Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1-1-2004 Contact stresses in conical shaped rollers Hashim Omar Al-Zain Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Recommended Citation Al-Zain, Hashim Omar, "Contact stresses in conical shaped rollers" (2004). Retrospective Theses and Dissertations. 20329. https://lib.dr.iastate.edu/rtd/20329 This Thesis is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Contact stresses in conical shaped rollers by Hashim Omar Al-Zain A thesis submitted to the graduate faculty in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Major: Mechanical Engineering Program of Study Committee: Donald Flugrad, Co-major Professor Abir Qamhiyah, Co-major Professor Loran Zachary Iowa State University Ames, Iowa 2004 Copyright © Hashim Omar Al-Zain, 2004. All rights reserved. 11 Graduate College Iowa State University This is to certify that the master's thesis of Hashim Omar Al-Zain has met the thesis requirements of Iowa State University Signatures have been redacted for privacy 111 Table of Contents Chapter 1: Introduction to Traction Drive Systems............................................................ 1 A Brief Historical Perspective on Traction Drives ....................................................... :: ~ 1 Background ..................................................................................................................... 8 Proof of Traction Concept ............................................................................................ 12 Chapter 2: Cylindrical Contact Stresses ........................................................................... 17 Cylindrical Contact ....................................................................................................... 17 Contact Stress ................................................................................................................ 18 Rolling Cylinders .......................................................................................................... 24 Pure Rolling: ............................................................................................................. 24 Rolling with Traction: ............................................................................................... 29 Where to Go From Here? .............................................................................................. 52 Chapter 3: Conical-Shaped Traction Drive Rollers .......................................................... 53 Why Conical Shaped Rollers? ...................................................................................... 53 Bevel gears in Differentials .......................................................................................... 56 Conical Shaped Rollers ................................................................................................. 62 Contact Stresses in Conical-Shaped Rollers ................................................................. 65 Pure Rolling: ............................................................................................................. 67 Rolling with Traction: ............................................................................................... 76 Chapter 4: Conclusion and Future Work .......................................................................... 97 Citations and References ............................................................................................. 100 IV Table of Figures FIGURE 1.1: SIMPLE TRACTION DRIVE SETUP ............................................................................... 6 FIGURE 1.2: OFFSET TRACTION DRIVE SYSTEM ........................................................................... 9 FIGURE 1.2A: FORWARD MOTION ...................................................................................................... 9 FIGURE 1.2B: REVERSE MOTION (NO ROTATION) ....................................................................... 10 FIGURE 1.3A: WEDGED ROLLER ARRANGEMENT....................................................................... 10 FIGURE 1.4: ROLLER TRACTION DRIVE FORCE DIRECTIONS ................................................ 11 FIGURE 1.5: MARS'S PATHFINDER (SOJOURNER) ROVER VEHICLE [4] ............................... 12 FIGURE 1.6: CAD MODEL OF THE TRACTION DRIVE SYSTEM (8) ........................................... 15 FIGURE 2.2A: EXTERNAL VIEW OF DEVELOPING SPALLS (11) ................................................ 21 FIGURE 2.2B: SECTION VIEW (SUBSURFACE) OF DEVELOPING SPALLS (11) ...................... 21 FIGURE 2.3: ARIAL VIEW OF RECTANGULAR CONTACT PATCH ........................................... 22 FIGURE 2.4: STRESS DISTRIBUTION AT SUBSURFACE [11) ........................................................ 23 FIGURE 2.5: ELLIPSOIDAL-PRISM PRESSURE DISTRIBUTION ................................................. 24 FIGURE 2.6A: TWO CYLINDERS IN CONTACT ............................................................................... 25 FIGURE 2.9: a-xn OR a- z,, VS. DISTANCE FROM CENTER OF CONTACT PATCH .................... 34 FIGURE 2.10: (Yxn vs. DISTANCE FROM CENTER OF CONTACT PATCH ................................ 35 FIGURE 2.11: a-x, VS. DISTANCE FROM CENTER OF CONTACT PATCH................................. 37 FIGURE 2.12: a-x VS. DISTANCE FROM CENTER OF CONTACT PATCH ................................. 39 FIGURE 2.13: a-z VS. DISTANCE FROM CENTER OF CONTACT PATCH .................................. 40 FIGURE 2.14: Txz vs. DISTANCE FROM CENTER OF CONTACT PATCH ................................. 41 FIGURE 2.15: a-y VS. DISTANCE FROM CENTER OF CONTACT PATCH ................................. 42 FIGURE 2.16: a-1 VS. DISTANCE FROM CENTER OF CONTACT PATCH .................................. 46 FIGURE 2.17: a-2VS. DISTANCE FROM CENTER OF CONTACT PATCH .................................. 47 v FIGURE 2.18: o-3 VS. DISTANCE FROM CENTER OF CONTACT PATCH .................................. 48 FIGURE 2.19: a' VS. DISTANCE FROM CENTER OF CONTACT PATCH .................................. 49 FIGURE 2.20: 'max vs. DISTANCE FROM CENTER OF CONTACT PATCH ............................... 50 FIGURE 3.1: BEVEL GEAR SETUP....................................................................................................... 54 FIGURE 3.2: HEAVY MACHINERY VEHICLE (BULLDOZER) ...................................................... 57 FIGURE 3.3: BOTTOM-VIEW OF A HEAVY MACHINERY VEHICLE (REAR-WHEEL DRNE) ............................................................................................................................................................. 58 FIGURE 3.4: OPEN DIFFERENTIAL .................................................................................................... 60 FIGURE 3.5: ISOMETRIC VIEW OF A BEVEL GEAR ARRANGEMENT ..................................... 62 FIGURE 3.6: AERIAL VIEW OF A BEVEL GEAR ARRANGEMENT ............................................. 62 FIGURE 3.7: FRONT VIEW OF A CONICAL TRACTION DRIVE ARRANGEMENT ................. 63 FIGURE 3.8: SIDE VIEW OF A CONICAL TRACTION DRIVE ARRANGEMENT ...................... 64 FIGURE 3.9: TRAPEZOID SHAPED CONTACT PATCH .................................................................. 65 FIGURE 3.10: RECTANGULAR SHAPED CONTACT PATCH ........................................................ 66 FIGURE 3.11: TRAPEZOIDAL-PRISM PRESSURE DISTRIBUTION ............................................. 67 FIGURE 3.12: SIDE CROSS-SECTIONAL VIEW IN A CONICAL SHAPED ROLLER ................ 69 FIGURE 3.13: CROSS-SECTIONAL AREA IN TWO CONES IN CONTACT ................................. 71 FIGURE 3.14: CLOSE-UP ON THE TRIANGLE DFC ......................................................................... 72 FIGURE 3.15: ISOMETRIC VIEW EXPLAINING THE 3D PROBLEM........................................... 73 FIGURE 3.16: AERIAL VIEW OF THE TRIANGLE NECESSARY TO CALCULATE B ............. 74 FIGURE 3.17: a x VS. DISTANCE FROM CENTER OF CONTACT PATCH ................................ 78 11 FIGURE 3.18: a x VS. DISTANCE FROM CENTER OF CONTACT PATCH................................. 80 1 FIGURE 3.19: a x VS. DISTANCE FROM CENTER OF CONTACT PATCH ................................. 81 FIGURE 3.20: a z VS. DISTANCE FROM CENTER OF CONTACT PATCH .................................. 82 FIGURE 3.21: ' xz vs. DISTANCE FROM CENTER OF CONTACT PATCH ................................. 83 FIGURE 3.22: a y VS. DISTANCE FROM CENTER OF CONTACT PATCH ................................. 84 Vl FIGURE 3.23: a 1 VS. DISTANCE FROM CENTER OF CONTACT PATCH .................................. 85 FIGURE 3.24: a 2 VS. DISTANCE FROM CENTER OF CONTACT PATCH .................................. 86 FIGURE 3.25: a 3 VS. DISTANCE FROM CENTER OF CONTACT PATCH .................................. 87 FIGURE 3.26: a VS. DISTANCE FROM CENTER OF CONTACT PATCH .................................. 88 FIGURE 3.27: rrnax VS. DISTANCE FROM CENTER OF CONTACT PATCH ............................... 89 FIGURE 3.28: PMAX VS. DISTANCE FROM APEX .............................................................................. 90 FIGURE 3.29: CONTACT