A COMPARISON OF THREE EXPERIMENTAL DESIGNS FOR TOLERANCE ALLOCATION A Thesis Presented to The Faculty ofthe Fritz J. and Dolores H. Russ College ofEngineering and Technology Ohio University In Partial Fulfillment Ofthe Requirement for the Degree Master ofScience By Ayman Eloseily March, 1998 11 ACKNOWLEDGEMENTS This thesis is dedicated to my wife Katrin, my mom and dad for their support and patience. A special thanks to my advisor Dr. Richard Gerth for his help and support with his knowledge and expertise through the thesis. Also, special thanks to the committee members, Dr. Patrick McCuistion and Dr. David Koonce, for providing their knowledge and expertise. III TABLE OF CONTENTS ACKNOWLEDGEMENTS ii TABLE OF CONTENTS iii LIST OF FIGURES vi LIST OF TABLES viii CHAPTER 1. INTRODUCTION 1 THESIS OBJECTIVE 4 SCOPE 5 2. LITERATURE REVIEW 7 FULL AND FRACTIONAL FACTORIALS 8 T AGUCHI' S INNER AND OUTER ARRAyS 10 BISGAARD'S PRE- AND POST-FRACTIONATION 13 3. CASE STUDY - AN IDLER WHEEL 16 PRODUCT CHARACTERISTIC....................................................................................................... 18 ASSEMBLY SEQUENCE 22 TOLERANCE SPECIFICATION 30 4. METHOD 32 IV HLM ANALySIS 33 THE EXPERIMENTAL DESIGNS 34 Standard Fractional Factorial 35 Double fractional factorial 39 Taguchi's Inner and Outer Arrays 43 EVALUATION MEASURES 46 5. RESULTS •••.•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••.••••••••••••.•••.••••••••.•••.•••••••.•••.••.••••.••..• 48 ANOVA RESULTS 48 NUMBER OF SIGNIFICANT INDEPENDENT VARIABLES: 52 NUMBER OF EXPERIMENTS & PARTS REQUIRED 55 Standard Design 55 Double Design (positive & negative designs) 56 Taguchi Inner and Outer Arrays Design 57 Bisgaard's pre andpost-.fractionation 58 6. DISCUSSIONS AND CONCLUSION 60 CONCLUSIONS 62 7. REFERENCES 64 APPENDICES 67 APPENDIX A. STANDARD FRACTIONAL FACTORIAL DESIGN 68 APPENDIX B. POSITIVE FRACTIONAL FACTORIAL DESIGN 69 APPENDIX C. NEGATIVE FRACTIONAL FACTORIAL DESIGN 70 v APPENDIX D. TAGUCHI'S INNER-OUTER ARRAY WITH DATA 71 APPENDIX E. IDLER WHEEL DIMENSIONS 72 VI List ofFigures Figure 1. Taguchi Inner and Outer Array for 14 Variables 11 Figure 2. Idler Wheel Assembly 17 Figure 3. Dependent Variable Position 19 Figure 4. The Fourteen Independent Variables 21 Figure 5. Assembly tree diagram ofidler wheel assembly. 22 Figure 6. Idler Wheel prior to Assembly 23 Figure 7. Base to Fixture Move Specification Window 24 Figure 8. Assembly after Base is Mounted on the Fixture 24 Figure 9. Assembly ofthe Journal to the Shaft 25 Figure 10. Assembly ofthe Shaft to the Base 26 Figure 11. Assembly ofthe Left Bushing to the Wheel. 27 Figure 12. Assembly ofRight Bushing to the Wheel. 27 Figure 13. Assembly ofthe Wheel Sub-Assembly to the Shaft 29 Figure 14. Assembly ofthe Hanger to the Base and the Shaft 30 Figure 15. Example Tolerance Specification Window. Lower and Upper Tolerances specified for each Independent Variable 31 Figure 16. Comparison ofStandard Fractional Factorial and Double Fractional Factorial for the Approximately Linear (a) and Non-Linear (b) Cases 40 Figure 17. Taguchi's Inner and Outer Array 45 vii Figure 18. Standard and Double Fractional Factorial. 61 Figure 19. Comparison ofStandard Fractional Factorial Array and Taguchi's Inner- Outer Array Design 62 VIII List ofTables Table 1. Independent Variables 20 Table 2. Independent variables and their values (Loose tolerance) 38 Table 3. Standard fractional factorial LI6 design 39 Table 4. L16 Array for Positive Design 41 Table 5. L16 Array for Negative Design 42 Table 6. Tight and Loose Tolerance Levels for Taguchi's Inner-Outer Array 44 Table 7. Pooled ANOVA for Standard Fractional Factorial. 49 Table 8. Pooled ANOVA for Positive Fractional Factorial. 50 Table 9. Pooled ANOVA for Negative Fractional Factorial. 50 Table 10. Pooled ANOVA for Taguchi Mean Response 51 Table 11. Pooled ANOVA for Taguchi Variance Response 51 Table 12. Summary Table ofSums of Squares and Percent Contributions ofthe Different Designs and the HLM Analysis 53 Table 13. Number ofParts Required in the Standard Fractional Factorial. 56 Table 14. Number ofParts Required in the Double Fractional Factorial. 57 Table 15. Number ofParts Required by the Taguchi Design 58 Table 16. Number ofParts Required by Bisgaard's Prefractionation 59 CHAPTERl INTRODUCTION This thesis examines three different experimental designs for their applicability to tolerance analysis. This chapter briefly describes the applied problem that motivated the investigation. The concepts oftolerance design and the inadequacies ofcurrent methods will be explained. Designed experiments are proposed as an alternative to the current methods. However, the best experimental design for attacking the tolerance design problem is yet to be determined. The thesis objective is determining the best ofthree proposed experimental designs. Finally the chapter closes with a briefdescription ofthe chapters to follow. Mass production is based upon the concept ofinterchangeable components (Bisgaard, 1993). Interchangeability is the concept that all components ofthe same type, which are within acceptable tolerances, can be interchanged in an assembled product without affecting product performance. Thus, acceptable product performance is a function ofthe individual component tolerances. Tolerances reflect a designer's intentions regarding product functional and behavioral requirements with corresponding implications for manufacturing and quality control. They specify the acceptable limits within which a component dimension is permitted to vary (ASME, 1994). This determines the choice ofmanufacturing processes 2 and is a factor in the final production cost. Tight tolerances can result in expensive manufacturing processes, increased inspection, and excessive processing cost; while loose tolerances may lead to increased product rejects, warranty returns, and dissatisfied customers. The engineer must design high quality products and processes at low cost, by specifying the allowable component dimension variation (i.e., tolerances) that will result in component interchangeability. A common tolerance design problem is that the product designer has identified the allowable product performance variation and must determine the allowable component variation, that when combined, will result in an acceptable product. Tolerance design is difficult because the designer must understand how the component variations combine to determine the final product performance variation. In practice, tolerances are most often assigned as an informal compromise between functionality, quality and manufacturing cost (Gerth, 1997). Thus, traditional tolerancing methods use past designs, handbooks, rules ofthumb, and recently CAD default settings. These methods can be imprecise, not based on relevant data, and/or insufficient to guarantee a cost effective, quality design. Tolerance design techniques can be classified into two categories: tolerance analysis and tolerance synthesis (Lee and Woo, 1993). Tolerance analysis determines the assembly distribution given the individual component dimension distributions. Common tolerance analysis methods are worst case analysis, statistical tolerancing (Evans, 1974; 3 Gerth, 1992), and Monte Carlo simulation (Evans, 1975; Araj and Ermer, 1989; Bjorke, 1989; Gerth, 1992). Tolerance allocation determines the individual dimension distributions given the specified assembly distribution. Tolerance allocation methods include standards, uniform and proportional scaling (Chase and Greenwood, 1988), various minimum cost optimization algorithms (Gerth, 94; Sayed and Kheir, 1985; Spotts, 1973), and Taguchi methods (Liou, et. aI., 1993). All current tolerancing algorithms, with the exception ofTaguchi methods, require the functional relationship between component dimensions and product performance to be known or specified. But, in many industrial cases the relationship is unknown. This often happens when physical models ofphenomena are not adequately understood, such as electromagnetic forces, frictional losses, wear phenomena, etc. For example, in inkjet printers and important performance characteristic is the skew angle: the angle at which the print is printed on the media. Factors such as width of the media tray, diameter ofthe pick and pinch rollers, and position ofthe chassis can affect the skew angle. Since these factors are geometric in nature, their relationship to the skew angle can be determined, and therefore, their tolerances determined using one ofthe above mentioned techniques. However, other important factors that affect the skew angle are deflection ofthe components under load (the components are not rigid) and roller slippage. Their effect on skew is not known and cannot be analytically determined. In 4 such a case, experimental design can be used to determine an empirical relationship between the component variation and performance measure. An experiment is a test in which purposeful changes are made to the input variables ofa process or a system so that we may observe and identify the reasons for changes in the output response (Montgomery, 1991). The input variables are called independent variables and the response is called the dependent variable. For tolerance design, the independent variables would be the component variation sources and the dependent variable is the product performance measure. The number ofexperiments conducted, the order in which they are conducted, and the manner in which in the independent variables are changed is called the experimental