Nonlinear Dynamics of Driveline Systems with Hypoid Gear Pair A dissertation submitted to the Division of Research and Advanced Studies of the University of Cincinnati in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in the Program of School of Dynamic Systems of the College of Engineering and Applied Science April 2012 by Junyi Yang M. S. Southeast University, Nanjing, P. R. China, 2007 B. S. Southeast University, Nanjing, P. R. China, 2004 Academic Committee Chair: Dr. Teik C. Lim Members: Dr. Dong Qian Dr. David F. Thompson Dr. Murali M. Sundaram ABSTRACT This dissertation research focuses on evaluating the nonlinear dynamics of driveline systems employed in motor vehicles with emphasis on characterizing the excitations and response of right-angle, precision hypoid-type geared rotor structure. The main work and contribution of this dissertation is divided into three sections. Firstly, the development of an asymmetric and nonlinear gear mesh coupling model will be discussed. Secondly, the enhancement of the multi-term harmonic balance method (HBM) is presented. Thirdly and as the final topic, the development of new dynamic models capable of evaluating the dynamic coupling characteristics between the gear mesh and other driveline structures will be addressed. A new asymmetric and nonlinear mesh model will be proposed that considers backlash, and the fact that the tooth surfaces of the convex and concave sides are different. The proposed mesh model will then be fed into a dynamic model of the right-angle gear pair to formulate the dimensionless equation of motion of the dynamic model. The multi-term HBM will be enhanced to simulate the right-angle gear dynamics by solving the resultant dimensionless equation of motion. The accuracy of the enhanced HBM solution will be verified by comparison of its results to the more computationally intensive direct numerical integration calculations. The stability of both the primary and sub-harmonic solutions predicted by applying multi-term HBM will be analyzed using the Floquent Theory. In addition, the stability analysis of the multi-term HBM solutions will be proposed as an approximate approach for locating the existence of sub- harmonic and chaotic motions. In this dissertation research, a new methodology to evaluate the dynamic interaction between the nonlinear hypoid gear mesh mechanism and the time-varying characteristics of the rolling element bearings will also be developed. The time-varying mesh parameters will be II obtained by synthesizing a 3-dimensional loaded tooth contact analysis (TCA) results. The time- varying stiffness matrix approach will be used to represent the dynamic characteristics of the rolling element bearings. An overall nonlinear dynamic model of the hypoid gear box considering elastic housing structure will be developed as well. A lumped parameter model of the flexible housing will be extracted form an appropriate set of frequency response functions through modal parameter identification method. In order to obtain the rotational coordinates, a rigid body interpolation of the translational responses at the bearing locations on the housing structure will be applied. The reduced model will be then coupled with the hypoid gear-shaft-bearing assembly model by applying a proposed dynamic coupling procedure. Finally, a hypoid geared rotor system model considering the propeller shaft flexibility will be established. The propeller shaft bending flexibility will be modeled as lumped parameter model through using the component mode synthesis (CMS). The torsional flexibility of propeller shaft will be simplified as a torsional spring connecting the inertia of moment of engine and pinion. Physically, the pinion input shaft is driven by the propeller shaft through a universal joint, which will be modeled as a flexible simple supported boundary condition as well as fluctuating rotation speed and torque excitation. III IV ACKNOWLEDGEMENTS I would like to thank Prof. Teik C. Lim, who is serving as my academic advisor and the chair of my academic committee, for his great instructions and support throughout my graduate study. I would also like to thank Dr. Dong Qian, Dr. David F. Thompson and Dr. Murali M. Sundaram for serving as my doctoral academic committee members. I wish to thank all my colleagues at the Vibro-Acoustic and Sound Quality Research Laboratory in University of Cincinnati for their friendship. I would like to express my special gratitude to Dr. Tao Peng for his valuable academic suggestions. Finally, I would like to thank my parents and my wife Si Chen for their support and patience during my graduate study. V TABLE OF CONTENTS ABSTRACT II ACKNOWLEDGEMENTS ........................................................................................................... V TABLE OF CONTENTS .............................................................................................................. VI LIST OF TABLES ......................................................................................................................... X LIST OF FIGURES ...................................................................................................................... XI LIST OF SYMBOLS ............................................................................................................... XVIII Chapter 1. Introduction ............................................................................................................... 1 1.1 Literature Review ................................................................................................................. 1 1.2 Scope and Objectives ............................................................................................................ 6 1.3 Organization .......................................................................................................................... 8 Chapter 2. A Review of Loaded Tooth Contact Analysis Approaches .................................... 12 2.1 Introduction ......................................................................................................................... 12 2.2 Gear Body Flexible Deflection ........................................................................................... 14 2.3 Gear Tooth Flexible Deflection .......................................................................................... 17 2.3.1 Cantilever Beam ........................................................................................................... 17 2.3.2 Cantilever Plate ............................................................................................................ 18 2.3.3 Shell .............................................................................................................................. 21 2.3.4 FEM Method ................................................................................................................ 23 2.3.5 Finite Strip Method....................................................................................................... 26 2.3.6 Finite Prism Method ..................................................................................................... 27 2.3.7 Combined Surface Integral and Finite Element Method .............................................. 27 2.4 Conclusion .......................................................................................................................... 29 VI Chapter 3. An Enhanced Multi-term Harmonic Balance Solution for Non-linear Period-one Dynamic Motions in Right-angle Gear Pairs ................................................................................ 31 3.1 Introduction ......................................................................................................................... 31 3.2 Mathematical Model ........................................................................................................... 34 3.3 Period-one Dynamics .......................................................................................................... 42 3.4 Parametric Studies .............................................................................................................. 46 3.4.1 Numerical Validation ................................................................................................... 47 3.4.2 Numerical Analysis ...................................................................................................... 50 3.5 Conclusion .......................................................................................................................... 58 Chapter 4. An Enhanced Multi-term Harmonic Balance Solution for Non-linear Period- Dynamic Motions in Right-angle Gear Pairs ................................................................................ 59 4.1 Introduction ......................................................................................................................... 59 4.2 Period- Sub-harmonic Response ...................................................................................... 62 4.3 Results and Discussion ....................................................................................................... 65 4.3.1 Comparison of HBM and numerical integration results ............................................... 66 4.3.2 Effect of Static Load ..................................................................................................... 75 4.3.3 Effect of Static Transmission Error Excitation ...........................................................