Vibrations of Rotating Machinery

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.

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 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.

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.

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.

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.

Torsional vibrations are often problematic in transient responses when a rotor-bearing system starts up. The vibration is more predominant and unavoidable in geared rotor systems. Though torsional vibration poses long-term problems, new methods of rotor design have been established by employing the transfer matrix method, which are capable of calculating precisely the rotor vibrations according to the shaft configuration. However, several coupled vibration problems have arisen as rotors have become lighter and faster. This chapter reviews conventional technologies including analysis and measurement of torsional vibration. It also explains the latest topics on coupled torsional vibration between a shaft and blade and another coupling effect between torsional and bending shaft vibrations.

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.

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.

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: Please try these exercises, which challenge readers to become world-leading experts in the rotor dynamic vibration field.