Characterization of Assembly. Variation Analysis Methods. A Thesis. Presented to the. Department of Mechanical Engineering. Brigham Young University

Transcription

1 Characterzaton of Assembly Varaton Analyss Methods A Thess Presented to the Department of Mechancal Engneerng Brgham Young Unversty In Partal Fulfllment of the Requrements for the Degree Master of Scence Robert Cvetko 997 By Robert Cvetko December 997

2 Ths thess by Robert Cvetko s accepted n ts present form by the Department of Mechancal Engneerng of Brgham Young Unversty as satsfyng the thess requrement for the degree of Master of Scence. Kenneth W. Chase, Commttee Char Carl D. Sorensen, Commttee Member Spencer P. Magleby, Commttee Member Date Crag Smth, Graduate Coordnator

3 Characterzaton of Assembly Varaton Analyss Methods Robert Cvetko Department of Mechancal Engneerng M.S. Degree, December 997 ABSTRACT Ths thess s a study and comparson of current assembly varaton analyss methods. New metrcs are presented for gaugng the accuracy of, not only the analyss methods, but also the problem nformaton wthout knowng the exact answer. A hybrd method usng Monte Carlo wth the method of system moments s also presented and evaluated. Matchng the accuracy of the problem and the analyss method allows for smple models and an estmate of confdence for the results. A framework for choosng the best method for dfferent analyss stuatons summarzes the fndngs. COMITTEE APPROVAL: Kenneth Chase, Commttee Char Carl D. Sorensen, Commttee Member Spencer P. Magleby, Commttee Member Date Crag Smth, Graduate Coordnator

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5 Table of Contents Lst of Fgures v Lst of Tables x Chapter. Introducton Secton.. Statement of the Problem Secton.. Analyss and Revew of Prevous Work n the Feld Secton.3. Contrbutons to be Made by ths Thess 3 Secton.4. Delmtatons of the problem 3 Secton.5. Defnton of General Terms and symbols 4 Chapter. Settng Up an Assembly Varaton Analyss Problem 5 Secton.. Defnng the Varaton Analyss Objectve 5 Secton.. Defnng an Assembly Functon 6 Secton.3. One-way Clutch Model 7 Secton.4. Vector Loop Model and Assembly Functon for the Clutch 0 Secton.5. Summary for the Varaton Analyss Problem Chapter 3. Monte Carlo Smulaton 3 Secton 3.. Background on Monte Carlo 3 Secton 3.. Random Number Generaton 5 Secton 3.3. Explct and Implct Assembly Functons 5 Secton 3.4. Methods to Calculate Senstvtes 6 Secton 3.5. The State-of-the-art n Estmatng the Effectveness of Monte Carlo 8 Secton 3.6. Error n Estmatng Moments wth Monte Carlo 0 Secton 3.7. Advantages and Drawbacks of Monte Carlo Smulaton

6 Secton 3.8. Sample Clutch problem Secton 3.9. Confdence Interval for PPM Rejects Estmated 5 Secton 3.0. Summary for Monte Carlo Smulaton 6 Chapter 4. Method of System Moments 7 Secton 4.. Background of Method of System Moments 7 Secton 4.. Second-order MSM Analyss 8 Secton 4.3. Lnear Varaton Analyss 9 Secton 4.4. Calculatng the Senstvtes for the Clutch Problem 9 Secton 4.5. Sample Clutch Problem Usng Method of System Moments 3 Secton 4.6. State of Art for Estmatng the Accuracy of MSM 3 Secton 4.7. Summary for the Method of System Moments 33 Chapter 5. RSS Frst-order Approxmaton 34 Secton 5.. Background of RSS Frst-order Approxmaton 34 Secton 5.. Frst-order versus Second-order 35 Secton 5.3. Lnear Explctzaton 35 Secton 5.4. Frst Order Varaton Analyss 37 Secton 5.5. Sample Clutch Problem usng Lnear Explctzaton and RSS 38 Secton 5.6. Summary of RSS Method 4 Chapter 6. Qualty Loss and Desgn of Experments 4 Secton 6.. The Taguch Qualty Loss Functon 43 Secton 6.. Desgn of Experments n General 47 Secton 6.3. Desgn of Experments n Varaton Analyss 47 Secton 6.4. Two-level Desgns 48 Secton 6.5. Three-level Desgns 54 v

7 Secton 6.6. Three-Level Weghted Desgn 58 Secton 6.7. Sample Clutch Problem DOE Results 6 Secton 6.8. Response Surfaces 63 Secton 6.9. Summary for DOE and Response Surfaces 66 Chapter 7. Lambda Dstrbuton 68 Secton 7.. Introducton to the Lambda Dstrbuton 68 Secton 7.. Fttng the Lambda Dstrbuton 7 Secton 7.3. Summary for the Lambda Dstrbuton 8 Chapter 8. Determnng the Effect of Errors n Moment Estmates 83 Secton 8.. Defnng Standard Moment Errors 84 Secton 8.. Defnng the One-Sgma Error Bounds 85 Secton 8.3. Estmatng the Effect of Input Varable Moment Uncertanty 87 Secton 8.4. Explctly Determnng the Senstvtes for Qualty Loss 9 Secton 8.5. Two-Level DOE to Determne the Effect of Errors n the Moments 93 Secton 8.6. Summary of Senstvty Results 00 Chapter 9. Estmatng the Effectveness of Monte Carlo Smulaton 04 Secton 9.. Graphcally Representng Error versus Sample Sze 04 Secton 9.. Error n Estmatng Moments wth Monte Carlo 06 Secton 9.3. The Error Inherent n Usng the Lambda Dstrbuton Secton 9.4. Total Error n Estmatng PPM Rejects wth the Moments and Lambda Dstrbuton 3 Secton 9.5. Comparson of Countng Rejects wth Usng the Lambda Dstrbuton and the Moments 7 Secton 9.6. Senstvtes and Percent Contrbuton from Monte Carlo 9 v

8 Secton 9.7. Qualty Loss estmated by Monte Carlo 0 Secton 9.8. Strengths/Weaknesses Summary for Monte Carlo Chapter 0. Estmatng the Effectveness of the MSM and RSS 3 Secton 0.. The Method of System Moments 3 Secton 0.. RSS Frst-order Approxmaton 30 Chapter. Estmatng the Effectveness of the Desgn of Experments and Response Surface Methods 40 Secton.. Performng Physcal Experments versus Computer Smulaton 40 Secton.. Accuracy versus Method and Mxng Desgns 4 Secton.3. Effcency of Response Surface Desgns to Estmate Senstvtes 45 Secton.4. Summary for Estmatng the Effectveness of DOE and Response Surface Methods 49 Chapter. Random Response Surface Hybrd Method 50 Secton.. Optmal Senstvtes for the Method of System Moments 50 Secton.. Random Response Surfaces for MSM Senstvtes 56 Secton.3. Estmatng the Error of the RRS Method 6 Secton.4. Summary of the Random Response Surface Method 73 Chapter 3. Matchng the Error n the Problem wth the Analyss Method 75 Secton 3.. Accuracy of Input Informaton versus PPM Rejects 77 Secton 3.. Accuracy of Input Informaton versus the Qualty Loss Functon 8 Secton 3.3. Matchng the Errors of the Analyss Methods 83 Chapter 4. Choosng between the Analyss Methods 85 Secton 4.. Drect Comparson of the Methods 85 Secton 4.. Table of Methods 90 v

9 Secton 4.3. Examples of Method Selecton over the Product Lfe Cycle 9 Secton 4.4. Summary for Choosng Between the Analyss Methods 95 Chapter 5. Concluson and Recommendatons for Further Research 96 Secton 5.. New Metrcs: Standard Moment Error and Quadratc Rato 97 Secton 5.. New Hybrd Method: Random Response Surface (RRS) 98 Secton 5.3. Framework for Choosng Accuracy and Method 99 Secton 5.4. Recommendatons for Further Research 99 Sources 0 v

10 Lst of Fgures Fgure -: Drawng of the One-way Clutch Assembly 7 Fgure -: Important Dmensons for Clutch 8 Fgure -3: Vector Loop for the Clutch Assembly 0 Fgure 3-: Graphcal Representaton of Monte Carlo Method 4 Fgure 3-: Hstogram of %Error for Lower Rejects for,000 Dfferent Runs (Smulatons of Monte Carlo), each wth 0,000 Samples 9 Fgure 3-3: Monte Carlo Output from Crystal Ball Fgure 3-4: Percent Change n Moments for Monte Carlo Smulaton as the Sample Sze Doubles 4 Fgure 6-: Comparson between Quadratc and Step Loss Functons 43 Fgure 6-: Combnng the Loss Functon wth a Probablty Dstrbuton 45 Fgure 6-3: Two-level Expermental Desgns for Three Factors 49 Fgure 6-4: Three-level Expermental Desgns for Three Factors 55 Fgure 6-5: Weghtngs for the Alternate Three-Level Desgn 58 Fgure 6-6: Popular Response Surface Desgns 63 Fgure 6-7: Response Surface for φ versus a and c 65 Fgure 7-: Comparson of Lambda to other Dstrbutons 70 Fgure 7-: Error n Modelng a Normal dstrbuton (left tal) wth the Lambda 7 Fgure 7-3: Functon to Mnmze 76 Fgure 7-4: Local Mnmums for Functon 77 Fgure 7-5: Skewness vs. λ 3 and λ 4 78 Fgure 7-6: Kurtoss vs. λ 3 and λ 4 79 Fgure 7-7: Curves for Skewness.0 and Kurtoss Fgure 7-8: Value of Kurtoss whle Followng the Skewness.0 Curve 8 Fgure 8-: Estmatng the Mean of the Standard Normal Dstrbuton versus Sample Sze 87 Fgure 8-: One-Sgma Error Bound for SER 8 at,000 Samples 89 Fgure 8-3: Senstvty Percent of PPM Rejects to Standard Moment Errors 0 Fgure 8-4: Percent Contrbuton of SER to Total Rejects versus Sgma 0 Fgure 8-5: Percent Contrbuton of SER to Upper/Lower Rejects versus Sgma 0 Fgure 9-: Percent Error for Countng Lower Rejects wth Monte Carlo Showng One-Sgma Bounds for the Error 05 Fgure 9-: Percent Error for Countng Rejects wth Monte Carlo versus Sample Sze 06 Fgure 9-3: Hstogram of SER4 for,000 Cycles of Monte Carlo, each wth 0,000 Samples 08 v

11 Fgure 9-4: Error n Moments versus Sample Sze and One-Sgma Bound 0 Fgure 9-5: The One-Sgma Bound on SER-4 versus Sample Sze Fgure 9-6: Percent Error n PPM Rejects for Fttng a Lambda Dstrbuton to a Normal and Exponental Dstrbuton Fgure 9-7: Makeup of Error for Estmatng PPM Rejects wth the Lambda Dstrbuton Versus Sample Sze 4 Fgure 9-8: Actual Versus Predcted Error for Usng Moments and the Lambda Dstrbuton to fnd Lower PPM Rejects for the Clutch 6 Fgure 9-9: Actual Versus Predcted Error for Lower PPM Rejects Usng Monte Carlo Smulaton 7 Fgure 9-0: Predcted Error n Lower PPM Rejects at a 4-Sgma Qualty Level Usng Monte Carlo Smulaton 8 Fgure 9-: Percent Error Estmatng Qualty Loss wth Monte Carlo Smulaton Fgure 0-: Comparng the Accuracy of MSM usng a Response Surface to fnd the Senstvtes 6 Fgure 0-: Estmatng SER-4 Usng the Monte Carlo Dfference Method 9 Fgure 0-3: Lnearzaton Error from the Hub Radus (determned by the QR) versus the Standard Devaton 38 Fgure -: Method Effcency for Calculatng Senstvtes 48 Fgure -: Conventonal Methods to Lnearze a Functon 5 Fgure -: Conventonal Methods to Approxmate a Functon by a Quadratc 53 Fgure -3: Lnear and Quadratc Random Response Surfaces 55 Fgure -4: Random Response Surface versus Sample Sze usng Least-Squares Ft 59 Fgure -5: Random Response Surface versus Sample Sze usng Least-Fourths Ft 60 Fgure -6: SER for the RRS Method: Actual Error versus MCD Estmate 63 Fgure -7: SER for the RRS Method: Actual Error versus MCD Estmate 64 Fgure -8: SER3 for the RRS Method: Actual Error versus MCD Estmate 66 Fgure -9: SER4 for the RRS Method: Actual Error versus MCD Estmate 67 Fgure -0: Comparng SER for RSS and Monte Carlo 69 Fgure -: Comparng SER for Adjusted RSS and Monte Carlo 70 Fgure -: Comparng SER3 for Adjusted RSS and Monte Carlo 7 Fgure -3: Comparng SER4 for Adjusted RSS and Monte Carlo 7 Fgure 3-: General Flowchart for Evaluatng the Error n the Problem Informaton 76 x

12 Lst of Tables Table -: Contact Angle Specfcatons 9 Table -: Independent Dmensons for the Clutch 9 Table 3-: Dstrbuton for %Error Countng Rejects for 0,000 Samples 0 Table 3-: Senstvty Informaton from Crystal Ball 3 Table 3-3: Benchmark Results from Monte Carlo Smulaton at One Bllon Samples 5 Table 4-: Analyss Results from the Second Order Method of System Moments 3 Table 5-: Calculaton of the Varance of the Contact Angle 39 Table 5-: Percent Contrbuton from RSS Lnear and Monte Carlo 40 Table 5-3: Analyss Results from RSS Lnear 40 Table 6-: The Requred Number of Runs for Typcal Two-Level Desgns 50 Table 6-: Two-level Expermental Desgn for the Clutch 5 Table 6-3: Two-level Expermental Results for the Clutch 5 Table 6-4: Calculatons of the Varatons for the Two-level Desgn 53 Table 6-5: Summary Table for Two-level Clutch Experment 54 Table 6-6: Number of Runs for Common Three-Level Orthogonal Desgns 55 Table 6-7: Three-level Expermental Desgn for the Clutch Assembly 56 Table 6-8: Calculatons for the Orthogonal Three-level Desgn 57 Table 6-9: Summary Table for the Three-level Orthogonal Desgn 57 Table 6-0: Frst Twelve (of Twenty-Seven) Runs for the Weghted Three-Level Desgn 59 Table 6-: Comparson of Weghted DOE wth Monte Carlo 60 Table 6-: Summary of Moment Results from DOE 6 Table 6-3: Summary of Percent Contrbuton Results from DOE 6 Table 7-: Standard Normal vs. Lambda 7 Table 8-: Senstvty of Output Moments to Input Moments 90 Table 8-: Expected Error n Moments from Errors of an Input Dstrbuton Estmated from only,000 Samples 9 Table 8-3: Determnng Levels for the Moments 95 Table 8-4: Two-Level Desgn for Moments Senstvty Estmaton 96 Table 8-5: Response of Runs for Total PPM Rejects 98 Table 8-6: Summary Table for Calculatng Senstvty of Total Rejects 99 Table 8-7: Summary Results for the Locaton of the Mode and Qualty Loss 03 Table 9-: Probablty Dstrbutons for Estmatng SER SER4 at 0,000 Samples 09 x

13 Table 0-: Comparson of Moment Errors due to Senstvtes Obtanes by Partal Dervatves and Response Surfaces 7 Table 0-: Requred Sample Sze to for a One-Sgma Confdence Level on the Error of MSM (Samplng Rato 0.) 8 Table 0-3: Total PPM Rejects for a Normal Dstrbuton versus the Sgma Qualty Level 3 Table 0-4: Calculatons for the Quadratc Ratos of the Varables for the Clutch Assembly 36 Table 0-5: Estmates for the Lnearzaton Error from Each of the Three Input Varables of the Clutch Assembly 36 Table 0-6: Summary for Estmates of Lnearzaton Error for the Clutch Assembly 37 Table -: Standard Moment Errors versus DOE Method for the Clutch Assembly 4 Table -: Equvalent Monte Carlo Sample Sze for Accuracy of DOE Methods on the Clutch Assembly 4 Table -3: Setup and Response for Mxed Desgn 44 Table -4: Relatve Error Comparson for Mxed Desgn 45 Table -5: Method to Fnd Senstvtes versus the Number of Runs 47 Table 4-: Recommended Methods for Balancng Input Informaton and Analyss Accuracy 9 x

14 Chapter. Introducton Varaton analyss, or tolerance analyss, s a valuable tool to desgn products for hgh qualty and low cost. Varaton analyss s used to predct the varaton n assembles due to the process varatons of component dmensons and other features. There are several methods avalable to perform assembly varaton analyss. The man methods are lsted below. Monte Carlo Smulaton: Approxmate the output dstrbuton by samplng the assembly functon many tmes (thousands or more dependng on desred certanty level) wth random nput values. Method of System Moments: Approxmate the frst four output dstrbuton moments by combnng the nput varable moments and the senstvtes (dervatves) of the assembly functon. RSS Frst Order Approxmaton: Approxmate the assembly varaton as the sumsquared product of each nput varable standard devaton and ts lnear senstvty. Desgn of Experments: Approxmate the total assembly varaton and the contrbuton from each nput varable by samplng the assembly functon at specfc combnatons of the nput varables. Secton.. Statement of the Problem Although there are several methods avalable for varaton analyss, many engneers do not understand the strengths and weaknesses of each. The accuracy of an analyss s seldom estmated. Even consderng the accuracy of the nput nformaton s not generally done. It s common for an engneer to learn and embrace only one of the methods, usng t n all varaton analyss stuatons.

15 The dfferent methods are based on dfferent assumptons, do not always yeld the same results, and offer dfferent knds of nformaton. Because dfferent knds of nformaton are needed at dfferent stages of product development and producton, all of the varaton analyss methods should be learned, and used where approprate. The methods can even be used together to ncrease effcency. Therefore the objectves of ths thess are: Compare the exstng varaton analyss methods to determne ther strengths and weaknesses. Identfy ways to estmate the accuracy of the methods wthout havng to perform a full Monte Carlo smulaton to benchmark. Present new hybrd methods to ncrease effcency. Present a framework for selectng the optmum analyss method. Secton.. Analyss and Revew of Prevous Work n the Feld The frst part of ths thess (Chapter through Chapter 6) demonstrates the state-of-theart n varaton analyss, and bulds a foundaton for the contrbutons. These chapters focus on the followng: Explanng the background for several varaton analyss methods Presentng current technques to estmate the accuracy of the methods Demonstratng the methods on a sample problem to allow for comparson of accuracy Because several varaton analyss methods are beng evaluated, the complete state-of-theart for each method can not be presented. However, the state-of-the-art n varaton analyss relatve to the contrbutons of ths thess wll be presented.

16 3 Secton.3. Contrbutons to be Made by ths Thess The rest of the chapters, Chapter 8 through Chapter 4, focus on the contrbutons made by ths thess. The man contrbutons are: Presentng and demonstratng new metrcs for estmatng the accuracy of the dfferent varaton analyss methods (Chapter 8 through Chapter ) Presentng and evaluatng a hybrd method that combnes the strengths of several dfferent analyss methods (Chapter 9 through Chapter ) Presentng a framework for choosng the level of accuracy, and the method, for varaton analyss (Chapter 3 and Chapter 4) The value of these contrbutons s that wth them an engneer can compare analyss needs to avalable analyss methods and select the optmum method, or even use the methods together as a hybrd. The accuracy of the analyss can be matched to the accuracy of the nput nformaton, allowng for effcency, model smplcty, and an understandng of the confdence of the results. One of the problems stated earler n ths thess s that many engneers do not understand the varous assembly varaton analyss methods but use only one n all crcumstances. Thus, even the frst several chapters of ths thess may help to correct that problem by demonstratng each of the methods all n one place. The value that these contrbutons make s helpng the varaton analyss process become more effcent and useful n many dfferent stuatons. Secton.4. Delmtatons of the problem Ths thess wll not be an exhaustve study of any one of the assembly varaton analyss methods, as all of the methods are covered and compared. The test problem, presented n Chapter, was chosen because t s known to be qute non-lnear and non-quadratc.

17 4 Addtonally, there wll be no expermental verfcaton of the results of ths thess. The results wll be compared to the Monte Carlo soluton for a very large sample of smulated assembles. Secton.5. Defnton of General Terms and symbols The followng terms are presented here for reference. They wll be referred to throughout the thess. µ st moment (or mean) of a statstcal dstrbuton n x n µ nd moment (or varance) of a statstcal dstrbuton n σ ( x µ ) n µ 3 rd moment (or skewness) of a statstcal dstrbuton n 3 3 ( x µ ) n µ 4 th moment (or kurtoss) of a statstcal dstrbuton n 4 4 ( x µ ) n α Standardzed skewness 3 3 µ 3 σ α Standardzed kurtoss 4 4 µ 4 σ

18 5 Chapter. Settng Up an Assembly Varaton Analyss Problem Ths chapter defnes what an assembly problem s, and the value of varaton analyss n brngng together the engneerng and manufacturng requrements. The engneerng requrements are the lmts on ft and functon crtcal to assembly performance. These assembly specfcatons are appled to the dependent dmensons (or dependent varables) n the assembly. The values and varatons of the part dmensons (or ndependent varables) determne the value and varaton of the dependent varables. The manufacturng requrements are the lmts to sze, shape, and locaton of part features. These requrements are normally appled to the assembly components n terms of tolerances. The varatons n the manufacturng processes to make the parts cause the varatons of the ndependent varables, and thus ndrectly cause the varatons n the dependent varables of the assembly. The dffculty les n tyng together the process varatons and the assembly varatons, n hopes of selectng the best processes to meet the engneerng requrements most effcently. Assembly varaton analyss s the tool to brng together the engneerng requrements and the manufacturng capabltes, allowng the manufacturng processes and requrements to be chosen. Before the analyss can be performed, the objectve for the analyss must be determned and the assembly functon defned. Secton.. Defnng the Varaton Analyss Objectve Defnng the varaton analyss objectve s an mportant step n analyzng the varaton n an assembly. The analyss objectve defnes the goals or reasons for performng the varaton analyss. The analyss objectve not only defnes the characterstc of the assembly that s crtcal for performance, but t should also specfy the desred output nformaton. Once the analyss objectve s clear, the varables that affect the objectve

19 6 should be dentfed and ther varaton defned. The lst below llustrates dfferent types of possble output nformaton from a varaton analyss. Estmatng the qualty levels expected n producton (upper, lower, or total rejects) Determnng the manufacturng requrements and processes for the parts n the assembly to meet the engneerng requrements Determnng the senstvty of the objectve functon to the ndependent varables to compare dfferent desgns or parameter values (robust desgn) Determnng the percent contrbuton of each of the nput varatons to the assembly varaton, to help focus mprovement efforts Weghng the savngs n qualty loss aganst the cost of nvestments or changes Once the problem objectve s understood, the assembly functon can be determned that wll help reach the analyss objectve. Secton.. Defnng an Assembly Functon In order to analyze assembly varatons, a relatonshp between the ndependent and dependent varables must be found. Ths relatonshp can be found through expermentaton, an explct functon, or a set of mplct equatons. Expermentaton means physcally buldng the parts of an assembly and measurng the desred assembly characterstcs. Expermentaton s a valuable tool when relatonshps are unknown, lke the effect of envronmental nose. If the assembly functon s smple enough, the objectve functon can be expressed as an explct equaton n whch the assembly objectve (dependent varable) can be determned from the ndependent varables. More complcated assembly functons are not easly reducble to an explct form. In these cases the assembly dmensons (dependent and

20 7 ndependent) are all ncorporated nto mplct equatons representng closure n each of the necessary translatonal and rotatonal drectons. Any assembly can be represented as a set of mplct equatons, but an teratve process s requred to solve them. Examples of explct and mplct equatons are presented n the next secton (see Eq. - and Eq. -). For the analyss n ths thess, the assembly functon that wll be used s the one-way clutch. Secton.3. One-way Clutch Model Fgure - below shows a one-way clutch assembly. The assembly conssts of four dfferent parts: one hub, one rng, four rollers, and four sprngs. The sprngs push the rollers out to reman n contact wth both the rng and the hub. If the hub s turned counter-clockwse the rollers bnd, causng the rng to turn wth the hub. Turnng the hub clockwse causes the rollers to slp and prevents the transmsson of torque to the rng. Sprng Roller Hub Rng Fgure -: Drawng of the One-way Clutch Assembly

21 8 The bndng of the rollers causes the transmsson of torque between the hub and rng. The angle of contact between the roller and the rng s very mportant for the one-way clutch to functon properly. The angle of contact and the dmensons that affect t are shown below n Fgure -. Independent Varables Dependent Varables φ a c e b Fgure -: Important Dmensons for Clutch The contact angle (φ) s the dependent varable n the assembly and determnes the performance of the clutch. The three ndependent dmensons affect the value of the contact angle and the varable b (locaton of contact between the roller and the hub). The contact angle for the roller, when all of the ndependent varables are at ther mean values, s degrees (nomnal angle). For proper performance, the upper and lower specfcaton lmts for φ are based upon ths nomnal angle and ± 0.60 degree desgn lmts. Table - below summarzes these values.

22 9 Table -: Contact Angle Specfcatons Contact Angle Value (degrees) Upper Lmt Nomnal Angle Lower Lmt If the contact angle s outsde of the lmts, the clutch does not perform correctly and must be reworked, or scrapped. The cost of ths rework, or the qualty loss, s $0. Ths loss due to poor qualty should be mnmzed. Varaton analyss can help to predct the qualty level of the clutches or, nversely, to determne the varatons for the ndependent dmensons that wll yeld the desred qualty level. Dmensons a, c, and e are ndependent varables, whch affect the contact angle φ. The varaton n the contact angle s a functon of the varatons of the ndependent varables as well as the assembly functon. The objectve of ths varaton analyss problem s to determne the varaton of the contact angle relatve to ts performance. Table - below shows the three ndependent varables, or dmensons, and the standard devaton of each. Each of the ndependent varables s assumed to be a statstcally ndependent (not correlated wth each other) and normally dstrbuted random varable. Table -: Independent Dmensons for the Clutch Varable Mean Standard Devaton a - hub radus mm mm c - roller radus.430 mm mm e - rng radus mm mm The contact angle for the one-way clutch s smple enough to be represented by the explct functon Eq. - (refer to Fgure - for an explanaton of the symbols). a + c φ arccos e c Eq. -

23 0 Most assembles are too complcated to be reduced nto an explct equaton. Thus, a vector loop method can be used to fnd mplct equatons [Chase 995, 67]. Implct equatons can always be found for an assembly usng the followng vector loop method. Secton.4. Vector Loop Model and Assembly Functon for the Clutch The vector loop method uses the assembly drawng as the startng pont. Vectors are drawn from part-to-part n the assembly, passng through the ponts of contact. The vectors represent the ndependent and dependent dmensons (varables) n the assembly. Fgure - below shows the resultng vector loop for the clutch assembly. φ Vector Loop c c a b φ e Fgure -: Vector Loop for the Clutch Assembly

24 The vectors pass through the ponts of contact between the three parts n the assembly. The vector b and the angles φ and φ are dependent varables. In ths case t easy to see that φ φ (the contact angle). Once the vector loop s defned, the mplct equatons for the assembly can easly be extracted. Eq. - shows the set of mplct equatons for the clutch assembly derved from the vectors. h h h x 0 b + c sn( φ) esn( φ) Eq. - y θ 0 a + c + c cos( φ ) ecos( φ ) φ 80 + φ + 90 φ + φ From the mplct equatons t s also easy to see that φ and φ are equal. Therefore, the mplct equatons can be smplfed nto just two equatons for two unknowns as shown below by Eq. -. h h x y 0 b + csn( φ) esn( φ) Eq. - 0 a + c + c cos( φ) ecos( φ) Each of these assembly equatons for the clutch equals zero because the vector loop s closed, and must mantan closure for the parts to assemble. However, f a vector loop s not closed (open loop vector to measure a gap for example) the equatons equal the value of the openng (gap length n the h y drecton for example). In ths case the mplct equaton h y contans only one dependent varable (the contact angle) and can be smplfed down to the explct equaton for the contact angle shown above by Eq. -. But most assembles are too complex to easly derve an explct equaton for the desred assembly functon. Secton.5. Summary for the Varaton Analyss Problem Defnng the problem well conssts of determnng the analyss objectves and sources of varaton and defnng the assembly functon. The one-way clutch assembly wll be

25 analyzed wth all of the analyss methods. The example problem, the one-way clutch assembly, was chosen for several dfferent reasons: The clutch can be represented wth a small number of varables (three ndependent and two dependent varables) for easy demonstraton and comprehenson. The clutch assembly s known to be qute non-lnear, and even shown to be very nonquadratc [Glancy 994, 43 45]. Therefore the output dstrbuton for the contact angle s skewed even though the nputs are normally dstrbuted. The contact angle for the clutch assembly can be represented explctly, and not just mplctly to allow for use wth any of the analyss methods. Now that the sample analyss problem s defned, the dfferent varaton analyss method can be explaned and demonstrated. The frst method demonstrated wll be Monte Carlo as t s wdely used as a benchmark for the accuracy of the other methods. The followng wll be evaluated because they are the ones commonly used for assembly varaton analyss: Chapter 3. Monte Carlo Smulaton Chapter 4. Method of System Moments Chapter 5. RSS Frst-order Approxmaton Chapter 6. Qualty Loss and Desgn of Experments Presentng the state-of-the-art wth these methods, and demonstratng ther use on the clutch sample problem wll provde a foundaton for understandng the value of contrbuton of ths thess.

26 3 Chapter 3. Monte Carlo Smulaton Ths chapter presents the Monte Carlo smulaton method and the state-of-the-art for estmatng ts accuracy. Ths state-of-the-art on Monte Carlo wll be used as the foundaton for the contrbutons made n Chapter 9. Monte Carlo Smulaton s a method of choosng nputs from the probablty dstrbutons of randomly varyng components, calculatng the outputs, and analyzng the output dstrbuton. In order to understand Monte Carlo smulaton, a custom smulaton program has been wrtten and used along wth the Excel add-n program Crystal Ball. Secton 3.. Background on Monte Carlo The Monte Carlo Method for varaton analyss conssts of settng the ndependent varables to random values accordng to ther respectve probablty dstrbutons. Fgure 3- below graphcally shows how the process works. The output dstrbuton s a functon of the dstrbutons of the nput varables and the assembly functon. Thousands of samples of the nput varables are combned to get a relable measure of the output dstrbuton.

27 4 Input Varables Output Dstrbuton Iteratve Assembly Functon Hstogram Mean Std. Dev. Skewness Fgure 3-: Graphcal Representaton of Monte Carlo Method The Monte Carlo method conssts of selectng random values for the ndependent varables, calculatng the assembly functon (an teratve process f the functon s mplct), and analyzng the output. Sometmes hundreds of thousands of samples are requred for accurate results. The hstogram of the outputs for the assembly functon can be plotted. The rejects (assembles that fall outsde the specfcaton lmts) can be counted durng the smulaton, f the lmts are known before the smulaton. Or a dstrbuton can be ft to the moments or percentles of the Monte Carlo output and the rejects estmated (see Chapter 7 for usng the Lambda dstrbuton). The number of samples requred depends on the desred accuracy of the output. Secton 3.5 wll present the state-of-the-art n estmatng the accuracy. Chapter 9 wll contrbute addtonal materal for estmatng the effectveness of Monte Carlo smulaton.

28 5 Secton 3.. Random Number Generaton One mportant aspect of a Monte Carlo smulaton s the generaton of the random numbers (or unform random devate) for the ndependent varables. The frst step s generatng unform random devate. Then they can be mapped onto the probablty densty functons of the ndependent varables, such as the normal dstrbuton. Ths mappng can be done through the equatons for the probablty densty functons f they are known, or by matchng a Lambda dstrbuton to data, dscussed n Chapter 7. To generate the unform random devate, the algorthm ran [Press 99, 8] was chosen because t has a perod greater than x 0 8 random numbers before t repeats. One of the thngs nvestgated n ths report s the requred number of samples for Monte Carlo to gve relable results. Ths random number generaton algorthm takes about twce as long to calculate than the standard random number generatng algorthm, ran0, but ran0 has a perod of only x 0 9. Unfortunately, the more random numbers that are needed, the longer t takes to calculate each random number. The nput varables for the clutch assembly are normally dstrbuted, not unformly dstrbuted. The algorthm gasdev [Press 99, 89] was used to map the unform random devate to the normal dstrbuton. After the random nputs are ready for Monte Carlo, the next step s to use them n the assembly functon. Secton 3.3. Explct and Implct Assembly Functons Once the random numbers are generated and mapped onto the probablty densty functons for each of the ndependent varables, the values for the dependent varables and objectve functons must then be calculated. As hundreds of thousands of samples may be requred when usng Monte Carlo, solvng mplct equatons may requre sgnfcantly more computaton tme.

29 6 Glancy used Newton s method for solvng systems of nonlnear assembly equatons. He found that the clutch assembly problem requred an average of 3.4 teratons to solve the mplct equatons for each Monte Carlo smulaton sample [Glancy 994, 48]. Usng explct equatons wth the Monte Carlo method s extremely advantageous, but often not possble. Secton 3.4. Methods to Calculate Senstvtes Senstvty nformaton s helpful when analyzng an assembly. The senstvtes reveal how the objectve functon changes as each of the ndependent varables change. For the clutch problem, the senstvty of the contact angle wth respect to the hub radus s the partal dervatve φ. The partal dervatves can be determned analytcally or wth a a fnte dfference method. The senstvty nformaton reveals the part features that mght requre more accurate processes. If the senstvty s large, then varaton of the nput varable wll be amplfed to the assembly functon. Percent contrbuton nformaton can easly be estmated from the senstvtes and tells whch varables wth ther varatons do contrbute the most to the overall assembly varance Senstvtes from the Correlaton Coeffcents When usng Monte Carlo smulaton to analyze an assembly, senstvty nformaton can be obtaned through usng correlaton coeffcents and ndvdual samplng. A lnear correlaton coeffcent tell how strongly the output functon changes as an nput changes. The correlaton coeffcents range from.0 to.0. A value of 0.0 ndcates that the output s completely uncorrelated to the nput. A value of.0 ndcates that there s an exact lnear relatonshp between the output and the nput varable, the output ncreasng as the nput varable ncreases. A value of.0 also ndcates an exact lnear relatonshp, but wth the output decreasng as the nput varable ncreases.

30 7 One method of calculatng the correlaton coeffcents s usng the actual values of the nput varables and the output to calculate the Pearson correlaton coeffcent (see Eq. 3-). But sometmes the Pearson correlaton coeffcent can conclude that a correlaton exsts even f one does not because of the dfferences n magntude between the nput and the output [Press 99, 640]. r p ( x x)( y y) Eq. 3- ( x x) ( y y) The Spearman rank-order correlaton coeffcent s a more robust method that uses the rank of the nput varables and the rank of the output functon (see Eq. 3-). Even though the Spearman coeffcents may be more robust (not measurng a correlaton f none exsts), there s a loss of nformaton, as the actual x and y data s converted nto R (the rank of the nput varable) and S (the rank of the output varable). All of the values for the nputs and outputs must be stored durng the smulaton. Then the sets of values are all ranked accordng to magntude. It s the rank of the varables used nstead of the actual values. r s ( R R)( S S) Eq. 3- ( R R) ( S S) The program Crystal Ball, by Decsoneerng Inc., uses the Spearman rank-order correlaton coeffcents to calculate the senstvty nformaton, and wll be used to fnd the senstvtes for the clutch problem. In order to estmate the percent contrbuton, the rank correlaton coeffcents are squared and normalzed to sum to 00%. Ths method s only an estmate, and not a true decomposton of varance as wll be presented wth Desgn of Experments, n Chapter 6.

31 Senstvtes from Indvdual Samplng Another method of calculatng the senstvtes wth the Monte Carlo s ndvdual samplng. Ths method conssts of holdng all of the ndependent varables constant except for the one beng studed. The value of the one varable s changed ether to specfc ponts (fnte dfference dervatve) or to many random ponts (comparng the varance of the output to the varance of the nput). Each of the other ndependent varables s analyzed smlarly. A fnte dfference approach s very smlar n that only one varable s changed at a tme. Secton 3.5. The State-of-the-art n Estmatng the Effectveness of Monte Carlo The error n usng Monte Carlo to count the parts per mllon (PPM) rejects s found by way of the bnomal dstrbuton [Shapro 98, 9]. Expressng the error n PPM rejects as a percent of the rejects s shown below n Eq. 3-. As the sample sze (n) doubles, the error estmatng the rejects decreases by about mllon rejects.. PPM s an estmate of the parts per ( σ ) σ PPM % PPM 6 PPM (0 PPM ) n σ PPM PPM 6 0 PPM PPM ( n ) Eq. 3- Where: σ PPM The one-sgma bound on the error of rejects (where PPM n the equaton s the ntal estmate of the PPM rejects) σ %PPM The one-sgma bound on percent error of PPM rejects The one-sgma confdence lmts calculated by Eq. 3- can be multpled by any value to obtan the confdence level desred for the estmate of accuracy. The fgures presented n ths thess wll use the one-sgma confdence lmts for the errors for easy comparson. To explan what the one-sgma error s, Fgure 3- below shows a hstogram of percent error for lower PPM rejects. Each of the calculatons of percent error represents one cycle (or

32 9 run) of 0,000 Monte Carlo samples. A total of,000 runs were performed to generate the hstogram. The assembly s the contact angle for the one-way clutch assembly, and the rejects correspond to assembles wth contact angles outsde of the specfcaton lmts presented n Secton.. The PPM rejects correspond to about a.6 sgma qualty level, or 4,600 PPM rejects Output Dstrbuton for %Error n Lower PPM Rejects One-Sgma Lmts % -36% -9% -3% -6% -0% -3% 3% 0% 6% % 9% 35% 4% 48% 55% Frequency %Error n Lower PPM Rejects Fgure 3-: Hstogram of %Error for Lower Rejects for,000 Dfferent Runs (Smulatons of Monte Carlo), each wth 0,000 Samples The one-sgma lmts for the percent error were calculated wth Eq. 3-. The mean of the percent error for all,000 runs (or cycles of Monte Carlo) was about two percent. Ths hstogram s portrayng the one-sgma error lmts, or bounds. The one-sgma lmts are at about sxteen percent. If the sample sze were doubled, then the one-sgma bound on the percent error (σ %PPM ) would be dvded by about the square root of two, and would then

33 be only eleven percent. Thus as the sample sze ncreases, the hstogram would become more narrow, and the accuracy ncrease. Table 3- shows the statstcs for the percent error of the lower, upper, and total PPM rejects for the,000 Monte Carlo smulatons. The estmates of standard devaton of the percent errors are also compared wth the estmates calculated based on Eq. 3-. Table 3-: Dstrbuton for %Error Countng Rejects for 0,000 Samples % Error n Lower PPM % Error n Upper PPM % Error n Total PPM Mean (hstogram).% -9.9% -.% Stdev (hstogram) 5.8% 0.0%.6% σ %PPM (equaton) 5.% 0.4%.% 0 The standard devaton of the percent error for the hstogram of,000 runs of 0,000 samples s very close to the estmate determned by Eq. 3-. Even at 0,000 samples the one-sgma bound on the percent error for the lower PPM rejects s stll about ffteen percent. Secton 3.6. Error n Estmatng Moments wth Monte Carlo The estmate for mean and varance of the output dstrbuton mproves as the number of runs ncreases. The accuracy of the estmate of the mean of the output dstrbuton s presented below n Eq. 3-. Where: ˆ µ µ Standard Normal Dstrbut σ on Eq. 3- n σ σ µ n µˆ The estmate of the th moment for a dstrbuton σ µ The one-sgma bound on the error of the estmaton of the mean, also called the standard error of the mean [Crevelng 997, 63]

34 The standard error for the estmate of the mean decreases wth the square root of n (sample sze). The error for the estmate of the varance s a lttle more complcated. The dstrbuton for estmatng the accuracy of the varance s presented below n Eq. 3-. ( n ) ˆ µ Eq. 3- Ch - Square Dstrbuton : µ mean ( n ), varance ( n ) The Ch-Square dstrbuton for varance can be approxmated by the normal dstrbuton (wth the same mean and varance) at large sample szes [Vardeman 994, 36], and Monte Carlo generally uses large sample szes. Thus wth the state-of-the-art, the accuracy of countng rejects, estmatng the mean, and estmatng the varance of a dstrbuton can be evaluated. Chapter 9 wll expand upon ths to estmate the accuracy of estmatng the skewness and kurtoss, as well as estmatng the PPM rejects usng the moments and the Lambda dstrbuton. Secton 3.7. Advantages and Drawbacks of Monte Carlo Smulaton Monte Carlo s extremely versatle. Input varables may be correlated to each other, or n other words, not ndependent. Input varables may be of any dstrbuton, even dscrete. The number of rejects (assembles outsde of the specfcaton lmts) can be counted drectly durng the smulaton, or the output dstrbuton can be ftted to a standard dstrbuton (normal) or a general dstrbuton (Lambda for example) and the PPM rejects estmated for any lmts. One of the drawbacks s that large sample szes are requred, whch may be tme consumng f the assembly functon s teratve. The smaller the percent rejects, the more samples are requred for an accurate estmate. Once the smulaton s performed, changes n the problem (changes n nput varable dstrbutons, changes n specfcaton lmts, etc.) most often requre that the smulaton be run agan. The estmaton of senstvtes s not a drect result of Monte Carlo. Although the varables could be changed one at a tme or the correlaton coeffcents calculated to estmate the senstvtes.

35 Secton 3.8. Sample Clutch problem The one-way clutch assembly problem s easly modeled n a spreadsheet program (as the equaton for the contact angle s explct) and analyzed wth Crystal Ball, a Monte Carlo smulaton program from Decsoneerng Inc. More dffcult assembles wth teratve assembly functons would be more dffcult to model, and would be mpractcal to use as an Excel model. Fgure 3- below shows sample output of a Monte Carlo smulaton on the clutch assembly. The hstogram estmates the probablty dstrbuton for the contact angle φ. 0,000 Trals Forecast:φ Frequency Chart Probablty Frequency Certanty s 99.8% from to Degrees Fgure 3-: Monte Carlo Output from Crystal Ball The hstogram shows an estmaton of the output dstrbuton for the contact angle. Wth the specfcaton lmts as ndcated, 99.8% of the sample assembles are good, and 0.7% of the assembles are outsde the lmts. The sample sze was 0,000 for ths run. The darkened areas of the tals of the hstogram show the samples that fell outsde the specfcaton lmts (rejects).

36 Usng the Spearman correlaton coeffcents as descrbed by Eq. 3-, Crystal Ball estmates the senstvtes and the percent contrbuton of the ndependent varables as shown below n Table 3-. Ths senstvty nformaton ndcates that the varable a (hub radus) contrbutes most of the varaton to the contact angle. A reducton n the varaton of varable a should reduce the varaton n the contact angle more than reductons n the varatons of the other two varables. Table 3-: Senstvty Informaton from Crystal Ball Varable Correlaton Coeffcent %Contrbuton a % c % e % 3 An estmaton of the mean and varance and other moments of the dstrbuton of the contact angle s also provded by Crystal Ball, but a custom Monte Carlo smulaton program, usng the random number generator descrbed earler, was used for flexblty n outputtng the dstrbuton moment nformaton. Fgure 3- shows how the estmates of the dstrbuton moments change durng the smulaton as the sample sze doubles. Ths smulaton of over a bllon samples, whch took 8.6 hours to complete, wll be used as the benchmark for accuracy of the other methods.

37 4 %Change n Moments for Monte Carlo 40% 30% 0% %Change- 0% 0% -0% µ σ α3 α4-0% -30% -40%.E+03.E+04.E+05.E+06.E+07.E+08.E+09 Sample Sze (log scale) Fgure 3-: Percent Change n Moments for Monte Carlo Smulaton as the Sample Sze Doubles The percent change for the mean s very lttle due to a large value of mean versus the standard devaton. The large change n skewness (α 3 ) s due to the small value of actual skewness. But t appears as though the error n estmatng the skewness s greater than for kurtoss (ths s not really the case though). A better method of calculatng errors wll be presented n Chapter 8. Usng a percent change n the raw moments or the percent error n the moments s not a good estmate or accuracy. As Fgure 3- shows, the percent change n the moments as the sample sze doubles s approxmately zero as the sample sze reaches a bllon. The true values for the objectve functons wll be estmated by the values from ths smulaton. These moments, rejects, and qualty loss for the clutch assembly are presented below n Table 3-.

38 5 Table 3-: Benchmark Results from Monte Carlo Smulaton at One Bllon Samples Contact Angle for the Clutch Value µ (mean) σ (Standard Devaton) α3 (Skewness) α4 (Kurtoss) Qualty Loss ($/part)... Lower Rejects (ppm)... Upper Rejects (ppm)... Total Rejects (ppm) ,406,66 6,57 The above values were obtaned by usng Monte Carlo Smulaton at 30 samples (over one bllon). A descrpton of how to calculate the qualty loss s gven n Chapter 6 as t s manly used wth the desgn of experments. Secton 3.9. Confdence Interval for PPM Rejects Estmated The σ %PPM s easly calculated from Eq. 3- to be , or 0.04 percent (usng the estmate of total PPM rejects n the table above). That means that the 68 percent confdence level (plus and mnus one σ PPM ) for the total PPM rejects s from 6,569 to 6,574. Smlarly, f greater confdence s desred n the total PPM rejects, the 99.7 percent confdence nterval (plus and mnus three σ PPM ) for the rejects s 6,564 to 6,579 rejects. At one bllon samples, the results are qute confdent. The estmaton of the confdence nterval for the mean and varance s just as easly calculated from Eq. 3- and Eq. 3-. Monte Carlo Smulaton s a very effectve way to smulate the randomness that occurs n nature. The requred sample sze for a Monte Carlo smulaton depends on the desred output of the analyss (mean, varance, qualty loss, rejects, etc.) and the desred accuracy of that estmate. More on the requred sample sze for the Monte Carlo smulaton method s presented n Chapter 9.

39 6 Secton 3.0. Summary for Monte Carlo Smulaton The research and expermentaton ndcates that Monte Carlo smulaton s the most flexble of the methods n terms of dfferent types of nput, output, and assembly functons. Input varables can be skewed or even correlated. The assembly functon can even be dscontnuous. Smulaton s defntely the easest method to understand. The major weaknesses of smulaton are the number of trals necessary for accuracy, teratve assembly functons, and calculatng the senstvtes of the nputs [Crevelng 997; Shapro 98]. Monte Carlo smulaton s a valuable tool n varaton analyss and s wdely used. It has many advantages as well as lmtatons. Monte Carlo s a smulaton method that attempts to model the randomness that occurs n nature. Because of ts use of random numbers, more state-of-the-art technques for estmatng ts accuracy are avalable than any other method, as wll be shown n later chapters. Ths chapter has not only provded the state-of-the-art on Monte Carlo and ts accuracy, but t also has demonstrated ts use on the sample clutch problem to allow for drect comparson wth the other methods.

40 7 Chapter 4. Method of System Moments Ths chapter ntroduces and demonstrates the method of system moments (MSM). Because MSM s not a very common method the background wll also be dscussed. The method of system moments s an advanced varaton analyss method that uses the frst and second dervatves of the assembly functon wth respect to the nput varables, along wth the frst eght moments of the nput varable probablty dstrbutons, to estmate the moments of the output dstrbuton. The Taylor seres s used to estmate how the dervatves and nput moments wll affect the resultant assembly dstrbuton. Often, only the frst order terms of the Taylor seres are used, but the second order terms can also be used to ncrease the accuracy of the output dstrbuton predcton [Cox 986; Shapro 98]. Once the output dstrbuton moments are estmated, a Lambda dstrbuton (dscussed n Chapter 7), a Pearson dstrbuton [Fremer 988, 3560], or another dstrbuton may be used to estmate percent rejects, qualty loss, or other objectve functons. Secton 4.. Background of Method of System Moments The method of system moments (MSM) estmates the assembly functon wth a multvarate Taylor seres approxmaton. Eq. 4- below shows the general form of a second order Taylor expanson. For lnear varaton analyss, the second order terms are assumed to be neglgble and thus elmnated [Cox 986, 4 5]. R R + 0 n h x ( x x ) + ( x x ) 0 n n h + 0 j j0 j > xx j + thrd order and hgher terms n h Eq. 4- x 0 ( x x )( x x ) Where: R h(x ) Value of the assembly functon value evaluated at the x

41 8 R 0 h(x 0 ) Value of the functon evaluated at the mean values of the nputs, x 0 h The assembly functon x The value of the th nput varable x 0 The th nput varable at ts mean value In order to smplfy the expressons for the fndng the moments of the assembly functon the followng substtutons are made: X b ( x x ) σ 0 h σ X b b j h Eq. 4- σ X h X X j σ σ j The reason for these substtutons s to transform the dstrbutons to non-dmensonal coordnates, where the mean s 0.0 and the varance s.0. Truncatng the terms of thrd order and hgher and makng the above substtutons, Eq. 4- smplfes nto Eq R R 0 + n ( b x + b x ) + n n j j > b x x j Eq. 4-3 The expected values of the moments of Eq. 4-3 are then the estmates for the output dstrbuton. Generally, ether the quadratc terms are used for a second-order analyss, or the just the lnear terms are used for a frst-order analyss. Secton 4.. Second-order MSM Analyss The resultng equatons for calculatng the mean and the varance of a quadratc assembly functon are found below n Eq. 4-. The equatons for calculatng the skewness and kurtoss of a quadratc equaton are qute complex, and although they were coded n the analyss program used for ths thess, they wll not be presented here. Refer to the book by Cox [Cox 986, 49 50] or by Shapro [Shapro 98, 357] for the full quadratc varaton analyss expressons for skewness and kurtoss.