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Dynamic Modeling of Double-Helical Planetary Gear Sets

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Dynamic Modeling of Double-Helical Planetary Gear Sets DYNAMIC MODELING OF DOUBLE-HELICAL PLANETARY GEAR SETS DISSERTATION DRAFT Presented in Partial Fulfillment of the Requirements for The Degree of Doctor of Philosophy in the Graduate School of The Ohio State University By Sondkar Prashant, M.Tech. Graduate Program in Mechanical Engineering The Ohio State University 2012 Dissertation Committee: Dr. Ahmet Kahraman, Advisor Professor Daniel Mendelsohn Professor Manoj Srinivasan Professor Junmin Wang © Copyright by Sondkar Prashant 2012 ABSTRACT This dissertation aims at investigating the dynamic response of double-helical planetary gear sets theoretically. A three-dimensional discrete dynamic model of a double-helical planetary gear set is proposed, including all gear mesh, bearing and support structure compliances. The model is presented in three levels of complexity: (i) a linear time-invariant (LTI) model, (ii) a LTI model with gyroscopic effects included, and (iii) a nonlinear time-varying (NTV) model with parametrically time-varying gear mesh stiffnesses and nonlinear tooth separation effects included. As the first step, a generic linear (no tooth separations), time-invariant (constant gear mesh stiffnesses) dynamic model is formulated to analyze any N-planet double- helical planetary gear system. The model includes any planet phasing conditions dictated by the number of planets, number of gear teeth and planet position angles as well as any phase shifts due to the designed stagger between the right and left sides of the gear set. The forced response due to gear mesh transmission errors excitations is computed by using the modal summation technique with the natural modes found from the corresponding Eigen value problem for the undamped system. The strain energies of the mode shapes are computed to identify the modes excitable by these excitations. Parametric studies are presented to demonstrate sizable influences of planet phasing, ii stagger conditions, gear and carrier support conditions as well as the number of planets on the steady-state forced response. In the second modeling step, the LTI model is modified to include a class of gyroscopic effects due to vibratory skew of spinning gears for the case of a stationary carrier. The complex Eigen solutions are examined to quantify the influence of rotational speed of the gear set through gyroscopic effects on the natural modes. A complex modal summation formulation is used to compute the forced response with gyroscopic effects. Results indicate that the influence of gyroscopic moments on natural frequencies is modest within typical speed ranges, with only a sub-set of modes exhibiting dominant tilting motions impacted by the gyroscopic effects. Effect of gyroscopic moments on forced response curves is found to be limited to slight changes in amplitudes and frequencies of certain resonance peaks. As the final step, mesh stiffness fluctuations due to change in number of tooth pairs are introduced as internal parametric excitation along with the transmission error excitations at the same phasing relations. Tooth separation functions are also applied to obtain a set of NTV equations of motion, which are solved by using direct numerical integration. Differences observed between the forced response curves for time-varying and time-invariant systems are characterized by additional resonance peaks and overall increases in response amplitudes while no signs of nonlinear behavior are noted. iii DEDICATION Dedicated to My dear family iv ACKNOWLEDGMENTS I would like to express my sincere gratitude to Prof. Ahmet Kahraman for the opportunity, guidance, and support throughout my research at The Ohio State University. I am grateful to Prof. Daniel Mendensohn, Prof. Junmin Wang, and Prof. Manoj Srinivasan for serving on my dissertation committee. Special thanks to Mr. Jonny Harianto for providing software support. I am thankful to Mr. Sam Shon and Dr. David Talbot for their technical expertise and willingness to share it. I would also like to extend my thanks to all my lab mates, including, but not limited to Devin Hilty and Mohammad Hotait for their friendship and support throughout my work at Gear Lab. I am thankful to Pratt & Whitney for sponsoring this research activity. I want to sincerely thank my dear family for their continuous support and encouragement without which this work would not have been possible. v VITA June 2004 ....................................................................................... Bachelor of Engineering Pune University, India June 2006 ........................................................................................... Master of Technology Indian Institute of Technology (IIT), Madras, India 2005-2006 ............................................................................................... DAAD Fellowship Technical University of Darmstadt, Germany 2006-2008 ............................................................................... Engineer (Engine Dynamics) General Electric, India 2008-2012 ............................................................................... Graduate Research Associate The Ohio State University, OH FIELDS OF STUDY Major Field: Mechanical Engineering Focus on Gear Dynamics. vi TABLE OF CONTENTS ABSTRACT ........................................................................................................................ ii DEDICATION ................................................................................................................... iv ACKNOWLEDGMENTS ...................................................................................................v VITA .................................................................................................................................. vi LIST OF TABLES ............................................................................................................. xi LIST OF FIGURES .......................................................................................................... xii NOMENCLATURE ..........................................................................................................xv CHAPTERS: 1. Introduction ................................................................................................................1 1.1 Background and Motivation .............................................................................1 1.2 Literature Review .............................................................................................6 1.3 Scope and Objectives ......................................................................................11 1.4 Dissertation Outline ........................................................................................13 References for Chapter 1 ..........................................................................................15 vii 2. A Linear Time-invariant Dynamic Model of a Double-Helical Planetary Gear Set .... ........................................................................................................................20 2.1 Introduction .....................................................................................................20 2.2 Discrete Model and its Assumptions ..............................................................21 2.2.1 A Sun-Planet i Pair Formulation .........................................................23 2.2.2 A Ring-Planet i Pair Formulation .......................................................29 2.2.3 A Carrier-Planet i Pair Formulation ...................................................34 2.2.4 Coupling of the Left and Right Sides .................................................38 2.2.5 The Overall System Equations ...........................................................41 2.2.6 Excitations ..........................................................................................43 2.3 Solution Methodology ....................................................................................48 2.4 An Example Simulation ..................................................................................51 2.4.1 Influence of Right-to-left Stagger .......................................................54 2.4.2 Influence of Planet Phasing Conditions ..............................................65 2.4.3 Influence of Number of Planets ..........................................................68 2.4.4 Influence of Radially Floating Sun Gear ............................................71 2.5 Mode Identification using Modal Strain Energy ............................................73 2.6 Summary .........................................................................................................80 References for Chapter 2 ..........................................................................................81 3. Influence of Gyroscopic Effects on Dynamic Behavior of Double-Helical Planetary Gear Sets ..................................................................................................83 3.1 Introduction .....................................................................................................83 3.2 Incorporation of Gyroscopic Moments in the Dynamic Model ......................84 3.2.1 A Sun-Planet i Pair with Gyroscopic Effects .....................................85 viii 3.2.2 A Ring-Planet i Pair with Gyroscopic Effects ....................................87 3.2.3 A Carrier-Planet i Pair with Gyroscopic Effects ................................88 3.2.4 Coupling Elements with Gyroscopic Effects ......................................89
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