I A DESIGN OF EXPERIMENT APPROACH TO *".. *".. -- TOLERANCE ALLOCATION/ A Thesis Presented to The Faculty of the Russ College of Engineering and Technology Ohio University In Partial Fulfillment of the Requirement for the Degree Master of Science Ziaul Islam/ d June, 1995 ACKNOWLEDGEMENTS I would like to express my sincere gratitude to my advisor Dr. Richard J. Gerth for his advice, direction, and encouragement during my pursuit of this degree. Working with him has been a learning experience, and I have derived a lot of inspiration fiom him. I also wish to thank the members of the graduate committee, Dr. David Koonce and Dr. Patrick McCuistion, for reviewing the thesis and offering suggestions for its improvement. TABLE OF CONTENTS ACKNOWLEDGEMENTS ........................................................................................... i LIST OF FIGURES ....................................................................................................... iv LIST OF TABLES .......................................................................................................... v LIST OF APPENDICES ................................................................................................ vi CHAPTER 1. INTRODUCTION .................................................................................. 1 2 . LITERATURE REVIEW ....................................................................... 6 Heuristic Approaches ................................................................... 6 Minimum Cost Optimization methods .......................................... 9 Lagrange Multiplier ........................................................ 11 Linear programming ........................................................ 13 Dynamic programming ................................................... 14 Non-linear programming ................................................. 15 Design of Experiments (DOE) ................................................... 19 Taguchi's Parameter Design ........................................... 22 Taguchi's Tolerance Design .......................................... 23 Tolerance Design using Taguchi's Parameter Design ......25 3 . CASE STUDY - A BENCH VICE ....................................................... 27 4 . EXPERIMENTAL DESIGN SETUP AND PROCEDURE ...................3 1 Inner and outer arrays .................................................................3 1 Designing an experiment using orthogonal arrays ........................32 Experimental variable and level selection ....................................34 Selection of orthogonal arrays and assignment of factors ............36 Conducting the experiment ......................................................... 37 5 . ANALYSIS AND RESULTS ............................................................... 39 The mean response analysis ........................................................ 40 The S/N response analysis .......................................................... 43 Confirmation experiment ............................................................49 Geometrical interpretation .......................................................... 54 6 . DISCUSSION AND CONCLUSIONS ................................................ 57 BIBLIOGRAPHY ........................................................................................................ 60 APPENDICES ............................................................................................................. 64 ABSTRACT LIST OF FIGURES Figure A detailed drawing of the bench vice assembly ..................................... ,29 Mean response graph ........................................................................... 44 S/N ratio response graph ....................................................................... 47 LIST OF TABLES Table 2.1 . Table of cost minimization techniques .................................................. 1 2.2 . Table of proposed cost tolerance relationships ....................................... 19 4.1 . An L16 InnerlOuter Orthogonal Array Structure .................................... 33 4.2 . Variables for the experiment .................................................................35 4.3 . An L 16 Orthogonal Array .................................................................... 37 5.1 . Response table ..................................................................................... 41 5.2 . Mean response table ............................................................................... 42 5.3 . S/N response table ............................................................................... 46 5.4 . Mean response ANOVA table ................................................................ 45 5.5 . S/N ratio ANOVA table ......................................................................... 48 5.6 . Confirmation experiment results ............................................................. 53 LIST OF APPENDICES Appendix A . List of figures of Bench Vice component and their Geometric Dimensioning & Tolerancing Drawings ................................................. 65 List of figures showing analysis of movable jaw and rod ........................ 70 List of formulae and calculations ........................................................... 72 Bench vice assembly dimensions ............................................................ 86 Bench vice assembly simulation program ............................................... 87 L16 orthogonal array data ..................................................................... 88 Mean response data analysis .................................................................. 90 ANOVA table on mean response ........................................................... 91 S/N ratio data analysis ........................................................................... 92 ANOVA table on S/N ratio ................................................................... 93 List of formulae used in simulation program of bench vice .....................94 Listing of Microsoft Excel Version 4.0 macro program ......................... 96 CHAPTER 1 INTRODUCTION Manufacturing is a sigdlcant portion of the U.S. economy and manufacturing has experienced a loss in its global competitive position ( Otto and Finnie, 1993). One of the key factors contributing to the loss of market share to foreign competitors is poor product quality (Juan, 1988; Deming, 1986). Therefore, much attention has been focused on the issue of producing quality products, which we will define as a product produced with minimum variation around a target dimensional value. If a product exhibits too much variation, it is a non-confoxming product or rejects. A reject can be defined as a part produced "out-of- specification", i.e. not within the acceptable tolerance region. Tolerancing, the method by which tolerances are assigned and their cumulative effects predicted, plays a key role in reducing rejects. A component tolerance is "the total amount by which a specific dimension in an engineering drawing is permitted to vary [ANSI, 19821". It is the difference between the upper and lower limits of the spdcation. Tolerances reflect a designer's intentions regarding product fbnctional and behavioral requirements with corresponding implications for manufacturing and quality control. They are important not only because they spec@ product performance, but also because they have a si@cant impact on the choice of manufacturing processes, which in turn determines the final production cost. Tight tolerances can result in excessive process cost, while loose tolerances may lead to increased rejects and assembly problems. The engineer must design high quality products and processes at low cost, by specdjkg the allowable variation that can be tolerated without loss of component interchangeability and functionality. The typical design problem is that the product designer has identified the allowable product performance variation and must determine the allowable component tolerances, that when combined, will result in an acceptable product. Analysis of the effects of tolerances on product performance is diicult because it requires determining the functional relationship between the component and assembly tolerances, which is often unknown or very complicated. Thus, traditional tolerancing methods are the use of past designs, handbooks, or "rules of thumb". These methods are, often imprecise, not based on relevant data, or insuflicient to guarantee a cost effective, quality design. Let Y, be the product or assembly response of interest which is a function of n component features Xi : Y, = f (XI, X2, ............. 9 X) [ll Equation [I] is known as the stackup equation I function, which may be linear or non-linear. Modern tolerancing is divided into two subareas: tolerance analysis and tolerance allocation or synthesis (Lee and Woo, 1986). Tolerance analysis means to determine the resulting assembly tolerance, Y, , when the individual component tolerances are, Xi , given; tolerance allocation means to determine the required component tolerances, X,, when the assembly tolerance, Y, , is given. Tolerance stackup or analysis methods are worst case analyses, statistical tolerancing (Evans, 1974; Gerth, 1992), and Monte Carlo simulation (Evans, 1975; Araj and Ermer, 1989; Bjorke, 1989). Tolerance allocation methods include standards, uniform and proportional scaling (Chase and Greenwood, 1988), various minimum cost opGmization algorithms (Gerth,