Search

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.

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.

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.

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 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 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.

Subsynchronous vibration of high-pressure steam turbine is one of the difficult problems to improve the reliability of power plant. Extensive work has been done to prevent the low frequency vibration of high-capacity steam turbine and most of the problems were practically solved (1). In the future, we must build up theoretical approach to design a new turbine operating under the steam condition of high-temperature and high-pressure. To design such an advanced steam turbine, it is necessary to solve the effect of partial admission on control stage of the steam turbine. This paper describes the experimental results from the scale model of the steam turbine and theoretical analysis of destabilizing forces. In theoretical analysis, destabilizing forces that take partial admission into consideration were analyzed. As a result, it was clarified that large destabilizing force was generated in partial admission condition and the destabilizing force varied periodically by the instant rotor position. In experimental study, we designed the new inlet seal for the prevention against the vibration. From the experimental result that increase of total flow rate leads the system stable, we showed the effectiveness of new inlet seal. And we paid attention to the step-up that is the step between blade and nozzle. As a result, it was experimentally proved that the step-up had a large influence on the rotor stability.

The effectiveness of squeeze-film dampers in controlling vibrations in rotating machinery may be limited by the nonlinear interactions between large rotor imbalance forces with fluid-film forces induced by dampers operating in cavitated conditions. From a practical point of view, the occurrence of nonsynchronous and chaotic motion in rotating machinery is undesirable and should be avoided as they introduce cyclic stresses in the rotor, which in turn may rapidly induce fatigue failure. The bifurcations in the response of a rigid rotor supported by cavitated squeeze-film dampers resulting from such interactions are presented in this paper. The effects of design and operating parameters, namely the bearing parameter (B), gravity parameter (W), spring parameter (S) and unbalance parameter (U), on the bifurcations of the rotor response are investigated. Spring parameter (S) values between 0 and 1 are considered. A spring parameter value of S=0 represents the special case of dampers without centering springs. With the exception of the case S=1, jump phenomena appeared to be a common bifurcation that occurred at certain combinations of B, W and U irrespective of the value of S. Period-doubling and secondary Hopf bifurcations which occurred for low values of S (⩽0.3) were not observed for the higher values S⩾0.5. For very low stiffness values (S

Numerical investigation on the unbalance response of a rigid rotor supported by squeeze-film dampers without centering springs revealed some complex bifurcation features that have not been previously reported in the literature. With the variation of the unbalance parameter (U), the period-1 solution was found to undergo a sequence of period-doubling bifurcations that eventually resulted in chaotic motion. The existence of a period-3 solution, which formed a closed bifurcation curve consisting of a pair of saddle nodes, was for the first time observed in such a system. The chaotic attractor arising from the period-doubling cascade of the period-1 solution, which was observed to co-exist with the period-3 attractor in a narrow range of U values, was eventually annihilated in a collision with the unstable period-3 orbit in a boundary crisis. Similar to the bifurcations of the period-1 solution, the period-3 solution was also found to bifurcate into solutions of period-6 and period-12, which eventually led to chaotic motion. A chaotic attractor was also observed to co-exist with a period-4 orbit. The period-4 orbit was found to undergo a sequence of reverse period-doubling bifurcations resulting in a large amplitude period-1 orbit. The occurrence of non-synchronous and chaotic motion in rotating machinery is undesirable and should be avoided as they introduce cyclic stresses in the rotor, which in turn may rapidly induce fatigue failure. The magnitude of rotor unbalance where non-synchronous and chaotic motion were observed in this study, although higher than the permissible unbalance level for rigid rotating machinery, may nevertheless occur with in-service erosion of the rotor or in the event of a partial or an entire blade failure.

The continuation method is employed to investigate the non-linear response of a rigid rotor supported by squeeze-film dampers without centering springs. The Floquet transition matrix method, which was used to examine the stability of the periodic response, indicated that the rotor may lose stability via period-doubling and saddle-node bifurcations at certain parameter values. The continuation method, which allows for switching from one solution branch to another, enabled the solution branches of period-two, four, eight and sixteen to be mapped out. By simultaneously varying the unbalance and bearing parameters, the combination of these two parameters that avoids the occurrence of non-synchronous vibration or jump phenomena in this system was determined.

This paper deals with a rotor balancing method based on influence coefficient method and LMI (Linear Matrix Inequality) optimization technique. The three rotor balancing objectives; (a) minimization of maximum residual vibration, (b) minimization of the least mean square of residual vibration and (c) minimization of the least mean square of correction masses, are considered. Since the conventional influence coefficient method cannot properly treat the objectives (a) and (c), these objectives (a) and (c) have been considered by heuristic trial & error method. In this paper, first, the LMI formulation of the rotor balancing objectives (a)-(c) are derived applying two basic LMI lemmas. Based on these LMI formulation of the objectives, a LMI optimization framework for rotor balancing is then constructed. The proposed method enables us to efficiently achieve the objectives (a)-(c) with the relative weight vector c using LMI optimization technique. Moreover, a reduction algorithm for the number of the correction planes, which is useful in the actual field balancing, is also considered. Finally, the effectiveness of the proposed rotor balancing method is evaluated by numerical simulations and experiments with a model rotor system.