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Variation Analysis of Involute Spline Tooth Contact Brigham Young University BYU ScholarsArchive Theses and Dissertations 2006-02-22 Variation Analysis of Involute Spline Tooth Contact Brian J. De Caires Brigham Young University - Provo Follow this and additional works at: https://scholarsarchive.byu.edu/etd Part of the Mechanical Engineering Commons BYU ScholarsArchive Citation De Caires, Brian J., "Variation Analysis of Involute Spline Tooth Contact" (2006). Theses and Dissertations. 375. https://scholarsarchive.byu.edu/etd/375 This Thesis is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of BYU ScholarsArchive. For more information, please contact [email protected], [email protected]. VARIATION ANALYSIS OF INVOLUTE SPLINE TOOTH CONTACT By Brian J. K. DeCaires A thesis submitted to the faculty of Brigham Young University in partial fulfillment of the requirements for the degree of Master of Science Department of Mechanical Engineering Brigham Young University April 2006 Copyright © 2006 Brian J. K. DeCaires All Rights Reserved BRIGHAM YOUNG UNIVERSITY GRADUATE COMMITTEE APPROVAL of a thesis submitted by Brian J. K. DeCaires This thesis has been read by each member of the following graduate committee and by majority vote has been found to be satisfactory. ___________________________ ______________________________________ Date Kenneth W. Chase, Chair ___________________________ ______________________________________ Date Carl D. Sorensen ___________________________ ______________________________________ Date Robert H. Todd BRIGHAM YOUNG UNIVERSITY As chair of the candidate’s graduate committee, I have read the thesis of Brian J. K. DeCaires in its final form and have found that (1) its format, citations, and bibliographical style are consistent and acceptable and fulfill university and department style requirements; (2) its illustrative materials including figures, tables, and charts are in place; and (3) the final manuscript is satisfactory to the graduate committee and is ready for submission to the university library. ___________________________ ______________________________________ Date Kenneth W. Chase Chair, Graduate Committee Accepted for the Department ______________________________________ Matthew R. Jones Graduate Coordinator Accepted for the College ______________________________________ Alan R. Parkinson Dean, Ira A. Fulton College of Engineering and Technology ABSTRACT VARIATION ANALYSIS OF INVOLUTE SPLINE TOOTH CONTACT Brian J. K. DeCaires Department of Mechanical Engineering Master of Science The purpose of this thesis is to provide an in-depth understanding of tooth engagement in splined couplings based on variations in clearances between mating teeth. It is standard practice to assume that 25-50% of the total spline teeth in a coupling are engaged due to variations from manufacture. Based on the assumed number of teeth engaged, the load capability of a splined coupling is determined. However, due to the variations in tooth geometry from manufacuture, the number of teeth actually engaged is dependent on the applied load and the tooth errors. The variations result in sequential tooth engagement with increasing load. To date, little work has been done to model tooth engagement and the stresses resulting from unequal load sharing among engaged teeth. A Statistical Tooth Engagement Model (STEM) has been developed which allows designers to estimate tooth engagement and resulting stress based on a statistical representation of the tooth errors. STEM is validated with finite element models as well as some preliminary experimental tests. Parametric studies are performed to determine the effect and sensitivities of variations in tooth parameters and tooth errors. ACKNOWLEDGMENTS Many people have provided me with much needed support and assistance throughtout the course of this research. I would like to thank my wife Brittany, for her love and encouragement, which helped me endure through the the long hours. Dr. Ken Chase provided me much needed guidance and always provided me with a challenge. I would also like to thank my parents for their support and assistance. Finally, I must express extreme gratification to the industrial sponser who made this research possible. Table of Contents Chapter 1: Introduction................................................................................................. 1 1.1 Background......................................................................................................... 2 1.1.1 Industrial Application ................................................................................. 4 1.2 Research Motivation ........................................................................................... 6 1.3 Thesis Objectives................................................................................................ 6 1.4 Scope Delimitations............................................................................................ 8 1.5 Thesis Summary.................................................................................................. 9 Chapter 2: Background Research ............................................................................... 11 2.1 Nomenclature.................................................................................................... 11 2.2 Spline Geometry ............................................................................................... 13 2.2.1 Geometry Modifications ........................................................................... 15 2.3 Tooth Errors...................................................................................................... 16 2.4 Hertz Contact Theory........................................................................................ 18 2.5 Hertz Contact Approximation........................................................................... 21 2.6 Tooth Stresses and Deflections......................................................................... 23 2.6.1 Tooth Stresses ........................................................................................... 23 2.6.2 Tooth Deflections Due to Shear and Bending Loads................................ 24 2.6.3 Tooth Deflections Due to Contact Loads.................................................. 25 2.7 Tooth Engagement............................................................................................ 26 vii 2.8 Chapter Summary ............................................................................................. 28 Chapter 3: Strength of Materials................................................................................. 29 3.1 Model Representation ....................................................................................... 29 3.2 Tooth Stresses ................................................................................................... 31 3.2.1 Bending Stress .......................................................................................... 32 3.2.2 Axial Stress ............................................................................................... 34 3.2.3 Transverse Shear Stress ............................................................................ 34 3.2.4 Contact Stress............................................................................................ 37 3.3 Tooth Deflections.............................................................................................. 39 3.3.1 Bending Deflection ................................................................................... 40 3.3.2 Transverse Shear Deflection..................................................................... 41 3.3.3 Contact Deflection .................................................................................... 42 3.4 Deflection Verification ..................................................................................... 42 3.5 Chapter Summary ............................................................................................. 43 Chapter 4: Estimating Tooth Engagement Analytically............................................. 45 4.1 Tooth Stiffness.................................................................................................. 46 4.2 Engagement....................................................................................................... 47 4.3 Engagement Model ........................................................................................... 51 4.4 Statistical Model ............................................................................................... 54 4.4.1 Tooth Clearance........................................................................................ 55 4.5 Effective Tooth Engagement ............................................................................ 56 4.6 Chapter Summary ............................................................................................. 58 Chapter 5: Model Verification.................................................................................... 59 viii 5.1 Case I: Two Parallel External Cylinders........................................................... 59 5.1.1 ANSYS Model.......................................................................................... 60 5.1.2 ANSYS Results......................................................................................... 61 5.1.3 Hertz Comparison ....................................................................................
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