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INFORMATION to USERS This Manuscript Has Been Reproduced INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly: to order. UMI University Microfilms International A Bell & Howell Information Company 300 North Zeeb Road. Ann Arbor, Ml 48106-1346 USA 313/761-4700 800:521-0600 Order Number 9307867 Design optimization procedure using robustness for minimizing transmission error in spur and helical gears Sundaresan, Sivakumar, Ph.D. The Ohio State University, 1992 Copyright ©1992 by Simdaresan, Sivakumar. All rights reserved. UMI 300N.ZeebRA Ann Arbor, MI 48106 DESIGN OPTIMIZATION PROCEDURE USING ROBUSTNESS FOR MINIMIZING TRANSMISSION ERROR IN SPUR AND HELICAL GEARS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Doctor of Philosophy in the Graduate School of the Ohio State University By Sivakumar Simdaresan, B.Tech., M.S. ***** The Ohio State University 1992 Approved by Dissertation Committee K. Ishii D.R. Houser G.L. Kinzel Co-advisers H.R. Busby Department of Mechanical Engineering Copyright © 1992 by Sivakumar Sundaresan To my wife, Latha and my parents ACKNOWLEDGMENTS I wish to thank my advisers, Dr. Kosuke Ishii and Dr. Donald R. Houser for their invaluable guidance and assistance throughout the course of this research. Not only did they act as mentors, directing my work, they created a congenial atmosphere for development and exchange of ideas. In addition I would like to express my gratitude to the other members of the dessertation commitee, Dr. Gary L. Kinzel and Dr. Henry R. Busby for their suggestions, review and evaluation of this thesis. I extend my appreciation to the Graduate School and the sponsors of the Gear Dynamics and Gear Noise Research Laboratory for supporting this research activity financially. I acknowledge the students of the Gear Dynamics and Gear Noise Research Laboratory and Life Cycle Engineering Group at Ohio State (LEGOS) for their helpful suggestions during the course of this work. Finally, I thank my wife, Latha and my parents for their support and encouragement. iii VITA January 2,1966 Bom - Madurai, Tamil Nadu, India. 06/1987 Bachelor of Technology, Indian Institute of Technology, Madras, India. 07/87 -12/87 Graduate Trainee Engineer, Hindustan Motors Limited, Hosur, India. 01/88 - 06/88 Graduate Teaching Associate, Dept. of Mechanical Engineering, The Ohio State University, Columbus, Ohio. 06/88 - 06/91 Graduate Research Associate, Gear Dynamics and Gear Noise Research Lab. The Ohio State University, Columbus, Ohio. 07/91 -12/91 University Presidential Fellow The Ohio State University, Columbus, Ohio. 01/92 - 06/92 College Intern Core Engineering Center of Integrated Products Eastman Kodak Company, Rochester, New York 07/92 - Present University Presidential Fellow The Ohio State University, Columbus, Ohio. IV PUBLICATIONS 1) Sundaresan. S.. Ishii, K., and Houser, D.R. (1991). A Procedure using Manufacturing Variance to Design Gears with Minimum Transmission Error, Journal Of Mechanical Design, Transactions of the ASME, September 1991, Vol.113, pp 318-324. 2) Sundaresan. S.. Ishii, K., and Houser, D.R. (1991). Design Optimization for Robustness using Performance Simulation Programs, Proceedings of the the American Society of Mechanical Engineers Design Automation Conference, DE-Vol. 32-1, pp 249-256, Miami, Florida, Sept 22-25. Also to appear in Journal of Engineering Optimization. 3) V. Ramamurti, H. Srinivasan, R. Sekhar, S.Sekar and S.Sivakumar (1989). Dynamic Analysis of Heavy Duty Wheels, Journal of Computers and Structures, Vol 31, No. 2 pp 275-285. 4) Sundaresan. S.. Ishii, K„ and Houser, D.R. (1990). A Procedure that Accounts for Manufacturing Errors in the Design Minimization of Transmission Error in Helical Gears, American Gear Manufacturers Association(AGMA) Fall Technical Meeting, Toronto, Canada. 5) Sundaresan. S.. Ishii, K., and Houser, D.R. (1991). Design of Helical Gears with Minimum Transmission Error under Manufacturing and Operational Variances, Proceedings of the Japan Society of Mechanical Engineers International Conference on Motion and Power Transmissions, Hiroshima, Japan, Nov 24-26,1991. 6) Bhakuni, N., Ishii, K., Tragesser, A. and Sundaresan. S. (1991). Structural Optimization methods for Aluminum Beverage Can Bottoms, Proceedings of the the American Society of Mechanical Engineers Design Automation Conference, DE-Vol. 32-1, pp 265-271, Miami, Florida, Sept 22-25. 7) Gear Design Program Overcomes Effects of Manufacturing Variations, edited by Joe Brown, Associate editor, Power Transmission Design, March 1991, pp 31-33, based on publications #1 and #4. FIELDS OF STUDY Major Field: Mechanical Engineering v TABLE OF CONTENTS DEDICATION ii ACKNOWLDEGEMENTS iii VITA iv TABLE OF CONTENTS vi LIST OF FIGURES ix LISTOFTABLES xii NOMENCLATURE xiii DISSERTATION ABSTRACT xvi CHAPTER I INTRODUCTION 1 1.1 Motivation 1 1.2 Design For Robustness (DFR) 2 1.2.1 Concept of Statistical Optimum 3 1.2.2 Literature review in statistical optimization 4 1.3 Gear design with minimum transmission error 11 1.3.1 Background 11 1.3.2 Literature review 13 1.4 Outline 18 II MINIMIZING TRANSMISSION ERROR USING TOOTH MODIFICATIONS 20 2.1 Introduction 20 2.2 Profile and Lead Modifications 20 2.3 Cross modification 28 2.4 Topographical modifications 31 2.5 Effect of Misalignment and Torque Variations 33 vi 2.6 Summary 34 IB UNCONSTRAINED STATISTICAL OPTIMIZATION 36 3.1 Introduction 36 3.2 Generalized Problem 36 3.3 Introduction to Design of Experiments 38 3.4 Procedure 42 3.5 Helical Gear Example 44 3.6 Discussion of Results 47 3.7 Summary 49 IV CONSTRAINED STATISTICAL OPTIMIZATION 51 4.1 Introduction 52 4.2 Background 50 4.3 Literature Review 57 4.4 Procedure 59 4.4.1 Method 1: Brute Force with Heuristics 59 4.4.2 Method 2: Constraints with built-in constraint variation 61 4.4.3 Method 3: Approach using KKT conditions 62 4.4.4 Discussion 65 4.5 Spur Gear Example 66 4.6 Summary 69 V PARAMETRIC STUDY USING ILLUSTRATIVE EXAMPLES 70 5.1 Introduction 70 5.2 Example #1 70 5.3 Example # 2 76 5.2.1 Effect of Hob shift 77 5.2.2 Optimization example 82 5.3 Example # 3 85 5.5 Example # 4 88 5.6 Example # 5 91 5.7 Summary 96 VI CONCLUSIONS 98 6.1 Concluding Remarks 98 vu 6.2 Contributions of this Research .• 100 6.2.1 Contributions in statistical optimization 100 6.2.2 Contributions in gear design 101 6.3 Recommendations for Future Work 102 BIBLIOGRAPHY 105 APPENDICES A INTRODUCTION TO ANALYSIS OF VARIANCE 110 A.l Introduction 110 A.2 Illustration 110 B ALGORITHMS USED IN OPTIMIZATION SCHEME 115 B.l Introduction 115 B.2 Golden Section Method 115 B.3 BFGS Variable Metric Method 118 B.4 Augmented Lagrange Multiplier Method 120 vui LIST OF FIGURES Figure Page 1.1 Concept of statistical optimum 4 1.2 Possible design space for two variables 9 1.3 Profile specification chart 17 2.1 Profile modification 21 2.2 Lead modification 21 2.3 Contour plots of equal PPTE in microinches for various parabolic profile modification with a fixed lead modification of 0.0001" 23 2.4 Contour plots of equal PPTE in microinches for various parabolic profile modification with a fixed corner modification of 0.0001" 24 2.5 Normalized load distribution over the contact zone for optimum parabolic profile with a fixed lead modification of 0.0001" 25 2.6 Optimum profile modification 26 2.7 Load distribution factors for various parabolic profile modifications with a fixed lead modification of 0.0001" 27 2.8 Cross modification 28 2.9 Contour plots of equal PPTE in microinches for various parabolic cross modifiations with a fixed lead modification of 0.0001" 29 2.10 Load distribution factors for various parabolic cross modifications with a fixed lead modification of 0.0001" 30 ix 2.11 Normalized load distribution over the contact zone for optimum parabolic profile with a fixed lead modification of 0.0001" 31 2.12 Contour plot of equal tooth modification in microinches over the contact zone 32 2.13 Contours of equal PPTE in microinches for a gear mesh with optimum profile modification for various values of misalignment and torque 33 2.14 Contours of equal PPTE in microinches for a gear mesh with optimum cross modification for various values of misalignment and torque 34 3.1 Full and fractional factorial experiments 39 3.2 Optimization results 46 3.3 Worst case of L8 for peak and statistical optimum 46
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