Masao Kobayashi

Basics for a Single-Degree-of-Freedom Rotor

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

Vibration Analysis of Blade and Impeller Systems

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.

Stability Problems in Rotor Systems

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.

Gyroscopic Effect on Rotor Vibrations

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.

Rotor System Evaluation Using Open-Loop Characteristics

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.

Approximate Evaluation for Eigenvalues of Rotor-Bearing Systems

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.

Bridge Between Inertial and Rotational Coordinate Systems

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:

Introduction of Rotordynamics

\lambda = \lambda_{r} + j\Omega