Mechanical Engineering Series

Frederick F. LingSeries Editor

Mechanical Engineering Series

Frederick F. LingSeries Editor

Advisory Board

Applied Mechanics F.A. LeckieUniversity of California,Santa Barbara

D. GrossTechnical University of Darmstadt

Biomechanics V.C. MowColumbia University

Computational Mechanics H.T. YangUniversity of California,Santa Barbara

Dynamical Systems and Control D. BryantUniversity of Texas at Austin

Energetics J.R. WeltyUniversity of Oregon, Eugene

Mechanics of Materials I. FinnieUniversity of California, Berkeley

Processing K.K. WangCornell University

Production Systems G.-A. KlutkeTexas A&M University

Thermal Science A.E. BerglesRensselaer Polytechnic Institute

Tribology W.O. WinerGeorgia Institute of Technology

Giancarlo Genta

Dynamics ofRotating Systems

With 260 Figures

Giancarlo GentaPolitecnico di TorinoTorino, Italy

Series EditorFrederick F. LingErnest F. Gloyna Regents Chair in EngineeringDepartment of Mechanical EngineeringThe University of Texas at AustinAustin, TX 78712-1063, USAand

Distinguished William Howard HartProfessor Emeritus

Department of Mechanical Engineering,Aeronautical Engineering and Mechanics

Rensselaer Polytechnic InstituteTroy, NY 12180-3590, USA

Library of Congress Cataloging-in-Publication DataOn file.

ISBN 0-387-20936-0 Printed on acid-free paper.

© 2005 Springer Science+Business Media, Inc.All rights reserved. This work may not be translated or copied in whole or in part without thewritten permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, NewYork, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis.Use in connection with any form of information storage and retrieval, electronic adaptation, com-puter software, or by similar or dissimilar methodology now known or hereafter developed is for-bidden.The use in this publication of trade names, trademarks, service marks, and similar terms, even ifthey are not identified as such, is not to be taken as an expression of opinion as to whether or notthey are subject to proprietary rights.

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Series Preface

Mechanical engineering, an engineering discipline born of the needs of theindustrial revolution, is once again asked to do its substantial share in thecall for industrial renewal. The general call is urgent as we face profoundissues of productivity and competitiveness that require engineering solu-tions, among others. The Mechanical Engineering Series is a series featur-ing graduate texts and research monographs intended to address the needfor information in contemporary areas of mechanical engineering.The series is conceived as a comprehensive one that covers a broad range

of concentrations important to mechanical engineering graduate educationand research. We are fortunate to have a distinguished roster of consultingeditors, each an expert in one of the areas of concentration. The names ofthe consulting editors are listed on page ii of this volume. The areas ofconcentration are applied mathematics, biomechanics, computational me-chanics, dynamic systems, and control, energetics, mechanics of materials,processing, thermal science, and tribology.

Preface

This book is the result of almost 30 years of work in the field of rotor-dynamics, which includes research, teaching, writing computer codes, andconsulting. It is the outcome of an interdisciplinary research team thatoperated, and still operates, in the Mechanics Department and in the In-terdepartmental Mechatronics Laboratory of Politecnico di Torino. Theaim is mostly to write in a systematic way what has been the subject ofa number of research papers, in such a way to give a consistent picture ofthe dynamic behavior of rotating machinery.The author must then give credits to many colleagues and Ph.D. students

who cooperated in various degree to this book: Much of the material theyproduced in their thesis work or in subsequent research found its way inthese pages. An even greater number of students cooperated to this work ina more subtle way: with their thesis work and their questions, but mainlywith their very presence that compels who tries to explain an involvedsubject to clarify his own ideas and to work out all details. To all of themgoes the gratitude of the author.A particular mention must be made to (in alphabetic order) Eugenio

Brusa, Stefano Carabelli, and Andrea Tonoli, who not only worked duringtheir doctoral thesis and subsequent research on these topics, but also whowere very helpful in the actual writing of this book, reading and correctingthe text, drawing figures, and doing much editorial work.As usual, a deep gratitude goes to Franca for her support in the long

work related with this book, and for the help in reading and correctingproofs.

viii Preface

As the title implies, this book is an attempt (only the reader can judgewhether it is successful) to go beyond what is usually referred to as ro-tordynamics. The aim is that of dealing with the dynamic behavior ofsystems having in common the feature of rotating. This definition includesobviously those systems, like transmission shafts, turbine rotors, and gy-roscopes, which are studied by rotordynamics, but also systems such asrotating blades (like in helicopter rotors) or flexible spinning spacecraft.Although rotordynamics usually deals only with the lateral behavior of ro-tors, some mention is made here also to torsional and axial vibration or tocases in which it is impossible to distinguish between them. However, theauthor imposed a limitation: No mention will be made of the dynamics ofmachines containing reciprocating parts, such as a crankshaft-connectingrod-piston mechanism. This arbitrary decision is based on the groundsthat their vibration (mainly torsional vibration, but also axial and lateralvibration) is a very specialized topic, dealt with in many handbooks andtextbooks and, above all, that to include it would have meant either to givea very insubstantial account or to double the size of the book.Another area in which a decision about where to stop was needed is

controlled rotors. A thorough study of the dynamics of many controlledrotors, like those running on active magnetic bearings or supplied withactive dampers, would have implied a detailed study of their control systems(hardware and, in case of digital systems, software) sensors and actuators(with the critical issue of the power amplifiers). As is typical of mechatronicsystems, only an integrated and interdisciplinary approach allows us toexploit the advantages of the potentialities modern technology has opened.As this would have lead too far from the main topics of this book, theseareas will be touched only marginally.The text is structured in two parts. The first one deals with what could

be defined as classic or basic rotordynamics. The contents are basically wellconsolidated, although some incorrect statements can be found even in re-cent papers published on well-known journals. The basic assumptions arelinearity, steady state operation, and at least some degree of axial symme-try.The second part, containing topics that are usually considered as special-

ized aspects of rotordynamics, could be titled advanced rotordynamics. Thementioned assumptions are dropped, and more detailed models are builtfor rotors departing from the classic configurations studied in rotordynam-ics. The contents of this part are more research topics than consolidatedapplications.The contents and the credits for the various chapters are the following:

• Chapter 1: Introduction. The basic concepts, graphical representa-tion, and methods of rotordynamics are illustrated in a qualitativeway. The expert reader, although familiar with these concepts, should

Preface ix

not skip it altogether because the basic notation and the viewpointthat will be followed in the whole text are described.

Part 1: Basic topics

• Chapter 2: Je cott rotor. The so-called Je cott rotor is the simplestrotor model that can be conceived. Although unable to account forsome typical phenomena linked with rotordynamics, like gyroscopice ect or centrifugal sti ening, it allows us to gain a good insightinto the peculiarities of rotating systems. In particular, it is essentialfor understanding the role of damping in rotordynamics. The topicsdealt with are as a whole standard, but the part on nonsynchronousdamping, studied together with E. Brusa and published in [1], is lesscommon.

• Chapter 3: Model with four degrees of freedom: Gyroscopic e ect. Asimple model in which a rigid body is substituted for the point massof the Je cott rotor is then studied, to allow the study of gyroscopice ects. This model is representative for the behavior of any rigid rotoron compliant bearings and allows us to define a modal gyroscopicsystem, on which modal decomposition of rotors can be based undersome assumptions.

• Chapter 4: Discrete multi-degrees-of-freedom rotors. The lateral be-havior of a flexible rotor modeled as a discrete parameter beamlike(1-D approach) system is then studied. Older approaches, like thetransfer matrices methods, are dealt with together with more modernones, like the finite element method (FEM). Some work on reductiontechniques by S. Carabelli and A. Tonoli [2] has been included.

• Chapter 5: Continuous systems: Transmission shafts. A short accounton modeling simple rotors as continuous system is then included. Thischapter can be considered more of academic rather than of practicalrelevance.

• Chapter 6: Anisotropy of rotors or supports. If either the rotor orthe stator are not isotropic, it is still possible to obtain a closed-formsolution for the linearized steady-state dynamics. Such systems arestudied with particular reference to the backward whirling causedby unbalance in isotropic rotors on asymmetric supports and to theinstability ranges of nonsymmetric rotors on isotropic supports.

• Chapter 7: Torsional and axial dynamics. The axial and torsionaldynamics of rotors is briefly dealt with. Considering that the torsionaland axial behavior is una ected by the rotation of the system (atleast if the basic assumptions of linearity and small displacementsare made), just a brief account is reported.

x Preface

• Chapter 8: Rotor-bearings interaction. The interaction between thebehavior of the rotor and of the bearing is a complex subject, mainlybecause of the nonlinear behavior of the latter. The approach here fol-lowed is the classic one: The nonlinearity of the bearings is accountedfor in computing their working conditions, and then the dynamic be-havior is linearized assuming small displacements about the staticequilibrium position (at speed). Rolling elements and lubricated andmagnetic bearings are dealt with.

Part 2: Advanced topics

• Chapter 9: Anisotropy of rotors and supports. The assumption thateither the stator or the rotor is isotropic is dropped. No closed-formsolution is any more possible, although a truncated series solutioncan be attempted.

• Chapter 10: Nonlinear rotordynamics. Here another assumption, thatof linearity, is dropped. The phenomena typical of nonlinear systems,like jumps and even chaotic behavior are discussed.

• Chapter 11: Nonstationary rotordynamics. The spin speed is no moreassumed to be constant, or other parameters, like unbalance, are al-lowed to change. In particular, the acceleration of the rotor througha critical speed and the occurrence of a blade loss are dealt with indetail. The work performed with C. Delprete [3] has been thoroughlyused.

• Chapter 12: Dynamic behavior of free rotors. Unconstrained rotatingobjects, like spinning celestial bodies or spacecraft, can be consideredas rotors. The main aim of this section is to show that the assump-tion of constant angular momentum, typical of the dynamic study offree rotors, and that of constant angular velocity, typical of classicrotordynamics, coincide when the small displacement and rotationsassumptions is made, so that the first can be approached with themethods of the latter. The chapter is based on the work performedwith E. Brusa [4], [5].

• Chapter 13: Dynamics of rotating beams and blades. The e ect ofrotation, about an axis perpendicular to their longitudinal axis, onthe dynamic behavior of beams and the blades-rotor interaction isstudied using simple models. The well-known phenomena related topropeller and helicopter rotors’ instability are dealt with, as well asother less-known phenomena regarding the e ects of blade dampingon the stability of a bladed rotor.

• Chapter 14: Dynamics of rotating discs and rings. Turbine and com-pressor discs are assumed, in classic rotordynamics, to behave as rigid

Preface xi

bodies. In this chapter, this assumption is dropped and the e ects ofthe flexibility of the discs are dealt with using simple models, startingfrom that introduced about 80 years ago by Southwell [6].

• Chapter 15: Three-dimensional modeling of rotors. This chapter dealswith numerical modeling, mostly based on the FEM, of complex ro-tors. The topics dealt with in Chapters 13 and 14 using simplifiedmodels are here treated with the aim of building more accurate mod-els, yielding precise quantitative results. The work performed with A.Tonoli [7, 8] and the models developed by M. Silvagni in his Ph.D.thesis are included [9].

• Chapter 16: Dynamics of controlled rotors. Active vibration controlis increasingly applied to rotors, either together with the use of activemagnetic suspension or with techniques using active dampers or thecontrol of more or less conventional bearings. As already stated, noattempt in modeling in detail the control, sensor or actuator dynam-ics is done, because it would lead too far from the central topics ofthis book. The work performed with S. Carabelli on sensor-actuatorcolocation [10] is reported.

• Appendix A: Vectors, matrices, and equations of motion. Some ba-sic topics of system dynamics, particularly for the peculiar aspectslinked with rotating systems, are summarized in this appendix, whichowes much to the specific viewpoint of control theory for which theauthor is indebted to S. Carabelli. The results on circulatory andnoncirculatory coupling published by Crandall [11] and relevant forrotordynamics are reported.

• Appendix B: An outline on rotor balancing. As many very good bookshave been written on rotor balancing, only a short account on thebasic topics are dealt with.

• Appendix E: Bibliography. Some of the books specifically devoted torotordynamics are listed in chronological order.

A CD-ROM comes with this book. It contains a simplified version of theDYNROT code and two short videos.DYNROT Finite Element code was developed by the author starting

in 1976. An initial version written in HPGL language for the early HPdesktop computers together with G. Brussino, then student at the Politec-nico di Torino, was followed by a version in BASIC written together withA. Gugliotta. In the subsequent years (almost 30), several versions havebeen developed, mostly using MatLab language with the help of countlessresearchers and students. The version reported here is DYNROT LIGHT,which contains only the basic elements and is based on the one-dimensional

xii Preface

approach seen in Chapter 4 (while the full-blown DYNROT has capabili-ties also for studying bladed disks dynamics, with the possibility of timedomain study of nonlinear and nonstationary rotors). A basic advantage ofa MatLab code is its openness, so that the reader can understand how itworks and can modify any points.The two videos, Gyroscopic E ect and Damping in Rotordynamics and

Dynamic Behaviour of Rotors on Anisotropic Supports, are based on exper-iments on simple demonstrators and were produced in an e ort of showingstudents some physical evidence of what can be interpreted as mathemat-ical divertissement more than the description of real-world machinery.

Giancarlo Genta

Torino, April 2004

Contents

Preface vii

Contents xiii

Symbols xxi

1 Introduction 1

1.1 Linear rotordynamics . . . . . . . . . . . . . . . . . . . . . . 41.1.1 Equation of motion . . . . . . . . . . . . . . . . . . . 51.1.2 Rotating systems . . . . . . . . . . . . . . . . . . . . 61.1.3 Complex coordinates . . . . . . . . . . . . . . . . . . 71.1.4 Free vibration . . . . . . . . . . . . . . . . . . . . . . 91.1.5 Forced response . . . . . . . . . . . . . . . . . . . . . 23

1.2 Nonlinear rotordynamics . . . . . . . . . . . . . . . . . . . . 291.3 Nonstationary rotordynamics . . . . . . . . . . . . . . . . . 301.4 Time domain versus frequency domain . . . . . . . . . . . . 31

I Basic topics 33

2 Je cott rotor 35

2.1 Undamped Je cott rotor . . . . . . . . . . . . . . . . . . . . 352.1.1 Equations of motion . . . . . . . . . . . . . . . . . . 362.1.2 Free whirling . . . . . . . . . . . . . . . . . . . . . . 39

xiv Contents

2.1.3 Unbalance response . . . . . . . . . . . . . . . . . . 422.1.4 Response to external forces in the frequency domain 44

2.2 Complex coordinates in rotordynamics . . . . . . . . . . . . 462.2.1 Free whirling . . . . . . . . . . . . . . . . . . . . . . 462.2.2 Unbalance response . . . . . . . . . . . . . . . . . . 48

2.3 Je cott rotor with shaft bow . . . . . . . . . . . . . . . . . 492.4 Je cott rotor with viscous damping . . . . . . . . . . . . . . 51

2.4.1 Equations of motion . . . . . . . . . . . . . . . . . . 512.4.2 Some considerations on rotating damping . . . . . . 552.4.3 Free whirling . . . . . . . . . . . . . . . . . . . . . . 582.4.4 Unbalance response . . . . . . . . . . . . . . . . . . 622.4.5 Response to a static force constant in time . . . . . 662.4.6 Shaft bow . . . . . . . . . . . . . . . . . . . . . . . . 662.4.7 Frequency response . . . . . . . . . . . . . . . . . . . 68

2.5 Je cott rotor with structural damping . . . . . . . . . . . . 702.5.1 Equation of motion . . . . . . . . . . . . . . . . . . . 702.5.2 Free whirling . . . . . . . . . . . . . . . . . . . . . . 712.5.3 Mixed damping . . . . . . . . . . . . . . . . . . . . . 762.5.4 Unbalance response . . . . . . . . . . . . . . . . . . 772.5.5 Dependence of the loss factor on frequency . . . . . 77

2.6 Je cott rotor with nonsynchronous damping . . . . . . . . . 772.7 E ect of the compliance of the bearings . . . . . . . . . . . 80

2.7.1 Unbalance response . . . . . . . . . . . . . . . . . . 822.7.2 Free whirling . . . . . . . . . . . . . . . . . . . . . . 84

2.8 Rotating coordinates . . . . . . . . . . . . . . . . . . . . . . 852.9 Stability in the supercritical field . . . . . . . . . . . . . . . 892.10 Drag torque at constant speed . . . . . . . . . . . . . . . . . 90

3 Model with four degrees of freedom: Gyroscopic e ect 93

3.1 Generalized coordinates and equations of motion . . . . . . 943.1.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . 943.1.2 Equations of motion in real coordinates . . . . . . . 983.1.3 Equations of motion in complex coordinates . . . . . 1013.1.4 Static and couple unbalance . . . . . . . . . . . . . . 102

3.2 Uncoupled gyroscopic system . . . . . . . . . . . . . . . . . 1033.2.1 Complex coordinates . . . . . . . . . . . . . . . . . . 1033.2.2 Real coordinates . . . . . . . . . . . . . . . . . . . . 106

3.3 Free whirling of the coupled, undamped system . . . . . . . 1073.4 Response to unbalance and shaft bow . . . . . . . . . . . . 1173.5 Frequency response . . . . . . . . . . . . . . . . . . . . . . . 1203.6 Unbalance response: modal computation . . . . . . . . . . . 1213.7 Modal uncoupling of gyroscopic systems . . . . . . . . . . . 123

3.7.1 Configuration-space approach . . . . . . . . . . . . . 1233.7.2 State-space, complex-coordinates approach . . . . . 1243.7.3 State-space, real-coordinates approach . . . . . . . . 127

Contents xv

4 Discrete multi-degrees-of-freedom rotors 139

4.1 Transfer matrices approach: the Myklestadt-Prohl method . 1414.1.1 Undamped systems . . . . . . . . . . . . . . . . . . . 1414.1.2 Damped systems . . . . . . . . . . . . . . . . . . . . 151

4.2 Lumped parameters sti ness method . . . . . . . . . . . . . 1554.3 The finite element method . . . . . . . . . . . . . . . . . . . 156

4.3.1 Timoshenko beam element for rotordynamic analysis 1594.3.2 Mass element . . . . . . . . . . . . . . . . . . . . . . 1654.3.3 Spring element . . . . . . . . . . . . . . . . . . . . . 1654.3.4 Assembling the structure . . . . . . . . . . . . . . . 1664.3.5 Constraining the structure . . . . . . . . . . . . . . . 1684.3.6 Damping matrices . . . . . . . . . . . . . . . . . . . 1694.3.7 Transfer matrices methods and the FEM . . . . . . 169

4.4 Real versus complex coordinates . . . . . . . . . . . . . . . 1704.5 Fixed versus rotating coordinates . . . . . . . . . . . . . . . 1724.6 Complex state-space equations . . . . . . . . . . . . . . . . 1734.7 Static solution . . . . . . . . . . . . . . . . . . . . . . . . . 1744.8 Critical-speed computation . . . . . . . . . . . . . . . . . . 1744.9 Computation of the unbalance response . . . . . . . . . . . 1764.10 Plotting the Campbell diagram and the roots locus . . . . . 1774.11 Reduction of the number of degrees of freedom . . . . . . . 183

4.11.1 Nodal reduction techniques . . . . . . . . . . . . . . 1844.11.2 Modal reduction . . . . . . . . . . . . . . . . . . . . 1914.11.3 Component mode synthesis . . . . . . . . . . . . . . 195

5 Continuous systems: Transmission shafts 201

5.1 The Euler-Bernoulli vibrating beam . . . . . . . . . . . . . 2015.2 Other boundary conditions . . . . . . . . . . . . . . . . . . 2095.3 E ect of the moments of inertia: Timoshenko beam . . . . . 2135.4 Dynamic sti ness matrix . . . . . . . . . . . . . . . . . . . . 219

6 Anisotropy of rotors or supports 227

6.1 Isotropic rotors on anisotropic supports . . . . . . . . . . . 2286.1.1 Je cott rotor on nonisotropic supports . . . . . . . . 2286.1.2 E ect of damping . . . . . . . . . . . . . . . . . . . 2336.1.3 System with many degrees of freedom . . . . . . . . 236

6.2 Nonisotropic rotors on isotropic supports . . . . . . . . . . 2466.2.1 Nonisotropic Je cott rotor . . . . . . . . . . . . . . . 2476.2.2 E ect of damping . . . . . . . . . . . . . . . . . . . 2516.2.3 Response to a static force . . . . . . . . . . . . . . . 2526.2.4 Anisotropic rotors with many degrees of freedom . . 256

7 Torsional and axial dynamics 265

7.1 Torsional free vibration . . . . . . . . . . . . . . . . . . . . 2657.1.1 Lumped parameters approach . . . . . . . . . . . . . 265

xvi Contents

7.1.2 Consistent parameters approach . . . . . . . . . . . 2707.1.3 Geared systems . . . . . . . . . . . . . . . . . . . . . 272

7.2 Forced vibrations . . . . . . . . . . . . . . . . . . . . . . . . 2757.3 Torsional critical speeds . . . . . . . . . . . . . . . . . . . . 2797.4 Axial vibration . . . . . . . . . . . . . . . . . . . . . . . . . 280

8 Rotor-bearings interaction 281

8.1 Rigid-body and flexural modes . . . . . . . . . . . . . . . . 2828.2 Linearization of the characteristics of the bearings . . . . . 2848.3 Rolling elements bearings . . . . . . . . . . . . . . . . . . . 2918.4 Fluid film bearings . . . . . . . . . . . . . . . . . . . . . . . 298

8.4.1 Forces exerted by the oil film on the journal in sta-tionary conditions . . . . . . . . . . . . . . . . . . . 298

8.4.2 Linearized dynamics of the bearing . . . . . . . . . . 3058.4.3 Stability problems linked with the use of lubricated

bearings . . . . . . . . . . . . . . . . . . . . . . . . . 3098.4.4 E ect of seals, clearances, and dampers . . . . . . . 313

8.5 Magnetic bearings . . . . . . . . . . . . . . . . . . . . . . . 3168.6 Bearing alignment in multibearing rotors . . . . . . . . . . . 327

II Advanced topics 329

9 Anisotropy of rotors and supports 331

9.1 Nonisotropic Je cott rotor . . . . . . . . . . . . . . . . . . . 3319.2 Equation of motion for an anisotropic machine with many

degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . 339

10 Nonlinear rotordynamics 347

10.1 Nonlinear isotropic Je cott rotor . . . . . . . . . . . . . . . 34810.1.1 Equation of motion . . . . . . . . . . . . . . . . . . . 34810.1.2 Unbalance response circular whirling . . . . . . . 351

10.2 Nonlinear isotropic Je cott rotor running on nonsymmetricsupports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

10.3 Nonlinear anisotropic Je cott rotor running on symmetricsupports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374

10.4 Systems with many degrees of freedom . . . . . . . . . . . . 377

11 Nonstationary rotordynamics 387

11.1 Nonstationary linear Je cott rotor . . . . . . . . . . . . . . 38711.1.1 Equations of motion . . . . . . . . . . . . . . . . . . 38711.1.2 Torsionally sti rotor with imposed acceleration . . 39011.1.3 Torsionally sti rotor with imposed torque . . . . . . 39311.1.4 Torsionally compliant rotor: small torsional vibra-

tions with imposed acceleration . . . . . . . . . . . . 394

Contents xvii

11.2 Nonstationary general Je cott rotor . . . . . . . . . . . . . 39711.3 Nonstationary rotor with four degrees of freedom . . . . . . 40111.4 Generic, torsionally sti , multi-degrees-of-freedom system . 40411.5 Blade loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

12 Dynamic behavior of free rotors 413

12.1 Single rigid-body rotor . . . . . . . . . . . . . . . . . . . . . 41412.1.1 General considerations . . . . . . . . . . . . . . . . . 41412.1.2 Equations of motion . . . . . . . . . . . . . . . . . . 419

12.2 Large amplitude whirling of a linearily constrained rigid rotor43112.3 Twin rigid-bodies free rotor . . . . . . . . . . . . . . . . . . 439

12.3.1 Linearized approach . . . . . . . . . . . . . . . . . . 44012.3.2 Nonlinear approach . . . . . . . . . . . . . . . . . . 447

12.4 Multibody free rotors . . . . . . . . . . . . . . . . . . . . . 456

13 Dynamics of rotating beams and blades 465

13.1 Rotating pendulum . . . . . . . . . . . . . . . . . . . . . . . 46613.2 Rotating pendulum constrained to oscillate in a plane . . . 47013.3 Spring-loaded rotating pendulum . . . . . . . . . . . . . . . 47213.4 Rotating string . . . . . . . . . . . . . . . . . . . . . . . . . 473

13.4.1 Rotating string constrained to oscillate in a plane . . 47813.4.2 Rotating beam . . . . . . . . . . . . . . . . . . . . . 479

13.5 Dynamics of a row of rotating pendulums . . . . . . . . . . 48413.5.1 Pendulums on a rigid support . . . . . . . . . . . . . 48413.5.2 In-plane oscillations of pendulums on elastic supports 48813.5.3 Spring-loaded pendulums on elastic supports . . . . 49513.5.4 Damped pendulums on elastic supports . . . . . . . 49813.5.5 Out-of-plane oscillations of pendulums on elastic sup-

ports . . . . . . . . . . . . . . . . . . . . . . . . . . . 50213.6 Interaction between the dynamics of the blades and the dy-

namics of the shaft . . . . . . . . . . . . . . . . . . . . . . . 509

14 Dynamics of rotating discs and rings 517

14.1 Rotating membranes . . . . . . . . . . . . . . . . . . . . . . 51714.2 Rotating circular plate . . . . . . . . . . . . . . . . . . . . . 52214.3 Disc-shaft interaction (modes with = 0 or = 1) . . . . 52514.4 Uncoupled modes (modes with 2) . . . . . . . . . . . . 52714.5 Vibration of rotating circular rings . . . . . . . . . . . . . . 528

14.5.1 Out-of plane flexural vibrations . . . . . . . . . . . . 53214.5.2 In-plane flexural vibrations . . . . . . . . . . . . . . 535

14.6 Vibration of thin-walled, rotating cylinders . . . . . . . . . 53814.7 Instability of rotating cylinders partially filled with liquid . 539

15 Three-dimensional modeling of rotors 541

15.1 Symmetry of the rotor . . . . . . . . . . . . . . . . . . . . . 542

xviii Contents

15.2 Simplified FEM elements for thin bladed-discs modeling . . 54815.2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . 54915.2.2 Shape functions . . . . . . . . . . . . . . . . . . . . . 55215.2.3 Kinetic and potential energy . . . . . . . . . . . . . 55515.2.4 Element matrices . . . . . . . . . . . . . . . . . . . . 557

15.3 General finite element discretization . . . . . . . . . . . . . 55815.3.1 Kinematics of the deformation of a rotating body . . 55915.3.2 Kinetic energy . . . . . . . . . . . . . . . . . . . . . 56115.3.3 Potential energy . . . . . . . . . . . . . . . . . . . . 56415.3.4 Equations of motion of the element . . . . . . . . . . 565

15.4 Equation of motion in the inertial frame . . . . . . . . . . . 56615.4.1 Velocity . . . . . . . . . . . . . . . . . . . . . . . . . 56615.4.2 Kinetic energy . . . . . . . . . . . . . . . . . . . . . 56715.4.3 Equations of motion of the element . . . . . . . . . . 568

15.5 Axi-symmetrical annular elements . . . . . . . . . . . . . . 56815.5.1 Shape functions . . . . . . . . . . . . . . . . . . . . . 56815.5.2 Kinetic and potential energy . . . . . . . . . . . . . 57015.5.3 Equations of motion . . . . . . . . . . . . . . . . . . 572

15.6 Axi-symmetrical shell element . . . . . . . . . . . . . . . . . 57315.6.1 Brick elements . . . . . . . . . . . . . . . . . . . . . 574

16 Dynamics of controlled rotors 583

16.1 Open-loop equations of motion . . . . . . . . . . . . . . . . 58416.1.1 Real coordinates . . . . . . . . . . . . . . . . . . . . 58416.1.2 Complex coordinates . . . . . . . . . . . . . . . . . . 585

16.2 Closed-loop equations of motion . . . . . . . . . . . . . . . 58616.2.1 Ideal proportional control . . . . . . . . . . . . . . . 58616.2.2 Ideal PID control . . . . . . . . . . . . . . . . . . . . 58716.2.3 Dynamics of the control system . . . . . . . . . . . . 591

16.3 Rigid rotor on magnetic linearized bearings . . . . . . . . . 59416.3.1 Equations of motion . . . . . . . . . . . . . . . . . . 59516.3.2 Symmetrical system . . . . . . . . . . . . . . . . . . 59916.3.3 Nonsymmetrical system . . . . . . . . . . . . . . . . 60016.3.4 Geometric re-colocation . . . . . . . . . . . . . . . . 603

16.4 Modal control of rotors . . . . . . . . . . . . . . . . . . . . . 608

A Vectors, matrices, and equations of motion 617

A.1 Equation of motion . . . . . . . . . . . . . . . . . . . . . . . 617A.1.1 Associated eigenproblem . . . . . . . . . . . . . . . . 618A.1.2 Free response . . . . . . . . . . . . . . . . . . . . . . 621A.1.3 Forced response . . . . . . . . . . . . . . . . . . . . . 622A.1.4 State-space representation . . . . . . . . . . . . . . . 623A.1.5 Frequency response . . . . . . . . . . . . . . . . . . . 624

A.2 Rotating systems . . . . . . . . . . . . . . . . . . . . . . . . 625A.2.1 Real coordinates . . . . . . . . . . . . . . . . . . . . 626

Contents xix

A.2.2 Complex coordinates . . . . . . . . . . . . . . . . . . 627A.3 Circulatory and noncirculatory coupling . . . . . . . . . . . 627

B An outline on rotor balancing 631

B.1 Rigid rotors . . . . . . . . . . . . . . . . . . . . . . . . . . . 632B.2 Flexible rotors . . . . . . . . . . . . . . . . . . . . . . . . . 635

B.2.1 Modal balancing . . . . . . . . . . . . . . . . . . . . 636B.2.2 Influence coe cients method . . . . . . . . . . . . . 639

C Rotordynamics videos 645

D DYNROT LIGHT rotordynamics code 647

E Books on rotordynamics 649

References 651

Index 657

Symbols

real part of a complex number, length, accelerationimaginary part of a complex number, shaft bow, lengthviscous damping coe cient, clearancestatic o set

f( ) generalized forces vectorf modal force vector

gravitational accelerationthicknessimaginary unit ( = 1), current

0 bias currentsti nesslengthmassth Lagrangian coordinate

( ) principal functioncomplex conjugate of

q( ) Vector of the generalized coordinates (real or complex)complex coordinate ( = + ), radius

r complex conjugate of r

Laplace variable, complex frequency (as in = 0 )0 complex frequency in the rotor fixed frame

s state vector (transfer matrices methods)time, air gap

, , , displacements in frame, components of the displacement

xxii Symbols

reference framez state vector

area of the cross sectionA dynamic matrix

magnetic fieldB input gain matrix, matrix for discretization of stressesC damping matrix, output gain matrixC modal damping matrix

Young’s modulusE sti ness matrix of the material

forceF Rayleigh dissipation function

Shear modulusG gyroscopic matrixG modal gyroscopic matrixH circulatory matrix, angular momentum

area moment of inertiaI identity matrixIm imaginary partJ inertia tensor

moment of inertiapolar moment of inertiatransversal moment of inertiasti ness, gain, constant

K sti ness matrixK modal sti ness matrixL Lagrangian functionM mass matrixM modal mass matrix

momentN matrix of the shape functions

load factor (Ockvirk number)quality factorth generalized force

Re real partradius

R rotation matrixSommerfeld number

T kinetic energyT transfer matrixU matrix of the right eigenvectors of the dynamic matrixU potential energy

velocity, volumephase angle, slenderness of a beamphase angle, ratio 2, attitude angle, ratiogeneric element of the compliance matrix phase angleshear strain, ratio

Symbols xxiii

ratio , ratio0 ratio ( )L virtual work

virtual displacementeccentricity

² strain vectordamping ratio, nondimensional coordinate ( = )rotationrotational sti ness, curvatureviscosity, coe cent of the nonlinear restoring forcemagnetic dipole momentPoissons’s ratiorotor-fixed reference framedensity, complex coordinate in the rotor fixed frame ( = + )

radius of inertia =p

radius of inertia =p

decay rate ( = Re( )), stressstress vectorcomplex coordinate ( = ), phase

, , rotation about , , axesangular misalignment, shear factor, nondimensional coordinate = 0

anglefrequency, whirl speed ( = Im( ))natural frequency

B compliance matrixrotational damping coe cientmatrix of the eigenvectorsrotational speedcritical speed

Subscriptsbearingnon synchronous rotating, deviatoric, dampingfieldfixed frameinnerjournalmeannonrotating, nodeouterpendulumrotating, restoring forcerotating framegloballeftright

xxiv Symbols

NotationScalar (real or complex)complex conjugate of

q Vector (implicit notation: bold and lower case){ } Vector (explicit notation: curl braces){ } Vector of dimensionM Matrix (implicit notation: bold and upper case)[ ] Matrix (explicit notation: square braces)[ ]

×Matrix of dimension ×

( ), q( ) Time dependance˙( ), q̇( ) Time derivative(̈ ), q̈( ) Time second order derivative( )

=1, (q )

=1, (M )

=1Collection of indexed elements

({q } )=1

Collection of indexed vectors of dimension