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In contrast to conventional rolling element and oil film bearings, magnetic bearings have had a relatively short history. The practical implementation of active magnetic bearings using electromagnets has been accelerated and driven recently by the development of digital control technology. Since 1988, the International Symposium on Magnetic Bearings (ISMB) series has been convened every two years and has been leading the studies in this field. In the industrial sector, the ISO TC108/SC2/WG7 “Working Group on Active Magnetic Bearings” started its activities in 1997 to develop standards to promote the adoption of rotating machinery with active magnetic bearings and to support the steady and smooth advancement of associated industries. The standards have been released as the ISO 14839 series. This chapter describes the basic specifications of magnetic bearings, along with the related vibration control technologies.

Oil-lubricated plain bearings are used widely to support the main rotors of rotating machinery, because the hydrodynamic oil films formed in the bearing clearances can guarantee smooth rotation of shafts and effectively prevent or suppress rotor vibrations. This chapter explains the principles of oil film formation in typical plain bearings, based on conventional hydrodynamic lubrication theory. The circumferential oil film shape at the steady-state equilibrium is determined, so that the hydrodynamic pressure generated in the oil film can balance the oil film reaction force vectorially with the applied bearing load. In other words, the oil film separates the journal from the bearing surface and the journal can float and rotate on the oil film. The oil film shape determines the journal center position represented by the journal eccentricity ratio and the attitude angle in polar coordinates. The steady-state journal center position and in turn the corresponding oil film shape vary with the operating condition of the journal bearing, that is, Sommerfeld number. When the journal center is slightly perturbed at the equilibrium, the oil film is slightly deformed, resulting in the slightly modified pressure distribution. Then, the corresponding oil film force is found to consist of the static reaction force at the equilibrium and also the two mutually perpendicular components of the small dynamic reaction force. Each of the two components is expressed by the linear sum of the two stiffness components and also the two damping components. In total, four stiffness components and four damping coefficients are determined at the equilibrium. These coefficients are the key factors that dominate the vibrational behavior of the rotor supported in the plain bearings. The coefficients obtained for various types of journal bearing are found to vary with the Sommerfeld number. Consequently, journal bearings need to be selected and designed so that the dynamic coefficients of the hydrodynamic oil films can enable satisfactory control of rotor vibration.

According to ISO 18436-2, entitled “Condition monitoring and diagnostics of machines—Requirements for training and certification of personnel—Part 2: Vibration condition monitoring and diagnostics,” the introduction of the certification examination for vibration experts is mentioned. The certification level is divided into four categories: 1—for beginners; 2—for elemental experts (apprentices); 3—for advanced experts; and 4—for super-experts. This chapter, including 100 questions, is prepared for ambitious experts challenging categories 3, 4, and higher. Every question requires knowledge strongly related to basic engineering mathematics, practical signal processing, vibrational dynamics theory, standard modal analysis, model order reduction, and so forth. It excludes knowledge gained purely from experience. If you are students, it is a good opportunity to know how to apply undergraduate knowledge to the field. If you are vibration consultants, apply these simple and understandable modeling techniques in troubleshooting for your customers. Based on the experience of one of the authors as a JSME (Japan) examiner, he found that questions of the examination are divided approximately into two groups. The first group includes questions related to knowledge of field experience in the maintenance service, for example, permissible vibration levels decided by API and/or ISO standards, how to measure and report vibration data, and so on. The second group involves questions concerning mathematics and dynamics, for example, rotor balancing, the FFT algorithm, modal analysis, applying complex numbers. Many examinees may have difficulty to recall and manage the theoretical background for questions if they graduated some time ago. A total of 100 questions have been selected, specifically to enrich an expert’s understanding of the mathematics and dynamics for them. The following advises the reader on how to use this chapter: (1) Questions 1–30: The first 30 are multiple choice questions with five options. It is in the same style as the certification exam; hence, examinees can check their level. The actual exam contains 100 questions in a 4–5 h period. Therefore, it is recommended that you complete these 30 questions within 60 min. Finally, check your answers with the correct answers in Table Q30, located in Sect. 12.1. (2) Questions 31–100: The aim of these questions is to sharpen examinees’ thinking, so the questions may be slightly beyond their knowledge. On average, the difficulty of these questions is almost equivalent to vibration category level 3 and 4; some may be near category 4 and higher. Do not be afraid to try these questions to test your knowledge. From Question 31 onwards, the content is divided into two ways: Questions stated in Sect. 12.2 with answers and hints associated with each question in Sect. 12.3. Please try these exercises, which challenge readers to become world-leading experts in the rotor dynamic vibration field.

This book aims to explain various phenomena and mechanisms of vibrations in rotating machinery based on theory and field experiences. It also aims to help engineers in carrying out diagnosis and in implementing countermeasures for various vibration problems in the field. It is normally easy to know the condition of machinery by means of measurement of vibrations and/or noise. However, it is rather difficult to interpret the observed phenomena correctly, derive real causes of the problems, and ascertain effective countermeasures, because a sufficient knowledge of case studies is needed. This chapter provides an overview of vibration problems in rotating machinery and general approaches for the countermeasures, and is intended to be helpful in solving vibration problems at the front line. Condition monitoring of a machine and vibration diagnostics including relevant ISO standards are illustrated. Case studies of vibration problems and the countermeasures in excess of 1000 cases, which have been collected until now by the Vibration Database (v_BASE) Committee under the Japan Society of Mechanical Engineers (JSME), are also referred to in this chapter.

This chapter explains the self-excited nature of vibration of a rotor caused by the destabilizing oil film force (oil whip or oil whirl); another self-excited vibration of a rotor caused by the destabilizing fluid force in seals and impellers (flow-excited vibration); and how to prevent the vibrations effectively by selecting appropriate specifications of plain bearings. Both of the unstable vibrations show two-dimensional whirl orbits of the rotor around the steady-state equilibrium position in the shaft rotating plane. The dominant component of the shaft whirl orbit is found to be the forward one, that is, orbiting in the same direction as the shaft rotation. The orbit size is sometimes enlarged when bending deformation of the rotating flexible shaft is added. In contrast with the unbalance vibration explained in Chap. 3, the whirl frequencies of the unstable vibrations are generally lower than the shaft rotating frequency, that is, subsynchronous, close to the natural frequency of the rotor-bearing system. These self-excited vibrations break out when the rotor-bearing systems exceed stability limits. When a linear vibration analysis is applied to the rotor-bearing system, the characteristic equation is derived in the form of an algebraic polynomial equation of the sixth degree from the equations of motion. When the Routh–Hurwitz criterion is applied to the characteristic equation, the stability limit of shaft speed can be obtained in the form of mathematical expression consisting of the rotordynamic coefficients of bearing oil film and the bending stiffness variable of shaft. The stability limit can be also found by means of eigenvalue analysis applied to the characteristic equation. In other words, when all the real parts of the eigenvalues have negative values for the given operating condition, the rotor-bearing system can remain stable. When at least one real part has a positive value, the system becomes unstable, starting oil whirl or oil whip. The effects of operating conditions and journal bearing configurations are demonstrated, and various effective countermeasures are derived. The flow-excited vibrations of a rotor by non-contact seals are characterized by dependence on load, because, with increase in load, the working fluid increases in pressure and flow rate, strengthening the destabilizing force (cross-coupled stiffness force) of the fluid flow. The magnitude of the destabilizing force is strongly dependent on the swirl velocity of the fluid in the direction of shaft rotation at the seal inlet. Consequently, one of the effective countermeasures against seal flow-excited vibration is to support the rotor by anisotropic bearings having oil film forces that give rise to elliptical whirl orbits. This is because the backward whirl component of the elliptical whirl orbit reduces the effect of the destabilizing seal force.

Oil-lubricated plain bearings used widely for large-sized rotating machinery can effectively suppress unbalance vibration amplitude of a rotor. This chapter outlines the nature of unbalance vibration amplitude of a rotor supported in plain bearings. The mathematical expression of unbalance vibration of a rotor is given in the rotating plane of the shaft. The two equations of motion given in the x–y-plane are converted into a single equation of motion derived with the complex expression of the displacement of the rotor. Then, the rotor is found to make an elliptical whirl orbit with major and minor axes around its equilibrium. The eight rotordynamic coefficients of a bearing oil film can be converted into the four effective coefficients expressed with the two coordinates of the major and minor axes. The vibration response and the whirl orbit are calculated and shown with varying shaft speed. The effects of bearing type and design variables on the unbalance vibration response are also calculated and compared. Furthermore, the effect of bearing pedestal characteristics on the unbalance vibration response is shown. Finally, it is shown how rotor-bearing systems can be designed to reduce the maximum amplitude of unbalance vibration when passing critical speeds.

This chapter is divided into three parts, which include our latest areas of interest. (1) Guyan reduction or mode synthesis: With respect to vibration analysis for actual machines, many types of codes, 3D-FEM and 1D-FEM (e.g., MyROT introduced in R1_Chap. 12), are available for engineering design based upon multi-DOF (degree of freedom) modeling. However, case studies of vibration problems show the benefits for understanding the vibration mechanisms and their practical and potential solutions using simplified models. In this book, Guyan reduction and the mode synthesis technique are strongly recommended for this purpose. (2) System instability in oil film bearing rotors: This model order reduction (MOR) is applied to a flexible rotor supported by oil film bearings to analyze system instability, often referred to as oil whip, oil whirl, and casing whirl. Stability analysis is generally performed by using complex eigenvalue codes in the 1D-FEM modeling. However, this chapter yields an alternative way without complex eigenvalue calculation, i.e., how to predict the limiting speed for stability and the resultant unstable frequency. (3) Blade modeling coupled with shaft vibration: The MOR by the mode synthesis technique is applied to a flexible blading system to analyze the vibration coupled with the rotating shafting system. Uncoupled eigen frequencies and modes may be analyzed initially for the rotating blading system only by a 3D-FEM code, based upon the rotating coordinates. As the result, the corresponding 2-DOF models for each eigen mode of blading are proposed by two methods depending upon the number of the nodal diameter, κ = 0 and κ = 1. The former is coupled with torsional and axial shaft vibration models and the latter with bending (lateral) shaft vibration models, respectively. These blade reduced models are combined into shaft vibration data by a 1D-FEM code for the evaluation of the coupling effect. The effectiveness of these procedures is confirmed by numerical examples.

Frequencies associated with self-excited vibrations are, in most cases, the natural frequencies of the system. The natural frequencies are not proportional to the rotational speed. Therefore, self-excited vibration is a form of non-synchronous vibration. In rotor systems, the lowest natural frequency is often below the rated rotational speed and it may become unstable, hence self-excited vibration is also known as sub-synchronous vibration. This chapter describes various case studies of self-excited vibrations, which are inherent in rotating machinery, as for a journal bearing, seal, centrifugal impeller, and blade for an axial flow machine. Also, the phenomena of internal friction, fluid trapped in a rotor, and rotor contacting with a stator, may produce strong self-excited vibration. While illustrating these unstable phenomena, the cause or mechanism of the instabilities and appropriate solutions are discussed by citing the v_BASE data. In addition, squeeze film dampers, which are used to stabilize the system by adding a damping effect, are explained.

The major problem in forced vibration response is resonance, where the frequencies of exciting forces coincide with the natural frequencies. Rotating machinery has different characteristics in resonance problems from general non-rotating structures. Therefore, the problems need be solved with this knowledge in mind, not only at the design stage, but also on-site. In the case of forced vibration, centrifugal forcing due to unbalance is a very typical example; however, various other external forces produced by different mechanisms may also become problematic and be experienced routinely. Examples include mechanically induced forces in gears, cross joints, and pulley belt systems; an electromagnetic force in a motor/generator; and rotating stall and impeller/blade interaction forces, which are induced by the fluid flow. Furthermore, torsional vibration is inevitable for rotating machinery and may become a large problem. In this chapter, these phenomena and appropriate countermeasures to implement are elucidated while referring to previously mentioned examples.

In the vibration diagnosis of rotating machines, the first attention is the evaluation of the unbalance vibration, i.e., the rotational synchronous point of vibration, noted as “1X” or “1N” vibration. A vector monitor (R1_Fig. 5.8) is a measuring instrument including a tracking filter that extracts only the 1X vibration component from the actual vibration waveform. In this chapter, the principle and the application of this vector monitor are explained. In the following, attention is given to the transformation from the waveforms in the time domain to amplitude (spectrum) in the frequency domain by using a FFT analyzer. We learn about the theory of signal processing related to FFT analysis, which is capable of providing numerous displays for easy understanding and effective diagnosis for vibration troubleshooting.

Rolling element bearings are used widely in rotating machinery. A significant variety of rolling element bearing types is available, and they are categorized by the shape of the ball or roller elements; the radial or axial loading direction; the ball type such as deep groove or angular contact; the roller type such as plain or tapered; etc. As most of them are standardized in ISO/JIS, interchangeability is guaranteed. This chapter introduces the dynamic properties of some rolling bearings, but does not make any reference to the static properties given in bearing manufacturers’ catalogues, such as load capacity, sizing, life estimation, and so on.

This chapter discusses vibrations of rotating structures such as blades in turbines and impellers in pumps or compressors. The natural frequencies of a rotating structure may be analyzed using the 3-D finite element method and classified by the number of the nodal diameters or circular nodal modes. These results are represented in the rotational coordinate system. The difference between the inertial coordinate system fixed to the stationary side and the rotational coordinate system fixed with the rotor must be taken into account in analysis of: (1) resonance caused by any static load distributed in the circumference direction of the stationary side facing blades or impellers, and (2) resonance caused by harmonic excitation at a certain point in the stationary side facing blades or impellers.

This chapter describes MyROT, a versatile software package for the rotor vibration analysis of rotating machinery. The calculation is based on a combination of beam elements according to finite element modeling , termed 1D-FEM, which is discretized by defining the mass, stiffness and damping matrices according to the actual geometry of the rotating shaft. These matrices are applied to calculations of natural frequencies, complex eigenvalues for stability analysis, unbalance vibration by frequency response analysis (FRA) and so on. Specifically, this chapter describes the theoretical background for formulation and presents outputs of typical rotor dynamic calculations in order to show how this program is convenient in analyzing rotordynamics. A trial version of the package is available at http://www.nda.ac.jp/cc/mech/member/osami.html. An input data manual and other documents related to the package may also be downloaded from the site.

This chapter discusses three typical topics of rotor dynamics problems: internal/external damping effects, vibration due to non-symmetrical shaft stiffness and thermal unbalance behavior. Though a rotor should rotate in a stable manner in a rotation test, problems are encountered in some cases. Most of the problems are related to unbalance, against which the countermeasure is balancing. However, more serious problems may occur that cannot be solved by balancing. In such cases other solutions must be sought. This chapter discusses the following three problems that may be encountered: (1) Internal damping: Loose fittings on the shaft cause damping due to sliding friction. It might seem that any damping is welcome, but this type of damping is rather a destabilizing factor at high speeds of rotation. (2) Asymmetric section of the rotor: Asymmetry in shaft stiffness, e.g. due to a key slot on the shaft often generates troublesome vibration. (3) Vibration due to thermal bow: The unbalance vibration vector of a rotor can be monitored during operation by a Nyquist plot. While the vector point normally remains unchanged during steady state operation, thermal deformation of the rotor, e.g., due to rubbing will move it. The mechanism of this phenomenon is described. For simplicity, a single-degree-of-freedom model is used in the following discussion.

This chapter specifies the definitions, calculation and measurement of basic vibration properties: natural frequency, modal damping, resonance and Q-value ( Q-factor).

The preceding chapter dealt with the basics of rotor vibrations concerning a single-degree-of-freedom (single-dof, 1-dof) system. An actual machine should, however, be analyzed as a multi-degree-of-freedom (multi-dof) system where multiple masses are arranged according to the shape of the rotor shafting. The equation of motion for such a system is represented using matrices. Eigenvalue analysis of the multi-dof system gives the natural frequencies and eigenmodes. These are important factors in rotor design because they represent the resonance frequencies and the vibration mode shapes at critical speeds. This chapter also discusses modal analysis, in which a multi-dof system is reduced to an assemblage of single-dof systems utilizing the orthogonal condition of the eigenmodes. In other words, a complicated actual system is simplified to a set of simple 1-dof systems corresponding to each mode. In addition, a simple estimation method of the natural frequency and the damping ratio is presented based on the orthogonality condition.

Modal analysis consists of superposing eigenmodes ϕ i , weighted according to the modal coordinates η i, through all modes. The modal coordinates no longer have the representation of the original physical coordinates. This chapter discusses methods to reduce the order of the system while preserving the physical coordinates as far as possible. One such method is based on Guyan reduction, in which only the “relatively important” nodes are chosen out of numerous nodes in a finely meshed model. The static deflection modes are developed to reduce the system matrices. The reduced system consists of the physical coordinates of the chosen nodes. Mode synthesis is another method. Here the “most important” nodes are treated with the Guyan reduction method, while the other nodes are considered as internal to the system and undergo modal analysis. Mode synthesis gives a model containing both physical and modal coordinates. Since, however, the mass matrix of this model is not diagonal, no equivalent model of the multiple mass system can be derived. The quasi-modal method is a solution that gives a physical model equivalent to the reduced model obtained by mode synthesis. A convenient model providing an appropriate physical meaning is obtained. In addition, a procedure will be presented in which the response of a bearing journal to a force acting on the rotor is created by the mode synthesis model as a plant transfer function.

This chapter discusses an approximate evaluation method to consider the effects of the dynamic characteristics of a bearing on the complex eigenvalues (damping characteristics as the real part and damped natural frequency as the imaginary part). The method consists basically of two steps: (1) System reduction down to a single-dof system is executed based on the orthogonality condition of modes in the conservative system, and the equation of motion of reduced system is expressed in the complex displacement form, and (2) Approximate analysis of the complex eigenvalues of the system is used to ascertain the effects of the bearing parameters on the natural frequencies and damping characteristics. This combination provides a simple model that helps understanding the phenomena of practical interest, such as the effects of the cross-stiffness of the bearing on the system instability or the stabilizing effect of anisotropy in the bearing stiffness. In addition, the shapes of resonance curves in unbalance vibration are discussed in relation to the dynamic characteristics of the bearing.

This chapter discusses a bridge for the knowledge with respect to the rotor-shaft vibration defined in an inertial coordinate system and the rotating structure vibration formulated in a rotating coordinate system. The equations of motion for rotor vibration discussed hitherto have been based on the description concerning the absolute complex displacement z = x + jy measured in an inertial (fixed, stationary) coordinate system. This description is requested from a practical viewpoint, because the vibrations measurement corresponds to displacement sensors (or gap sensors, displacement meters) placed on a stationary part of machine. Alternatively, this vibration can be measured by strain gauges fixed at a rotational coordinate system, as written by the displacement z r . These variables are mutually related by: \( z = z_{r} {\text{e}}^{{j\Omega t}} \) (Ω = rotational speed) Therefore, if an eigenvalue is λ in the inertial coordinate system and \( \lambda_{r} \) in the rotational coordinate system, these entities are mutually related by: $$ \lambda = \lambda_{r} + j\Omega $$ This chapter moves the viewpoint concerning vibration measurement from z to z r .

This chapter discusses the gyroscopic effect characterizing rotordynamics as being different from the structural dynamics, associated with the non-rotating parts of the rotor system, such as the casing and foundation. A top spinning at a high speed whirls slowly in a tilted position. Similarly, a rotor of a rotating machine whirls while rotating around the driven shaft axis. The spinning top does not fall due to a moment, generated by the gyroscopic effect, which is proportional to the rotational speed. This gyroscopic effect of a rotor system appears as the self-centering tendency during rotation, which may be considered as an increase in the centering stiffness. It is absolutely essential to understand the influence of the gyroscopic effect on the natural frequency and the resonances in the frequency response in rotating machinery vibrations.

This chapter discusses an evaluation method for rotor vibration characteristics by utilizing the open-loop frequency response of the system, instead of conventional eigenvalue analysis. The vibration characteristics of a rotor system are represented by the (damped) natural frequency and damping ratio. They have been estimated in the previous chapters from the viewpoint of the eigenvalue solution, the impulse response waveform, and the resonance curve (FRA) under harmonic excitation. The open-loop characteristics are from a concept in control engineering. A rotor-bearing system can be conceived as a control system as shown in Fig. 8.1, of which the open-loop characteristics are related to the vibration characteristics: the gain cross-over frequency is an estimate of the natural frequency, and the phase margin is an indicator for the damping ratio. Details of estimation are described below.

This book explains various phenomena and mechanisms that induce vibrations in rotating machinery, based on theory and field experiences (Chaps. 1– 12 in volume 1). It also provides guidance in undertaking diagnosis and implementing effective countermeasures against various vibration problems in the field (in volume 2). Consequently, volume 1 is intended mainly for beginners and students, while volume 2 is mainly for design engineers and practitioners. This chapter of volume 1 emphasizes the subtlety of vibration problems in rotating machinery and the importance of reliable technologies that help to stabilize and reduce vibrations. It also outlines a wide variety of rotating machinery, vibration problems found in the field, and mathematical approaches to analyze vibration problems. In high-speed rotating machinery, the steady rotating state corresponds to a stationary equilibrium condition with a high rotational energy. Vibration brings the machine into a “dynamic” state. If the rotating system becomes unstable in the dynamic state, resulting in self-excited vibration, the machine enters a very dangerous operational condition. Since the energy source of self-excited vibration in a rotating system is provided by the spin of the rotor, the only way to avoid this dangerous situation is to stop the energy source, for example, by shutting down the power source in the case of a motor driven system. Vibrations caused by an external force, unless kept small enough, may also lead to a serious problems through contact between the rotor and the stationary part (stator).

Most of the cases of rotor vibration problems come from excessive vibration or resonance due to unbalance. A quick remedy to compensate for it is balancing. Balancing is based on the assumptions of a linear relationship between input (unbalance) and output (vibration), namely, The amplitude of unbalance vibration is proportional to the level of unbalance, and A shift in the angular position of unbalance on a rotor results in a corresponding phase shift of the vibration waveform. While this linearity is the sole theoretical concept to explain how balancing works, practical methods of balancing include various alternatives based on operators’ experiences. Readers are expected to explore the numerical examples prepared so as to experience a wide variety of techniques.