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Ohio University Library Acknowledgements COST TOLERANCE OPTIMIZATION FOR PIECEWISE CONTINUOUS COST TOLERANCE FUNCTIONS A Thesis Presented to The Faculty of the Fritz J. and Dolores H. Russ College of Engineering and Technology Ohio University In Partial Fulfillment of the Requirement for the Degree Master of Science by Murtaza Kaium Shehabi June, 2002 OHIO UNIVERSITY LIBRARY ACKNOWLEDGEMENTS The definition of a 'Thesis' was once preached to me as a work in one's field of academic interest. It is a contribution, which addresses a problem that has not been resolved to date. In my endeavor to accomplish what no person has, I am truly indebted to my advisors Dr. Gerth and Dr. Masel. Their constant encouragement and belief in my capabilities was invaluable. A great portion of my research involved programming in Visual Basic. I was extremely fortunate to have received guidance from Dr. Dhamija through some of the most critical phases of my research. Dr. Masel was very supportive, especially towards the end stages of my work. During the course of my efforts I found rejuvenating strength in the saying that, "Hope is a good thing to have, perhaps the best of things - and good things never die". My parents and brother have been very instrumental in shaping my life and thoughts. Every friend I have known who has inspired me to believe in myself; this is my opportunity to thank each one of you. Even as I write this, perceiving the end of my academic career, temporarily, the famous words from Robert Frost ring through my mind - "Miles to go before I sleep, Miles to go before I sleep" Table of Contents LIST OF TABLES .................................................................................vi . LIST OF FIGURES ...............................................................................VII 1. INTRODUCTION.......................................................................................................... 1 1.1 Tolerance Analysis ................................................................................................2 1.2 Min-Cost Tolerancing ............................................................................................5 2 . LITERATURE SEARCH ...............................................................................................8 2.1 Cost Tolerance functions .......................................................................................8 2.2 Optimization Techniques .....................................................................................10 2.3 Heuristics .............................................................................................................11 2.3.1 Simulated Annealing ............................................................................. .... 1 1 2.3.2 Tabu Search ...................................................................................................13 2.4 Modified NLP .....................................................................................................14 3 . PROBLEM STATEMENT & RESEARCH OBJECTIVE ........................................ 16 3.1 Problem Statement ...............................................................................................16 3.2 Research Objective ..............................................................................................16 4 . METHODOLOGY .......................................................................................................17 4.1 Casestudy ...........................................................................................................17 4.1.1 C-T functions ................................................................................................19 4.1.2 DOE setup .....................................................................................................20 4.1.3 Discussion of Case Study ..............................................................................21 4.2 Total Enumeration Method (TEM) ...................................................................... 24 v 4.2.1 Global Optimization ......................................................................................26 4.3 Research Methodology - Hypothesis .............................................................26 4.4 TEM compared to Case Study Method ...............................................................27 5 . TEM .SOFTWARE ARCHITECTURE ......................................................................29 5.1 Architecture ........................................................................................................29 5.2 User Interface ....................................................................................................3 0 6 . RESULTS .....................................................................................................................36 6.1 Case A - The Gerth and Pfieffer method ............................................................36 6.2 Case B - Case A augmented with Visual Inspection ...........................................37 6.3 Case C - Automated TEM ............................................................................. ...... 39 7 . SUMMAIIY.................................................................................................................. 42 7.1 Limitations and Disadvantages of TEM ..............................................................43 7.2 Significant Observations ......................................................................................44 7.3 Future Scope .......................................................................................................45 8 . REFERENCES .............................................................................................................47 APPENDICES ...................................................................................................................50 8.1 Appendix A ..........................................................................................................51 8.2 Appendix B ..........................................................................................................54 8.3 Appendix C ..........................................................................................................57 List of Tables Table 1. Table of component features. tolerances. and partial derivatives for Transmission Case Study ...................................................................19 Table 2 . Fractional Factorial Experimental Design Matrix .................................21 Table 3 . Number of CT curve segments for Experiment 1................................. 28 Table 4 . Case A - Gerth and Pfeiffer method ................................................37 Table 5 . Case B - Gerth and Pfeiffer method with visual inspection .....................38 Table 6 . Case C - Automated TEM ............................................................39 List of Figures Figure 1. Influence of dimensional tolerance on cost of manufacture .....................2 Figure 2 . Example of a continuous cost-tolerance function .................................9 Figure 3 . Example of a discrete cost-tolerance function .....................................9 Figure 4 . Example of a piecewise continuous cost-tolerance function, discontinuous at two points .......................................................................................9 Figure 5. Excel spreadsheet of TEM for Experiment 1...................................... 22 Figure 6 . Flow chart of the TEM System .....................................................30 Figure 7 . Sheet 1: Input Sheet ..................................................................31 Figure 8 . Sheet 2: CT Data Sheet .............................................................32 Figure 9. Sheet 3: Output sheet ................................................................34 Figure 10. Graphical representation of the results from the three cases ....................41 Figure 11 . A convex function ....................................................................51 Figure 12. Example of a convex set ............................................................52 As an integral part of mechanical design, tolerances have a profound influence on the functional performance and manufacturing costs of the designed product. Tolerance specification is a complex and demanding task and is carried out traditionally on a trial- and-error basis [5]. The problem of specifying component tolerances to produce the least expensive system satisfying the performance requirements is of great importance to design engineers. Tolerance synthesis jointly models, evaluates, and aims to optimize the functional performance and manufacturing costs of a mechanical product [I]. One such tolerance synthesis technique is called 'minimum manufacturing cost design7 or 'minimum cost tolerancing' [5]. Minimum cost tolerancing (MCT) attempts to reduce the overall cost of a product by widening the tolerances on the more expensive component features and reducing the tolerances on the less expensive component features [14]. In order for MCT to function, one must know the cost - tolerance (C-T) relationship of each component feature. The C-T functions can generally be discrete, continuous, or piecewise continuous. Figure 1 shows a typical continuous function, where cost reduces non-linearly with the widening of the tolerance like a convex curve. Rough Stages of Processes Grinding 1Machining .c, U q 16 - .I .c, cJ m 12 - 2a 4- ii 'G 8 - 2 L 9 4 - I I 0.004 0.012 0.020 Tolerance, in +- Figure 1. Influence of dimensional
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