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- Herausgeber: Wiley-VCH
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
A wide-ranging treatment of fundamental rotordynamics in order to serve engineers with the necessary knowledge to eliminate various vibration problems.
New to this edition are three chapters on highly significant topics:
Vibration Suppression - The chapter presents various methods and is a helpful guidance for professional engineers.
Magnetic Bearings - The chapter provides fundamental knowledge and enables the reader to realize simple magnetic bearings in the laboratory.
Some Practical Rotor Systems - The chapter explains various vibration characteristics of steam turbines and wind turbines.
The contents of other chapters on Balancing, Vibrations due to Mechanical Elements, and Cracked Rotors are added to and revised extensively.
The authors provide a classification of rotating shaft systems and general coverage of key ideas common to all branches of rotordynamics. They offers a unique analysis of dynamical problems, such as nonlinear rotordynamics, self-excited vibration, nonstationary vibration, and flow-induced oscillations. Nonlinear resonances are discussed in detail, as well as methods for shaft stability and various theoretical derivations and computational methods for analyzing rotors to determine and correct vibrations.
This edition also includes case studies and problems.
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Seitenzahl: 627
Veröffentlichungsjahr: 2013
Ähnliche
Table of Contents
Related Titles
Title Page
Copyright
Dedication
Foreword to the First Edition
Preface to the First Edition
Preface to the Second Edition
Acknowledgments
Chapter 1: Introduction
1.1 Classification of Rotor Systems
1.2 Historical Perspective
References
Chapter 2: Vibrations of Massless Shafts with Rigid Disks
2.1 General Considerations
2.2 Rotor Unbalance
2.3 Lateral Vibrations of an Elastic Shaft with a Disk at Its Center
2.4 Inclination Vibrations of an Elastic Shaft with a Disk at Its Center
2.5 Vibrations of a 4 DOF System
2.6 Vibrations of a Rigid Rotor
2.7 Approximate Formulas for Critical Speeds of a Shaft with Several Disks
References
Chapter 3: Vibrations of a Continuous Rotor
3.1 General Considerations
3.2 Equations of Motion
3.3 Free Whirling Motions and Critical Speeds
3.4 Synchronous Whirl
References
Chapter 4: Balancing
4.1 Introduction
4.2 Classification of Rotors
4.3 Balancing of a Rigid Rotor
4.4 Balancing of a Flexible Rotor
References
Chapter 5: Vibrations of an Asymmetrical Shaft and an Asymmetrical Rotor
5.1 General Considerations
5.2 Asymmetrical Shaft with a Disk at Midspan
5.3 Inclination Motion of an Asymmetrical Rotor Mounted on a Symmetrical Shaft
5.4 Double-Frequency Vibrations of an Asymmetrical Horizontal Shaft
References
Chapter 6: Nonlinear Vibrations
6.1 General Considerations
6.2 Causes and Expressions of Nonlinear Spring Characteristics: Weak Nonlinearity
6.3 Expressions of Equations of Motion Using Physical and Normal Coordinates
6.4 Various Types of Nonlinear Resonances
6.5 Nonlinear Resonances in a System with Radial Clearance: Strong Nonlinearity
6.6 Nonlinear Resonances of a Continuous Rotor
6.7 Internal Resonance Phenomenon
References
Chapter 7: Self-Excited Vibrations Due to Internal Damping
7.1 General Considerations
7.2 Friction in Rotor Systems and Its Expressions
7.3 Self-Excited Vibrations Due to Hysteretic Damping
7.4 Self-Excited Vibrations Due to Structural Damping
References
Chapter 8: Nonstationary Vibrations during Passage through Critical Speeds
8.1 General Considerations
8.2 Equations of Motion for Lateral Motion
8.3 Transition with Constant Acceleration
8.4 Transition with Limited Driving Torque
8.5 Analysis by the Asymptotic Method (Nonlinear System, Constant Acceleration)
References
Chapter 9: Vibrations due to Mechanical Elements
9.1 General Considerations
9.2 Ball Bearings
9.3 Bearing Pedestals with Directional Difference in Stiffness
9.4 Universal Joint
9.5 Rubbing
9.6 Self-Excited Oscillation in a System with a Clearance between Bearing and Housing
References
Chapter 10: Flow-Induced Vibrations
10.1 General Considerations
10.2 Oil Whip and Oil Whirl
10.3 Seals
10.4 Tip Clearance Excitation
10.5 Hollow Rotor Partially Filled with Liquid
References
Chapter 11: Vibration Suppression
11.1 Introduction
11.2 Vibration Absorbing Rubber
11.3 Theory of Dynamic Vibration Absorber
11.4 Squeeze-Film Damper Bearing
11.5 Ball Balancer
11.6 Discontinuous Spring Characteristics
11.7 Leaf Spring
11.8 Viscous Damper
11.9 Suppression of Rubbing
References
Chapter 12: Some Practical Rotor Systems
12.1 General Consideration
12.2 Steam Turbines
12.3 Wind Turbines
References
Chapter 13: Cracked Rotors
13.1 General Considerations
13.2 Modeling and Equations of Motion
13.3 Numerical Simulation (PWL Model)
13.4 Theoretical Analysis (PS Model)
13.5 Case History in Industrial Machinery
References
Chapter 14: Finite Element Method
14.1 General Considerations
14.2 Fundamental Procedure of the Finite Element Method
14.3 Discretization of a Rotor System
14.4 Free Vibrations: Eigenvalue Problem
14.5 Forced Vibrations
14.6 Alternative Procedure
References
Chapter 15: Transfer Matrix Method
15.1 General Considerations
15.2 Fundamental Procedure of the Transfer Matrix Method
15.3 Free Vibrations of a Rotor
15.4 Forced Vibrations of a Rotor
References
Chapter 16: Measurement and Signal Processing
16.1 General Considerations
16.2 Measurement and Sampling Problem
16.3 Fourier Series
16.4 Fourier Transform
16.5 Discrete Fourier Transform
16.6 Fast Fourier Transform
16.7 Leakage Error and Countermeasures
16.8 Applications of FFT to Rotor Vibrations
References
Chapter 17: Active Magnetic Bearing
17.1 General Considerations
17.2 Magnetic Levitation and Earnshaw's Theorem
17.3 Active Magnetic Levitation
17.4 Active Magnetic Bearing
References
Appendix A: Moment of Inertia and Equations of Motion
Appendix B: Stability above the Major Critical Speed
Appendix C: Derivation of Equations of Motion of a 4 DOF Rotor System by Using Euler Angles
Appendix D: Asymmetrical Shaft and Asymmetrical Rotor with Four Degrees of Freedom
D.1 4 DOF Asymmetrical Shaft System
D.2 4 DOF Asymmetrical Rotor System
Reference
Appendix E: Transformation of Equations of Motion to Normal Coordinates: 4 DOF Rotor System
E.1 Transformation of Equations of Motion to Normal Coordinates
E.2 Nonlinear Terms
References
Appendix F: Routh–Hurwitz Criteria for Complex Expressions
References
Appendix G: FFT Program
References
Index
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The Authors
Prof. Yukio Ishida
308 Nenokami, Nagakuto
Aichi-ken 480-1141
Japan
Prof. Toshio Yamamoto
414 Takama-cho, Meito-ku
Nagoya 465-0081
Japan
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Dedicated to the memory of Professor Yamamoto
Foreword to the First Edition
The dynamics of rotating machinery has been extensively investigated during the past century. Many people in England, Germany, and the United States, in the late nineteenth and early twentieth centuries, studied the fundamental concepts associated with rotordynamic systems and investigated the effects of many types of mechanisms on these systems. The published material in this area diminished significantly between the start of World War I and the end of World War II. With the development of the gas turbine as a commercially viable engine after World War II, the need to better understand the dynamics of high-speed rotating systems became critical. Subsequent development of the digital computer assisted the development of many highly sophisticated procedures for analyzing, simulating, designing, and testing rotor systems. Many talented and dedicated people, throughout the world, have contributed to a better understanding of the dynamics of high-speed machinery during the last half of the twentieth century.
During the approximately 20-year interval defined by the end of World War II and the beginning of the digital computer revolution, three people stand out as significant contributors to the rotordynamic literature. F. M. Dimentberg in Russia documented his work in 1961 with the publication of a book entitled Flexural Vibrations of Rotating Shafts. In Czechoslovakia, Alex Tondl documented his studies in 1965 with the publication of a book entitled Some Problems of Rotor Dynamics. During the same period, Toshio Yamamoto worked independently and conscientiously at his Nagoya University laboratory on numerous topics related to rotordynamic systems. His work focused on fundamental concepts related to the dynamics of high-speed machinery and included meticulous laboratory test rigs to back up his analytical predictions. All of this work was done without the benefit of a high-speed digital computer or sophisticated electronic test equipment. Some of his work was documented in 1954 and 1957 with the publication of two Nagoya University Memoirs entitled “On the Critical Speed of a Shaft” and “On the Vibrations of a Rotating Shaft.” These works and other publications of Dr Yamamoto, were not, however, circulated widely outside Japan. Thus it has taken the engineering world a little longer to recognize the genius of this dedicated and talented researcher. This book includes presentations on several of the original topics investigated by Dr Yamamoto and Dr Ishida, and also many important contributions by several other rotating machinery analysts and researchers around the world.
In 1975, Yukio Ishida graduated as one of Toshio Yamamoto's doctoral students at Nagoya University. Since that time, Dr Ishida has both independently and in collaboration with Dr Yamamoto made many additional and significant contributions to the area of rotordynamics. In particular, he and Dr Yamamoto have paid special attention to the effects of various nonlinear mechanisms on the dynamic behavior of rotor systems. Dr Ishida has also extensively investigated the use of modern digital signal processing as a valuable tool for the analytical and experimental investigation of vibrations in rotordynamic systems. Several topics associated with these more recent studies are included in this work. It is highly fitting that Dr Yamamoto and Dr Ishida are finally documenting a significant portion of their half a century of rotordynamic expertise with the publication of this book.
Harold D. Nelson, Ph.D.
Professor and Chair
Department of Engineering
Texas Christian University
1991–1998
Preface to the First Edition
Rotating machinery, such as steam turbines, gas turbines, internal combustion engines, and electric motors, are the most widely used elements in mechanical systems. As the rotating parts of such machinery often become the main source of vibrations, correct understanding of the vibration phenomena and sufficient knowledge of rotordynamics are essential for considering adequate means to eliminate vibrations. However, compared to rectilinear vibrations, the whirling motions of rotors seem difficult for students and engineers to understand in the beginning.
Studies of rotordynamics started more than 100 years ago. At that early stage, the primary concern was prediction of the resonance rotational speeds called critical speeds. As the normal operation speed increased above the critical speeds, engineers encountered various kinds of new problems, and rotordynamics developed through their efforts to overcome these technical difficulties. It is a very difficult task to master the entire range of rotordynamics by reading various technical papers. Although many standard books on vibrations contain a chapter explaining rotordynamics, the content is insufficient for practicing engineers. Recently, many excellent books on rotordynamics have been published by distinguished researchers. However, some of them are too practically oriented and others contain only recent technical topics.
The scope of this book includes most branches of rotordynamics. But it is intended especially to provide a detailed explanation of the basic concepts of rotordynamics because correct understanding becomes a strong tool with which to tackle vibration problems.
In Chapter 1, a classification of rotating shaft systems and a historical perspective are given.
In Chapter 2, the fundamentals of rotordynamics are explained using rotor models consisting of a massless elastic shaft and a rigid disk. The key ideas common to all branches of rotordynamics, such as critical speed, gyroscopic moment, whirling motion, and frequency diagrams, are explained. Balancing a rigid rotor is also discussed.
In Chapter 3, the dynamic analysis of a rotor with distributed mass is presented. Gyroscopic moment inertia and rotary are considered in the equation of the motion. Balancing a flexible rotor is also explained.
In Chapter 4, vibrations of an asymmetrical shaft with unequal stiffness and an asymmetrical rotor with unequal moments of inertia are discussed. Such systems with rotating asymmetry in stiffness or inertia belong to the category of parametrically excited systems, and the coefficients of the equations of motion are functions of time. The most distinguished feature of these systems is the appearance of an unstable zone at the major critical speed. Analysis of free vibrations, forced vibrations, and unstable vibrations is explained in this chapter.
In Chapter 5, various types of nonlinear resonances are considered. Rotating shaft systems have many elements that can cause nonlinearity in the shaft restoring forces. Several methods to obtain solutions for subharmonic resonances, combination resonances, and chaotic vibrations are presented. Nonlinear phenomena, such as jump phenomena, hysteresis phenomena, and period doublings, are shown. The dynamic behavior of a cracked shaft, which has attracted the interest of researchers in the field of vibration diagnosis, is also explained.
In Chapter 6, the effects of internal damping on shaft stability are explained. Owing to internal friction in the shaft material and dry friction between rotating components, self-excited vibrations appear in a wide speed range above the major critical speed. Expressions for such internal damping forces and the characteristics of self-excited vibrations are investigated.
In Chapter 7, nonstationary vibrations during transition through critical speeds are explained. Such nonstationary phenomena become a matter of great interest when the rated speed of a rotating machine is above the major critical speeds. Nonstationary phenomena are highly dependent on the magnitude of the driving torque of a motor. Interaction of the oscillating system with the energy source occurs, especially when the driving torque is small.
In Chapter 8, vibrations due to various machine elements are outlined. Ball bearings, bearing pedestals, universal joints, and couplings are the elements widely used in rotating machinery and may cause vibrations. In practical operations, an understanding of the characteristics of these machine elements is indispensable.
In Chapter 9, various kinds of flow-induced vibrations are explained. Oil whip in journal bearings, steam whirls in turbines, and vibrations of a hollow rotor partially filled with liquid are typical phenomena caused by liquid flow.
In Chapter 10, the finite element method is explained. In the analysis of a complex-shaped rotor in a practical machine, theoretical derivation of the equations of motion is impossible. Therefore, numerical procedures suitable for computer calculation are adopted instead. One of the most widely used means in the analysis of complex structures is the finite element method. This chapter describes how to apply this method to analyze rotating shaft systems.
In Chapter 11, another representative computational method called the transfer matrix method is explained. This method is especially suitable for the analysis of rotor systems.
In Chapter 12, a digital signal processing technique called complex fast Fourier transform (complex-FFT) is explained. In contrast to the ordinary FFT, this technique can distinguish between forward and backward whirling motions. The complex-FFT method is useful especially for the experimental study of rotor whirling motions.
Nagoya, Japan
January 2001
Toshio Yamamoto
Yukio Ishida
Preface to the Second Edition
Ten years have passed since the first edition of this book was published in 2001. I was very pleased to learn that the first edition has come to be used as a text in some graduate schools, and I am grateful to have received valuable comments and questions from many readers about the contents. The highly constructive advice from readers has helped us to polish the interpretations and explanations contained in this text and motivated us to publish this second edition. The major objective of our revisions is to first discuss additional subject matter of interest to engineers in the field of rotordynamics. Furthermore, we also decided to include additional case histories in order to illustrate the importance of the contents; exercises have also been included to aid readers' understanding of the text. The additional chapters cover topics such as balancing (Chapter 4), vibration suppression (Chapter 11), steam turbines and windmills (Chapter 12), cracked rotors (Chapter 13), and magnetic bearings (Chapter 17). In addition, Chapter 9 on vibrations of mechanical elements is extensively revised. It is hoped that these revisions will assist readers in their endeavor to develop a holistic understanding of rotordynamics.
In the second half of the twentieth century, the power of turbines increased very rapidly, and accordingly, the theory of rotordynamics developed remarkably during this time. Now, many of the researchers and engineers who first engaged in the design of these turbines are gradually retiring. I am afraid that their technology and “know-how,” which have been accumulated through many failures and successes, may be lost over time. Owing to the commercial demand in recent times for the shortening of the development period, many engineers have come to depend on commercial software, and thus opportunities to learn from practice seem to me to be decreasing. It is our greatest wish that this book should help to transfer the contributions of our predecessors to the next generation of rotordynamists.
Finally, Professor Toshio Yamamoto, the coauthor of the first edition of this book, regrettably passed away in 2007 at the age of 86. I believe that he would be pleased at the publication of this new edition, although he was unable to make these changes by his own hand. With this in mind, I dedicate this book to Professor Yamamoto with respect to and in memory of his great achievements in the field of rotordynamics.
Nagoya, Japan
January, 2012
Yukio Ishida
Acknowledgments
The first edition and this second edition were developed out of lecture notes used in our undergraduate and postgraduate courses. The information contained in this book is based not only on our own work but also on the work of many excellent pioneers and leaders in the field of rotordynamics. We hereby record our sincere gratitude to all these distinguished scholars. We also wish to express our special thanks to Professor Harold D. Nelson, formerly of Texas Christian University, for writing the Foreword and to Professor Ali H. Nayfeh, of Virginia Polytechnic Institute and State University, for recommending and acting as series editor for this publication. Further, we would like to thank Professor Takashi Ikeda of Hiroshima University, Professor Imao Nagasaka of Chubu University, Professor Takao Torii of Shizuoka University, Professor Tsuyosi Inoue of Nagoya University and Dr. Kentaro Takagi of Nagoya University for their review and kind suggestions. Finally, we would like to thank our former and present students for their enthusiasm and contributions to our research work at Nagoya University.
1
Introduction
1.1 Classification of Rotor Systems
In general, rotating machinery consists of disks of various shapes, shafts whose diameters change depending on their longitudinal position, and bearings situated at various positions.1 In vibration analyses, such a complex rotor system is simplified and a suitable mathematical model is adopted. In this modeling process, we must know which parameters are important for the system.
Rotating machines are classified according to their characteristics as follows: If the deformation of the rotating shaft is negligible in the operating speed range, it is called a rigid rotor. If the shaft deforms appreciably at some rotational speed in the operating speed range, it is called a flexible rotor. We cannot determine to which of these categories the rotor system belongs by considering only its dimensions. In rotordynamics, the rotating speeds that produce resonance responses due to mass eccentricity are called critical speeds. The deformation of a rotor becomes highest in the vicinity of the critical speed. Therefore, the range of the rated speed relative to these critical speeds determines whether the rotor is rigid or flexible.
Figure 1.1 called a critical speed diagram shows variations of critical speeds and vibration modes versus the stiffness of the supports for a symmetrical rotor. The left part of this figure represents values for rotors that are supported softly. In the first and second modes, the rotors do not deform appreciably but the supporting parts deflect. In this case, the rotor is considered to be a rigid rotor. As the stiffness of the supports decreases, the natural frequency of these modes approaches zero. In the third mode, the rotor deforms and it is considered to be a flexible rotor. Depending on the type of mode to be discussed, the same system may be considered as a rigid or a flexible rotor. On the right part of this figure, deformation occurs in all three modes and therefore the rotor is considered to be flexible in every mode.
Figure 1.1 Critical speeds and mode shape versus the stiffness of the bearing support.
In some models, disks are considered to be rigid and the distributed mass of an elastic shaft is concentrated at the disk positions. Such a model is called a lumped-parameter system. If a flexible rotor with distributed mass and stiffness is considered, this model is called a distributed-parameter system or continuous rotor system. The mathematical treatment of the latter is more difficult than that of the former because it is governed by the partial differential equations.
Rotors are sometimes classified into vertical shaft systems and horizontal shaft systems. We mainly discuss the former model, however, the latter model tends to be considered in cases in which we must clarify the effect of gravity.
1.2 Historical Perspective
The evolution of research in the field of rotordynamics is shown in Figure 1.2. Research on rotordynamics spans at least a 140-year history, starting with Rankine's paper on whirling motions of a rotor in 1869. Rankine discussed the relationship between centrifugal and restoring forces and concluded that operation above a certain rotational speed is impossible. Although this conclusion was wrong, his paper (refer to the “Topic: The First Paper on Rotordynamics”) is important as the first publication on rotordynamics. The research progressed significantly at the end of the nineteenth century with contributions by de Laval and others. De Laval, an engineer in Sweden, invented a one-stage steam turbine and succeeded in its operation. He first used a rigid rotor, but later used a flexible rotor and showed that it was possible to operate above the critical speed by operating at a rotational speed about seven times the critical speed (Stodola, 1924).
Figure 1.2 History of rotordynamics.
In the early days, the major concern for researchers and designers was to predict the critical speed, because the first thing that had to be done in designing rotating machinery was to avoid resonance. Dunkerley (1894) derived an empirical formula that gave the lowest critical speed for a multirotor system. He was the first to use the term “critical speed” for the resonance rotational speed. The word “critical,” which refers to a state or a value at which an abrupt change in a quality or state occurs, was coined possibly based on Rankine's conclusion mentioned above. Holzer (1921) proposed an approximate method to calculate the natural frequencies and mode shapes of torsional vibrations.
The first recorded fundamental theory of rotordynamics can be found in a paper written by Jeffcott (1919) . We can appreciate Jeffcott's great contributions if we recall that a shaft with a disk at the midspan is called the Jeffcott rotor, especially among researchers in the United States. This simplified fundamental rotor system is also called the Laval rotor, named after de Laval.
The developments made in rotordynamics up to the beginning of the twentieth century are detailed in the masterpiece written by Stodola (1924). This superb book explains nearly the entire field related to steam turbines. Among other things, this book includes the dynamics of elastic shafts with disks, the dynamics of continuous rotors without considering the gyroscopic moment, the balancing of rigid rotors, and methods for determining approximate values of critical speeds of rotors with variable cross sections.
Thereafter, the center of research shifted from Europe to the United States, and the scope of rotordynamics expanded to consider various other phenomena. Campbell (1924) at General Electric investigated vibrations of steam turbines in detail. His diagram, representing critical speed in relation to the cross points of natural frequency curves and the straight lines proportional to the rotational speed, is now widely used and referred to as the Campbell diagram. As the rotational speed increased above the first critical speed, the occurrence of self-excited vibrations became a serious problem. In the 1920s, Newkirk (1924) and Kimball (1924) first recognized that internal friction of shaft materials could cause an unstable whirling motion. These phenomena, in which friction that ordinarily dampens vibration causes self-excited vibration, attracted the attention of many researchers. Newkirk and Taylor (1925) investigated an unstable vibration called oil whip, which was due to an oil film in the journal bearings. Rotor is generally surrounded by a stator such as seals with a small clearance. Newkirk (1926) showed a forward whirl induced by a hot spot on the rotor surface, which was generated by the contact of the rotor and the surroundings. This hot spot instability is called the Newkirk effect.
About a decade later, the study of asymmetrical shaft systems and asymmetrical rotor systems began. The former are systems with a directional difference in shaft stiffness, and the latter are those with a directional difference in rotor inertia. Two pole generator rotors and propeller rotors are examples of such systems. As these directional differences rotate with the shaft, terms with time-varying coefficients appear in the governing equations. These systems therefore fall into the category of parametrically excited systems. The most characteristic property of asymmetrical systems is the appearance of unstable vibrations in some rotational speed ranges. Smith (1933)'s report is a pioneering work on this topic. Various phenomena related to the asymmetries of rotors were investigated actively in the middle of the twentieth century by Taylor (1940) and Foote, Poritsky, and Slade (1943), Brosens and Crandall (1961), and Yamamoto and Ota (1963a, 1963b, 1964).
Nonstationary phenomena during passage through critical speeds have been studied since Lewis reported his investigation on the Jeffcott rotor in 1932. Numerous reports on this topic are classified into two groups. One group classifies nonstationary phenomena that occur in a process with a constant acceleration and the other classifies phenomena that occur with a limited driving torque. In the latter case, mutual interaction between the driving torque and the shaft vibration must be considered. As the theoretical analysis of such transition problems is far more difficult than that of stationary oscillations, many researchers adopted numerical integrations. The asymptotic method developed by the Russian school of Krylov and Bogoliubov (1947) and Bogoliubov and Mitropol'skii (1958) considerably boosted the research on this subject.
The vibrations of rotors with continuously distributed mass were also studied. The simplest continuous rotor model corresponding to the Euler beam was first studied in the book by Stodola (1924). In the 1950s and 1960s, Bishop (1959), Bishop and Gladwell (1959), and Bishop and Parkinson (1965) reported a series of papers on the unbalance response and the balancing of a continuous rotor. Eshleman and Eubanks (1969) derived more general equations of motion considering the effects of rotary inertia, shear deformation, and gyroscopic moment, and investigated these effects.
The most important and fundamental procedure to reduce unfavorable vibrations is to eliminate geometric imbalance in the rotor. The balancing technique for a rigid rotor was established relatively early. A practical balancing machine based on this technique was invented by Lawaczeck in 1907 (Miwa and Shimomura, 1976). In 1925, Suehiro invented a balancing machine that conducts balancing at a speed in the postcritical speed range (Miwa and Shimomura, 1976). And in 1934, Thearle developed the two-plane balancing (Thearle, 1934). The arrival of high-speed rotating machines made it necessary to develop a balancing technique for flexible rotors. Two representative theories were proposed. One was the modal balancing method proposed in the 1950s by Federn (1957) and Bishop and Gladwell (1959). The other was the influence coefficient method proposed in the early 1960s and developed mainly in the United States alone with the progress of computers. Goodman (1964) improved this method by taking into the least square methods.
In the latter half of the twentieth century, various vibrations due to fluid were studied. The above-mentioned oil whip is a representative flow-induced vibration of rotors. In the middle of the twentieth century, Hori (1959) succeeded in explaining various fundamental characteristics of oil whip by investigating the stability of shaft motion and considering pressure forces due to oil films. At almost the same time, other types of flow-induced vibrations attracted the attention of many researchers. Seals that are used to reduce the leakage of working fluids through the interface between rotors and stators sometimes induce unstable vibrations. In 1964, Alford reported accidents due to labyrinth seals. Another one was a self-excited vibration called thesteam whirl. The mechanism of this vibration in turbines was explained by Thomas (1958) and that in compressors was explained by Alford (1965). These phenomena are still attracting the interest of many researchers for practical importance. The vibration of a hollow rotor containing fluid is a relatively new problem of flow-induced vibrations. In 1967, Ehrich reported that fluid trapped in engine shafts induced asynchronous vibrations. A noteworthy paper on this phenomenon is that of Wolf (1968). He succeeded in explaining the appearance of an unstable speed range in a postcritical region of a rotor system containing inviscid fluid.
As rotors became lighter and their operational speeds higher, the occurrence of nonlinear resonances such as subharmonic resonances became a serious problem. Yamamoto (1955, 1957a) studied various kinds of nonlinear resonances after he reported on subharmonic resonances due to ball bearings, in 1955. He discussed systems with weak nonlinearity that can be expressed by a power series of low order. Aside from subharmonic resonances, he also investigated combination resonances (he named them summed-and-differential harmonic oscillations) and combination tones. In the 1960s, Tondl (1965) studied nonlinear resonances due to oil films in journal bearings. Ehrich (1966) reported subharmonic resonances observed in an aircraft gas turbine with squeeze-film damper bearings. The cause of strong nonlinearity in aircraft gas turbines is the radial clearance of squeeze-film damper bearings. Later, Ehrich (1988, 1991) reported the occurrence of various types of subharmonic resonances up to a very high order and also chaotic vibrations in practical engines.
In the practical design of rotating machinery, it is necessary to know accurately the natural frequencies, modes, and forced responses to unbalances in complex-shaped rotor systems. The representative techniques used for this purpose are the transfer matrix method and the finite-element method. Prohl (1945) used the transfer matrix method in the analysis of a rotor system by expanding the method originally developed by Myklestad (1944). This analytical method is particularly useful for multirotor-bearing systems and has developed rapidly since the 1960s by the contribution of many researchers such as Lund and Orcutt (1967) and Lund (1974). The finite-element method was first developed in structural dynamics and then used in various technological fields. The first application of the finite-element method to a rotor system was made by Ruhl and Booker (1972). Then, Nelson and McVaugh (1976) generalized it by considering rotating inertia, gyroscopic moment, and axial force.
From the 1950s, cracks were found in rotors of some steam turbines (Ishida, 2008). To prevent serious accidents and to develop a vibration diagnosis system for detecting cracks, research on vibrations of cracked shafts began. In the 1970s, Gasch (1976) and Henry and Okah-Avae (1976) investigated vibrations, giving consideration to nonlinearity in stiffness due to open–close mechanisms. They showed that an unstable region appeared or disappeared at the major critical speed, depending on the direction of the unbalance. The research is still being developed and various monitoring systems have been proposed.
The latest topics in rotordynamics are magnetic bearings that support a rotor without contacting it and active control. This study has received considerable attention since Schweitzer (1975) reported his work in 1975. Nonami (1985) suppressed an unbalance response of a rotor controlling the bearing support actively using the optimal regulator theory.
In this chapter, the history of rotordynamics has been summarized briefly. The authors recommend the readers to read excellent introductions on the history of rotordynamics by Miwa (1991), Dimarogonas (1992), and Nelson (1994, 1998). The following are representative books on rotordynamics, some of which have detailed bibliographies: Stodola (1924), Kimball (1932), Biezeno and Grammel (1939), Dimentberg (1961), Tondl (1965), Gunter (1966), Roewy and Piarulli (1969), Eshleman, Shapiro, and Rumabarger (1972), Gasch and Pfutzner (1975), Miwa and Shimomura (1976), Dimarogonas and Paipetis (1983), Vance (1988), Darlow (1989), Lalanne and Ferraris (1990), Zhang (1990), Rao (1991), Ehrich (1992), Lee (1993), Childs (1993), Kramer (1993), Gasch, Nordmann, and Pfützner (2002), Genta (2005), Muszynska (2005), and Bachschmid, Pennacchi, and Tanzi (2010).
Figure 1.3 Rankine and his paper. (a) The first article on rotordynamics (Rankine, (1869)). (b) Portrait of Rankine. (Courtesy of the University of Glasgow.)
Notes
1 Rotor is often used as the general term for the rotating part of a rotating machine. The opposite term is stator, which means the static part of the machine.
References
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Bachschmid, N., Pennacchi, P., and Tanzi, E. (2010) Cracked Rotors, Springer.
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2
Vibrations of Massless Shafts with Rigid Disks
2.1 General Considerations
In practical rotating machinery, various disks and blades are mounted on a shaft whose diameter changes in its longitudinal direction. In vibration analysis, such a rotating machine is replaced by a simple mathematical model. Figure 2.1 illustrates several mathematical models that are often adopted in the analyses of rotor vibrations. In Figure 2.1, a–c are lumped-parameter systems in which a rigid disk is mounted on a massless elastic shaft with a circular cross section. When a rotor is mounted at the center of an elastic shaft supported at both ends, the lateral deflection r and the inclination θ of the rotor are independent of each other and therefore this system can be decomposed into two rotor systems with two degrees of freedom (2 DOF). shows a 2 DOF model executing a deflection motion. This simplified mathematical model called is the most widely used model in the theoretical analysis of rotors. is a 2 DOF model executing an inclination motion. Different from the system shown in , the gyroscopic moment acts in this system and, as a result, natural frequencies change as a function of the rotational speed. This is the simplest model that has characteristics unique to rotor systems. For some types of vibration problems, this model is more suitable than the Jeffcott rotor. shows 4 DOF models where the deflection motion and the inclination motion couple with each other. shows a rigid rotor that is supported elastically. The movement of this rotor is governed by the same equations of motion as those for the model in . In this chapter, the fundamentals of rotordynamics are explained using these mathematical models.
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