dynamics of rotating machines

Cambridge University Press 978-0-521-85016-2 - Dynamics of Rotating Machines Michael I. Friswell, John E. T. Penny, Seamus D. Garvey and Arthur W. Lees Frontmatter More information

DYNAMICS OF ROTATING MACHINES This book equips the reader to understand every important aspect of the dynamics of rotating machines. Will the vibration be large? What influences machine stability? How can the vibration be reduced? Which sorts of rotor vibration are the worst? The book develops this understanding initially using extremely simple models for each phenomenon, in which (at most) four equations capture the behavior. More detailed models are then developed based on finite element (FE) analysis, to enable the accurate simulation of R is available the relevant phenomena for real machines. Analysis software compatible with MATLAB for download from the book’s Web site, www.cambridge.org/friswell, and novices to rotordynamics can expect to make good predictions of critical speeds and rotating mode shapes within days. The book is structured more for self-study than as a reference handbook and, as such, provides readers with more than 100 worked examples and more than 100 problems and solutions. Professor Michael I. Friswell joined Aston University as a Lecturer in 1987, after five years with the Admiralty Research Establishment in Portland. He moved to Swansea in 1993 and was promoted to a personal chair in 2000. Between 2002 and 2008, he was the Sir George White Professor of Aerospace Engineering at Bristol University before returning to Swansea in 2009 as Professor of Aerospace Structures. He received an EPSRC Advanced Research Fellowship (1996–2001), a Royal Society–Wolfson Research Merit Award (2002–2007), and an EC Marie Curie Excellence Grant (2005–2008). Professor Friswell has a wide range of research interests, primarily involving rotordynamics and structural dynamics, including inverse methods, condition monitoring, damping, nonlinear dynamics, and model-reduction methods. Professor Friswell’s recent associate editorships include the Journal of Intelligent Material Systems and Structures, Structural Health Monitoring, and the Journal of Vibration and Acoustics. He is a Fellow of the Institute of Mathematics and Its Applications and the Institute of Physics and a Member of the American Society of Mechanical Engineers. Professor John E. T. Penny served an apprenticeship with the English Electric Co. and worked for that company as a development engineer for three years. He then joined the staff at Aston University, initially as a Research Fellow, then as a Lecturer and Senior Lecturer, and became Head of the Mechanical and Electrical Engineering Department. Following this, Professor Penny became Director of Research at the School of Engineering and Applied Science. He has taught bachelor- and master’s-level students in vibration and rotordynamics and related topics, such as numerical analysis and instrumentation. His research interests include topics in structural dynamics and rotordynamics. He has published in journals including the Journal of Sound and Vibration, Mechanical Systems and Signal Processing, and AIAA Journal. He is now an Emeritus Professor at Aston University but is still teaching and doing research. Professor Penny is a Fellow of the Institute of Mathematics and Its Applications. Professor Seamus D. Garvey began his career with six years at GEC Large Electrical Machines Ltd., Rugby, and his first rotordynamics experience was acquired there. When he left the company in 1990, he was Principal Engineer for Mechanical Analysis and had written the computer program that has been used ever since for rotordynamics analysis. He then spent 10 years at Aston University, after which he joined Nottingham University as a Professor of Dynamics. He remains active in rotordynamics research – especially in the areas of active control and developing control forces through the airgaps of electrical machines – and serves on the organizing committees of both the IFToMM Rotordynamics conference and the IMechE Conference on Vibrations in Rotating Machines. He is currently Director of the RollsRoyce University Technology Centre in Gas Turbine Transmissions at Nottingham University. Professor Garvey is a Fellow of the Institution of Mechanical Engineers and a Member of the Institute of Engineering and Technology. Professor Arthur W. Lees has spent most of his career in the power-generation industry. After completing his PhD in physics, he joined the Central Electricity Generating Board, initially developing FE codes and later resolving plant problems. After a sequence of positions, he was appointed head of the Turbine Group for Nuclear Electric Plc. He moved to Swansea University in 1995, where his position was jointly funded by British Energy Plc and BNFL until August 2000. He was then appointed to a permanent chair within Swansea University. He is a regular reviewer of many technical journals and is currently on the editorial boards of the Journal of Sound and Vibration and Communications on Numerical Methods in Engineering. His research interests include structural dynamics, rotordynamics, and heat transfer. Professor Lees is a Fellow of the Institution of Mechanical Engineers and a Fellow of the Institute of Physics.

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Dynamics of Rotating Machines Michael I. Friswell Swansea University

John E. T. Penny Aston University

Seamus D. Garvey Nottingham University

Arthur W. Lees Swansea University


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cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, ˜ Paulo, Delhi, Dubai, Tokyo Sao Cambridge University Press 32 Avenue of the Americas, New York, NY 10013-2473, USA www.cambridge.org Information on this title: www.cambridge.org/9780521850162 C Michael I. Friswell, John E. T. Penny, Seamus D. Garvey, and Arthur W. Lees 2010

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2010 Printed in the United States of America A catalog record for this publication is available from the British Library. Library of Congress Cataloging in Publication data Fundamentals of rotor dynamics / Michael Friswell . . . [et al.]. p. cm. – (Cambridge aerospace series ; 26) Includes bibliographical references and index. ISBN 978-0-521-85016-2 (hardback) 1. Rotors – Dynamics. 2. Rotors – Vibration. I. Friswell, M. I. Title. II. Series. TJ1058.F86 2010 621.8 2 – dc22 2009042020 ISBN 978-0-521-85016-2 Hardback Additional resources for this publication at www.cambridge.org/friswell Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.


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Contents

Preface Acronyms

page xiii xv

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Overview 1.2 Rotating Machine Components 1.2.1 Features of Rotors 1.2.2 Features of Bearings and Rotor–Stator Interactions 1.2.3 Stators and Foundations 1.3 Aspects of Rotating Machine Behavior 1.3.1 Lateral Vibrations 1.3.2 Axial Vibrations 1.3.3 Torsional Vibrations 1.4 Examples of Rotating Machines 1.4.1 Electrical Machines 1.4.2 Turbo-Generator Sets 1.4.3 Gas Turbines 1.4.4 Vacuum Pumps 1.4.5 Vertical-Axis Pumps 1.5 Scope and Structure of the Book 1.6 Required Background Knowledge 1.7 Developing a Course of Instruction Using this Book 1.8 Software

1 2 3 3 4 5 5 6 6 7 7 10 11 11 13 13 15 15 15

2 Introduction to Vibration Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1 Introduction 2.2 Linear Systems 2.3 Single Degree of Freedom Systems 2.3.1 The Equation of Motion 2.3.2 Free Vibrations of a Single Degree of Freedom System 2.3.3 Forced Vibrations

17 18 19 20 22 25 vii


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viii

CONTENTS

2.4

2.5

2.6

2.7 2.8 2.9

2.3.4 Nonviscous Damping 2.3.5 Forced Vibration: Periodic Excitation 2.3.6 Forced Vibration: Arbitrary Excitation Multiple Degrees of Freedom Systems 2.4.1 System Equations 2.4.2 Free Vibrations of a Multiple Degrees of Freedom System 2.4.3 The Influence of Damping on the Free Response 2.4.4 Forced Vibrations of a Multiple Degrees of Freedom System 2.4.5 Computing the Receptance of an Undamped System by Modal Decomposition 2.4.6 Computing the Receptance of a Damped System by Modal Decomposition 2.4.7 Modal and Proportional Damping 2.4.8 Operating Deflection Shapes Imposing Constraints and Model Reduction 2.5.1 Model Reduction 2.5.2 Component Mode Synthesis Time Series Analysis 2.6.1 Simulation of a System Response 2.6.2 The Fourier Transform 2.6.3 The Discrete Fourier Transform Nonlinear Systems Summary Problems

28 30 31 32 32 34 38 41 43 45 49 51 51 52 55 58 58 60 61 64 70 70

3 Free Lateral Response of Simple Rotor Models . . . . . . . . . . . . . . . . 76 3.1 3.2 3.3 3.4 3.5

Introduction Coordinate Systems Gyroscopic Couples Dynamics of a Rigid Rotor on Flexible Supports A Rigid Rotor on Isotropic Flexible Supports 3.5.1 Neglecting Gyroscopic Effects and Elastic Coupling 3.5.2 Neglecting Gyroscopic Effects but Including Elastic Coupling 3.5.3 Including Gyroscopic Effects 3.5.4 Complex Coordinates 3.6 A Rigid Rotor on Anisotropic Flexible Supports 3.6.1 Forward and Backward Whirl 3.7 Natural Frequency Maps 3.8 The Effect of Damping in the Supports 3.8.1 Rigid Rotor on Isotropic Supports with Damping 3.8.2 Anisotropic Support Damping 3.9 Simple Model of a Flexible Rotor 3.10 Summary 3.11 Problems

76 77 78 80 83 84 87 90 95 96 98 103 107 108 109 112 116 117

4 Finite Element Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.1 Introduction


124

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CONTENTS

4.2 4.3 4.4 4.5 4.6

Defining Generalized Coordinates Finite Element Modeling of Discrete Components Axial Deflection in a Bar Lateral Deflection of a Beam Developing General Element Matrices 4.6.1 Axial Bar Element 4.6.2 Torsion Element 4.7 Assembling Global Matrices 4.8 General Finite Element Models 4.9 Summary 4.10 Problems

ix

126 127 129 134 137 138 140 142 143 151 151

5 Free Lateral Response of Complex Systems . . . . . . . . . . . . . . . . . . 155 5.1 5.2 5.3 5.4

Introduction Coordinate Systems Disk Elements Shaft Elements 5.4.1 Euler-Bernoulli Beam Theory 5.4.2 Including Shear and Rotary Inertia Effects 5.4.3 The Effect of Axial Loading 5.4.4 Mass and Stiffness Matrices for Shaft Elements 5.4.5 Gyroscopic Effects 5.4.6 The Effect of Torque 5.4.7 Tapered-Shaft Elements 5.4.8 Rotor Couplings 5.5 Bearings, Seals, and Rotor–Stator Interactions 5.5.1 Hydrodynamic Journal Bearings 5.5.2 Hydrostatic Journal Bearings 5.5.3 Rolling-Element Bearings 5.5.4 Magnetic Bearings 5.5.5 Rigid Bearings 5.5.6 Seals 5.5.7 Alford’s Force 5.5.8 Squeeze-Film Dampers 5.5.9 Unbalanced Magnetic Pull 5.6 Modeling Foundations and Stators 5.7 Assembly of the Full Equations of Motion 5.7.1 Speed Dependence of the System Matrices 5.7.2 Branching 5.8 Free Response of Complex Systems 5.8.1 Features of Eigenvalues and Eigenvectors 5.8.2 Number of Degrees of Freedom Required in a Model 5.8.3 The Effect of Shear and Rotary Inertia 5.8.4 Modeling the Shaft and Disk Interface 5.9 Modeling Examples 5.10 Summary 5.11 Problems


155 155 156 158 159 162 166 167 169 171 172 174 175 176 182 182 185 187 188 189 189 191 194 195 196 197 198 199 200 202 204 205 221 221

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x

CONTENTS

6 Forced Lateral Response and Critical Speeds . . . . . . . . . . . . . . . . . 228 6.1 Introduction 6.2 Simple Models of Rotors 6.2.1 Modeling Out-of-Balance Forces and Moments 6.2.2 Response of a Rigid Rotor on Isotropic Supports to Out-of-Balance Forces 6.2.3 Response of a Jeffcott Rotor to Out-of-Balance Forces 6.2.4 Response of an Isotropic Rotor System to Out-of-Balance Moments 6.2.5 Response of a Rigid Rotor on Anisotropic Supports to Out-of-Balance Forces and Moments 6.2.6 Forward- and Backward-Whirl Orbits 6.2.7 Response of Bent Rotors 6.3 Complex Rotor Models 6.3.1 Response of Rotors to Out-of-Balance Forces and Moments 6.3.2 Harmonic or Sub-Harmonic Response of Rotors to Sinusoidal Forces 6.3.3 Response of Bent Rotors 6.3.4 Response to Forces Applied through Auxiliary Bearings 6.4 Forces on the Supports due to Rotor Vibration 6.5 Response to Ground Vibration 6.6 Co-axial Rotors 6.7 Formal Definitions of Critical Speeds 6.8 Computing Critical Speeds 6.8.1 A Direct Approach 6.8.2 An Iterative Approach 6.8.3 Features of Critical Speeds 6.9 Mode Shapes Associated with Critical Speeds 6.10 Maps of Critical Speeds and Mode Shapes 6.11 Running through Critical Speeds 6.12 Stresses in a Rotor 6.12.1 Radial and Hoop Stresses due to Spin 6.12.2 Axial Stresses due to Lateral Deformation of the Rotor 6.13 Summary 6.14 Problems

228 230 230 234 241 242 245 246 248 251 251 258 259 263 266 267 271 275 276 277 278 278 281 281 282 288 288 289 291 291

7 Asymmetric Rotors and Other Sources of Instability . . . . . . . . . . . . 296 7.1 Introduction 7.2 Rotating Coordinate Systems 7.3 Rotor Asymmetry with Isotropic Supports: Simple Rotors 7.3.1 Relating Frequencies in the Stationary and Rotating Coordinate Systems 7.3.2 Stability of Asymmetric Rotors 7.3.3 The Effect of External Damping on the Asymmetric Rotor 7.3.4 Unbalance Response 7.3.5 The Gravity Critical Speed


296 297 299 301 303 305 307 311

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CONTENTS

7.3.6 Response to Sinusoidal Excitation in the Stationary Frame 7.3.7 Response to General Excitation in the Stationary Frame 7.4 Asymmetric Rotors Supported by Anisotropic Bearing: Simple Rotors 7.5 Internal Rotor Damping: Simple Rotors 7.6 Rotor Asymmetry with Isotropic Supports: Complex Rotors 7.6.1 Disks 7.6.2 Shaft Elements 7.6.3 Bearings and Foundations 7.6.4 The Equations of Motion 7.7 Internal Rotor Damping: Complex Rotors 7.8 Internal Cross-Coupling in the Bearing: Simple Rotors 7.9 Internal Cross-Coupling in the Bearing: Complex Rotors 7.10 Summary 7.11 Problems

xi

312 313 315 318 321 322 323 325 326 330 334 335 336 336

8 Balancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 8.1 8.2 8.3 8.4

Introduction Balancing Rigid Rotors at the Design Stage The Shaft Marker and the Phase of Response Signals Field Balancing of Rigid Rotors 8.4.1 Single-Plane Balancing 8.4.2 Two-Plane Balancing 8.5 Field Balancing of Flexible Rotors 8.5.1 The Influence-Coefficient Method 8.5.2 Modal Balancing 8.6 Balancing Machines without a Phase Reference 8.7 Automatic Balancing Methods 8.8 Issues in Balancing Real Machines 8.9 Summary 8.10 Problems

339 340 342 343 345 347 351 351 363 369 372 373 376 376

9 Axial and Torsional Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 9.1 9.2 9.3 9.4 9.5 9.6

Introduction Simple System Models for Axial Vibrations Shaft-Line Finite Element Models for Axial Vibrations Simple System Models for Torsional Vibrations Shaft-Line Finite Element Analysis of Torsional Motion Geared and Branched Systems 9.6.1 Applying Constraints for Geared Systems 9.6.2 A More Formal Approach to Geared Systems 9.6.3 Developing a Transformation to Effect Constraints 9.7 Axial and Torsional Vibration with External Excitation 9.7.1 Force-Driven Excitation of Torsional Vibration 9.7.2 Force-Driven Excitation of Axial Vibration 9.7.3 Displacement-Driven Excitation of Torsional Vibration 9.8 Parametric Excitation of Torsional Systems


383 384 386 387 389 390 390 392 398 400 401 409 410 413

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xii

CONTENTS

9.9 Summary 9.10 Problems

415 415

10 More Complex Rotordynamic Models . . . . . . . . . . . . . . . . . . . . . . 420 10.1 Introduction 10.2 Simple Rotating Elastic Systems 10.2.1 Stress and Geometric Stiffening 10.2.2 Damping in a Spinning Rotor 10.3 Finite Element Analysis of Rotors with Deformable Cross Sections 10.3.1 General Finite Element Models 10.3.2 Axisymmetric Finite Element Rotor Models 10.4 Rotor with Flexible Disks 10.4.1 Analysis of a Single Flexible Disk 10.4.2 Analysis of Rotor–Disk Assemblies 10.5 Detailed Models for Axial Vibration 10.6 Detailed Models for Torsional Vibration 10.7 Rotors Consisting of a Flexible Cylinder 10.8 Bending Vibrations of Blades Attached to Rotors 10.9 Coupled Systems 10.10 Rotor–Stator Contact in Rotating Machinery 10.11 Alignment 10.12 Nonlinear Bearings, Oil Whirl, and Oil Whip 10.12.1 Oil Whirl 10.12.2 Nonlinear Bearing Models and Oil Whip 10.13 The Morton Effect 10.14 Cracked Rotors 10.15 Summary

420 421 424 430 431 431 436 440 440 442 446 448 449 453 456 459 466 472 473 477 481 481 489

Solutions to Problems

491

Appendix 1 Properties of Solids

499

Appendix 2 Stiffness and Mass Coefficients for Certain Beam Systems

500

Appendix 3 Torsional Constants for Shaft Sections

505

Bibliography

507

Index

519


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Preface

This book addresses the dynamics of rotating machines, and its purpose may be considered threefold: (1) to inform readers of the various dynamic phenomena that may occur during the operation of machines; (2) to provide an intuitive understanding of these phenomena at the most basic level using the simplest possible mathematical models; and (3) to elucidate how detailed modeling may be achieved. This is an engineering textbook written for engineers and students studying engineering at undergraduate and postgraduate levels. Its aim is to allow readers to learn and gain a comprehensive understanding of the dynamics of rotating machines by reading, problem solving, and experimenting with rotor models in software. The book deliberately eschews any detailed historical accounts of the development of thinking within the dynamic analysis of rotating machines, focusing exclusively on modern matrix-based methods of numerical modeling and analysis. The structure of the book (described in Chapter 1) is driven largely by the desire to introduce the subject in terms of matrix formulations, beginning with the exposition of the necessary matrix algebra. All of the authors are avid devotees of matrix-based approaches to dynamics problems and all are constantly inspired by the intricacy and detail that emerge from even relatively simple numerical models. The emergence of software packages such as MATLAB that enable what would once have been considered large matrix computations to be conducted easily on a personal computer is one of the most exciting and important innovations in dynamics in the past two decades. With such a package, sophisticated models of machines can be assembled “from scratch” using only a few prewritten functions, which are available from the Web site associated with this book. This book was written in a period of several years and, during that time, the single remark that emerged most often among the authors is this: “There is always more to discover about the dynamics of rotating machines”; this remark is usually exclaimed in wonder. It has been a pleasure to write this book and we hope that this pleasure is visible to and shared by readers. We thank our respective wives, Wendy, Wendy, Antonia, and Rita, for their patience, and the publishers for their considerable forbearance. During the preparation of the manuscript, we drew on the knowledge and insight of many other seasoned practitioners in the field – too many to thank individually – but a collective acknowledgment is entirely appropriate because it is heartfelt. xiii


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Acronyms

BSF

ball spin frequency

DFT

discrete Fourier transform

FE

finite element

FEA

finite element analysis

FEM

finite element method

FFT

fast Fourier transform

FRF

frequency response function

FTF

fundamental train frequency

IRS

Improved Reduced System

ISO

International Organization for Standardization

MMF

magneto-motive force

ODE

ordinary differential equation

ODS

operating deflection shape

SEREP

System Equivalent Reduction Expansion Process

UMP

unbalanced magnetic pull

xv


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dynamics of rotating machines

dynamics of rotating machines

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