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Analytical Study on Compound Planetary Gear Dynamics

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Analytical Study on Compound Planetary Gear Dynamics Analytical Study On Compound Planetary Gear Dynamics DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Yichao Guo, B.S., M.S. Graduate Program in Mechanical Engineering The Ohio State University 2011 Dissertation Committee: Prof. Robert G. Parker, Advisor Prof. Daniel A. Mendelsohn, Co-advisor Prof.StephenE.Bechtel Dr. Sandeep M. Vijayakar c Copyright by Yichao Guo 2011 Abstract Noise and vibration are major concerns in applications of compound planetary gears, and dynamic analysis is essential to their reduction. This work conducts a series of analytical investigations on several problems in compound planetary gear dynamics. A purely rotational model for compound planetary gear is developed, and the unique modal properties for the natural frequency spectra and vibration modes are presented and analytically proved. This model aims to simplify subsequent analyses on compound planetary gear dynamics while keeping the main dynamic behavior generated by tooth mesh forces. A systematic study on general compound planetary gear eigensensitivities is per- formed by utilizing the rotational-translational model in [53]. The eigensensitivities are derived in compact, closed-form expressions for all parameter variations in both tuned (each stage in the system retains cyclic symmetry) and mistuned systems. The resultant formulae also suggest that modal strain and kinetic energy distribution plots are effective and straightforward means to identify the system parameters that have the greatest impact on adjusting the related natural frequency. ii Compound planetary gear natural frequency veering and crossing phenomena are also systematically examined. By grouping all system parameters into tuned and mis- tuned parameters, the veering/crossing patterns with respect to each system param- eter are determined. These patterns provide critical information on dramatic mode shape changes when tuning a compound planetary gear during the design stage. Gear mesh phases that are critical for analytical or computational study on com- pound planetary gear dynamics are defined and calculated analytically. All the mesh phases are grouped into a hierarchical structure of system-level, stage-level, and train- level mesh phases to simplify the subsequent analytical investigations. In addition to providing a complete procedure to determine all the necessary relative phases, the spe- cific relations between train-level relative phases are derived by applying the assembly conditions of compound planetary gears. Such relations, together with the systemat- ically defined mesh phases, provide the foundation for the general rules to suppress selected dynamic responses of a general compound planetary gear through proper mesh phasing. The resultant mesh phasing rules are crucial for the troubleshooting of the vibration and noise problems in compound planetary gear applications. Compound planetary gear parametric instabilities caused by mesh stiffness vari- ations are analytically explored. Systems with single or multiple mesh frequencies are investigated. The instability boundaries are derived for different cases depending on the degeneracy of the natural frequencies. Application of the well-defined modal properties yields simple, closed-form expressions for instability boundaries. Some instability boundaries vanish under specific mesh phasing conditions. The back-side mesh stiffness variation is inspected in this work to address the needs for gear vibration models that consider the gear tooth contacts on the back iii side. The results reveal the inherent relationship between the back-side and drive- side mesh stiffnesses. The impact of backlash on the back-side mesh stiffness variation function is also quantified. iv Dedicated to my wife, Zongcui Mu to my son, Kevin Guo and to my mother, Jinmei Wang v Acknowledgments I would like to express my sincere thanks to my advisor, Robert G. Parker, for the opportunity to pursue this Ph.D. degree, his generous academic guidance, the precious opportunities to explore the academic world through numerous conferences, and the chance to understand the engineering world through practical projects and trainings. In addition, his invaluable suggestions during manuscript preparations has helped me greatly improve my academic/technical writing skills. I want to express my sincere thanks to Prof. Mendelsohn for being my co-advisor. Without his help, the defense is not possible to happen. I sincerely thank Prof. Bechtel for his kindness of being the committee member of my Ph.D. dissertation. I am thankful to Dr. Vijayakar for providing exceptional finite element analysis capability that is used extensively in this work and for being the committee member of my Ph.D. dissertation. I also want to thank Ms. Sands for her kind help throughout my graduation and my lab mates in the Dynamics and Vibrations Lab for their help during my study in The Ohio State University. I owe my thanks to my wife and my son for their support and, particularly, I want to thank my mother, who recently injuried in a car accident while still supporting me to finalize my Ph.D. study. vi Vita July, 1999 ..................................B.S.,MechanicalEngineering, Shanghai Jiao Tong University, Shang- hai, China July, 2002 ..................................M.S.,VehicleEngineering, Shanghai Jiao Tong University, Shang- hai, China May, 2004 ..................................M.S.,MechanicalEngineering, State University of New York at Stony Brook, NY, USA September 2004 - September 2008 ..........University/GraduateFellowship, The Ohio State University September 2008 - Present ...................Powertrain System Engineer, Delphi Publications Research Publications Guo, Y., and Parker, R. G. “Natural Frequency Veering and Crossing Patterns for General Compound Planetary Gears”. In preparation, 2011. Guo, Y., and Parker, R. G. “Suppression of Various Modal Responses in General Compound Planetary Gears through Mesh Phasing”. In preparation, 2011. Guo, Y., and Parker, R. G. “Parametric Instabilities of General Compound Planetary Gear Caused by Mesh Stiffness Variations”. In preparation, 2011. Guo, Y., and Parker, R. G. “Back-side Contact Gear Mesh Stiffness”. 11th In- ternational ASME Power Transmission and Gearing Conference (DETC2011/PTG- 48055), accepted, Washington, DC. vii Guo, Y., and Parker, R. G. “Analytical Determination of Mesh Phase Relations in General Compound Planetary Gears”. Mechanism and Machine Theory, accepted. Guo, Y., and Parker, R. G. “Sensitivity of General Compound Planetary Gear Natural Frequencies and Vibration Modes to Model Parameters”. Journal of Vibration and Acoustics, 132(1):1–13, 2010. Guo, Y., and Parker, R. G. “Purely Rotational Model and Vibration Modes of Compound Planetary Gears”. Mechanism and Machine Theory, 45:365–377, 2010. Guo, Y., and Parker, R. G. “Mesh Phase Relations of General Compound Planetary Gears”. Proceedings of 10th International ASME Power Transmission and Gearing Conference (IDETC2007-35799), Las Vegas, NV. Guo, Y., and Parker, R. G. “Sensitivity of General Compound Planetary Gear’s Nat- ural Frequencies and Vibration Modes to Model Parameters”. Proceedings of 10th In- ternational ASME Power Transmission and Gearing Conference (IDETC2007-35802, Las Vegas, NV. Guo, Y., and Parker, R. G. “Sensitivity of Tuned General Compound Planetary Gear’s Natural Frequencies and Vibration Modes to Model Parameter”. Proceedings of 2006 International Mechanical Engineering Congress and Exposition (IMECE2006- 14978), Chicago, IL. Fields of Study Major Field: Mechanical Engineering viii Table of Contents Page Abstract....................................... ii Dedication...................................... v Acknowledgments.................................. vi Vita......................................... vii ListofTables.................................... xiii ListofFigures................................... xvi 1. Introduction.................................. 1 1.1MotivationandObjectives....................... 1 1.2LiteratureReview........................... 5 1.3ScopeofInvestigation......................... 11 2. Compound Planetary Gear Models and Associated Modal Properties . 14 2.1 Purely Rotational Model and Vibration Modes of Compound Plan- etaryGears............................... 14 2.1.1 Purely Rotational Model of Compound Planetary Gears . 15 2.1.2 Characteristics of Natural Frequencies and Vibration Modes 27 2.1.3 Summary for Purely Rotational Compound Planetary Gear Model.............................. 32 2.2 Rotational-translational Model of Compound Planetary Gears and theAssociatedModalProperties................... 35 2.2.1 Rotational-translational Model of Compound Planetary Gears 35 2.2.2 Modal Properties of General Compound Planetary Gears . 38 2.3Conclusion............................... 40 ix 3. Sensitivity of General Compound Planetary Gear Natural Frequencies and VibrationModestoModelParameters................... 42 3.1Introduction.............................. 42 3.2EigensensitivityCalculation...................... 44 3.3EigensensitivityofTunedSystems.................. 47 3.3.1 CalculationofEigensensitivityofTunedSystems...... 47 3.3.2 Application of the Modal Strain/Kinetic Energy Distribution Plots............................... 55 3.4EigensensitivityofMistunedCompoundPlanetaryGears...... 59 3.5Conclusion............................... 67 4. Natural Frequency Veering and Crossing Patterns for General Compound PlanetaryGears...............................
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