Coupled Belt-Pulley Mechanics in Serpentine

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Coupled Belt-Pulley Mechanics in Serpentine COUPLED BELT-PULLEY MECHANICS IN SERPENTINE BELT DRIVES DISSERTATION Presented in Partial Fulfillment of the Requirements of the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Lingyuan Kong, B.S., M.S. ***** The Ohio State University 2003 Dissertation Committee: Prof. Robert G. Parker, Advisor Approved by Prof. Stephen E. Bechtel Prof. Chia-Hsiang Menq _________________________ Advisor Prof. Rajendra Singh Department of Mechanical Engineering ABSTRACT Belt vibration and slip are primary concerns in the design of serpentine belt drives. Belt-pulley coupling is essential for the analysis. This work investigates issues to advance the understanding of belt-pulley mechanics. Closed-form eigensolution approximations for an axially moving beam with small bending stiffness are given. This model is the first order approximation for the transverse vibration of each span in a serpentine belt drive. Perturbation techniques for algebraic equations and the phase closure principle are used. The eigensolutions are interpreted in terms of propagating waves. For a complete serpentine belt drive, a hybrid continuous-discrete model is built. Incorporation of belt bending stiffness introduces linear belt-pulley coupling. This model can explain the transverse span vibrations caused by crankshaft pulley fluctuations at low engine idle speeds where other coupling mechanisms do not. For the steady state analysis, a novel transformation of the governing equations to a standard ODE form for general-purpose BVP solvers leads to numerically exact steady solutions. A closed-form singular perturbation solution is developed for the small bending stiffness case. A coupling indicator based on the steady state is defined to quantify the undesirable belt- pulley coupling. A spatial discretization is developed to find the free vibration ii eigensolutions. In contrast to prior formulations, this discretization is numerically robust and free of missing/false natural frequency concerns. New dynamic properties induced by bending stiffness are characterized. Dynamic response calculations using the discretized model follow naturally. The effects of major design variables are investigated. This provides knowledge to help optimize structural design, especially to reduce large belt transverse vibration. Finally, to better predict the belt-pulley contact interactions applicable to serpentine belt drives an improved model is established for the steady state mechanics. Bending stiffness is considered while other factors in the literature such as belt-pulley friction and belt inertia are retained. An iterative solution based on general-purpose BVP solvers is presented to determine the belt deflections and the distributions of speed, tension, and friction along the belt as well as the belt-pulley contact points and adhesion/slip zones on the pulleys. Key design criteria like maximum transmissible moment and power efficiency are examined. iii Dedicated to my wife, Hong Chi iv ACKNOWLEDGMENTS I wish to express my sincere thanks to Dr. Robert G. Parker for his assistance and guidance throughout the research project upon which this dissertation is based. Especially, I am grateful for his help in continuously correcting and modifying my writing, from which I benefited greatly. I also thank Dr. Stephen E. Bechtel, Dr. Chia- Hsiang Menq, and Dr. Rajendra Singh, who served as members of my dissertation committee. I would like to acknowledge the generous financial support given to this project by Mark IV/Dayco Corporation. Furthermore, I thank my colleagues for their suggestions regarding my dissertation. Finally I thank my wife who constantly supported me throughout this endeavor. v VITA March, 1972…..……………….Born Guizhou, China 1993……………………………B.S., Shanghai Jiaotong University 1999……………………………M.S., Tsinghua University 1999 – present …………………Graduate Research Assistant, The Ohio State University PUBLICATIONS 1. L. Kong and R. G. Parker, 2003, “Equilibrium and Belt-Pulley Vibration Coupling in Serpentine Belt Drives,” ASME Journal of Applied Mechanics, 70(5), pp.739-750. FIELDS OF STUDY Major Field: Mechanical Engineering vi TABLE OF CONTENTS Abstract ...............................................................................................................................ii Dedication .......................................................................................................................... iv Acknowledgments............................................................................................................... v Vita..................................................................................................................................... vi List of Figures ....................................................................................................................ix List of Tables....................................................................................................................xiii Chapters CHAPTER 1 INTRODUCTION ........................................................................................ 1 1.1 Motivation and Objectives ........................................................................................ 1 1.2 Literature Review...................................................................................................... 6 1.2.1 Vibration of axially moving materials ............................................................... 7 1.2.2 Serpentine belt drives......................................................................................... 8 1.2.3 Belt-pulley steady state contact mechanics...................................................... 12 1.3 Scope of Investigation............................................................................................. 13 CHAPTER 2 APPROXIMATE EIGENSOLUTIONS OF AXIALLY MOVING BEAMS WITH SMALL BENDING STIFFNESS.......................................................................... 17 2.1 Introduction ............................................................................................................. 18 2.2 Model Equations ..................................................................................................... 22 2.3 Application of the Phase Closure Principle............................................................. 26 2.4 Other Boundary Conditions .................................................................................... 32 CHAPTER 3 MODELING AND STEADY STATE ANALYSIS OF SERPENTINE BELT DRIVES WITH BENDING STIFFNESS..............................................................36 3.1 Introduction ............................................................................................................. 37 3.2 System Model.......................................................................................................... 40 vii 3.3 Numerical Solution ................................................................................................. 49 3.4 Numerical Results and Discussion.......................................................................... 52 3.5 Approximate Closed-Form Solution ....................................................................... 64 CHAPTER 4 DYNAMIC ANALYSIS OF SERPENTINE BELT DRIVES WITH BENDING STIFFNESS.................................................................................................... 79 4.1 Introduction ............................................................................................................. 80 4.2 Linearization of Equations of Motion ..................................................................... 83 4.3 Extended Operator Formulation.............................................................................. 88 4.4 Galerkin Discretization ........................................................................................... 94 4.5 Results and Discussion............................................................................................ 96 CHAPTER 5 STEADY STATE MECHANICS OF BELT-PULLEY SYSTEMS ........113 5.1 Introduction ........................................................................................................... 114 5.2 Nonlinear Equations of a Moving Curved Beam.................................................. 116 5.3 BVP Solver Based Method for Problem with Unknown Boundaries ................... 119 5.4 Steady State Analysis of Belt Pulley Drives: an Iteration Method ....................... 122 5.4.1 Regular Moment Transmission Problem ....................................................... 125 5.4.2 Maximum Transmissible Moment Problem................................................... 131 5.5 Results and Discussion.......................................................................................... 134 5.5.1 Example of regular moment transmission problem ....................................... 136 5.5.2 Example of maximum transmissible moment problem ................................. 140 CHAPTER 6 SUMMARY AND FUTURE WORK....................................................... 145 6.1 Summary ............................................................................................................... 145 6.2 Future Work .......................................................................................................... 150 REFERENCE……………… …………………………………………………………..158 viii LIST OF FIGURES Figure Page Figure 1.1 Installed serpentine belt drive............................................................................ 3 Figure 1.2 Sketch of
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