Rotordynamics - Agnieszka Muszynska

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Rotordynamics

2005 by Taylor & Francis Group, LLC

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DK3162_title 3/30/05 9:58 AM Page 1

Rotordynamics

Boca Raton London New York Singapore

A CRC title, part of the Taylor & Francis imprint, a member of theTaylor & Francis Group, the academic division of T&F Informa plc.

Agnieszka (Agnes) MuszynskaA. M. Consulting

Minden, Nevada, U.S.A.


Published in 2005 byCRC PressTaylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300Boca Raton, FL 33487-2742

2005 by Taylor & Francis Group, LLCCRC Press is an imprint of Taylor & Francis Group

No claim to original U.S. Government worksPrinted in the United States of America on acid-free paper10 9 8 7 6 5 4 3 2 1

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Muszynska, AgnieszkaRotordynamics / by Agnieszka (Agnes) Muszynska.

p. cm. -- (Mechanical engineering ; 188)Includes bibliographical references and index.ISBN 0-8247-2399-6 (alk. paper)1. Rotors--Dynamics. I. Title. II. Mechanical engineering (Marcel Dekker, Inc.) ; 188.

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Dedication

In memory of Bently Rotor Dynamics Research Corporation


Foreword Bently

I came to know Dr. Agnieszka (Agnes) Muszynska in 1980, when she was working as avisiting scientist at the University of Dayton on an unclassified contract of Wright-PattersonAir Force Base.

Earlier in the seventies, Dr. Czeslaw Broniarek showed me an original copy ofDr. Muszynskas book On Rotor Dynamics, which was a review of several hundred papers onthis subject. I later obtained my own copy of her book. This book was translated into Englishand I studied it; however, when I referenced this book in a paper, published by the AmericanSociety of Mechanical Engineers (ASME), I stated author unknown because I thoughtAgnieszka Muszynska was the name of a committee. Only a few months later I learned ofmy misunderstanding.

In 1974, when almost 200 of the worlds top rotordynamic experts were meeting inDenmark, I was teaching at a machinery-monitoring seminar in Frankfurt, Germany at thesame time so I could not attend this conference. A couple of years later, my friend Dr. EdGunter told me about a pretty blonde Polish lady who attended this conference in Denmark(she was the only woman among 200 men!). Shortly thereafter, I met Dr. Muszynska at aworkshop on instability problems in turbomachinery sponsored by the National Aeronauticsand Space Administration (NASA), which was held at Texas A&M University. As a result ofthis meeting, she came to work with me on rotor dynamic problems, first at Bently Nevada,then at Bently Rotor Dynamics Research Corporation in Minden, Nevada.

It is my belief that we did about 50 years worth of work inside about a 20-year period,since she is a very independent thinker and orderly researcher. Largely I worked in thelaboratory running the experiments and Dr. Muszynska did the theoretical work writingthe equations. I brought to the table the machinery problems as observed in the field; shebrought the theory and knowledge. For many years, we both lectured on modern rotordynamics all around the world.

Dr. Muszynska is credited in the dedication of my book, Fundamentals of RotatingMachinery Diagnostics (BPB Press, 2002), for the development of many of the equations andmethodology used in modern rotor dynamics. Her current book presents more theory,whereas mine is more practical. You should consider having both books for your library.

Of special importance is the application on more modern modal equations of rotordynamics and the close relationship between control theory, vibration theory, and rotatingmachinery theory. The work of Dr. Walter R. Evans is strongly emphasized in both books.Each book represents a step forward in our general knowledge of rotating machinery. Due tothe natural inertia of human beings, it is always extremely difficult to introduce newmethodology and concepts; however, it is very important to advance and learn, therefore, weboth teach these modern 21st century techniques.

Donald E. BentlyBently Pressurized Bearing Company

Minden, Nevada


Foreword Jones

As an engineer with many years of experience in several engineering and scientificdisciplines, but not in rotordynamics, I found reading the draft manuscript of this book tobe something of a revelation. The author has many years of experience in the discipline ofrotordynamics, both theoretical and experimental, 18 of which were spent confrontinga wide variety of real-world problems at Bently Rotor Dynamics Research Corporation, asubsidiary of Bently Nevada Corporation in Minden, Nevada. This brings a unique approachto the subject in depth and breadth.

The book addresses the general problem of analytically modeling and predicting thedynamic response and instability of many types and sizes of rotating machines, with a view tounderstanding in detail a wide variety of observed phenomena, including miscellaneous typesof malfunctions. The approach is based on modal analysis generally concentrating on themost important low order modes, but by no means limited in principle to those modes. Manyforcing and feedback mechanisms are addressed, and a unique emphasis is applied tocorrelating observed behavior with analytical models. In fact, some models were developedusing modal identification techniques with emphasis on the model adequacy relative toobserved phenomena. This is not only to vindicate the approach but also to hold out thepromise of applying parameter variation to identify the source of measured responseirregularities and hence prepare appropriate corrective measures in time to avoidunscheduled shutdowns or even catastrophic failures of machines.

For a great variety and number of facilities around the world which apply large (usually)rotating machines for power generation or material handling, for example, the economicconsequences of undetected or misdiagnosed malfunctions, resulting in unscheduledshutdowns, can be serious. Catastrophic failures, while rare, do occur and the costs indamage, injury, and liability can be even greater. The need for means of identifyingimpending malfunctions and determining appropriate corrective measures has beenrecognized for a long time.

Practicing engineers, who are responsible for running major facilities using rotatingmachines, are undoubtedly well aware of their responsibility to avoid such dire situations.This is why condition monitoring and diagnosis has gained such wide acceptance in recenttimes, especially in view of the currently available technology for on-line detection ofvibratory data, and computerized processing and display of the measured signals for humanevaluation. The signals and trends obtained by current monitoring systems provide theearliest available evidence of any impending malfunctions before they more openly manifestthemselves. The problem is to correctly interpret the observed data, so that appropriatedecisions can be made in response to important questions such as, can it wait until the nextscheduled shutdown?, should the bearing oil temperature be raised or lowered?, shouldwe shut down right now and look for a crack, and where should we look? There must bemany more such questions.

I believe that this book offers a unique new approach to these issues, based on rationalmodal models with analytical descriptions of a number of internal and external forces thatcan result in instabilities of rotating machines, with application to recognizing and identifyingof the observed behavior of machines. Instabilities are not welcome events in machines. Abetter understanding of mechanisms leading to instabilities may sensitize and stimulatemachine designers and developers in order to prevent these mechanisms from occurring.The better understanding among machine users will prevent them from purchasing faultymachines, susceptible to malfunctions.


The book is not written for beginners a number of such books are already available but if it is studied with care, attention, and diligence, it will provide readers with ideasand concepts which are capable of further development in research centers. Particularlyattractive may be the development of software, based on the modal models advanced in thisbook, to interpret signals provided by the monitoring systems of specific machines,perhaps specifically perturbed while in normal operation. Such software would lead to anexpert system pertaining to that particular machine or type of machines. The softwaredevelopment costs would be far less than those involved in failing to catch a major problemin time.

In summary, this book may represent a new paradigm in the understanding ofrotordynamic phenomena and malfunctions. I highly recommend it for all readers interestedin rotordynamics in general, as well as those with specific technical goals for which the bookmight provide some directions.

David I.G. JonesConsulting EngineerChandler, Arizona


Preface

Rotating machines represent the largest and most important class of machinery usedfor fluid media transportation, for metal working and forming, for energy generation, forproviding aircraft propulsion, and for other purposes. Rotors equipped with bladeddisks or impellers rotating at high speeds in the fluid environment allow rotating machinesto produce, absorb, transform, or condition an amazing amount of energy often incomparatively small, compact packages. Increasing economic demands for larger capacity,higher quality, and environmental acceptance in production and transportation, as well asinevitably growing user expectations, place stringent requirements on the performance ofmachines.

Rotordynamics is an extremely important branch of the discipline of dynamics thatpertains to the operation and behavior of a huge assortment of rotating machines. Thismachine behavior encompasses a wide variety of physical phenomena, all of which caninterfere with the proper functioning of machines and can even lead to catastrophic failuresif not properly identified and corrected.

This book represents the culmination of many years work by the author to contribute tothe knowledge on rotor dynamic behavior and in particular to apply and further develop themodal methodology for modeling the dynamic behavior of rotating machines of varioustypes, and under a range of conditions. The modeling is confronted with realistically obtainedvibration data from machines. The modal equations are relatively simple, incorporatingseveral parameters, which are identified from diagnostic tests obtained under normaloperating conditions. The theory is classically phenomenological, in the same sense as currentexperimental modal analysis techniques, applied for identification and diagnosis of non-rotating structures. This is a significant advantage. The time for first principles is at thedesign stage, not when the machine is in operation and critical day-to-day decisions haveto be made with no room for error. What is needed during the life of the machine is notthe last word in analytical capability, involving large finite element models and massivecomputing power, but rather a relatively simple means of replicating the essential featuresof the observed, measurable behavior which may contain the telltale signs of specificimpending problems, and varying relevant parameters to determine the most effectivecorrective measures. Potentially, equations such as described in the book could be interfacedwith the monitoring system computer codes so as to respond to particular, previouslyidentified changes in the monitored data. This would be cost-effective, particularly for verylarge machine systems which have been in service for some time in a number of locations, sothat some accumulated history of actual problems would be available for imbedding in theanalysis. It is to be hoped that many practicing engineers will make the effort to examine thisbook in that light.

Condition monitoring systems are designed to measure vibration and other data invarious critical parts of the machine on a continuous or near-continuous basis. The taskof operating engineers is to interpret the results provided by the monitoring system in orderto accurately identify impending problems and recommend proper corrective actions in timeto prevent these problems from reaching a critical stage, which would lead to unscheduledshutdowns or even to catastrophic failure of the system. Correct diagnosis is clearly essentialif this task is to be accomplished successfully. Trial-and-error modifications are oftenattempted, but are seldom effective if the problem is misdiagnosed. One must keep in view therelatively sophisticated level to which condition monitoring has progressed in recent decades,


especially in view of the currently available computational capability to process the largevolume of measured data in real time; what is needed is a relatively simple but adequateanalytical approach. The approach such as modal expansion method, which would satis-factorily model the known phenomena affecting measurable machine performance, therebypermitting variation of appropriate parameters helps interpret the measured data andallows rational conclusions to be drawn. In this way, hopefully, any impending problemscan be identified early, and proper corrective steps can be scheduled. An early diagnosisis particularly important for large and critical rotating machinery, for which unscheduleddown-time can be very expensive. Such a modeling approach is offered in this book.

Most results presented in this book were obtained at Bently Rotordynamics ResearchCorporation (abbreviation BRDRC which phonetically reads Bird Rock). BRDRC wasfounded in 1982 as a subsidiary of Bently Nevada Corporation (BNC). The primary objectiveof BRDRC was to expand the current body of knowledge relating to rotating machinerydynamics, including machine malfunction diagnostic techniques. To this end, BRDRCaccumulated and evaluated pertinent knowledge from external sources, and developed newknowledge from internal research. Since isolated knowledge is useless, it was made availableto the greater community through various means. Among them there were published papersand reports (over 300 publications), conference presentations, worldwide seminars andcourses on rotating machine dynamics and diagnostics, academic lectures at universities andresearch centers, patents, consulting services to industry customers, collaboration with otherscientific organizations, donation of equipment, training interns and college students,conference sponsorship, and other means. In addition to all this, BRDRC providedfundamental and extended knowledge to BNC in order to improve the performance andincrease the value of company products for its customers.

In January 2002, BNC and BRDRC were sold to the General Electric Corporation(GE) by the sole owner of both corporations, Donald E. Bently. A few months later, GEdissolved BRDRC. The 20-year career of BRDRC had ended. The accumulated knowledgebase was dispersed, with no possibility of continuation by another generation of researchers.

This book, which acts in parallel to classical treatments of rotordynamic problems,presents major achievements on theoretical and experimental rotating machinery dynamicsand diagnostics obtained at BRDRC. It may be viewed as an epitaph to BRDRC.

Perhaps a few words of history would help explain the uniqueness of BRDRC. The talestarts with a short story about Bently Nevada Corporation. This company was founded byDonald E. Bently in Berkeley, California, in 1955, as Bently Scientific Company. In 1961,Don Bently moved the company to its present location in Minden, Nevada, and renamed itBently Nevada Corporation. Although Don Bently was not the first to invent the principleof noncontacting eddy-current displacement transducers, he did pioneer their practicalapplication for measuring mechanical vibration and static position in machinery. For the firsttime, these noncontacting transducers provided, to manufacturers and users of machinery,clear and accurate information about the actual dynamic behavior of mechanical elementswithin the machine. In particular, they provided direct measurements of static positionsand vibratory motion, as well as centerline average positions during vibrations of the mostimportant elements of machines, namely, the rotors. Previously, these measurements couldonly be inferred from indirect measurements on the machine casing or by the use of muchless accurate, often unreliable, mechanical devices (as shaft riders), which required directphysical contact with the rotating rotors.

Starting from these fundamental noncontacting eddy-current displacement transducersand taking advantage of progress in electronic and computer technology, Bently Nevadagave birth to an entire new industry. Today, this industry produces sophisticated monitoringsystems for machinery protection, including on-line software tools and embedded-knowledge


software capable of analyzing machine malfunctions, and providing expert advice whichmimics the thought processes of experienced machinery specialists. The basic eddy currenttransducer, however, remains at the heart of all monitoring systems of rotating machines.

Donald E. Bently created BRDRC in 1982, one year after he had hired me as an engineerat BNC Mechanical Engineering Services. I became, and remained, the BRDRC ResearchManager for the next seventeen years. In 1999, I left the company and started my ownconsulting business.

At the beginning, at both BNC and BRDRC, I worked alone, learning experimentaltechniques which were entirely new to me, and providing analytical results to confront withexperimental data. Mr. Bently was anxious to keep track of progress (when I was hired, heexpected from me something like a great symphony in rotordynamics. . .), and often helpedpersonally by means of experiments demonstrating various physical phenomena, which hehad previously observed using his proximity transducers, newly introduced to the world.Small portable rotor rigs imitating rotating machines, which Mr. Bently himself designed,made a career of their own. During the past 25 years, Mr. Bently has donated hundreds ofsuch rigs to universities and research centers around the world, as basic learning toolsto demonstrate various rotor dynamic phenomena and to interactively teach about thedynamic behavior of rotors under various states of operation and when specific malfunctionswere introduced.

During my first years at work, Mr. Bently and I had many discussions about the subjectof mathematical modeling. His electronic engineering background often challenged my ownstrong theoretical mathematical and mechanical background. For me, everything in thelaboratory was new and exciting. I had discovered the world of experimentation, full ofundocumented and poorly understood phenomena. Since I was familiar with the worldliterature on rotor dynamics, having worked in this area of science for the previous 20 yearsor so, I understood that electronic instrumentation had become available only recently at thattime. This instrumentation led to new and fresh tools for observing and simulating thebehavior of mechanical systems ranging from large rotating machinery to simple rotor rigs.These new tools offered unlimited possibilities to access previously unexplored areas.

Our first paper, coauthored with Don Bently, was presented at the Second Texas A&MWorkshop on Instability in High Performance Rotating Machinery in 1982, and was later

Mr. Bently was always very busy with BNC business matters, but he tried to participatepersonally in all research projects in his spare time, providing invaluable suggestionspertaining to the experimental procedures, while I conducted the research and didthe preliminary writing of the co-authored papers. In the early years, I worked atBRDRC, practically alone, consulting daily with Mr. Bently, the president. As time passed,we added one, then two, technicians and one, then a couple, of engineers. In the last decadethe group, while still small, had grown and reached a peak of 14 people in 1996. This groupincluded two technicians and two secretaries (BRDRCs and Mr. Bentlys personalsecretary). Yet even so, we became quite famous (or infamous) in the world of rotordynamicsthrough an abundance of publications, conference presentations, and lectures. We generatedsome quite large controversies, since we had dared to shake up the old classical theories byour new discoveries and new theories. In particular, our new fluid force model in rotor-to-stationary part clearances, known today as the Bently/Muszynska (B/M) model, which weidentified using specific modal testing procedures, created a lot of discussions (the B/M modelis presented in Chapter 4). I was always extremely busy with lecturing, writing papers,documenting our research results, and creating a unique and unprecedented database.BRDRC worked together with BNC Engineering and Diagnostic Services, as well as incooperation with BNC customers, to diagnose malfunctions of various machines in the


published by NASA (see references to Chapter 4, Bently et al., 1982).

field. We simulated, in our laboratory, many phenomena which were reported to occur inmachinery. We identified specific machine signals and then tested them on our rotor rigs toassociate these signals with particular malfunctions induced in the machines.

During these times, I never asked Mr. Bently what to do next, nor did he bother toassign projects, or work with the customary company bureaucracy. There was always a longlist of projects in front of my eyes, which I regarded as waiting to be attacked. Each day didnot have enough hours to accomplish all the tasks, which I assigned for myself. Mr. Bentlywas briefed on the daily progress and was always actively involved in the experiments and inthe final stages of paper writing, providing suggestions and alterations.

Although the BRDRC budget was very limited, I greatly appreciated the minimum ofbeaurocracy and the freedom, efficiency, and elasticity this environment provided for theresearch. New rotor rigs for specific research projects were built in a few days. The necessaryparts were designed as simple sketches, with details being discussed directly with themachinists. For acquisition and processing of vibration data, we used all the newest BNCinstrumentation every year in the process evaluating the performance and quality of thisequipment and giving suggestions for improvements and additions for next generations ofinstruments.

Sometimes difficulties occurred, which can be exemplified in the following story. At oneinstance while working in the laboratory with another engineer, we discovered a totally newphenomenon the second mode of fluid whirl and fluid whip instability of the rotor (see

my associate and me. He claimed that these phenomena would not be interesting to BNCcustomers. Risking our jobs, we ignored Mr. Bentlys threats and continued to work on theproject after regular hours, under rather uncomfortable conditions and in spite of our bosssadverse attitude. As a result of these clandestine efforts, we completed the research andwrote a paper. A few months later, we were presented with the American Society ofMechanical Engineers Award for the best-published paper of the year 1991, which describedthese phenomena! After that, Mr. Bently promptly forgot about his earlier position, andthereafter said WE got an award!, and he was very proud of it.

I had intended to write a book on rotordynamics over fifteen years ago. At that timethere was no single volume available on the subject. I even received contract proposals fromtwo publishers. However, Mr. Bently persuaded me to publish a book internally, throughBNC. Therefore, we started working on that book. Around that time, computers becameoverwhelming gadgets so that it seemed nobody would read archaic paper books anymore. Our book project was consequently suddenly diverted into a CD ROM MachineLibrary. This electronic book on compact disk (CD) was completed in 1995 and in factbecame a magnificent learning tool for people interested in practical rotordynamics. Withanimation and user-interactive features, it provided and still provides a refreshing newdimension in the learning process.

Even so, a more traditional paper book was still considered necessary. Last year, DonBently finished his own book Fundamentals of Rotating Machinery Diagnostics (BentlyPressurized Bearing Press, 2002). Then I began working on this book.

Rotors are the most important parts of rotating machinery. Through their rotationalmotion, rotors are designed to perform the primary work of machines. Being the hearts ofmachines, rotors are also most prone to malfunctions. Very high levels of rotational energyare accumulated, and this energy may easily be diverted into other unwelcome forms ofenergy: vibratory energy, in particular. If anything goes wrong in the machine, theconsequences can be catastrophic. Portions of a broken rotor can act as high velocityprojectiles, causing enormous destruction. Prevention of such occurrences is possible only


Section 4.9 of Chapter 4). I started working on the matter, but Mr. Bently threatened to fire

through an understanding of rotor operation and of all the potential malfunctions that mayoccur. With early recognition, such malfunctions can be corrected quite easily.

It is my intention that this book present basic rotordynamic problems through theapplication of mathematical models. In particular, these models are based on rotor modalbehavior. The emphasis is on understanding rotor-dynamic physical phenomena, as descri-bed through mathematical models, and their correlation with measured data. Consequently,this book is designed to be of most interest to readers who are somewhat familiar with thetheory of mechanical vibration and who have some background in linear and nonlineardifferential equations. Less mathematically confident readers may find useful information

Several of the mathematical models described in the book were first identified throughmodal perturbation tests. Therefore, their further application reflects observable andmeasurable phenomena occurring in rotating machines. A good understanding of thephysical phenomena which take place in rotating machines permits better, more educateddiagnosis and correction of machine malfunctions. This book sets out the fundamentals ofvibration monitoring and diagnosis of rotating machines. However, the book is not intendedto provide information on rotor design for specific applications, nor does it provide computerprograms for rotordynamic calculations. Many versions of such software are available on thecommercial market. Such programs are used routinely, often without adequate under-standing of the physical phenomena which stand behind the numbers produced by thecomputer codes. Consequently, the designer is often unable to check the validity orreasonableness of the numbers. This book emphasizes, using simple but adequate models forthe lowest modes, the understanding of the phenomena taking place in the rotor and itsenvironment, thereby providing a tool for qualitatively evaluating the range of computer-generated numerical results. Multitudes of experiments, described in detail, not only supportthe presented analytical material but perhaps may also stimulate an enthusiasm to repeatand extend these experiments.

The book provides an insight into nonlinear vibrations of rotor systems. Complexequations are often solved analytically with a certain degree of approximation. The emphasisis on qualitative representation of observable physical phenomena. Symbols used in mostmodels discussed in the book have physical meaning and are the same throughout the text.Only when using the method of small parameter to solve nonlinear equations, nondimen-

Alongside the mainstream of research on mechanical system modeling leading towardimproved adequacy of models of real systems and their higher accuracy by trading offwith the computational burden, there exists an equal need to develop models which, in spiteof their relative simplicity, reflect basic properties of real systems within a limited range offrequencies. These models represent useful tools in qualitative analyses of the system dynamicresponses to additional, often nonlinear, factors. Their solutions also better compare withpractical, limited measurement data. Finally, such models represent useful educational toolsdesigned for better understanding of system dynamics.

One such simple model is the Jeffcott model of a rotor (1919). In spite of its age of over80 years, the Jeffcott rotor is still widely used for the above-mentioned purposes. Manyresearchers who base their considerations and results on Jeffcott models are often subject tocriticism that these models are not realistic enough for analysis. The Jeffcott rotor in itsoriginal form is, however, nothing less than a simplified version of the one-lateral-mode

As applied throughout this book, the extensions of the Jeffcott rotor toward the modalmodels embrace incorporation of rotor support stiffness as parts of the total system stiffness,support anisotropy and/or rotor cross-sectional asymmetry and various nonlinearities.


on rotating machine monitoring and diagnostics in Chapter 2 and Chapter 7.

modal model of the rotor (see Chapter 1).

sional ratios of variables were introduced for convenience (Section 5.7 of Chapter 5).

Further modifications of models, which include additional masses (such as the mass of thejournal), are considered as a direct step into multi-mode modeling.

In spite of being widely used and proving their usefulness in a variety of applications, themodels based on the Jeffcott rotor suffer from prejudiced opinions about their unrealisticsimplification and inferior features. This criticism would always be justified if such modelswere not appropriately linked to the modal behavior of rotors. With the modal adequacyassured, the lumped mass models have gained a new career. The modal testing, which is nowextensively used, provides means for identification of multimode parameters. Being directlyrelated to the modal characteristics, with practically identifiable parameters, multimodemodels now have a solid base for further application in such areas as stability and post-stability self-excited vibrations, sensitivity analysis, active control, solidfluid interaction,local/global dynamic effects, fractional resonances, chaotic vibrations, and many others.Several topics from this list are discussed in this book.

The book is arranged in seven chapters.

In this model, which is based on practical measurements of lateral vibrations of rotors, a veryimportant innovation is introduced. The physical models, which form the basis for themathematical models, are not abstract combinations of massless springs and heavy, rigiddisks (or other elements), but instead correspond to measurable modal parameters for theparticular systems under consideration. This modal approach to modeling is applied for mostmathematical models discussed throughout the book. In the first chapter, instead of limitingthe analysis of the simplest model of a rotor to consideration of the classical effects of anunbalance force, an external non-synchronously rotating force is introduced. This permitsexplanation of the application of basic modal procedures for identification of system modalparameters. In these procedures, as generally in vibration measurements, an accuratemeasurement of the vibration phase is highly emphasized. In fact, the phase measurement isoften more important than the measurement of the amplitude. Actually, both of these partsare lumped into one measurement parameter, the vibration response vector.

This chapter also introduces orbits, which represent a magnified path of the actualmotion of the rotor centerline during rotor lateral vibrations. Machine rotor orbits can beobserved online on an oscilloscope or through a computerized data acquisition andprocessing system. From the shape of an orbit, some information on forces as primarysources of the rotor motion can be obtained. This information is valuable for rotatingmachine diagnostics.

During the past several decades, there has been significant progress in mechanicalvibration measurement and vibration monitoring equipment. This progress is reflected on thequality and sophistication of the available instrumentation and has led to an abundance ofaccumulated results, including the outcome of dedicated research in rotor dynamics andevaluation of case histories gathered from malfunctions of various machines.

rotating machinery, and reviews trends in machinery management and monitoring programs.The presentation is inevitably biased towards Bently Nevada philosophy in vibrationmeasurement and surveillance of machinery. However, Bently Nevada Corporation wasthe world pioneer of major vibration measuring instrumentation on machinery, and set anumber of well-acknowledged standards.

were discussed in Chapter 1. This part of the book is essentially classical, although theintroduction of a nonsynchronously rotating external exciting force is seldom seen in therotordynamics literature. In this chapter, the torsional and coupled lateral/torsionalvibrations and their significant role in rotating machine dynamics are presented.


In Chapter 1, the fundamental two-lateral-mode isotropic model of a rotor is introduced.

Chapter 2 presents an introduction to vibration monitoring and data processing for

Chapter 3 presents extended rotor models, which include more modes and forces than

problems arising from rotor/fixed structure clearances in rotating machines. The subjectmatter of this chapter is unique and original, being based on numerous investigationsreported by the author and Donald E. Bently and their associates at BRDRC. Fluid-inducedforces, which act on rotors operating in a fluid environment, have been recognized for over70 years. These forces are infamous for their rotor-destabilizing effects. The resultingunwelcome rotor vibrations are known by several names, most frequently associated withterms such as whirl and whip (e.g., oil whirl/whip, steam whirl/whip, aerodynamicwhirl/whip, etc.), or more simply known as rotor fluid-induced instabilities. Sincethese vibrations can be sustained over a wide range of rotational speeds, they may seriouslyperturb normal operation, often causing severe damage to the machine and even leading tocatastrophic failure. Until recently, the mechanisms leading to these vibrations were not wellunderstood, so that measures taken for their correction, elimination, or prevention wereoften inappropriate, inefficient, or even counterproductive.

In Chapter 4, dynamic phenomena induced by interactions between the rotor and thesurrounding fluid, such as in bearings, seals and, more generally, in any rotor/stator radialor axial clearances of fluid handling machines, are modeled with the support of dataacquired from identification procedures. The B/M model of the fluid forces in rotorclearances is based on the strength of the circumferential fluid flow along with itsrepresentative function, the fluid circumferential average velocity ratio (denoted by ,lambda). The B/M model adequately represents the observed phenomena and providesanalytical tools for the control of these undesirable rotor-destabilizing phenomena. The fluidwhirl and fluid whip were identified as limit cycles of self-excited vibrations, after theinstability threshold was exceeded. New phenomena such as higher mode fluid whirl andfluid whip are discussed, experimentally demonstrated, and adequately modeled. The theorybehind the model, including general-type nonlinearities, is consistent. In practicalapplications, this model helps to predict conditions when the fluid surrounding the rotormay cause rotor instability and thus predict conditions of the machine malfunction. Themodel offers measures to prevent and ultimately cure the fluid-induced instabilitymalfunctions in rotating machines.

The chapter also introduces a formal but unconventional derivation of the fluid forces,starting from the classical Reynolds equation. The derivation leads to the extraction of theimportant parameters in the fluid force model, including the parameter . The presentationalso clarifies some paradoxes, which are described in the rotordynamic literature.Comparisons are also presented between the results of the B/M model and the well knownclassical bearing coefficients, thus providing a tool for extracting the value of and otherparameters of the B/M model from the bearing coefficient tables. In Chapter 4, the conceptof dynamic stiffness is extended. It is shown that the quadrature part of the dynamic stiffness,which in non-rotating structures is limited to damping, in rotors is complemented by thefluid-related tangential force component. This is not a new concept, but the latter componentnow has a new, more adequate look. It has been identified in thousands of experimental testsand machinery data.

A large section of Chapter 4 is devoted to identification of rotor/bearing/support systemsand identification techniques. The identification procedures of system parameters areextremely important. Any new machine should be a subject of such identification proce-dures, at least at the prototype level and/or acceptance testing stage. The identificationprocedures are also essential as the first step in any experimental research involving dynamicsof mechanical systems, and rotors in particular.

stator dry contact-related rubbing. In the introduction to the chapter, many occurrences of


Chapter 4 is the most comprehensive in the book, and discusses important fluid-related

Chapter 5 discuses another rotating machinery malfunction problem, namely rotor-to-

rotor rub-related dynamic phenomena in machinery are described. Among these is the self-excited rotor vibration known as dry whip, which is one of the more serious anddestructive of the malfunctions that occur in rotating machinery. In seven sections of thischapter, experimental results, mathematical models of rubbing rotors, their solutions, anddiagnostic recognition patterns are presented. The contents of this chapter material areoriginal, elaborated at BRDRC.

meaningful dynamic phenomena occurring in rotating machinery, which have to berecognized, predictable, and controlled to protect machinery. Among these topics is anintroduction to balancing, with emphasis on understanding these often automatic,computerized procedures. The topic of rotor coupled lateral/torsional vibrations is discussed

In Chapter 6, an introduction to multi-mode modal modeling is also presented. Themodal approach facilitates development of relatively closed-form expressions which are, infact, the lowest order terms of what in general would be modal expansions having manyterms, representing a large number of modes of vibration of the rotor system. By truncatingthe expansions to encompass only the lowest order modes, which are usually of mostpractical interest, and transforming modal variables into variables related to measurableones, the analysis can tractably represent a wide variety of physical phenomena that areobserved in rotating machines.

Other sections of Chapter 6 include discussions of loose rotating parts malfunctionand early detection of rotor crack(s) using vibration data. There is also a section on stressesin rotating and laterally vibrating rotors, which emphasizes the fact that it is not thevibrations but the stresses and deformations which break rotors. Measurable vibrationsmay not necessarily directly reflect high rotor stress conditions. The chapter also includessections on dynamics of rotors with anisotropic supports, and discusses the specific role ofdamping in rotating structures.

illustrated by means of basic simplified mathematical rotor models, which were presented inmore depth in previous chapters. Machine vibration data and specific machine case historiescomplement the discussed subject.

References and a list of mathematical notations follow each chapter.

The author realizes that in such long monograph, it is very difficult to avoid mistakesand repetitions (although the latter are sometimes intentional for educational purpose).In advance, the author apologizes for the mistakes and omissions, and will be pleased tocommunicate with the readers regarding specific problems.


Chapter 6 presents a series of selected topics in rotordynamics relating to various other

in greater depth than in Chapter 3.

Chapter 7 outlines vibration diagnosis of particular malfunctions in rotating machines,

Ten Appendices and a Glossary of terminology complete the book.

The Author

Agnieszka (Agnes) Muszynska, Ph.D. isa native of Warsaw, Poland. She receivedher B.S. and M.S. degrees in MechanicalEngineering from the Technical Universityof Warsaw (M.S. in March 1960). Twoyears of her undergraduate studies werecompleted in Moscow, USSR, at BaumanTechnical University. Dr. Muszynskareceived her Ph.D. in technical sciences(October 1966) and the second level Ph.D.(habilitation, May 1977), both fromthe Polish Academy of Sciences. In 1998,she was awarded the highest professionaldegree, Professor of Technical Sciences, bythe President of Poland, AleksanderKwasniewski. Dr. Muszynska is fluent inPolish, English, Russian, and French.

In February 2000, Dr. Muszynskastarted her own business she createdA.M. Consulting. From October 2000through April 2001, she worked on acontract at the Institute of Robotics ofthe Swiss Federal Institute of Technologyin Zurich, Switzerland. Her consultingwork there concerned rotor/retainer bear-ing dynamics. In August 2001, Dr. Muszynska presented a keynote address on rotor/fluidinteraction problems to the participants of the International Conference ISCORMA-1(Lake Tahoe, August 2024, 2001). From September through December 2001, Dr.Muszynska worked as a visiting professor and consultant at the Laboratory of AppliedMechanics of the University of Franche Comte in Besancon, France, lecturing on rotordynamics.

From 1981 to 1999 Dr. Muszynska worked as a senior research scientist and researchmanager at Bently Nevada Corporation (BNC) and its subsidiary, Bently Rotor DynamicsResearch Corporation (BRDRC). During these 18 years, Dr. Muszynska conductedtheoretical and experimental research on rotating machine dynamics, participated asa lecturer in BNC technical training programs, and was a member of BRDRC Boardof Directors. In 1997, she served also as a member of the BNC Board of Directors.

Bently Nevada Corporation, created in 1961 by Donald E. Bently, is a manufacturerof electronic hardware and software instrumentation for vibration monitoring on machinery.Its subsidiary, BRDRC, was created in 1982 to enhance theoretical knowledge on machinedynamic behavior leading to mechanical vibrations. The vibrations occur as side effects ofthe main machine processes. The enhancements developed by Dr. Muszynskas contributionsled BNC instrumentation to more efficient technological definition: today the instrumenta-tion not only serves for measuring machine vibration, but also as the diagnostic andprognostic tool in machine maintenance. The knowledge-based link between vibrationcauses and effects led to preventive measures, thus to the development of machine vibration


control technology. Dr. Muszynska brought to BNC an academic excellence in the areaof mechanical engineerings dynamics of rotating machinery. Dr. Muszynska maintainsfriendly relations with BNC and the new company created by Donald E. Bently, BentlyPressurized Bearings.

Prior to joining BNC, Dr. Muszynska held an associate professorship at the Instituteof Fundamental Technological Research of the Polish Academy of Sciences, where sheconducted research on vibrations and machine dynamics. She also taught postgraduateclasses on vibration and mechanical system stability from 1967 through 1979.

From 1975 to 1977, she was visiting professor at the National Institute of AppliedSciences in Lyon, France, where she taught mechanics and machine dynamics and wrote astudent manual on this subject. She was a visiting scientist at the University of Dayton, Ohiofrom January 1980 through June 1981, during which time she worked on a contract forWright Patterson Air Force Base involving vibration control of turbomachinery blades. Shealso taught the class on dynamics of rotating machinery to the University of Dayton graduatestudents.

Dr. Muszynska has authored or co-authored over 250 technical papers on mechanicalvibration theory, nonlinear vibrations, vibration control, and rotating machine dynamics andvibrational diagnostics. Her major contributions consist of the introduction of modalmodeling to such systems as machinery rotors in fluid environment. Her other contributionsare in stability theory of mechanical systems, vibration control, and bladed disk dynamics.She introduced adequate models of such phenomena as rotor-to-stator rubbing, looseness inrotor systems, and lateral/torsional vibrations of rotors. Based on experimental resultsobtained together with Donald E. Bently, in 1986 she published the consistent theory ofinstability of machine shafts rotating in fluid environment. Its simplified version is nowaccepted as Bently Nevada training standard. Also working with Bently, she formalized andpopularized application of modal testing of rotating systems with fluid interaction, as well asthe implementation of the solid/fluid system dynamic stiffness concept. The experimentaldiscovery and adequate modeling of the second and higher modes of the rotor instabilityphenomena, fluid whirl and fluid whip, is one of her significant achievements.

Several of Dr. Muszynskas publications have been nationally recognized. Her paper onmodal analysis of rotating machines received the Best Paper of the Year 1986 award from theAmerican Society for Experimental Mechanics. The report from research on influence ofrubbing on rotordynamics, on which Dr. Muszynska was principal researcher, has been givenan award by NASA in the category of Invention/New Technology. Dr. Muszynskas paper,Stability and Instability of a Two-Mode Rotor Supported by Two Fluid-LubricatedBearings, co-authored by J. Grant, received the Best Paper of the Year 1991 award from theAmerican Society of Mechanical Engineers (ASME) Gas Turbine Division, Structures andDynamics Committee.

Dr. Muszynska has served as the scientific/technical editor of several books, such as thePolish Academy of Sciences yearly journal Nonlinear Vibration Problems (19671980),Machine Dynamics (PASci, 1974), Vibration Control (PASci, 1978), Instability in RotatingMachinery (NASA, 1985), Rotating Machinery Dynamics (ASME, 1987), and Don Bentlythrough the Eyes of Others (Bird Rock Publishing, 1995).

Dr. Muszynska has traveled extensively, actively participating in numerous scientificconferences, giving lectures at courses and university seminars in Europe, North America,Asia, Africa, and Australia. She has also organized or coorganized many internationalscientific meetings, such as the Second and the Sixth International Conferences onNonlinear Oscillations (Warsaw 1962, Poznan 1972, Poland), workshops on machinedynamics (Jablonna, Poland, 1978, 1979), the International Symposium on Instability inRotating Machinery (Carson City, Nevada, June 1985), the session on rotating machinery


dynamics at the 11th Biennial ASME Design Engineering Division Conference on Vibrationand Noise (Boston, Massachusetts, September 1987), the Second, Third, Fourth, Fifth, andSixth International Symposia on Transport Phenomena, Dynamics, and Design of RotatingMachinery (Honolulu, Hawaii, 1988, 1990, 1992, 1994, 1996), and the rotor dynamicsession at the ASME Turbo Expo 1994 Land, Sea, and Air (Hague, The Netherlands, 1994).In 1996, Dr. Muszynska became the chairperson of the organizing committee for theSeventh International Symposium on Transport Phenomena and Dynamics of RotatingMachinery, Honolulu, Hawaii, 1998. She organized a well-received and very successfulsymposium for over 200 international participants, and was the scientific editor of the morethan 1800-page symposium proceedings (print and CD ROM versions). In 2002 and 2003,she was a member of the organizing committee of the Second International Symposium onStability of Rotating Machinery (ISCORMA-2), Gdansk, August, 48, 2003. She wasco-editor of the proceedings of the ISCORMA-2. Currently she is a member of the ScientificCommittee and Organizational Commitee of ISCORMA-3.

In 1985, she was a part-time associate professor at the University of Nevada in Reno(UNR) and a faculty member of the UNR College of Engineering (19851989). In 1985, shelectured to UNR students on mechanical system vibrations.

Dr. Muszynska is a member of the ASME, the Polish Institute of Arts & Sciences ofAmerica, and Rotary International Club. From 1986 to 1988, she was a member of theASME Technical Committee on Vibration and Sound. From 1988 to 1994, Dr. Muszynskaserved as an associate editor of the Transactions of the ASME Journal of Vibrations andAcoustics. In 1994, Dr. Muszynska received the prestigious grade of Fellow in the AmericanSociety of Mechanical Engineers.

Dr. Muszynska received the 1996 Distinguished Research Award for researchachievements in the field of rotating machinery from the Pacific Center of Thermal-Fluid Engineering.

Dr. Muszynska was honored as the Woman Entrepreneur of the Year 1997 by theDouglas County Republican Womens Club.

Dr. Muszynska was honored by the International Biographical Centre, Cambridge, UK,as an International Woman of the Year 1999/2000. Dr. Muszynska is also listed in thepublications Marquis Whos Who in American Women, Whos Who in Polish-American,Marquis Whos Who in America: Science and Engineering, and Marquis Whos Who in theWorld and in American Registry of Outstanding Professionals 2004. In 2004, three significantperpetual foundations for permanent Endowed Chair Professorship were created at thethree following universities: Cleveland University, Cleveland, Ohio, entitled Donald Bentlyand Agnes Muszynska Endowed Chair in Rotordynamics (one million dollars); KoreaAdvanced Institute of Science and Technology (KAIST) in Daejeon (Korea), entitled Bentlyand Muszynska Endowed Chair in Energy (one million dollars); and Korea Universityin Seoul (Korea), entitled Bently and Muszynska Endowed Chair in Life Sciences(500,000 dollars).

Dr. Muszynska has one son and two grandchildren.


Acknowledgment

I would like to thank Donald E. Bently for creating a wonderful research environment atBently Rotor Dynamics Research Corporation, the subsidiary of the Bently NevadaCorporation, and for his continuous friendship. I am also very grateful for his recentlyprovided assistance through the Bently Pressurized Bearing Company. I am also indebted toJeff Jarboe, Neil Bishop, and Tom Frey.

I extend my thanks to all my colleagues at Bently Rotor Dynamics Research Corporationand Bently Nevada Corporation, with whom I worked during 18 years. In particular, I amindebted to Jeanette Cox, who was my administrative assistant for 13 years and helped mereduce my English language handicap considerably, while typing my technical papers.Thanks to Bob Grissom, who is always very helpful in solving a multitude of problems.I would like to thank Bently Nevada Diagnostic Services and Bently Nevada TrainingDepartment for very fruitful and efficient cooperation throughout the years.

My sincere gratitude is also extended to Dr. David I.G. Jones, who was always verysupportive, and who recently read the manuscript of this book and provided me withextremely valuable and helpful suggestions.

Most sections of this book are adapted from my previously published papers. I would liketo extend my thanks to a number of technical societies for permission to reprint some figures.In particular, warm thanks go to the American Society of Mechanical Engineers, to theInstitute of Mechanical Engineers in the United Kingdom, and to the Pacific Center ofThermalFluid Engineering.

Finally, I would like to thank my family and friends for their moral support andforbearance during over a year of intensive work on this book, when I did not have so muchtime for them.

Agnieszka (Agnes) Muszynska


Chapter 1 Basic Rotordynamics: Two Lateral Mode Isotropic Rotor ............................. 11.1 Introduction ................................................................................................................. 11.2 Mathematical Model of Two Lateral Mode Isotropic Rotor ..................................... 81.3 Eigenvalue Problem Rotor Free Response Natural

Frequencies ................................................................................................................ 131.4 Rotor Static Displacement......................................................................................... 141.5 Rotor Nonsynchronous Vibration Response ............................................................ 15

1.5.1 Forced Response to Forward Circular Nonsynchronous Excitation............ 151.5.2 Complex Dynamic Stiffness Diagram Based on Equation (1.15) ................. 17

1.5.2.1 Low excitation frequency, ! 0................................................... 171.5.2.2 Response at direct resonance, ! K=Mp . Case of low

damping, 51 ............................................................................... 181.5.2.3 Response at high excitation frequency, !!1 ........................... 201.5.2.4 Rotor response for the case of high damping, 1 .................... 211.5.2.5 Rotor nonsynchronous amplification factor ................................. 21

1.6 Unidirectional Harmonic, Nonsynchronous Excitation ............................................ 221.7 Rotor Synchronous Excitation Due to Unbalance Force ......................................... 23

1.7.1 Rotor Response to Unbalance Force............................................................ 231.7.2 Differential Technique ................................................................................... 25

1.8 Complex Dynamic Stiffness as a Function of Nonsynchronous PerturbationFrequency: Identification of the System Parameters. Nonsynchronous andSynchronous Perturbation ......................................................................................... 26

1.9 Closing Remarks........................................................................................................ 28References .......................................................................................................................... 29

Chapter 2 Vibration Monitoring of Rotating Machinery ............................................... 312.1 Trends in Machinery Management Programs ........................................................... 312.2 Trends in Vibration Monitoring Instrumentation ..................................................... 342.3 Trend in the Knowledge on Rotating Machine Dynamics ....................................... 372.4 Rotating Machine Vibration Monitoring and Data Processing Systems.................. 39

2.4.1 Vibration Transducers ................................................................................... 392.4.1.1 Accelerometers............................................................................... 392.4.1.2 Velocity transducer........................................................................ 402.4.1.3 Applicability of accelerometers and velocity transducers on

rotating machinery ........................................................................ 402.4.1.4 Displacement transducer ............................................................... 412.4.1.5 Dual transducer ............................................................................. 442.4.1.6 Keyphasor transducer ................................................................. 44

2.4.2 Transducer Selection...................................................................................... 472.4.3 Machine Operating Modes for Data Acquisition and Data

Processing Formats........................................................................................ 482.4.4 Modal Transducers Virtual Rotation of Transducers Measurement

of Rotor Torsional Vibrations ...................................................................... 562.4.5 Application of Full Spectrum and Complex Variable Filtering in

Rotor Health Diagnostics.............................................................................. 62


Contents

2.4.6 Measurement and Documentation Conventions ........................................... 692.4.7 Recommendations for Monitoring of Rotating Machines............................ 712.4.8 Instruments for Data Processing and Displaying in Real Time ................... 71

2.4.8.1 Oscilloscope ................................................................................... 722.4.8.2 Monitors ........................................................................................ 722.4.8.3 Filters............................................................................................. 732.4.8.4 FFT Spectrum Analyzer................................................................ 73

2.4.9 Computerized Data Acquisition and Processing Systems ............................. 742.4.10 Incorporation of Machine Modeling into Data Processing Systems ............ 74

2.5 Closing Remarks........................................................................................................ 76References .......................................................................................................................... 76

Chapter 3 Basic Rotordynamics: Extended Rotor Models ............................................. 793.1 Introduction ............................................................................................................... 793.2 Rotor Modes.............................................................................................................. 79

3.2.1 Introduction................................................................................................... 793.2.2 Lateral Modes of a Two-Disk Isotropic Rotor............................................. 813.2.3 Modes of a Flexible Rotor in Flexible Supports .......................................... 883.2.4 Modes of an Overhung Rotor in Flexible Supports ..................................... 893.2.5 Modes of a Multi-Rotor Machine: Example A Turbogenerator Set ....... 893.2.6 Other Modes of Rotor Systems..................................................................... 90

3.3 Model of the Rotor with Internal Friction ............................................................... 903.3.1 Introduction: Role of External and Internal Damping in Rotors ................ 903.3.2 Transformation to the Rotating Coordinates Attached to the Rotor .......... 923.3.3 Rotor Response ............................................................................................. 95

3.3.3.1 Rotor free response, natural frequencies, instability threshold..... 953.3.3.2 Rotor static displacement.............................................................. 98

3.3.3.2.1 Experimental demonstration of the attitude angle...... 993.3.3.3 Rotor nonsynchronous vibration response: forced response

for forward circular excitation ..................................................... 1003.3.3.4 CDS diagram................................................................................ 102

3.3.4 Isotropic Rotor Model with Nonlinear Hysteretic Internal Friction.......... 1073.3.5 Rotor Effective Damping Reduction Due to Internal Friction .................. 1093.3.6 Internal Friction Experiment....................................................................... 1103.3.7 Instability of an Electric Machine Rotor Caused by Electromagnetic

Field of Rotation......................................................................................... 1133.3.8 Summary...................................................................................................... 115

3.4 Isotropic Rotor in Flexible Anisotropic Supports: Backward Orbiting.................. 1173.4.1 Rotor Model and Rotor Forced Response to External Nonsynchronous

Rotating Force Excitation........................................................................... 1173.4.2 Constant Amplitude Rotating Force Excitation ......................................... 1213.4.3 Rotating Force Excitation with Frequency-Dependent Amplitude ............ 1233.4.4 Final Remarks ............................................................................................. 124

3.5 Anisotropic Rotor in Isotropic Supports ................................................................ 1243.5.1 Anisotropic Rotor Model............................................................................ 1243.5.2 Eigenvalue Problem: Rotor Natural Frequencies and Stability

Conditions ................................................................................................... 1253.5.3 Rotor Response to a Constant Radial Force.............................................. 1293.5.4 Rotor Vibration Response to a Rotating Force ......................................... 134


3.5.4.1 A general case of nonsynchronous frequency excitation ............. 1343.5.4.2 Excitation by rotor unbalance force ............................................ 135

3.6 Angular Momentum Model of an Isotropic Rotor................................................. 1403.6.1 Rotor Model Derivation ............................................................................. 1403.6.2 Eigenvalue Problem and Resonance Speeds in Case without Damping ..... 1433.6.3 Rotor Response to Unbalance .................................................................... 144

3.7 Angular Momentum Model of an Anisotropic Rotor with Anisotropic Disk ....... 1473.7.1 Rotor Model Derivation ............................................................................. 1473.7.2 Eigenvalue Problem, Rotor Free Vibrations, and Stability Conditions...... 1483.7.3 Rotor Response to Skewed Disk Unbalance-Related Excitation ............... 149

3.8 Model of Coupled Transversal and Angular Motion of the IsotropicRotor with Axisymmetric Disk and Anisotropic Supports ..................................... 1513.8.1 Rotor Model................................................................................................ 1513.8.2 Eigenvalue Problem and Rotor Free Vibrations......................................... 1523.8.3 Rotor Response to Constant Unidirectional Force .................................... 1543.8.4 Rotor Forced Response to Unbalance........................................................ 155

3.9 Model of Coupled Lateral Transversal and Lateral Angular Motion ofan Anisotropic Rotor with Unsymmetric Disk ....................................................... 1553.9.1 Rotor Model................................................................................................ 1553.9.2 Eigenvalue Problem: Natural Frequencies and Stability Conditions .......... 1563.9.3 Rotor Response to Unbalance .................................................................... 159

3.10 Torsional and Torsional/Lateral Vibrations of Rotors ........................................... 1603.10.1 Introduction: Role of Damping in the Torsional Mode ........................... 1603.10.2 Model of Pure Torsional Vibrations of Rotors......................................... 1613.10.3 Model of Pure Torsional Vibrations of a Two-Disk Rotor

and its Solution.......................................................................................... 1643.10.4 Model of Coupled Lateral and Torsional Vibrations of an Anisotropic

Rotor with One Massive Disk................................................................... 1663.10.4.1 Rotor model .............................................................................. 1663.10.4.2 Eigenvalue problem: natural frequencies and stability

conditions .................................................................................. 1693.10.4.3 Rotor forced response to unbalance ......................................... 1713.10.4.4 Rotor forced response to gravity force ..................................... 1713.10.4.5 Rotor forced response to a variable torque.............................. 173

3.10.5 Torsional/Lateral Cross Coupling due to Rotor Anisotropy:Experimental Results ................................................................................. 1753.10.5.1 Experimental rotor rig .............................................................. 1763.10.5.2 Experimental results .................................................................. 1763.10.5.3 Discussion ................................................................................. 180

3.10.6 Summary and Conclusions ........................................................................ 1813.11 Misalignment Model................................................................................................ 184

3.11.1 Introduction............................................................................................... 1843.11.2 Mathematical Model of Misaligned Rotor................................................ 185

3.11.2.1 Rotor nonlinear model.............................................................. 1853.11.2.2 Harmonic balance solution for the rotor forced response........ 1863.11.2.3 Approximate solution ............................................................... 189

3.11.3 Case History on Nonlinear Effects of a Side-Loaded Rotor Supportedin One Pivoting Bronze Bushing and One Fluid Lubricated Bearing....... 1913.11.3.1 Introduction .............................................................................. 1913.11.3.2 Description of the rotor rig ...................................................... 192


3.11.3.3 Static load testing...................................................................... 1933.11.3.4 Rotor lateral response rata during start-up with

concentric journal...................................................................... 1933.11.3.5 Rotor lateral response data during start-up with

side-loaded journal .................................................................... 1953.11.3.6 Discussion ................................................................................. 202

3.11.4 Closing Remarks........................................................................................ 203References ........................................................................................................................ 205

Chapter 4 Fluid-Related Problems in Rotor/Stator Clearances .................................... 2094.1 Introduction ............................................................................................................. 209

4.1.1 Some Personal Remarks .............................................................................. 2094.1.2 What This Chapter Presents........................................................................ 210

4.2 Fluid Whirl and Fluid Whip: Rotor Self-Excited Vibrations ................................. 2144.2.1 Description of the Startup Vibration Behavior of a

Rotor/Bearing System ................................................................................. 2144.2.2 Fluid-Related Natural Frequency of the Rotor/Fluid System.................... 2224.2.3 Stability versus Instability. Practical Stability of a

Rotating Machine........................................................................................ 2264.2.4 Fluid Whirl and Fluid Whip in Seals and in Fluid-Handling

Machines...................................................................................................... 2264.2.5 Summary...................................................................................................... 227

4.3 Mathematical Model of Fluid Forces in Rotor/Stator Clearances ......................... 2274.3.1 Fluid Force Model ...................................................................................... 2274.3.2 Experimental Results ................................................................................... 235

4.3.2.1 Impulse testing: fluid circumferential average velocity ratioas a decreasing function of journal eccentricity ........................... 235

4.3.2.2 Fluid starvation lowers the fluid circumferential averagevelocity ratio value ....................................................................... 236

4.3.2.3 Conclusions from experiments ..................................................... 2424.3.3 Summary...................................................................................................... 246

4.4 Response of Two Lateral Mode Isotropic Rotor with Fluid Interaction toNonsynchronous Excitation. Introduction to Identification of Rotor/FluidCharacteristics.......................................................................................................... 2474.4.1 Introduction............................

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Rotordynamics - Agnieszka Muszynska