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Dynamics of Rotating Machines 1st Edition Michael I.
Friswell Digital Instant Download
Author(s): Michael I. Friswell, John E. T. Penny, Seamus D. Garvey, Arthur
W. Lees
ISBN(s): 9780521850162, 0521850169
Edition: 1
File Details: PDF, 29.83 MB
Year: 2010
Language: english
CAMBRIDGE

AEROSPACE

S E R I E S

DYNAMICS
OF ROTATING
MACHINES
Mtchae! I. Friswell
John E. T. Penny
Seamus D. Garvey
Arthur W. Lees

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i l a k ' Ol P h y s ic s .

Cam b r id g e
U N IV E R S IT V 1‘ RliSS
www.rambri dye org

■I I I H I M I ■■ ■ I III I I ■ ■ ■ ■ ■ I I ■■ II
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 sim­
ple 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
the relevant phenomena for real machines. Analysis software compatible with M A T L A B ® is available
for download from the book’s W eb 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 Struc­
tures. H e received an EPSRC Advanced Research Fellowship (1996-2001), a Royal Society-W olfson
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 dynam­
ics, including inverse methods, condition monitoring, damping, nonlinear dynamics, and model-reduction
methods. Professor Friswell’s recent associate editorships include the Journal o f Intelligent Material Sys­
tems and Structures, Structural Health Monitoring, and the Journal o f 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 o f 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 o f Engineering and Applied Science. He has taught bachelor- and masler’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. H e has published in journals
including the Journal o f Sound and Vibration, Mechanical Systems and Signal Processing, and A IA A
Journal. H e 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 G EC 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. H e 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 Rolls-
Royce University Technology Centre in Gas Turbine Transmissions at Nottingham University. Professor
Garvey is a Fellow of the Institution o f Mechanical Engineers and a Member of the Institute of Engi­
neering and Technology.

Professor Arthur W. Lees has spent most of his career in the power-generation industry. After complet­
ing 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 Pic. He moved to Swansea University in 1995, where his position was jointly
funded by British Energy Pic 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 o f the Journal o f Sound and Vibration and Communications on Numerical Methods in
Engineering. His research interests include structural dynamics, rotordynamics, and heal transfer. Pro­
fessor Lees is a Fellow o f the Institution of Mechanical Engineers and a Fellow of the Institute of Physics.
Cambridge A erospace Series

Editors: Wei Shyy and Michael J. Rycroft

1. J. M. Rolfe and K. J. Staples (eds.): Flight Simulation

2. P. Berlin: The Geostationary Applications Satellite
3. M. J. T. Smith: Aircraft Noise
4. N. X. Vinh: Flight Mechanics of High-Performance Aircraft
5. W. A. Mair and D. L. Birdsall: Aircraft Performance
6. M. J. Abzug and E. E. Larrabee: Airplane Stability and Control
7. M. J. Sidi: Spacecraft Dynamics and Control
8. J. D. Anderson: A History of Aerodynamics
9. A. M. Cruise, J. A. Bowles, C. V. Goodall, and T. J. Patrick: Principles of Space
Instrument Design
10. G. A. Khoury and J. D. Gillett (eds.): Airship Technology
11. J. Fielding: Introduction to Aircraft Design
12. J. G. Leishman: Principles of Helicopter Aerodynamics, 2nd Edition
13. J. Katz and A. Plotkin: Low-Speed Aerodynamics, 2nd Edition
14. M. J. Abzug and E. E. Larrabee: Airplane Stability and Control: A History of the
Technologies that Made Aviation Possible, 2nd Edition
15. D. H. Hodges and G. A. Pierce: Introduction to Structural Dynamics and Aeroelasticity
16. W. Fehse: Automatic Rendezvous and Docking of Spacecraft
17. R. D. Flack: Fundamentals of Jet Propulsion with Applications
18. E. A. Baskharone: Principles of Turbomachinery in Air-Breathing Engines
19. D. D. Knight: Numerical Methods for High-Speed Flows
20. C. Wagner, T. Hiittl, and P. Sagaut: Large-Eddy Simulation for Acoustics
21. D. Joseph, T. Funada, and J. Wang: Potential Flows of Viscous and Viscoelastic Fluids
22. W. Shyy, Y. Lian, H. Liu, J. Tang, and D. Viieru: Aerodynamics of Low Reynolds
Number Flyers
23. J. H. Saleh: Analyses for Durability and System Design Lifetime
24. B. K. Donaldson: Analysis of Aircraft Structures, 2nd Edition
25. C. Segal: The Scramjel Engine: Processes and Characteristics
26. M. I. Friswell, J. E. T. Penny, S. D. Garvey, and A. W. Lees: Dynamics of Rotating
Machines
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

C a m b r id g e
U N IV E R SIT Y PRESS
C A M B R ID G E U N IV E R S IT Y P R E S S
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore,
Sao Paulo, Delhi, Dubai, Tokyo

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

© 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 fo r this publication is available from the British Library.

Library o f Congress Cataloging in Publication data

Fundamentals o f rotor dynamics / Michael Friswell. .. [et al.J.

p. cm. - {Cambridge aerospacc series ; 26)
Includes bibliographical references and index.
ISBN 978-0-521-85016-2 (hardback)
1. Rotors - Dynamics. 2. Rotors - Vibration. I. Friswell, M. 1. 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.
Contents

Preface page xiii

Acronyms xv

1 Introduction............................................................................................... .. 1
1.1 Overview 1
1.2 Rotating Machine Components 2
1.2.1 Features of Rotors 3
1.2.2 Features of Bearings and Rotor-Stator Interactions 3
1.2.3 Stators and Foundations 4
1.3 Aspects of Rotating Machine Behavior 5
1.3.1 Lateral Vibrations 5
1.3.2 Axial Vibrations 6
1.3.3 Torsional Vibrations 6
1.4 Examples of Rotating Machines 7
11.4.1 Electrical Machines 7
1.4.2 Turbo-Generator Sets 10
1.4.3 Gas Turbines 11
1.4.4 Vacuum Pumps 11
1.4.5 Vertical-Axis Pumps 13
1.5 Scope and Structure of the Book 13
1.6 Required Background Knowledge 15
1.7 Developing a Course of Instruction Using this Book 15
1.8 Software 15

2 Introduction to Vibration A nalysis.................................................................. 17

2.1 Introduction 17
2 .2 Linear Systems 18
2.3 Single Degree of Freedom Systems 19
2.3.1 The Equation of Motion 20
2.3.2 Free Vibrations of a Single Degree of Freedom System 22
2.3.3 Forced Vibrations 25
 2.3.4 Nonviscous Damping 28
2.3.5 Forced Vibration: Periodic Excitation 30
2.3.6 Forced Vibration: Arbitrary Excitation 31
2.4 Multiple Degrees of Freedom Systems 32
2.4.1 System Equations 32
2.4.2 Free Vibrations of a Multiple Degrees of Freedom System 34
2.4.3 The Influence of Damping on the Free Response 38
2.4.4 Forced Vibrations of a Multiple Degrees of Freedom System 41
2.4.5 Computing the Receptance of an Undamped System
by Modal Decomposition 43
2.4.6 Computing the Receptance of a Damped System
by Modal Decomposition 45
2.4.7 Modal and Proportional Damping 49
2.4.8 Operating Deflection Shapes 51
2.5 Imposing Constraints and Model Reduction 51
2.5.1 Model Reduction 52
2.5.2 Component Mode Synthesis 55
2.6 Time Series Analysis 58
2.6.1 Simulation of a System Response 58
2.6.2 The Fourier Transform 60
2.6.3 The Discrete Fourier Transform 61
2.7 Nonlinear Systems 64
2.8 Summary 70
2.9 Problems 70

3 Free Lateral Response of Simple Rotor M o d e ls......................................... 76

3.1 Introduction 76
3.2 Coordinate Systems 77
3.3 Gyroscopic Couples 78
3.4 Dynamics of a Rigid Rotor on Flexible Supports 80
3.5 A Rigid Rotor on Isotropic Flexible Supports 83
3.5.1 Neglecting Gyroscopic Effects and Elastic Coupling 84
3.5.2 Neglecting Gyroscopic Effects but Including Elastic Coupling 87
3.5.3 Including Gyroscopic Effects 90
3.5.4 Complex Coordinates 95
3.6 A Rigid Rotor on Anisotropic Flexible Supports 96
3.6.1 Forward and Backward Whirl 98
3.7 Natural Frequency Maps 103
3.8 The Effect of Damping in the Supports 107
3.8.1 Rigid Rotor on Isotropic Supports with Damping 108
3.8.2 Anisotropic Support Damping 109
3.9 Simple Model of a Flexible Rotor 112
3.10 Summary 116
3.11 Problems 117

4 Finite Element Modeling ............................................................................... 124

4.1 Introduction 124
 4.2 Defining Generalized Coordinates 126
4.3 Finite Element Modeling of Discrete Components 127
4.4 Axial Deflection in a Bar 129
4.5 Lateral Deflection of a Beam 134
4.6 Developing General Element Matrices 137
4.6.1 Axial Bar Element 138
4.6.2 Torsion Element 140
4.7 Assembling Global Matrices 142
4.8 General Finite Element Models 143
4.9 Summary 151
4.10 Problems 151

5 Free Lateral Response of Complex System s.............................................. 155

5.1 Introduction 155
5.2 Coordinate Systems 155
5.3 Disk Elements 156
5.4 Shaft Elements 158
5.4.1 Euler-Bernoulli Beam Theory 159
5.4.2 Including Shear and Rotary Inertia Effects 162
5.4.3 The Effect of Axial Loading 166
5.4.4 Mass and Stiffness Matrices for Shaft Elements 167
5.4.5 Gyroscopic Effects 169
5.4.6 The Effect of Torque 171
5.4.7 Tapered-Shaft Elements 172
5.4.8 Rotor Couplings 174
5.5 Bearings, Seals, and Rotor-Stator Interactions 175
5.5.1 Hydrodynamic Journal Bearings 176
5.5.2 Hydrostatic Journal Bearings 182
5.5.3 Rolling-Element Bearings 182
5.5.4 Magnetic Bearings 185
5.5.5 Rigid Bearings 187
5.5.6 Seals 188
5.5.7 Alford's Force 189
5.5.8 Squeeze-Film Dampers 189
5.5.9 Unbalanced Magnetic Pull 191
5.6 Modeling Foundations and Stators 194
5.7 Assembly of the Full Equations of Motion 195
5.7.1 Speed Dependence of the System Matrices 196
5.7.2 Branching 197
5.8 Free Response of Complex Systems 198
5.8.1 Features of Eigenvalues and Eigenvectors 199
5.8.2 Number of Degrees of Freedom Required in a Model 200
5.8.3 The Effect of Shear and Rotary Inertia 202
5.8.4 Modeling the Shaft and Disk Interface 204
5.9 Modeling Examples 205
5.10 Summary 221
5.11 Problems 221
6 Forced Lateral Response and Critical S p e e d s............................................228
6.1 Introduction 228
6.2 Simple Models of Rotors 230
6.2.1 Modeling Out-of-Balance Forces and Moments 230
6.2.2 Response of a Rigid Rotor on Isotropic Supports
to Out-of-Balance Forces 234
6.2.3 Response of a Jeffcott Rotor to Out-of-Balance Forces 241
6.2.4 Response of an Isotropic Rotor System to Out-of-Balance
Moments 242
6.2.5 Response of a Rigid Rotor on Anisotropic Supports
to Out-of-Balance Forces and Moments 245
6.2.6 Forward- and Backward-Whirl Orbits 246
6.2.7 Response of Bent Rotors 248
6.3 Complex Rotor Models 251
6.3.1 Response of Rotors to Out-of-Balancc Forces and Moments 251
6.3.2 Harmonic or Sub-Harmonic Response of Rotors to
Sinusoidal Forces 258
6.3.3 Response of Bent Rotors 259
6.3.4 Response to Forces Applied through Auxiliary Bearings 263
6.4 Forces on the Supports due to Rotor Vibration 266
6.5 Response to Ground Vibration 267
6.6 Co-axial Rotors 271
6.7 Formal Definitions of Critical Speeds 275
6.8 Computing Critical Speeds 276
6.8.1 A Direct Approach 277
6.8.2 An Iterative Approach 278
6.8.3 Features of Critical Speeds 278
6.9 Mode Shapes Associated with Critical Speeds 281
6.10 Maps of Critical Speeds and Mode Shapes 281
6.11 Running through Critical Speeds 282
6.12 Stresses in a Rotor 288
6.12.1 Radial and Hoop Stresses due to Spin 288
6.12.2 Axial Stresses due to Lateral Deformation of the Rotor 289
6.13 Summary 291
6.14 Problems 291

7 Asymmetric Rotors and Other Sources of Instability.............................. 296

7.1 Introduction 296
7.2 Rotating Coordinate Systems 297
7.3 Rotor Asymmetry with Isotropic Supports: Simple Rotors 299
7.3.1 Relating Frequencies in the Stationary and Rotating
Coordinate Systems 301
7.3.2 Stability of Asymmetric Rotors 303
7.3.3 The Effect of External Damping on the Asymmetric Rotor 305
7.3.4 Unbalance Response 307
7.3.5 The Gravity Critical Speed 311
 7.3.6 Response to Sinusoidal Excitation in the Stationary Frame 312
7.3.7 Response to General Excitation in the Stationary Frame 313
7.4 Asymmetric Rotors Supported by Anisotropic Bearing:
Simple Rotors 315
7.5 Internal Rotor Damping: Simple Rotors 318
7.6 Rotor Asymmetry with Isotropic Supports: Complex Rotors 321
7.6.1 Disks 322
7.6.2 Shaft Elements 323
7.6.3 Bearings and Foundations 325
7.6.4 The Equations of Motion 326
7.7 Internal Rotor Damping: Complex Rotors 330
7.8 Internal Cross-Coupling in the Bearing: Simple Rotors 334
7.9 Internal Cross-Coupling in the Bearing: Complex Rotors 335
7.10 Summary 336
7.11 Problems 336

8 B alan cin g.......................................................................................................... 339

8.1 Introduction 339
8.2 Balancing Rigid Rotors at the Design Stage 340
8.3 The Shaft M arker and the Phase of Response Signals 342
8.4 Field Balancing of Rigid Rotors 343
8.4.1 Single-Plane Balancing 345
8.4.2 Two-Plane Balancing 347
8.5 Field Balancing of Flexible Rotors 351
8.5.1 The Influence-Coefficient Method 351
8.5.2 Modal Balancing 363
8.6 Balancing Machines without a Phase Reference 369
8.7 Automatic Balancing Methods 372
8.8 Issues in Balancing Real Machines 373
8.9 Summary 376
8.10 Problems 376

9 Axial and Torsional V ibration.......................................................................383

9.1 Introduction 383
9.2 Simple System Models for Axial Vibrations 384
9.3 Shaft-Line Finite Element Models for Axial Vibrations 386
9.4 Simple System Models for Torsional Vibrations 387
9.5 Shaft-Line Finite Element Analysis of Torsional Motion 389
9.6 Geared and Branched Systems 390
9.6.1 Applying Constraints for Geared Systems 390
9.6.2 A More Formal Approach to Geared Systems 392
9.6.3 Developing a Transformation to Effect Constraints 398
9.7 Axial and Torsional Vibration with External Excitation 400
9.7.1 Force-Driven Excitation of Torsional Vibration 401
9.7.2 Force-Driven Excitation of Axial Vibration 409
9.7.3 Displacement-Driven Excitation of Torsional Vibration 410
9.8 Parametric Excitation of Torsional Systems 413
 9.9 Summary 415
9.10 Problems 415

10 More Complex Rotordynamic M odels...........................................................420

10.1 Introduction 420
10.2 Simple Rotating Elastic Systems 421
10.2.1 Stress and Geometric Stiffening 424
10.2.2 Damping in a Spinning Rotor 430
10.3 Finite Element Analysis of Rotors with Deformable
Cross Sections 431
10.3.1 G eneral Finite Elemen t Models 431
10.3.2 Axisymmetric Finite Element Rotor Models 436
10.4 Rotor with Flexible Disks 440
10.4.1 Analysis of a Single Flexible Disk 440
10.4.2 Analysis of Rotor-Disk Assemblies 442
10.5 Detailed Models for Axial Vibration 446
10.6 Detailed Models for Torsional Vibration 448
10.7 Rotors Consisting of a Flexible Cylinder 449
10.8 Bending Vibrations of Blades Attached to Rotors 453
10.9 Coupled Systems 456
10.10 Rotor-Stator Contact in Rotating Machinery 459
10.11 Alignment 466
10.12 Nonlinear Bearings, Oil Whirl, and Oil Whip 472
10.12.1 Oil Whirl 473
10.12.2 Nonlinear Bearing Models and Oil Whip 477
10.13 The Morton Effect 481
10.14 Cracked Rotors 481
10.15 Summary 489

Solutions to Problems 491

Appendix 1 Properties o f Solids 499
Appendix 2 Stiffness and Mass Coefficients fo r Certain Beam Systems 500
Appendix 3 Torsional Constants fo r Shaft Sections 505
Bibliography 507
Index 519
Preface

This book addresses the dynamics of rotating machines, and its purpose may be con­
sidered 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 mathemat­
ical 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 develop­
ment of thinking within the dynamic analysis of rotating machines, focusing exclu­
sively 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 in­
troduce 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 emer­
gence 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 appropri­
ate because it is heartfelt.
Acronyms

BSF ball spin frequency

DPT 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

 Introduction

1.1 Overview

The aim of this book is to introduce readers to modem methods of modeiing and
analyzing rotating machines to determine their dynamic behavior. This is usually re­
ferred to as rotordynamics. The text is suitable for final-year undergraduates, post­
graduates, and practicing engineers who require both an understanding of modern
techniques used to model and analyze rotating systems and an ability to interpret
the results of such analyses.
Before presenting a text on the dynamics of rotating machines, it is appropriate
to consider why one would wish to study this subject. Apart from academic interest,
it is an important practical subject in industry, despite the forbidding appearance
of some of the mathematics used. There are two important application areas for the
techniques found in the following pages. First, when designing the rotating parts of a
machine, it is clearly necessary to consider their dynamic characteristics. It is crucial
that the design of a machine is such that while running up to and functioning at its
operating speed(s), vibration does not exceed safe and acceptable levels. An unac-
ceptably high level of rotor vibration can cause excessive wear on bearings and may
cause seals to fail. Blades on a rotor may come into contact with the stationary hous­
ing with disastrous results. An unacceptable level of vibration might be transmitted
to the supporting structure and high levels of vibration could generate an excessive
noise level. The second aspect of importance is the understanding of a machine’s be­
havior when circumstances change, implying that a fault has occurred in the rotating
parts of the machine. This understanding is needed for the diagnosis of the fault and
for the formulation of repair strategies involving important decisions, such as “Is it
safe to run?” and “How long can it be run?” These questions concern personnel and
machine safety, legal issues, and, in many cases, very large sums of money.
After accepting that the dynamics of rotors is an important subject worthy of
study, it is necessary to explain why a book on this topic is required. Are not the
dynamics of rotors simply particular cases of the more general dynamics of struc­
tures? In fact, whereas in many respects, a rotor system behaves dynamically like a
fixed structure, there are some important differences because of the rotation. The
most fundamental of these differences is that fixed structures do not have inherent

1
forcing, whereas rotating machines do. The following is a list of some, but by no
means all, of the phenomena unique to rotating systems:
• When a rotor spins, lateral forces and moments may be generated, These so-
called unbalance forces and moments are always present due to limitations in
machining and assembly accuracy. These forces and moments give rise to vibra­
tion at the same frequency as the rotational speed.
• Gyroscopic moments also act on the spinning rotor and cause its natural fre­
quencies to change with rotational speed.
• The stiffness and damping properties of some types of bearings vary with rotor
speed; these changes also influence the system’s natural frequencies.
• Centrifugal forces acting on a blade attached to a spinning rotor cause the blade
stiffness to increase with rotational speed.
• Errors in gear profiles generate forces on the rotor, which are generated by the
imposed motion introduced by geometric errors.
• Not all rotors are perfectly symmetric, and even minor asymmetries can have
significant effects. When an asymmetric rotor spins, its stiffness changes period­
ically at the rotational speed or a multiple thereof, when viewed in a fixed set of
coordinates. This can cause instability.
• The damping in rotating machines, in some circumstances, may be relatively
high compared with that found in most fixed structures. In other circumstances,
the effective damping may be negative, causing instability.
Thus, there is a variety of respects in which the analysis of rotating machines dif­
fers from that of a normal fixed structure. Study of the behavior of such systems is
the science of rotordynamics and, whereas great reliance is placed on techniques of
analysis from structural dynamics, rotordynamic analysis represents a considerable
extension in scope and complexity. Given the wide application of rotating machin­
ery, the field is one of great practical importance.

1 .2 Rotating Machine Components

Any attempt to describe a typical rotating machine is inevitably fraught with diffi­
culty in view of the wide range of machine sizes, duties, and speeds. Nevertheless,
the techniques of modeling and analysis described in the following chapters can
be applied to a wide variety of machines. The methods are equally applicable to
small motors with rotors of, for instance, 40 mm in length with a mass of 0.04 kg
(or smaller), or to turbo-generators, in which a typical large 660 MW machine has a
total length of 50 m and a mass of 250,000 kg.
Every rotating machine consists of three principal components - namely, the ro­
tor, the bearings, and their supporting structure. Some type of rationalization may
be reached by considering the features that are pertinent to each component, which
has its own properties that range enormously in complexity. Despite this, however,
there are some generic methods to aid analysis that can be applied to gain insight
into the behavior of a machine. It is important to emphasize this concept of insight:
It is frequently more important to gain an understanding of a machine’s behavior
than it is to calculate precise numerical values of its response. Very often, the level
of forces in service are unknown or known only approximately; therefore, detailed
response calculations cannot be performed. Nevertheless, an understanding of the
behavior in such circumstances can lead an analyst to propose appropriate courses
of action to ensure the safe operation of a machine. We now consider each compo­
nent in turn.

1.2.1 Features of Rotors

The rotor is at the heart of any rotating machine; therefore, the first part of any anal­
ysis must address the dynamic properties of the rotating element. In most cases, the
rotor is relatively simple insofar as it often can be represented as a single beam or as
a series of beam elements and rigid disks. The rotor, of course, may be complicated
by numerous changes of section and the need for some treatment of effects, such
as the shrink-fitting of disks. However, because the rotor is conceptually simple, it
is a straightforward matter to check the accuracy of the rotor model by compar­
ing the predicted behavior of the rotor alone with measured data, for example, by
performing an impulse test on the rotor freely suspended.
The rotor of a small machine is usually rigid but, with increasing machine size,
it generally becomes more flexible, and this must be accounted for in the analysis.
However, there are exceptions. For example, the large electrical motors that drive
rolling mills have rotors that are essentially rigid.
In many machines, the lateral stiffness of the rotor is the same in all planes con­
taining the axis of rotation; in other cases, this is not so. For example, the rotors of
many two-pole electrical machines have a lateral stiffness in one plane that is lower
than in a perpendicular plane because of groups of slots that are cut into the rotor
to carry the electrical conductors. Another feature of rotors is that they sometimes
carry components or subassemblies with dynamic characteristics of their own. For
example, a helicopter rotor consists of a relatively short, rigid vertical shaft carrying
three very flexible blades. Finally, a rotor may have internal damping; contrary to
intuition, internal damping can cause instability.

1 .2 .2 Features of Bearings and Rotor-Stator Interactions

The rotor is connected to the supporting structure by means of bearings, which may
be of several types. For small machines with light loads, the bearing may take the
form of a simple bush in which the rotor runs. However, as bearing loads increase,
such a simple arrangement becomes inadequate; rotors are then mounted on ball
or rolling-element bearings. These provide greater load capacity and stiffness but
because they have internal moving components, they can make significant contri­
butions to the dynamics of the overall machine. For large, heavy machines, such as
turbo-generators, the bearings are almost invariably of the journal type, of which
there are several forms. In a journal, or hydrodynamic, bearing, there is a film of
oil in a smail clearance between the static and rotating elements. The rotor creates
a hydrodynamic pressure distribution within the oil film, which supports the weight
and unbalance forces of the rotor.
In the past two decades, magnetic bearings have been introduced for some ma­
chine types. In these bearings, the rotor and stator are held apart by a magnetic
field. The advantage of this system is the complete elimination of contact and the
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Review 1: Literature review and discussion

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- Example: Practical application scenario
- Note: Important consideration
Formula: [Mathematical expression or equation]
Key Concept: Fundamental concepts and principles
• Best practices and recommendations
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Remember: Historical development and evolution
• Comparative analysis and synthesis
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Important: Assessment criteria and rubrics
• Assessment criteria and rubrics
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Formula: [Mathematical expression or equation]
Key Concept: Ethical considerations and implications
• Study tips and learning strategies
- Sub-point: Additional details and explanations
- Example: Practical application scenario
Definition: Comparative analysis and synthesis
• Research findings and conclusions
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Note: Best practices and recommendations
• Current trends and future directions
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Practice Problem 79: Current trends and future directions
• Assessment criteria and rubrics
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Chapter 9: Fundamental concepts and principles
Key Concept: Statistical analysis and interpretation
• Assessment criteria and rubrics
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Key Concept: Theoretical framework and methodology
• Theoretical framework and methodology
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Key Concept: Study tips and learning strategies
• Historical development and evolution
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
[Figure 83: Diagram/Chart/Graph]
Important: Best practices and recommendations
• Best practices and recommendations
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Remember: Study tips and learning strategies
• Learning outcomes and objectives
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
[Figure 85: Diagram/Chart/Graph]
Definition: Literature review and discussion
• Experimental procedures and results
- Sub-point: Additional details and explanations
- Example: Practical application scenario
Formula: [Mathematical expression or equation]
[Figure 86: Diagram/Chart/Graph]
Practice Problem 86: Historical development and evolution
• Critical analysis and evaluation
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Definition: Problem-solving strategies and techniques
• Practical applications and examples
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Key Concept: Assessment criteria and rubrics
• Case studies and real-world applications
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Important: Practical applications and examples
• Problem-solving strategies and techniques
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Formula: [Mathematical expression or equation]
[Figure 90: Diagram/Chart/Graph]
Appendix 10: Fundamental concepts and principles
Note: Learning outcomes and objectives
• Key terms and definitions
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Definition: Current trends and future directions
• Ethical considerations and implications
- Sub-point: Additional details and explanations
- Example: Practical application scenario
Important: Key terms and definitions
• Current trends and future directions
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Definition: Study tips and learning strategies
• Problem-solving strategies and techniques
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Formula: [Mathematical expression or equation]
Example 94: Interdisciplinary approaches
• Key terms and definitions
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
[Figure 95: Diagram/Chart/Graph]
Key Concept: Fundamental concepts and principles
• Case studies and real-world applications
- Sub-point: Additional details and explanations
- Example: Practical application scenario
Example 96: Historical development and evolution
• Statistical analysis and interpretation
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Formula: [Mathematical expression or equation]
Important: Study tips and learning strategies
• Critical analysis and evaluation
- Sub-point: Additional details and explanations
- Example: Practical application scenario
Example 98: Critical analysis and evaluation
• Case studies and real-world applications
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Formula: [Mathematical expression or equation]
Definition: Key terms and definitions
• Best practices and recommendations
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Background 11: Problem-solving strategies and techniques
Example 100: Practical applications and examples
• Historical development and evolution
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
[Figure 101: Diagram/Chart/Graph]
Note: Case studies and real-world applications
• Current trends and future directions
- Sub-point: Additional details and explanations
- Example: Practical application scenario
Formula: [Mathematical expression or equation]
Example 102: Comparative analysis and synthesis
• Critical analysis and evaluation
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
[Figure 103: Diagram/Chart/Graph]
Remember: Fundamental concepts and principles
• Case studies and real-world applications
- Sub-point: Additional details and explanations
- Example: Practical application scenario
Important: Statistical analysis and interpretation
• Critical analysis and evaluation
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Formula: [Mathematical expression or equation]
Definition: Key terms and definitions
• Ethical considerations and implications
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Remember: Problem-solving strategies and techniques
• Current trends and future directions
- Sub-point: Additional details and explanations
- Example: Practical application scenario
[Figure 107: Diagram/Chart/Graph]
Practice Problem 107: Literature review and discussion
• Ethical considerations and implications
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
[Figure 108: Diagram/Chart/Graph]
Remember: Best practices and recommendations
• Assessment criteria and rubrics
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Example 109: Fundamental concepts and principles
• Learning outcomes and objectives
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Section 12: Study tips and learning strategies
Remember: Study tips and learning strategies
• Critical analysis and evaluation
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Example 111: Problem-solving strategies and techniques
• Key terms and definitions
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Example 112: Practical applications and examples
• Case studies and real-world applications
- Sub-point: Additional details and explanations
- Example: Practical application scenario
Definition: Problem-solving strategies and techniques
• Problem-solving strategies and techniques
- Sub-point: Additional details and explanations
- Example: Practical application scenario
Key Concept: Problem-solving strategies and techniques
• Current trends and future directions
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Example 115: Assessment criteria and rubrics
• Best practices and recommendations
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
[Figure 116: Diagram/Chart/Graph]
Important: Current trends and future directions
• Problem-solving strategies and techniques
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Definition: Study tips and learning strategies
• Literature review and discussion
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Remember: Problem-solving strategies and techniques
• Problem-solving strategies and techniques
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Example 119: Historical development and evolution
• Statistical analysis and interpretation
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Formula: [Mathematical expression or equation]
Abstract 13: Literature review and discussion
Key Concept: Critical analysis and evaluation
• Key terms and definitions
- Sub-point: Additional details and explanations
- Example: Practical application scenario
Example 121: Problem-solving strategies and techniques
• Current trends and future directions
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Formula: [Mathematical expression or equation]
[Figure 122: Diagram/Chart/Graph]
Practice Problem 122: Research findings and conclusions
• Problem-solving strategies and techniques
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Note: Current trends and future directions
• Study tips and learning strategies
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Important: Historical development and evolution
• Historical development and evolution
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Important: Statistical analysis and interpretation
• Theoretical framework and methodology
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Formula: [Mathematical expression or equation]
Important: Study tips and learning strategies
• Research findings and conclusions
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Practice Problem 127: Experimental procedures and results
• Problem-solving strategies and techniques
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Key Concept: Problem-solving strategies and techniques
• Experimental procedures and results
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Formula: [Mathematical expression or equation]
Practice Problem 129: Literature review and discussion
• Problem-solving strategies and techniques
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Formula: [Mathematical expression or equation]
Exercise 14: Learning outcomes and objectives
Note: Literature review and discussion
• Critical analysis and evaluation
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Note: Study tips and learning strategies
• Fundamental concepts and principles
- Sub-point: Additional details and explanations
- Example: Practical application scenario
Note: Literature review and discussion
• Problem-solving strategies and techniques
- Sub-point: Additional details and explanations
- Example: Practical application scenario
Formula: [Mathematical expression or equation]
Remember: Learning outcomes and objectives
• Case studies and real-world applications
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Definition: Experimental procedures and results
• Critical analysis and evaluation
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Remember: Ethical considerations and implications
• Comparative analysis and synthesis
- Sub-point: Additional details and explanations
- Example: Practical application scenario
Definition: Problem-solving strategies and techniques
• Fundamental concepts and principles
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
[Figure 137: Diagram/Chart/Graph]
Definition: Current trends and future directions
• Practical applications and examples
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Practice Problem 138: Theoretical framework and methodology
• Theoretical framework and methodology
- Sub-point: Additional details and explanations
- Example: Practical application scenario
Formula: [Mathematical expression or equation]
Important: Comparative analysis and synthesis
• Research findings and conclusions
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
[Figure 140: Diagram/Chart/Graph]
Topic 15: Critical analysis and evaluation
Key Concept: Historical development and evolution
• Study tips and learning strategies
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Definition: Ethical considerations and implications
• Practical applications and examples
- Sub-point: Additional details and explanations
- Example: Practical application scenario
[Figure 142: Diagram/Chart/Graph]
Remember: Fundamental concepts and principles
• Statistical analysis and interpretation
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Note: Research findings and conclusions
• Statistical analysis and interpretation
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Formula: [Mathematical expression or equation]
Important: Literature review and discussion
• Learning outcomes and objectives
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Important: Critical analysis and evaluation
• Best practices and recommendations
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Formula: [Mathematical expression or equation]
Note: Key terms and definitions
• Current trends and future directions
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
Practice Problem 147: Interdisciplinary approaches
• Historical development and evolution
- Sub-point: Additional details and explanations
- Example: Practical application scenario
- Note: Important consideration
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