside the slot #07 #06.x-410,0-(bslotwidth2/2),lS; Illpoint inside the slot #08 #07.x,0+(bslotwidth2/2),15; Illpoint inside the slot bonus tol = HOLE sizeC/2 - (9.938/2); #03.x = -250 + (DOE(k,l) * (HOLE_radC + bonus tal)); //lvariation along x axis for datum C hole #04.x = -250 + (DOE(k,l) * (HOLE_radC + bonus tol)); II I variation along x axis for datum C hole if (DOE(k,3)==1) { bonus tal = HOLE sizeD/2 - ((8.2-0.062)/2); MMC =-bonus tal * bonus prob2; base.OS = p;int varied irom point base.05 by HOLE_radD+r'.lMC radially at angle 90 in standard csystem zx; base.06 point varied from point base.06 by HOLE_radD+NMC radially at angle 90 in standard csystem zx; } if ((DOE(k,3)==0.5) & (DOE(k,3)==-0.S)) { bonus tal = HOLE sizeD/2 - ((8.2-0.062)/2); MMC =-bonus tol * bonus prob2; base.05 = p;int varied irom point base.OS by HOLE_radD+~1MC radially at angle 45 in standard csystem zx; base.06 point varied from point base.06 by HOLE_radD+~1MC radially 200 at angle 45 in standard csystem zx; } if (DOE(k,3)==O) { bonus tal = HOLE sizeD/2 - ((8.2-0.062)/2); MMC =-bonus tal * bonus prob2; base.OS = point varied from point base.OS by 0 radially at angle 0 in standard csystem zx; base.06 point varied from point base.06 by 0 radially at angle a in standard csystem zx; } if (DOE(k,3)==-1) { bonus tol = HOLE sizeD/2 - ((8.2-0.062)/2); MMC =-bonus_tol * bonus_prob2; base.OS = point varied from point base.OS by HOLE_radD+MMC radially at angle a in standard csystem zx; base.06 point varied from point base.06 by HOLE_radD+MMC radially at angle a in standard csystem zx; bonus_tol = 14.87 - (-#OS.y + #06.y); MMC = bonus_tal * bonus_prob3; base.05.y base.OS.y + float2var + MMC; //Ipoint #OS with TP and MMC base.06.y = base.06.y + float2var + MMC; //Ipoint #06 with TP and MMC if (DOE(k, 4) ==1) { bonus tol = HOLE sizeE/2 - ((8.2-0.062)/2); base.07 = point ;aried from point base.07 by DOE(k,2)*(HOLE_radE+ bonus_tal) radially at angle 90 in standard csystem zx; base.OB point varied from point base.08 by DOE(k,4)*(HOLE_radE+ bonus tal) radially at angle 90 in standard csystem zx; } if ((DOE(k,4)==-O.S) & (DOE(k,4)==0.5)) { bonus tol = HOLE_sizeE/2 - ((8.2-0.062)/2); base.07 = point varied from point base.07 by DOE(k,2) * (HOLE_radE+ bonus tal) radially at angle 45 in standard csystem zx; base.OB point varied from point base.08 by DOE(k,2)*(HOLE_radE+ bonus tal) radially at angle 4S in standard csystem zx; } if (DOE(k,4)==O) { bonus tol = HOLE sizeE/2 - ((8.2-0.062)/2); base.07 = point ;aried from point base.07 by 0 radially 201 at angle 0 in standard csystem zx; base.OB point varied from point base.OB by 0 radially at angle 0 in standard csystem zx; } if (DOE(k,4)==-1) { bonus tal = HOLE sizeE/2 - ((8.2-0.062)/2); base.07 = point ;aried from point base.07 by DOE(k,2)*(HOLE_radE+ bonus tal) radially at angle 0 in standard csystem zx; base.OB point varied from point base.08 by DOE(k,2)*(HOLE_radE+ bonus tol) radially at angle ° in standard csystem zx; bonus tal = 14.87 - (-#07.y + #08.y); 11 I bonus tal = Actual size (radius) _. MMC condition MMC = bonus_tol * bonus_probS; base.07.y base.07.y + float3var + MMC; Illpoint #07 with TP and MMC base.OB.y base.08.y + float3var + MMC; I/Ipoint #08 with TP and MMC object top(6)i context top; value tHOLE_sizeB; tHOLE sizeB = 8.2 + (DOE(k,S)*0.062); #01 0,O,-tthickness/2; //Ibottom of the datum B hole #02 0,0,tthickness/2; III top of the datum B hole #03 #01.x+307.5,0,-tthickness/2; I/Ibottom of the datum C hole #04 #02.x+307.S,O,tthickness/2; //Itop of the datum C hole #05 #01.x+424,0,-tthickness/2; //Ibottom of the measurement hole #06 #02.x+424,0,tthickness/2i III top of the measurement hole bonus tol = tHOLE_sizeC/2 - (8.2-0.062)/2; MMC = bonus_tol * bonus_prob6; top.03.x top.03.x + (tHOLE_radC + MMC); top.04.x = top.04.x + (tHOLE_radC + MMC); bonus_tal = tHOLE_sizeM/2 - (8.2-0.062)/2; MMC = bonus_tol * bonus_prob7; top.OS = point varied from point top.OS by tHOLE_radmes + MMC radially at angle tHOLE_angmes in standard csystern xy; top.06 point varied from point top.06 by tHOLE_radmes + MMC radially at angle tHOLE_angmes in standard csystem xy; object support(6); context support; value sHOLE sizeCi sHOLE sizeC-= (DOE(k,6)*0.062) + 8.2; 202 #01 0,0,-sthicknessl/2; Ilibottom of the datum B hole #02 0,0,sthicknessl/2; Illtop of the datum B hole #03 512.5,0,floatl.z - sthickness2/2; Illpoint #03 wrt support part size #04 512.5,0,floatl.z + sthickness2/2; Illpoint #04 wrt support part size #05 512.5,0, floatl.z - slotwidth/2; Illpoint #05 wrt slot size #06 512.5,0,floatl.z + slotwidth/2; Illpoint #06 wrt slot size bonus tal = sHOLE sizeC/2 - (8.2-0.062)/2; Illbonus tolerance for dtmC hole MMC =-bonus tol *-bonus prob8; support.03.~ support.03.x + (sHOLE_radC+MMC); Illbonus tal on C wrt x-axis support.04.x support.04.x + (sHOLE_radC+MMC); II Ibonus tal on C wrt x-axis support.05.x support.05.x + (sHOLE_radC+MMC); Illbonus tal on C wrt x-axis support.06.x support.06.x + (sHOLE_radC+MMC); Ilibonus tal on C wrt x-axis bonus tal = 35.15 - (-#03.z + #04.z); Illbonus tal for support part MMC =-bonus_tol * bonus_prob9; support.03.z support.03.z + floatvarl + MMC; Illpoint #03 with TP and MMC support.04.z = support.04.z + floatvarl + MMC; Illpoint #04 with TP and MMC bonus_tal = (-#05.z + #06.z) - 12.87; Illbonus tal for slot width MMC = bonus_tal * bonus_probIO; support.05.z support.05.z + floatvar2 + MMC; Illpoint #05 with TP and MMC support.06.z = support.06.z + floatvar2 + MMC; Illpoint #06 with TP and MMC FFD validation result Session: 20010529-092515 dist_ mes2 (FFD val idation) -> 50000 samples Sample Est. of (VarianceY'0.5 90% Conf. Interval Statistics Pop. in Est. of Pop. in Est. of Pop. Paramo Nominal: 0.0000 Mean: 0.1086 0.1086 0.0003 0.1081 0.1092 Std Dev : 0.0722 0.0722 0.0004 0.0719 0.0726 Cp: N/A N/A N/A N/A N/A Cpk: N/A N/A N/A N/A N/A Skew: 1.0347 LDL/UDL : N/A I N/A Kurt : 1.0768 Distribution: Pearson-Beta Sample Est. Sample ** Est. 0/0 < Low Limit N/A N/A Low Val 0.0003 0.0040 % > High Limit N/A N/A High Val 0.5465 0.4030 % Out of Spec N/A N/A Range 0.5462 0.3990 203 DFFD validation program code (declaration only) 1*********************************************************************1 1* Filename: bracket.vsl *1 1* *1 1* Written by: Misako Hata *1 1* Date: 06/14/2000 *1 1* *1 1* PURPOSE: Phase II programming *1 1* "DOE" simulation *1 1* *1 1* Assumptions: Components variations are set as if brackets were *1 1* manufactured and assembled for DOE prototype, which is considered*1 1* as tested APM response. *1 1* *1 1* Revisions: *1 1* Date Modeler Rev Nature and Purpose of Change *1 1* *1 I************************~********************************************1 II I #include 'dec_hata.inc' II Ideclarations repeat for k=l to 5 by 1 { #include 'base hata.tol' II Ibase part description #include 'top_hata.tol' II Itop part description #include 'spt hata.tol' II Isupport part description #include 'pinAhata.tol' II Isupport part description #include 't asmhata.asm' II I assembly operation for top bar #include 's asmhata.asm' II lassembly operation for top bar } II I II I II I I************************~********************************************1 1********************* END OF FILE *******************************1 1*********************************************************************1 II IDeclarations constant x=l, y=2, z=3; II Icoordinate system constant xy 1,yz = 2,zx 3; II Icoordinate system variation Illvariation for the base HOLE sizeC normal 10 +- 0.062, Illdia 10 + 0.062 hole C size HOLE sizeD normal 8.2 +- 0.062, Illdia 8 + 0.062 hole D size HOLE sizeE normal 8.2 +- 0.062, Illdia 8 + 0.062 hole E size bonus_probl normal 0 +- 1, bonus_prob2 normal 0 +- 1, bonus_prob3 normal 0 +- 1, bonus_prob4 normal 0 +- 1, bonus_probS normal 0 +- 1, float2var 0 +- 0.1, Illtrue position 0.2 MMC (RHS) float3var 0 +- 0.1, Illtrue position 0.2 MMC (LHS) bslotwidth1 normal 15 +- 0.13, III size of the base slot along y axis (RHS) bslotwidth2 normal 15 +- 0.13, III size of the base slot along y axis (LHS) HOLE radC normal = 0 +- 0.15, III (POSldia 0.3 MMCIAIB MMCI dtmCJ hole C I*HOLE radD DOE(k,l) normal o +- 0.15, 111[POSldia 0.3_MMCIAIB_MMCIC_MMCldtmDJ hole D*/ - I*HOLE_radE DOE(k,2) normal o +- 0.15, III [POSldia 0.3_MMCIAID_MMCIC_MMCl hole E*/ HOLE_angC uniform = 0 +- 180, III [POSldia 0.3 MMCIAIB MMCldtmCJ hole C HOLE angD uniform = 0 +- 180, III (POSldia 0.3 MMCIAIB MMCIC-MMCldtmDJ hole D /*HOLE_angE = DOE(k,3) uniform o +- 180; III [POSldia O~3_MMCTAID_MMCIC_MMC] hole E: */ 204 Illvariation for the top bar l*tHOLE_sizeB = DOE(k,4) normal = 8.2 +- 0.062, Illdia 8.2 +- 0.062 top hole B size*1 tHOLE sizeC normal 8.2 +- 0.062, Illdia 8.2 +- 0.062 top hole C size tHOLE sizeM normal 8.2 +- 0.062, Illdia 8.2 +- 0.062 top measurement hole bonus_prob6 normal o +- 1, bonus_prob7 normal o +- 1, tthickness normal = 13 +- 0.15, 11113 +- 0.15 bar thickness l*tHOLE_radC = DOE(k,5) normal = 0 +- 0.15, III [POSldia 0.3_MMCIAIB_MMCldtmC] top hole C*I tHOLE radmes normal = 0 +- 0.15, III [POSldia 0.3_MMCIAIB_MMCIC_MMC] roes hole tHOLE_angC uniform = 0 +- 180, III [POSldia 0.3 MMCIAIB MMCldtmC] top hole C tHOLE_angmes uniform = 0 +- 180, III [POSldia 0.3_lMMCIAIB_MMCIC_MMC] mes hole I*group tHOLE C is level 2 tHOLE radC levelS tHOLE_angC '[POSldia 0.3_MMCIA_MMCIBldtmC] top hole C'i group tHOLE mes is level 2 tHOLE radmes levelS tHOLE_angmes '[POSldia 0.3_MMCIB_MMCIC_MMC]mes hole'i group MMC tC is level 3 tHOLE_sizeC, bonus_prob6 'MMC on top hole C'i group MMC tM is level 3 tHOLE_sizeM, bonus_prob7 'MMC on top mes hole';*1 Ilvariation for the support bar sHOLE sizeB normal 8.2 +- 0.062, Illdia 8.2 +- 0.062 top hole B size sHOLE sizeC normal 8.2 +- 0.062, Illdia 8.2 +- 0.062 top hole C size bonus_probS normal 0 +- 1, bonus_prob9 normal 0 +- 1, bonus_problO normal = 0 +- 1, floatvar1 o +- 0.05, Illtrue position 0.1 MMC floatvar2 o +- 0.05, II/true position 0.1 MMC sthicknessl normal = 13 +- 0.15, III size of the bar along z axis sthickness2 normal = 35 +- 0.15, III size of the support part along z axis slotwidth normal = 13 +- 0.15, III size of the slot along z axis sHOLE radC normal = 0 +- 0.15, II I [POS I dia 0.3 MMCIAIB MMCI dtmC] support hole C sHOLE_angC uniform 0 +- 180, II I [POS I dia 0.3=MMCIAIB=MMCldtmC] support hole C I*group sHOLE C is level 2 sHOLE radC levelS sHOLE_angC '[POSldia 0.3_MMCIA_MMCIBldtmCJ support hole C'i group MMC sC is level 3 sHOLE_sizeC, bonus_prob8 'MMC on top hole C'i group MMC s34 is level 3 floatvarl, bonus_prob9 'MMC on spt pt 3,4'i group MMC s56 is level 3 floatvar2, bonus_prob10 'MMC on spt pt 5,6'i*1 III variation for pin sizes pinA_size normal = 7.9375 +- 0.035, Illdia 7.9375 +- 0.035 pin A size l*pinB size normal 7.9375 +- 0.035, Illdia 7.9375 +- 0.035 pin B size*1 l*pinC=size normal 7.9375 +- 0.035, /iidia 7.9375 +- 0.035 pin C size*/ III variation for t and s asmhata.asm probuniforml uniform 0 +- 1, probuniform2 uniform = 0 +- 1; I*group b s asm is level 3 bslotwidth1, sthicknessl, probuniform2 'clearance bwn spt and base'; group b t asm is level 3 bslotwidth2, tthickness, probuniforml 'clearance bwn base and top' i*l- - III .tol variables value bonus_tal; value MMC; 205 Illassembly variables value clr asm; III clearance (w/prob) between top or support bar and baseslot. gravity effect of pins on top or support bar side holes value dist_mes2; value dist; value i; point topfloat, sptfloat; point transstop, rotstop; point float1; floatl = 512.5,0,0; III float point for support bar point float2, float3; III float point for base slots point tar1, tar2, tar3; tar1 0,0,0; tar2 1,0,0; tar3 0,250,0; value k; array DOE(5,7); k- 1; repeat for k = 1 to 5 by l{ if (k == 1) { DOE (1,1) -1; 00E(1,2) 1; 00E(1,3) -1; 00E(1,4) 1; 00E(1,5) -1; DOE (1, 6) 1; 00E(1,7) -1; } if (k == 2) { 00E(2,1) 1; 00E(2,2) -1; 00E(2,3) 1; 00E(2,4) -1; 00E(2,5) 1; DOE (2,6) -1; 00E(2,7) 1; } if (k == 3) { DOE(3,1) -0.5; 00E(3,2) 0.5; 00E(3,3) -0.5; 00E(3,4) 0.5; 00E(3,5) -0.5; 00E(3,6) 0.5; 00E(3,7) -0.5; } if (k == 4) { DOE(4,l) 0.5; DOE(4,2) -0.5; 00E(4,3) 0.5; DOE(4,4) -0.5; 00E(4,5) 0.5; DOE (4,6) -0.5; 00E(4,7) 0.5; } if (k == 5) { 00E(5,1) = 0; 206 DOE(5,2) 0; DOE(5,3) 0; DOE(5, 4) 0; DOE(5,S) 0; DOE(S,6) 0; DOE(5,7) 0; } } print DOE; object base(8); context base; value HOLE radD, HOLE_radE; HOLE radD 0.15; HOLE radE = 0.15; #01 0,0,0; //lbottom of datum B (DRF) #02 0,0,30; //ltop of the datum B hole #03 -250,0,0; //Ibottom of datum C #04 -250,0,30; //Itop of the datum C hole #05 80,O-(bslotwidthl/2),15; //Ipoint inside the slot #06 80,0+(bslotwidthl/2),15; //Ipoint inside the slot #07 #06.x-410,0-(bslotwidth2/2),15; //Ipoint inside the slot #08 #07.x,0+(bslotwidth2/2),15; //Ipoint inside the slot bonus tal = HOLE sizeC/2 - (9.938/2); MMC =-bonus_tol * bonus_probl; #03.x -250 + HOLE_radC + MMC; //Ivariation along x axis for datum C hole #04.x = -250 + HOLE radC + MMC; //!variation along x axis for datum C hole bonus tol = HOLE_sizeD/2 - ((8.2-0.062)/2); MMC = bonus tol * bonus_prob2; base.05 = point varied from point base.05 by DOE(k,l)*(HOLE_radD+MMC) radially at angle HOLE_angD in standard csystem zx; base.06 point varied from point base.06 by DOE(k,l)*(HOLE_radD+MMC) radially at angle HOLE_angD in standard csystem zx; bonus tol = 14.87 - (-#05.y + #06.y); MMC =-bonus_tol * bonus_prob3; base.05.y base.05.y + float2var + MMC; //Ipoint #05 with TP and MMC base.06.y = base.06.y + float2var + MMC; //Ipoint #06 with TP and MMC if (DOE (k, 3) ==1) { bonus tal = HOLE sizeE/2 - ((8.2-0.062)/2); base.07 = point ;aried from point base.07 by DOE(k,2)*(HOLE_radE+ bonus tal) radially at angle 90 in standard csystem zx; base.08 point varied from point base.08 by DOE(k,2)*(HOLE_radE+ bonus_tal) radially at angle 90 in standard csystem zx; } if ((DOE(k,3)==-0.5) & (DOE(k,3)==0.5)) { bonus tol HOLE sizeE/2 - ((8.2-0.062)/2); 207 base.07 point varied from point base.07 by DOE(k,2)*(HOLE_radE+ bonus tal) radially at angle 45 in standard csystem zx; base.08 point varied from point base.08 by DOE(k,2)*(HOLE_radE+ bonus_tal) radially at angle 45 in standard csystem zx; } if (DOE(k,3)==0) { bonus tal = HOLE sizeE/2 - ((8.2-0.062)/2); base.07 = point ~aried from point base.07 by 0 radially at angle 0 in standard csystem zx; base.08 point varied from point base.08 by 0 radially at angle 0 in standard csystem zx; } if (DOE(k,3)==-1) { bonus tal = HOLE sizeE/2 - ((8.2-0.062)/2); base.07 = point ~aried from point base.07 by DOE(k,2)*(HOLE_radE+ bonus tal) radially at angle 0 in standard csystem zx; base.08 point varied from point base.08 by DOE(k,2)*(HOLE_radE+ bonus_tal) radially at angle 0 in standard csystem zx; bonus tal = 14.87 - (-#07.y + #08.y); Illbonus tal = Actual size(radius) - MMC condition MMC = bonus tal * bonus_probS; base.07.y base.07.y + float3var + MMC; /Ilpoint #07 with TP and MMC base.08.y base.08.y + float3var + MMC; /Ilpoint #08 with TP and MMC print base; object top(6); context top; value tHOLE sizeB, tHOLE radC; tHOLE_sizeB-= 8.2 + (DOE(k,4)*0.062); tHOLE radC = 0.15; #01 0,0,-tthickness/2; I/Ibottom of the datum B hole #02 O,O,tthickness/2; III top of the datum B hole #03 #01.x+307.5,0,-tthickness/2; I/Ibottom of the datum C hole #04 #02.x+307.5,O,tthickness/2; I/Itop of the datum C hole #05 #01.x+424,0,-tthickness/2; Illbottom of the measurement hole #06 #02.x+424,0,tthickness/2; III top of the measurement hole bonus tal = tHOLE_sizeC/2 - (8.2-0.062)/2; top.03.x top.03.x + (DOE(k,S)*(tHOLE_radC + bonus_tol)); top.04.x = top.04.x + (DOE(k,S)*(tHOLE_radC + bonus_tal)); bonus tal = tHOLE_sizeM/2 - (8.2-0.062)/2; MMC = bonus_tol * bonus_prob7; 208 top.05 point varied from point top.05 by tHOLE_radmes + MMC radially at angle tHOLE_angmes in standard csystem xy; top.06 point varied from point top.06 by tHOLE_radmes + MMC radially at angle tHOLE_angmes in standard csystem xy; object support(6); context support; #01 0,0,-sthickness1/2; Illbottom of the datum B hole #02 0,0,sthickness1/2; Illtop of the datum B hole #03 512.5,0,float1.z - sthickness2/2; Illpoint #03 wrt support part size #04 512.5,0,floatl.z + sthickness2/2; Illpoint #04 wrt support part size #05 512.5,0,floatl.z - slotwidth/2; Ilipoint #05 wrt slot size #06 512.5,0,float1.z + slotwidth/2; Illpoint #06 wrt slot size bonus tol = sHOLE sizeC/2 - (8.2-0.062)/2; Illbonus tolerance for dtmC hole MMC =-bonus tol *-bonus probS; support.03.x support.03.x + (sHOLE_radC+MMC); Illbonus tol on C wrt x-axis support.04.x support.04.x + (sHOLE_radC+MMC); Illbonus tol on C wrt x-axis support.05.x support.05.x + (sHOLE_radC+MMC); Illbonus tol on C wrt x-axis support.06.x support.06.x + (sHOLE radC+MMC); Illbonus tol on C wrt x-axis bonus_tol = 35.15 - (-#03.z + #04.z); Illbonus tal for support part MMC = bonus tal * bonus prob9; support.03.z support.03.z + floatvar1 + MMC; Illpoint #03 with TP and MMC support.04.z = support.04.z + floatvar'l + MMC; Illpoint #04 with TP and MMC bonus_tol = (-#05.z + #06.z) - 12.87; Illbonus tal for slot width MMC = bonus_tal * bonus_probIO; support.05.z support.05.z + floatvar2 + MMC; Illpoint #05 with TP and MMC support.06.z = support.06.z + floatvar2 + MMC; Illpoint #06 with TP and MMC DFFD validation result Session: 20010529-101542 dist_mes2 (DFFD validation) -> 50000 samples Sample Est. of (Variance}AO.5 900/0 Conf. Interval Statistics Pop. in Est. of Pop, in Est. of Pop. Paramo Nominal: 0.0000 Mean: 0.0521 0.0521 0.0001 0.0519 0.0523 Std Dev : 0.0329 0.0329 0.0002 0.0328 0.0331 Cp: N/A N/A N/A N/A N/A Cpk: N/A N/A N/A N/A N/A Skew: 0.9818 LDL/UDL: N/A / N/A Kurt: 1.0602 Distribution: Pearson-Beta Sample Est. Sample ** Est. % < Low Limit N/A N/A Low Val 0.0001 0.0004 0/0> High Limit N/A N/A High Val 0.2764 0.1869 % Out of Spec N/A N/A Range 0.2763 0.1865 ** Estimated range of 99.7300% 209 FeeD validation program code (declaration onlyj) 1*********************************************************************1 1* Filename: bracket.vsl */ /* */ 1* Written by: Misako Hata */ 1* Date: 06/14/2000 */ 1* */ 1* PURPOSE: Phase II programming */ 1* "DOE" simulation */ 1* */ 1* Assumptions: Components variations are set as if brackets were */ 1* manufactured and assembled for DOE prototype, which is considered*/ 1* as tested APM response. */ /* */ 1* Revisions: */ 1* Date Modeler Rev Nature and Purpose of Change */ 1* */ 1***************************************************** ****************/ 1/ I #include 'dec_hata.inc' II Ideclarations repeat for k=l to 5 by 1 { #include 'base hata.tol' 1/ Ibase part description #include 'top_hata.tol' /1 Itop part description #include 'spt_hata.tol' II I support part description #include 'pinAhata.tol' II Isupport part description #include 't asmhata.asm' II I assembly operation for top bar #include 's asmhata.asm' II lassembly operation for top bar } /1 I II I II I 1***************************************************** ****************/ 1***-****************** END OF FILE *******************************1 1***************************************************** ****************/ II I Declarations constant x=l, y=2, z=3; II Icoordinate system constant xy l,yz = 2,zx 3; II Icoordinate system variation Illvariation for the base HOLE sizeC normal 10 +- 0.062, Illdia 10 + 0.062 hole C size HOLE sizeD normal 8.2 +- 0.062, Illdia 8 + 0.062 hole 0 size HOLE sizeE normal 8.2 +- 0.062, Illdia 8 + 0.062 hole E size bonus_prob1 normal 0 +- 1, bonus_prob2 normal 0 +- 1, bonus_prob3 normal 0 +- 1, bonus_prob4 normal 0 +- 1, bonus_probS normal 0 +- 1, float2var 0 +- 0.1, I/Itrue position 0.2 MMC (RHS) float3var 0 +- 0.1, Illtrue position 0.2 MMC (LHS) bslotwidth1 normal 15 +- 0.13, III size of the base slot along yaxis (RHS) bslotwidth2 normal 15 +- 0.13, I/Isize of the base slot along y axis (LHS) HOLE radC normal = 0 +- 0.15, /11 (POSldia 0.3 MMCIAIB MMCldtmC] hole C I*HOLE_radD = DOE(k,l) normal = 0 +- 0.15, 111[POSldia 0.3_MMCIAIB_MMCIC_MMCldtmD] hole 0*1 HOLE radE normal = 0 +- 0.15, //1 (POSldia 0.3_MMCIAID_MMCIC_MMC] hole E HOLE_angC uniform = 0 +- 180, III (POSldia 0.3_MMCIAIB_MMCldtmC] hole C 210 I*HOLE angD DOE(k,2) uniform ° +- 180,1/1 [POSldia 0.3_MMCIAIB MMCIC_MMCldtmD]hole 0*1 I*HOLE angE DOE(k,3) uniform o +- 180;1/1 [POSldia 0.3_MMCIAID_MMCIC_MMCJ hole E*/ Illvariation for the top bar l*tHOLE_sizeB = DOE(k,4) normal = 8.2 +- 0.062, Illdia 8.2 +- 0.062 top hole B size*1 tHOLE sizeC normal 8.2 +- 0.062, Illdia 8.2 +- 0.062 top hole C size tHOLE sizeM normal 8.2 +- 0.062, Illdia 8.2 +- 0.062 top measurem~nt hole bonus_prob6 normal o +- 1, bonus_prob7 normal o +- 1, tthickness normal = 13 +- 0.15, 11113 +- 0.15 bar thickness l*tHOLE_radC = DOE(k,5) normal = 0 +- 0.15, III [POSldia 0.3_MMCIAIB_MMCldtmC] top hole c*1 l*tHOLE_radmes = DOE(k,6) normal 0 +- 0.15, III [POSldia 0.3_MMCIAIB_MMCIC_MMC] roes hole*1 tHOLE_angC uniform = 0 +- 180, III [POSldia 0.3_MMCIAIB_MMCldtmC] top hole C tHOLE_angroes uniform = 0 +- 180, III [POSldia 0.3_MMCIAIB_MMCIC_MMC] mes hole I*group tHOLE C is level 2 tHOLE_radC levelS tHOLE_angC '[POSldia 0.3_MMCIA_MMCIBldtmC] top hole C'; group tHOLE mes is level 2 tHOLE_radmes levelS tHOLE_angmes '[POSldia 0.3_MMCIB_MMCIC_MMC]mes hole'; group MMC tC is level 3 tHOLE_sizeC, bonus_prob6 'MMC on top hole C'; group MMC tM is level 3 tHOLE_sizeM, bonus_prob7 'MMC on top mes hole';*1 Ilvariation for the support bar sHOLE sizeB normal = 8.2 +- 0.062, Illdia 8.2 +- 0.062 top hole B size l*sHOLE_sizeC = DOE(k,9) normal = 8.2 +- 0.062, Illdia 8.2 +- 0.062 top hole C size*1 bonus_prob8 normal = 0 +- 1, bonus_prob9 normal = 0 +- 1, bonus_prob10 normal = 0 +- 1, floatvar1 o +- 0.05, II/true position 0.1 MMC floatvar2 o +- 0.05, Illtrue position 0.1 MMC sthickness1 normal = 13 +- 0.15, III size of the bar along z axis sthickness2 normal = 35 +- 0.15, III size of the support part along z axis slotwidth normal = 13 +- 0.15, III size of the slot along z axis l*sHOLE_radC= DOE(k,10) normal = 0 +- 0.15, III [POSldia 0.3_MMCIAIB_MMCldtmC] support hole C*I sHOLE angC uniform 0 +- 180, III [POSldia 0.3_MMCIAIB_MMCldtmCJ support hole C I*group sHOLE C is level 2 sHOLE radC level 5 sHOLE_angC 1 [POSldia 0.3_MMCIA_MMCIBldtmC] support hole C'; 1 group MMC sC is level 3 sHOLE_sizeC, bonus_prob8 'MMC on top hole C ; 1 group MMC s34 is level 3 floatvar1, bonus_prob9 'MMC on spt pt 3,4 ; group MMC s56 is level 3 floatvar2, bonus_prob10 'MMC on spt pt 5,6 1;*1 III variation for pin sizes l*pinA_size = DOE(k,ll) normal = 7.9375 +- 0.035, Illdia 7.9375 +- 0.035 pin A size*1 l*pinB size = DOE(k,12) normal = 7.9375 +- 0.035, Illdia 7.9375 +- 0.035 pin B size*/ pinC size normal = 7.9375 +- 0.035, Illdia 7.9375 +- 0.035 pin C size //\ variation for t_ and s asrnhata.asm probuniform1 uniform 0 +- 1, probuniform2 uniform = 0 +- 1; 211 I*group b_s_asm is level 3 bslotwidthl, sthicknessl, probuniform2 'clearance bwn spt and bas e ". group b_t_asm is level bslotwidth2, tthickness, probuniforml 'clearance bwn base and top';*1 III .tol variables value bonus tol; value MMCi Illassembly variables value clr asm; III clearance (w/prob) between top or support bar and baseslot. gravity effect of pins on top or support bar side holes value dist_mes2; value dist; value i; point topfloat, sptfloat; point transstop, rotstop; point floatl; floatl = 512.5,0,0; I/Ifloat point for support bar point float2, float3; //Ifloat point for base slots point tarl, tar2, tar3; tar1 0,0,0; tar2 1,0,0; tar3 0,250,0; value k; array DOE(5,12); k = 1; repeat for k = 1 to 5 by 1{ if (k == 1) { DOE(I,I) -1; DOE(1,2) 1; DOE (1, 3) -1; DOE(I,4) 1 ; DOE(1,5) -1; DOE (1,6) 1; DOE(1,7) -1; DOE (1, 8) 1; DOE (1, 9) -1; DOE(I,10) 1; DOE(I,II) -1; DOE(I,12) 1 ; } if (k == 2) { DOE(2,I) 1; DOE(2,2) -1; DOE(2,3) 1 ; DOE(2,4) -1; DOE(2,5) 1; DOE (2,6) -1; DOE(2,7) 1; DOE(2,8) -1; DOE(2,9) 1; DOE(2,10) -1; DOE (2,11) 1 ; DOE(2,12) -1; } if (k == 3) { DOE(3,1) -0.5; DOE(3,2) 0.5; DOE(3,3) -0.5; DOE(3,4) 0.5; 212 DOE(3,5) -0.5; DOE(3,6) 0.5; DOE(3,7) -0.5; DOE(3,8) 0.5; DOE(3,9) -0.5; DOE(3,10) 0.5; DOE(3,11) -0.5; DOE(3,12) 0.5; } if (k == 4) { DOE(4,1) 0.5; DOE(4,2) -0.5; DOE(4,3) 0.5; DOE{4,4) -0.5; DOE(4,5) 0.5; DOE (4, 6) -0.5; DOE{4,7) 0.5; DOE(4,8) -0.5; DOE(4,9) 0.5; DOE(4,10) -0.5; DOE{4,11) 0.5; DOE(4,12) -0.5; } if (k == 5) { DOE(5,1) 0; DOE(5,2) 0; DOE{5,3) 0; DOE(5,4) 0; DOE(5,5) 0; DOE(5,6) 0; DOE(5,7) 0; DOE(5,8) 0; DOE(5,9) 0; DOE(5,10) 0; DOE(5,11) 0; DOE(5,12) 0; } } print DOE; object base(8); context base; value HOLE_radD; HOLE_radD = 0.15; #01 0,0,0; Illbottom of datum B (DRF) #02 0,0,30; Illtop of the datum B hole #03 -250,0,0; Illbottom of datum C #04 -250,0,30; III top of the datum C hole #05 80,0-(bslotwidthl/2),lS; Illpoint inside the slot #06 80,0+(bslotwidthl/2),15; Illpoint inside the slot #07 #06.x-4l0,0-(bslotwidth2/2) ,15; Illpoint inside the slot #08 #07.x,0+(bslotwidth2/2) ,15; Illpoint inside the slot bonus_tal = HOLE_sizeC/2 - (9.938/2); MMC = bonus_tal * bonus_probl; #03.x -250 + HOLE radC + MMC; Illvariation along x axis for datum C hole #04.x = -250 + HOLE radC + MMC; Illvariation along x axis for datum C hole if (DOE (k, 2) ==1) t bonus tal = HOLE_sizeD/2 - ((8.2-0.062)/2); base.05 = point varied from point base.05 by DOE(k,l)*(HOLE_radD+bonus_tol) radially at angle 90 213 in standard csystem zx; baseo06 point varied from point base.06 by DOE(k,l) * (HOLE_radD+bonus tal) radially at angle 90 in standard csystem zx; } if ((DOE(k,2)==0.5) & (DOE(k,2)==-0.S)) { bonus tal = HOLE sizeD/2 - ((8.2-0.062)/2); base.OS = point ;aried from point base.OS by DOE(k,l)*(HOLE_radD+bonus tal) radially at angle 4S in standard csystern zx; base.06 point varied from point base.06 by DOE(k,l)*(HOLE_radD+bonus tal) radially at angle 45 in standard csystern zx; } if (DOE(k,2)==0) { bonus tal = HOLE_sizeD/2 - ((8.2-0.062)/2); base.05 = point varied from point base.OS by 0 radially at angle 0 in standard csystern zx; base.06 point varied from point base.06 by 0 radially at angle 0 in standard csystem zx; } if (DOE(k,2)==-1) { bonus tal = HOLE sizeD/2 - ((8.2-0.062)/2); base.05 = point ~aried from point base.OS by DOE(k,l)*(HOLE_radD+bonus tal) radially at angle 0 in standard csystem zx; base.06 point varied from point base.06 by DOE(k,l)*(HOLE_radD+bonus tal) radially at angle 0 in standard csystem zx; bonus_tal = 14.87 - (-#05.y + #06.y); MMC = bonus_tal * bonus_prob3; baseo05.y base.05.y + float2var + MMC; Illpoint #05 with TP and MMC base.06.y = base.06.y + float2var + MMC; //Ipoint #06 with TP and MMC if (DOE(k f 3 ) == 1 ) { bonus_tal = HOLE_sizeE/2 - ((8.2-0.062)/2); MMC = bonus_tol*bonus_prob4; base.07 = point varied from point base.07 by HOLE_radE + MMC radially at angle 90 in standard csystern zx; base.08 point varied from point base.08 by HOLE_radE + MMC radially at angle 90 in standard csystem zx; 214 if ((DOE(k,3)==-0.5) & (DOE(k,3)==0.5)) { bonus tol = HOLE sizeE/2 - ((8.2-0.062)/2); MMC =-bonus tol*bonus prob4; base.07 = p~int varied from point base.07 by HOLE_radE + MMC radially at angle 4S in standard csystem zx; base.08 point varied from point base.08 by HOLE_radE + MMC radially at angle 45 in standard csystem zx; } if (DOE(k,3)==O) { bonus_tal = HOLE_sizeE/2 - ((8.2-0.062)/2); MMC = bonus tol*bonus prob4; base.07 = point varied from point base.07 by 0 radially at angle 0 in standard csystem zx; ba _.~ . 08 point varied from point base.OB by 0 radially at angle ° in standard csystem zX; } if (DOE(k,3)==-1) { bonus_tal = HOLE_sizeE/2 - ((8.2-0.062)/2); MMC = bonus tol*bonus prob4; base.07 = point varied from point base.07 by HOLE_radE + MMC radially at angle ° in standard csystem zx; base.08 point varied from point base.OB by HOLE_radE + MMC radially at angle ° in standard csystem zx; bonus tal = 14.87 - (-#07.y + #08.y); Illbonus tol = Actual size(radius) - MMC condition MMC = bonus tal * bonus_probS; base.07.y base.07.y + float3var + MMC; Illpoint #07 with TP and MMC base.08.y base.08.y + float3var + MMC; Illpoint #08 with TP and MMC object top(6); context top; value tHOLE sizeB, tHOLE radC, tHOLE radmes; tHOLE_sizeB-= 8.2 + (DOE(k,4)*0.062); tHOLE radC = 0.15; tHOLE radrnes = 0.15; #01 0,0,-tthickness/2; Illbottom of the datum B hole #02 0,0,tthickness/2; III top of the datum B hole #03 #Ol.x+307.5,0,-tthickness/2; Illbottom of the datum C hole #04 #02.x+307.S,0,tthickness/2; III top of the datum C hole #OS #01.x+424,O,-tthickness/2; Illbottom of the measurement hole #06 #02.x+424,0,tthickness/2; Illtop of the measurement hole if (DOE(k, 7) ==1) { bonus tal tHOLE sizeC/2 - (8.2-0.062)/2; 215 top.03.x top.OJ.x + (DOE(k,5)*(tHOLE_radC + bonus_tol)); top.04.x top.04.x t (DOE(k,5)*(tHOLE_radC + bonus tol)); } if ((DOE(k,7)==O.5) & (DOE(k,7)==-0.5)) { bonus_tol = tHOLE_sizeC/2 - (8.2-0.062)/2; value temp; temp = DOE(k,S)*(tHOLE_radC + bonus_toll; temp = (temp*temp)/2; top.03.x top.03.x + sqrt(temp); top.04.x top.04.x + sqrt(temp); top.03.y top.03.y + sqrt(temp); top.04.y top.04.y + sqrt(temp); } if (DOE(k,7)==0) { bonus_tal = tHOLE_sizeC/2 - (8.2-0.062)/2; top.03.x top.03.x; top.04.x = top.04.x; } if (DOE(k,7)==-1) { bonus_tal = tHOLE sizeC/2 - (8.2-0.062)/2; top.03.y top.03.y + (DOE(k,S)*(tHOLE_radC + bonus_tol)); top.04.y = top.04.y + (DOE(k,S)*(tHOLE_radC + bonus toll); } if (DOE(k,8)==1) { bonus_tal = tHOLE sizeM/2 - (8.2-0.062)/2; top.OS = point varied from point top.05 by DOE(k,6) * (tHOLE_radrnes + bonus tal) radially at 90 in standard csystem xy; top.06 point varied from point top.06 by DOE(k,6)*(tHOLE_radrnes + bonus toll radially at 90 in standard csystem xy; } if ((DOE(k,8)==O.S) & (DOE(k,8)==-O.5)) { bonus_tal = tHOLE_sizeM/2 - (8.2-0.062)/2; top.OS = point varied from point top.OS by DOE(k,6)*(tHOLE_radrnes + bonus tal) radially at 45 in standard csystem xy; top.06 point varied from point top.06 by DOECk,6)*(tHOLE_radrnes + bonus tal) radially at 45 in standard csystem xy; } if (DOE(k,8)==0) { bonus_tol = tHOLE sizeM/2 - (8.2-0.062)/2; top.OS = point varied from point top.OS by 0 radially at 0 in standard csystem xy; top.06 point varied from point top.06 by 0 radially at 0 in standard csystem xy; } if (DOE(k,8)==-1) { 216 bonus tal = tHOLE sizeM/2 - (8.2-0.062)/2; top.OS = point va~ied from point top.05 by DOE(k,6)*(tHOLE_radrnes + bonus tol) radially at 0 in standard csystem xy; top.06 point varied from point top.06 by DOE(k,6)*(tHOLE_radrnes + bonus tal) radially at 0 in standard csystem xy; object support(6); context support; value sHOLE_sizeC, sHOLE_radC; sHOLE_sizeC = 8.2 + (DOE(k,9)*O.062); sHOLE radC = 0.15; #01 O,O,-sthicknessl/2; //Ibottom of the datum B hole #02 O,O,sthickness1/2; Illtop of the datum B hole #03 512.5,0,float1.z sthickness2/2; I/Ipoint #03 wrt support part size #04 512.5,O,float1.z + sthickness2/2; //Ipoint #04 wrt support part size #05 512.5,0,float1.z slotwidth/2; I/Ipoint #05 wrt slot size #06 512.5,O,float1.z + slotwidth/2; Illpoint #06 wrt slot size bonus tol = sHOLE sizeC/2 - (8.2-0.062)/2; Illbonus tolerance for dtmC hole support.03.x support.03.x + DOE(k,10)*(sHOLE- radC+bonus- tal); /Ilbonus tal on C wrt x-axis support.04.x support.04.x + DOE(k,10)*(sHOLE- radC+bonus tal) ; /Ilbonus tol on C wrt x-axis support.05.x support.05.x DOE(k,10)*(sHOLE radC+bonus tal) ; + - - //Ibonus tol on C wrt x-axis support.06.x support.06.x + DOE(k,10)*(sHOLE- radC+bonus tol) ; /Ilbonus tol on C wrt x-axis bonus_tol = 35.15 - (-#03.z + #04.z); /Ilbonus tol for support part MMC = bonus_tol * bonus_prob9; support.03.z support.03.z + floatvar1 + MMC; Illpoint #03 with TP and MMC support.04.z = support.04.z + floatvarl + MMC; Illpoint #04 with TP and MMC bonus_tol = (-#05.z + #06.z) - 12.87; Illbonus tol for slot width MMC = bonus_tol * bonus_prob10; support.05.z support.05.z + floatvar2 + MMC; Illpoint #05 with TP and MMC support.06.z = support.06.z t floatvar2 + MMC; /Ilpoint #06 with TP and MMC 217 FeeD validation result dist_mes2 (FCCD validation) -> 50000 samples Sample Est. of (Variancey'0.5 900/0 Conf. Interval Statistics Pop. in Est. of Pop. in Est. of Pop. Paramo Nominal: 0.0000 Mean: 0.0338 0.0338 0.0001 0.0336 0.0339 Std Dev : 0.0223 0.0223 0.0001 0.0222 0.0224 Cp: N/A N/A N/A N/A N/A Cpk: N/A N/A N/A N/A N/A Skew: 1.1347 LDL/UDL : N/A / N/A Kurt : 1.5474 Distribution: Pearson-Beta Sample Est. Sample ** Est. %