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Transmission Error in Spur Gears: Static and Dynamic Transmission Error in Spur Gears: Static and Dynamic Finite-Element Modeling and Design Optimization By Raul Tharmakulasingam BEng. MSc. (Eng) Submitted in accordance with the requirements for the degree of Doctor of Philosophy School of Engineering and Design Brunel University United Kingdom October 2009 ACKNOWLEDGEMENTS “Success is sweet: the sweeter if long delayed and attained through manifold struggles and defeats.” I would like to take this opportunity to thank everyone who has helped me in my pursuit of success in my PhD. Firstly; I would like to thank my supervisor Dr. Giulio Alfano for helping me shape my research and motivating me at the right times with a timely reminder of how much we have achieved. I would also like to Dr Mark Atherton for bringing a fresh perspective to everything we did during the research, and always providing a kind ear and an objective view to all our problems. I am also very grateful to Prof. Luiz Wrobel for providing me with the opportunity to undertake my PhD research, and also helping me with my numerous problems while carrying out my research. To the colleagues who helped me with moments of respite during a hard day’s work, inspiration by example, and lasting friendships, I thank you from the bottom of my heart and look forward to keeping in touch. My family who have endured my worst and celebrated my best of times during my PhD also receive a heartfelt embrace and acknowledgement for everything they have done for me. My dear friends who constantly reminded me of my duty to hand in my Thesis by the incessant question “Have you submitted yet?”. I thank you for the motivation and the celebrations afterwards! Finally, to the one person who understands me better than I do myself, Jonit, you are my life, my love and my soul. I could not have completed my PhD without you. You helped me in so many ways that I cannot begin to describe them on this finite piece of paper. I look forward to being there for you when you write up your PhD Thesis, and help you in every way I can. I just want to thank you and let you know that I will always remember and be there for you, always… ~ II ~ ABSTRACT The gear noise problem that widely occurs in power transmission systems is typically characterised by one or more high amplitude acoustic signals. The noise originates from the vibration of the gear pair system caused by transmission error excitation that arises from tooth profile errors, misalignment and tooth deflections. This work aims to further research the effect of tooth profile modifications on the transmission error of gear pairs. A spur gear pair was modelled using finite elements, and the gear mesh was simulated and analysed under static conditions. The results obtained were used to study the effect of intentional tooth profile modifications on the transmission error of the gear pair. A detailed parametric study, involving development of an optimisation algorithm to design the tooth modifications, was performed to quantify the changes in the transmission error as a function of tooth profile modification parameters as compared to an unmodified gear pair baseline. The work also investigates the main differences between the static and dynamic transmission error generated during the meshing of a spur gear pair model. A combination of Finite-Element Analysis, hybrid numerical/analytical methodology and optimisation algorithms were used to scrutinise the dynamic behaviour of the gear pairs under various operating conditions. ~ III ~ PUBLICATIONS Static and dynamic transmission error of spur gear pair, R. Tharmakulasingam, G. Alfano, M. Atherton, Submitted to Journal of Sound and Vibration: (July 2010). Reduction of gear pair transmission error with tooth profile modification, R. Tharmakulasingam, G. Alfano, M. Atherton, Proceedings of International conference on Sound and Vibration: (September 2008). ~ IV ~ Table of Contents 1. Introduction ............................................................................................................... 1 1.1 Background .......................................................................................................... 1 1.2 Scope and objectives ............................................................................................ 6 1.2.1 Comparison between STE and DTE ........................................................... 7 1.2.2 Full non-linear dynamic finite-element analyses of gear pair interaction... 7 1.2.3 Validation of our FE model with the Hybrid Numerical model ................. 8 1.2.4 Application of the automated profile modification tool to reduce TE ........ 8 1.3 Outline of Thesis .................................................................................................. 9 2. Fundamentals of gear design ................................................................................. 12 2.1 Introduction ............................................................................................................ 12 2.2 Types of gears ........................................................................................................ 13 2.2.1 Spur gears ....................................................................................................... 13 2.2.2 Helical gears ................................................................................................... 14 2.2.3 Bevel gears ..................................................................................................... 15 2.2.4 Worm gears ................................................................................................... 16 2.3 Gear selection criteria ............................................................................................ 17 2.4 Spur gear nomenclature ......................................................................................... 18 2.5 Velocity ratio .......................................................................................................... 20 2.6 Conjugate action..................................................................................................... 21 2.7 Gear tooth profile ................................................................................................... 22 2.8 Standardisation of gears ......................................................................................... 24 2.9 Contact ratio of gears ............................................................................................. 25 2.10 Interference in gears ............................................................................................. 27 2.11 Manufacturing of gear teeth ................................................................................. 28 2.11.1 Milling .......................................................................................................... 30 2.11.2 Shaping ......................................................................................................... 30 2.11.3 Hobbing ........................................................................................................ 30 2.12 Spur gears ............................................................................................................. 31 3. Literature review ..................................................................................................... 34 3.1 Introduction ............................................................................................................ 34 3.2 Engine related gear noise ....................................................................................... 34 3.3 Main types of gear noise ........................................................................................ 35 3.3.1 Gear rattle in gear transmissions .................................................................... 36 3.3.2 Gear whine in gear dynamics ......................................................................... 37 ~ V ~ 3.4 General gear noise research ................................................................................... 40 3.5 Introduction to transmission error .......................................................................... 42 3.5.1 Sources of transmission error ......................................................................... 43 3.5.2 Types of transmission error ............................................................................ 47 3.5.2.1 Manufacturing transmission error ........................................................... 47 3.5.2.2 Static transmission error .......................................................................... 48 3.5.2.3 Kinematic transmission error .................................................................. 48 3.5.2.4 Dynamic transmission error .................................................................... 49 3.6 Methods to estimate transmission error ................................................................. 49 3.7 Mesh stiffness modelling ....................................................................................... 51 3.8 Gear dynamic modelling ......................................................................................... 52 3.9 Non-linear finite element analysis of gear pairs .................................................... 53 3.10 Optimisation of gears ........................................................................................... 56 3.10.1 Optimisation criteria .................................................................................... 56 4. Spur Gear methodology and data validation
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