Dynamical Contact Problems with Friction

Walter Sextro

Dynamical Contact Problems with MODELS, Friction
METHODS, EXPERIMENTS AND APPLICATIONS
With 133 Figures

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Prof. Dr.-Ing. habil. Walter Sextro
Graz University of Technology Institute of Mechanics Kopernikusgasse 24/III A-8010 Graz Austria E-mail: sextro@tugraz.at

Originally published as volume 3 of the series Lecture Notes in Applied Mechanics, Springer-Verlag Berlin Heidelberg 2002. Library of Congress Control Number: 2006940067

ISBN-10 3-540-69535-4 Springer Berlin Heidelberg New York ISBN-13 978-3-540-69535-6 Springer Berlin Heidelberg New York
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Preface

Friction contacts are used to transmit forces or to dissipate energy. A better understanding of friction phenomena can result in improvements like the reduction of noise and maintenance costs, increased life time of machines and improved energy efficiency. There exists a rich literature on friction. Depending on the features of the friction contact, different contact models are applied and dependent on the contact model, different solution methods are preferred. The aim of this book is to describe an efficient procedure to model dynamical contact problems with friction. This procedure is applied to different practical problems and verified by experiments. The dynamics of the elastic bodies in contact are described by a reduced order model through the so called modal description, to speed up calculations. This description is presented in Chap. 2. In Chap. 3 the generalized contact model is developed, which includes the main physical effects like contact elasticity, roughness, friction characteristics etc.. The contact planes are discretized and a point contact model is applied to each area element. The thermomechanics of the contact is investigated, as well as the calculation of wear. The application of the combined procedure of the point contact model and the modal description of the elastic components are illustrated by three different examples. An impact and friction oscillator, see Chap. 4, is investigated in the time domain, while the friction damping of elastic structures with expanded friction contacts is analyzed within the frequency domain, see Chap. 5. The stationary rolling contact is presumed to calculate the wear of wheel-rail-systems, see Chap. 6. This work arose during my research at the Institute of Mechanics of the University of Hannover in Germany. Part of the work was supported by the “Forschungsvereinigung Verbrennungskraftmaschinen e.V. (FVV, Frankfurt)” and was sponsored by the “Bundesministerium für Wirtschaft” through the “Arbeitsgemeinschaft industrieller Forschungsvereinigungen e.V. (AiF, Köln), (AiF Nr. 10684)”, a federal collaboration of the turbomachinery-industry and the “Deutsche Forschungsgemeinschaft (Projekt Nr. SE 895/3-1)”. This book is based on the script that leads to my “Habilitation” in Mechanics. The “Habilitation” marks the end of the education as lecturer. In this context I would like to thank Prof. Dr.-Ing. habil. K. Popp, Prof. Dr.-Ing. habil. P. Wriggers and Prof. Dr.-Ing. habil. G.-P. Ostermeyer for carefully reading the script and for their support. Furthermore, I would like to thank all of my colleagues at the Institute of Mechanics for the open discussion of any problems and the successful cooperation. I

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Preface

would especially like to emphasize Prof. Dr.-Ing. habil. K. Popp. During my time at the institute, he always supported me and therefore most of my thanks belong to him. Hannover, 2002 Walter Sextro

Preface to the Second Edition

Since the last edition of this book the knowledge about friction has increased. Therefore several new results have been added like the wear calculation of a wheel-rail system as well as the efficient calculation of multi-coupled bladed disc assemblies with friction contacts. This book can be seen as the result of more than ten years research at the Institute of Mechanics (now Institute of Dynamics and Vibration) at the University of Hannover (now Leibnitz University Hannover). Again I have to thank Prof. Dr.-Ing. habil. Karl Popp for the good collaboration and his support. He passed away in April 2005 and therefore I would like to dedicate this book to him. Furthermore I would like to thank the “Deutsche Forschungsgemeinschaft (DFG)” for the financial support of the project of the “Forschergruppe: Dynamische Kontaktprobleme mit Reibung bei Elastomeren”. Regarding Chapter 3.6 I have to acknowledge Dr.-Ing. Markus Lindner, Dipl.-Ing. Patrick Moldenhauer and Dipl.-Ing. M. Wangenheimof of the Institute of Dynamics and Vibration, Leibnitz University Hannover, for their work done with regard to the friction characteristics of rubber. With regard to the results presented in Chapter 6.10 dealing with the instationary rolling contact I have to thank Dipl.-Ing. Florian Gutzeit. Furthermore I have to thank Dr. Jaroslaw Szwedowicz, ABB Turbo Systems, Baden, Switzerland, who carried out spin pit tests with regard of bladed disc assemblies with shrouds to validate the developed method as presented in Chapter 5. The numerical investigations in chapter 5.4 due to the multi-coupling of bladed disc was supported by the “Forschungsvereinigung Verbrennungskraftmaschinen e.V. (FVV, Frankfurt)” and was sponsored by the “Bundesministerium für Wirtschaft (BMWi)” through the “Arbeitsgemeinschaft industrieller Forschungsvereinigungen e.V. (AiF, Köln), (AiF Nr. 12565)”. Here, I have to thank the corresponding working group and the chairman Dr.-Ing. Karl Urlichs, Siemens Power Generation AG, Nürnberg for the good collaboration. The application of the theory to a system with extended friction contacts was carried out by Dipl.-Ing. Alexander Genzo, Volkswagen, Wolfsburg. I have to thank him for this investigation presented in Chapter 5.5 and Volkswagen for their support. Furthermore I would like to thank Dipl.-Ing. Ingo Kaiser, German Aerospace Center (DLR), Wesseling for his support in the application of the developed module to calculate the wear of railway wheels as presented in Chapter 6.9.

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Last but not least I would like to thank my wife Ursula and my children Marvin and Rocco for their moral support in writing this book. Graz, 2006 Walter Sextro

Contents

1 Introduction ......................................................................................................... 1 1.1 Problem Description ..................................................................................... 1 1.2 Review .......................................................................................................... 6 1.3 Aim of this Work ........................................................................................ 14 2 Dynamical Descriptions of Elastic Bodies....................................................... 17 2.1 Kinematics of Elastic Bodies...................................................................... 17 2.2 Governing Equations for Linear Elasticity................................................. 19 2.3 Equation of Motion of an Elastic Body ...................................................... 22 2.4 Modal Description of Elastic Bodies.......................................................... 27 3 Contact Model.................................................................................................... 31 3.1 Contact Stiffnesses and Microscopic Contact ............................................ 31 3.2 Limits of Application of the Contact Model .............................................. 36 3.3 Mesoscopic Contact Laws for Rough Surfaces.......................................... 40 3.4 Hyperbolic Contact Laws for Rough Surfaces ........................................... 49 3.5 Experimental Validation of the Contact Laws .......................................... 51 3.6 Friction Characteristics............................................................................... 53 3.7 Three-dimensional Point Contact Element................................................. 59 3.8 Microslip Effects due to Rough Surfaces ................................................... 69 3.9 Wear of Rough Surfaces............................................................................. 76 4 Oscillators with Elastic Contact and Friction ................................................ 81 4.1 System Description..................................................................................... 81 4.2 Oscillator with Elastic Contact ................................................................... 84 4.3 Friction Oscillator....................................................................................... 86 4.4 Bifurcations in Dynamical Systems with Friction...................................... 88 5 Friction Damping of Elastic Multibody Systems............................................ 93 5.1 Forced Vibrations of Elastic Structures...................................................... 93 5.2 Macroscopic Contact Model....................................................................... 97 5.3 Experimental Validation of the Contact Model........................................ 106 5.4 Spin Pit Tests of Bladed Disc Assemblies ............................................... 113 5.5 Optimization of Tuned Bladed Disc Assemblies with Friction Contacts 116 5.6 Vibration of Detuned Bladed Disc Assemblies with Friction Contacts... 122 5.7 Elastic Multibody Systems with Extended Friction Contacts .................. 128

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Contents

6 Rolling Contact................................................................................................ 131 6.1 Motivation ................................................................................................ 131 6.2 Normal Contact Kinematics ..................................................................... 132 6.3 Tangential Contact Kinematics ................................................................ 136 6.4 Contact Stiffnesses ................................................................................... 139 6.5 Generalized Contact Forces...................................................................... 144 6.6 Validation of the Rolling Contact Model ................................................. 149 6.7 Contact Temperature Distribution............................................................ 154 6.8 Wear Calculation ...................................................................................... 160 6.9 Wear of Railway Wheels.......................................................................... 163 6.10 Instationary Rolling Contact Tyre-Road ................................................ 164 7 Conclusion........................................................................................................ 169

References ........................................................................................................... 173 Nomenclature...................................................................................................... 185 Index .................................................................................................................... 189