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Impact Damping and Friction in Non-Linear Mechanical Systems with Combined Rolling-Sliding Contact

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Impact Damping and Friction in Non-Linear Mechanical Systems with Combined Rolling-Sliding Contact Impact damping and friction in non-linear mechanical systems with combined rolling-sliding contact Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Sriram Sundar, B. E. Graduate Program in Mechanical Engineering The Ohio State University 2014 Dissertation Committee: Prof. Rajendra Singh, Advisor Prof. Dennis A Guenther Prof. Ahmet Kahraman Prof. Vishnu Baba Sundaresan Dr. Jason T Dreyer Copyright by Sriram Sundar 2014 ABSTRACT This research is motivated by the need to have a better understanding of the non- linear contact dynamics of systems with combined rolling-sliding contact such as cam- follower mechanism, gears and drum brakes. Such systems, in which the dominant elements involved in the sliding contact are rotating, have unique interaction among contact mechanics, siding friction and kinematics. Prior models used in the literature are highly simplified and do not use contact mechanics formulation hence the dynamics of the system are not well understood. The main objective of this research is to gain a fundamental understanding of the non-linearities and contact dynamics of such systems, for which a cam-follower mechanism is used as an example case. Specifically, the non- linearities, impact damping and coefficient of friction are analyzed in this study. The problem is examined using a combination of analytical, experimental, and numerical methods. First, the various non-linearities (kinematic, dry friction, and contact) of the cam- follower system with combined rolling-sliding contact are investigated using the Hertzian contact theory for both line and point contacts. Alternate impact damping formulations are assessed and the results are successfully compared with experimental results as available in the literature. The applicability of the coefficient of restitution model is also critically analyzed. Second, a new dynamic experiment is designed and instrumented to ii precisely acquire the impact events. A new time-domain based technique is adopted to accurately calculate the system response by minimizing the errors associated with numerical integration. The impact damping force is considered in a generalized form as a product of damping coefficient, indentation displacement raised to the power of damping index, and the time derivative of the indentation displacement. A new signal processing procedure is developed (in conjunction with a contact mechanics model) to estimate the impact damping parameters (damping coefficient and index) from the measurements by comparing (on the basis of three residues) them to the results from the contact mechanics model. Also few unresolved issues regarding the impact damping model are addressed using the experimental results. Third, the coefficient of friction under lubrication is estimated using the same experimental setup (operating under sliding conditions). A signal processing technique based on complex-valued Fourier amplitudes of the measured forces and acceleration is proposed to estimate the coefficient of friction. An empirical relationship for the coefficient of friction is suggested for different surface roughnesses based on a prior model under lubrication. Possible sources of errors in the estimation procedure are identified and quantified. Some of the major contributions of this research are as follows. First, impact damping model was determined experimentally and related unresolved issues were addressed. Second, coefficient of friction for a cam-follower system with point contact under lubricated condition was estimated. Finally, better understandings of the effect of non-linearities and shortcomings of coefficient of restitution formulation are obtained. iii Dedication To the lotus feet of my spiritual master His Holiness Sri Rangaraamaanuja Mahaadesikan iv ACKNOWLEDGEMENTS First, I would like to thank my advisor, Prof. Rajendra Singh, for his patience and guidance throughout my graduate study. His tremendous experience and knowledge has been helped me overcome the difficulties faced during this process. I also would like to express my deepest appreciation to Dr. Jason Dreyer for his extremely valuable support in the experimental work and many technical discussions. I would like to thank my committee members, Prof. Guenther, Prof. Kahraman and Prof. Sundaresan for their time to review my work. I also would like to thank Caterina Runyon-Spears for her careful reviews of this work and all the members of Acoustics and Dynamics Lab for their providing with an amicable atmosphere over the past four years. Special thanks to Laihang for helping me record the experimental data. I would like to thank the Vertical Lift Consortium, Inc., Smart Vehicles Concept Center (www.SmartVehicleCenter.org) and the National Science Foundation Industry/University Cooperative Research Centers program (www.nsf.gov/eng/iip/iucrc) for partially supporting this research. I am most grateful to my parents, brother, fiancée and other family members for their constant faith, support and patience. I would like to thank all my friends especially, Adarsh, Ranjit, Sriram, Saivageethi and Darshan who made my graduate life, away from home, a memorable one. Also a special thanks to Sriram’s mom, for her mother-like care during all her visits in these four years. v VITA December 25, 1985……………………………… Born - Chennai, India 2003……………………………………………… B. E. Mechanical Engineering Anna University, Chennai, India 2009 – Present…………………………………… University Fellow/ SVC Fellow Graduate Research Associate The Ohio State University Columbus, OH PUBLICATIONS 1. S. Sundar, J. T. Dreyer and R. Singh, Rotational sliding contact dynamics in a non- linear cam-follower system as excited by a periodic motion, Journal of Sound and Vibration, (2013). 2. S. Sundar, J. T. Dreyer and R. Singh, Effect of the tooth surface waviness on the dynamics and structure-borne noise of a spur gear pair, SAE Technical Paper 2013-01- 1877, 2013, SAE Noise and Vibration Conference. FIELDS OF STUDY Major Field: Mechanical Engineering Main Study Areas: Mechanical Vibrations, Nonlinear Dynamics, Sliding Contact Systems, Contact Mechanics. vi TABLE OF CONTENTS Page ABSTRACT……………………………………………………………...……………… ii DEDICATION…………………………………………………………………………... iv ACKNOWLEDGEMENTS……………………………………………………...………. v VITA……………………………………………………………………………..…….... vi LIST OF TABLES ............................................................................................................ xi LIST OF FIGURES ........................................................................................................ xiii LIST OF SYMBOLS ...................................................................................................... xix CHAPTER 1: INTRODUCTION....................................................................................... 1 1.1 Motivation ........................................................................................................ 1 1.2 Literature review............................................................................................... 2 1.3 Problem formulation......................................................................................... 4 References for Chapter 1 ..................................................................................... 10 CHAPTER 2: ROTATIONAL SLIDING CONTACT DYNAMICS IN A NON-LINEAR CAM-FOLLOWER SYSTEM AS EXCITED BY A PERIODIC MOTION………..… 16 2.1 Introduction .................................................................................................... 16 2.2 Problem formulation………………………………………………………... 17 2.3 Analytical model……………………………………………………….…… 20 2.3.1 Relationship between the coordinate systems……………….……. 20 vii 2.3.2 Equations of motion………………………………………………. 21 2.3.3 Static equilibrium and linearized natural frequency……………… 24 2.3.4 Contact damping and dry friction models……………………….... 25 2.4 Examination of the contact non-linearity and alternate damping models…... 27 2.5 Assessment of the coefficient of restitution ( ξ) concept……………………. 33 2.6 Study of the line and point contact models in the sliding contact regime….. 38 2.7 Analysis of the friction non-linearity……………………………………….. 40 2.7.1 Effect of direction………………………………………………… 40 2.7.2 Dynamic bearing and friction forces……………………………… 40 2.8 Study of kinematic non-linearity…………………………………………… 47 2.9 Conclusion………………………………………………………………….. 49 References for Chapter 2 ..................................................................................... 53 CHAPTER 3: ESTIMATION OF IMPACT DAMPING PARAMETERS FROM TIME- DOMAIN MEASUREMENTS ON A MECHANICAL SYSTEM……………...…….. 58 3.1 Introduction…………………………………………………………...…….. 58 3.2 Problem formulation……………………………………………………...… 59 3.3 Design of the laboratory experiment and instrumentation………………….. 61 3.4 Analytical model……………………………………………………………. 62 3.4.1 Kinematics of the system……………………………………….… 62 3.4.2 Non-contact regime……………………………………………..… 65 3.4.3 Contact regime…………………………………………………..... 66 3.5 Estimation of the impact damping parameters ( β and n)…………………… 68 viii 3.5.1 Time-domain based technique to estimate the system response….. 68 3.5.2 Signal processing procedure to estimate β and n…………………. 70 3.6 Error and sensitivity analyses on the estimation procedure……………….... 72 3.6.1 Error analysis…………………………………...………………… 72 3.6.2 Sensitivity analysis……………………………………..…………. 76 3.7 Estimation of the impact damping from the measurements………………… 79 3.8 Equivalent coefficient of restitution…………………………………………
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