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A Study of Geometry and Deformable-Body Characteristics of Non-Right Angle Worm Gear Pairs THESIS Presented in Partial Fulfillme

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A Study of Geometry and Deformable-Body Characteristics of Non-Right Angle Worm Gear Pairs THESIS Presented in Partial Fulfillme A Study of Geometry and Deformable-body Characteristics of Non-right Angle Worm Gear Pairs THESIS Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Sriram Madhavan, B.E. Graduate Program in Mechanical Engineering The Ohio State University 2012 Master's Examination Committee: Dr. Ahmet Kahraman, Advisor Dr. Donald Houser Copyright by Sriram Madhavan 2012 Abstract In this study, a formulation to define the three-dimensional geometry of worm gear drives having non-right angle shafts is developed. The geometry of the worm is determined by defining the geometry of the cutter and solving the corresponding equation of meshing between the worm and the cutter. The geometry of the worm gear is then defined by using a cutter which has the exact shape of the worm and solving the corresponding equation of meshing between the worm and the worm gear. Both right- and left-hand, single enveloping worm drives of ZK type with any number of worm threads are included in this formulation. With the tooth surface geometries defined, a commercial finite element gear analysis package with specific worm mesh generators is used to develop a deformable-body model of a non-right angle worm gear pair A parametric design sensitivity study is performed by using this deformable-body model to quantify the effects of basic geometric parameters including the shaft cross angle, lead angle, pressure angle, and addendum and dedendum coefficients on the maximum contact stress and mechanical efficiency of the gear pairs. In addition, variations to shaft center distance and cross angle are introduced to investigate their influence on gear pair performance. iii Dedication This thesis is dedicated to my family and friends for their love and support. iv Acknowledgments I would like to express my sincere gratitude to my advisor Professor Dr. Ahmet Kahraman for providing the research opportunity, guidance throughout my research at OSU and his effort in reviewing this thesis. I am grateful to Dr. Donald Houser for accepting to be part of my Master’s examination committee. I am also thankful to Jonny Harianto for the support through the graduate program and while being a GRA at Gearlab. I would like to thank Honda R&D Americas, Ohio for providing financial support throughout my study. I would like to thank Dr. Sandeep Vijayakar for providing the CALYX package and support throughout my research. I am also thankful to Karthikeyan Marambedu of Advanced Numerical Solutions, Inc. for helping me out with computer programming and software troubleshooting required for the research. Finally, I deeply appreciate the support and love of my parents, brother and friends. I would like to specially thank all my friends in Columbus for making my stay memorable and all my lab mates for their help and friendship throughout my masters program at OSU. v Vita May, 2010 ......................................................B.E. Mechanical Engineering College of Engineering Guindy Anna University Chennai, India Oct, 2010- Present ..........................................Graduate Research Associate, Gear and Power Transmission Research Laboratory, Department of Mechanical Engineering Ohio State University Fields of Study Major Field: Mechanical Engineering vi Table of Contents Abstract .............................................................................................................................. iii Dedication .......................................................................................................................... iv Acknowledgments............................................................................................................... v Vita ..................................................................................................................................... vi List of Tables ...................................................................................................................... x List of Figures .................................................................................................................... xi Nomenclature……………………………………………………………………………xiv Chapter 1 Introduction……………………………………………………………………………… 1 1.1 Background and Motivation……………………………………………………. 1 1.2 Literature Survey……………………………………………………………….. 4 1.3 Scope and Objective……………………………………………………………. 6 1.4 Thesis Outline ………………………………………………………………….. 7 vii Chapter 2 Definition of the Geometry of a Non-Right Angle Worm Gear Pair…………………. 9 2.1 Introduction…………………………………………………………………… 9 2.2 Definition of Cutter Geometry ……………………………...................... ........... 10 2.3 Definition of Worm Geometry……………………………………………….. 15 2.3.1 Cutter Installment and relative Motion between Cutter and Worm…….. 15 2.3.2 Equation of Meshing……………………………………………………. 21 2.3.3 Worm Surface Equations……………………………………………….. 23 2.3.4 Geometric Model of the Worm…………………………………………. 26 2.4 Geometric Model of the Worm Gear…………………………………………. 29 Chapter 3 Computational Model and Parametric Study…………………………………………. 34 3.1 Introduction…………………………………………………………………… 34 3.2 Deformable Body Finite Element Analysis of Worm Gear Pair……………… 35 3.2.1 Mesh Generation………………………………………………………... 35 3.2.2 Pre-processing in Hypoid K program…………………………………… 38 3.2.3 Post-processing in Hypoid K program………………………………….. 38 viii 3.3 Design Parameter Sensitivity Study………………………………………….. 44 3.3.1 Effect of Cross Angle and Lead Angle on and …………. 50 max 3.3.2 Effect of Pressure Angle c on and Effect of Addendum a and Dedendum d Coefficients on and ……………………………………………………………………. 68 3.4 Manufacturing Variability Study……………………………………………… 68 Chapter 4 Conclusions…………………………………………………………………………… 76 4.1 Summary……………………………………………………………………… 76 4.2 Conclusions…………………………………………………………………… 77 4.3 Recommendations for Future Work………………………………………….. 79 References……………………………………………………………………………. 81 ix List of Tables Table Page 2.1 An Example set of user-defined and calculated cutter parameters. Parameters with an asterisk next to it are user-defined ones………………… 16 3.1 Input design parameters to define a worm drive……………………………... 39 3.2 Design constraints imposed in the parametric study......................................... 40 3.3 Material and grease parameters………………………………………………. 49 3.4 Geometric parameters used for the vs. study…………………………… 51 3.5 Geometric parameters used for the c study………………………………… 63 3.6 Geometric parameter ranges suitable for addendum and dedendum coefficient study…………………………………………………………………………... 70 3.7 Addendum and dedendum coefficients for worm and worm gear…………..... 71 3.8 Parameters considered in the manufacturing variability study……………….. 73 x List of Figures Figure Page 1.1 Worm gear drive consisting of worm and worm gear (worm wheel)…….......... 2 2.1 (a) Axial section of the cutter and (b) generating cone surface………………… 11 2.2 Definition of cutter geometry parameters……………………………………… 14 2.3 Installment of the cutter on the worm surface…………………………………. 17 2.4 (a) Coordinate systems applied for generation of K type worms and (b) definition of angle c ……………………………………………………… 18 2.5 Relative motion between the cutter and the worm………………...................... 20 3.1 Definition of gear surfaces in CALYX………………………………................ 37 3.2 (a) Right angle 90 and (b, c) non-right angle design configurations with (b) 90 and (c) 90 ........................................................................... 41 3.3 FE mesh model of a worm gear pair with m21=31, n =1, =110 , = 7.5 and c =15 …………………………………………………………………… 42 3.4 FE meshes and cross-sectional views of (a) a worm and (b) its worm gear mate…………………………………………………………. 43 3.5 Contact Patterns on (a) the worm gear and (b) the worm for a pair For parameters m21 31, n=1, =100 , = and = ……………….. 45 3.6 Instantaneous load intensities on the worm gear of a pair for parameters xi m21=31, =1, = , = and =15 at 5 different mesh steps……… 46 3.7 Contact pattern and load distribution on worm gear for parameters =23, max =2, =19.5 and =15 ; (a) = 70 (b) 110 ………………………. 47 3.8 Design configurations considered to study of effect of and on max and with =1……………… ……………………………………….. 52 3.9 (a) Effect of on max at different values, and (b) effect of on at different values for = 15 and = 1……………………………. 53 3.10 (a) Effect of on at different values, and (b) effect of on at different values for = and = 1………………………….......... 54 3.11 Design configurations considered to study of effect of and on and with =2………………………………………………………….. 57 3.12 (a) Effect of on at different values, and (b) effect of on at different values for = and = 2…………………………….. 58 3.13 (a) Effect of on at different values, and (b) effect of on at different values for = and m=21 2………n .................................110 7.5 ...... 59 3.14 Comparison between single thread (n=1) and double thread designs c (n=2) (a) Effect of on max , and (b) effect of on for =110 ………... 60 3.15 Tooth size comparison of (a) single thread and (b) double thread worm gear design options……………………………………………………… 61 3.16 Design configurations considered to study the effect of c and on and at =105 ………………………………………………………. 64 xii 3.17 (a) Effect of on at different values, and (b) effect of on c max at different values for =105 ………………………… ……………. 65 3.18 (a) Effect of on at different values, and (b) effect of on at different values for 110 …………………………………… 66 3.19 (a) Effect of on at different values for = , and (b) effect of on at different values for =110 ……………………... 67 3.20 Effect of a and d on (a)
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