Vibrations of Rotating Machinery

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).

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

Basic properties featuring in a vibrating system, which are obtained from the free vibration waveform, are:

Using these parameters, the resonance caused by forced excitation can be predicted with

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.

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,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.

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 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: 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 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 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: This chapter moves the viewpoint concerning vibration measurement from z to z _r .

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:

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: For simplicity, a single-degree-of-freedom model is used in the following discussion.

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.