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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 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.

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

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 describes an active jiggle control of hung structure using a control moment gyro (CMG) which has two gimbals for one flywheel. The double gimbal CMG enables us to simultaneously control the vibrations for two different dirctions by only one flywheel. Due to the nonlinearlity and interference, it is however difficult to construct the control system for the double gimbal CMG. First, we propose a simple control system design method for the double gimbal CMG based on H2 optimal control theory. In order to improve the gimbal stability, which is necessary for avoiding the interference, we then propose an another design method based on H2/H∞ control using LMI optimization. The performance of all designed control system are evaluated by simulations and experiments.

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.

In this paper, the static characteristics of the four-nozzle pintle type servovalve are studies theoretically, under constant supply pressure and constant supply flow conditions.The study is on an eye to the effect of the valve improper configuration, the required pintle driving torque, and the effect of using unsymmetrical loads on the static characteristics.Errors in valve control gaps resulting from inaccurate initial adjustment, wear of else showed to affect slightly the load flow at each load pressure and input displacement.Units driven by unsymmetrical actuators showed to have lower valve flow forces and higher valve efficiency than symmetrical actuators.The minimum flow gain and flow pressure coefficient are obtained when the load is symmetric.The study showed that the flow forces have two instability sources, decentering nature and similar values at different input displacements.The study discussed the selection of the pintle centering spring to overcome the problems of instability.

In parallel mechanisms with redundant actuators, actuator forces are underspecified because of redundancy. In this paper, the method to minimize a maximum actuator force was proposed. At first, parallel mechanism with two redundant actuators and Stewart platform model using the linear shaft motor were developed. This redundant parallel mechanism was consists of Stewart platform and additional 2 redundant actuators. Inverse dynamics of each mechanisms were formulated. Secondly, effectiveness of dynamic control of Stewart platform was experimentally confirmed. Finally, the method to minimize a maximum actuator force using Linear Matrix Inequality (LMI) was proposed. And the effectiveness of the proposed method was confirmed by simulation.

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][2]. 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 Alford force considered partial admission condition to solve the problem. (1) Subsynchronous vibration was reproduced in the scale model test. (2) Partial admission gave larger destabilizing force compared with full admission condition for same total flow rate. (3) Initial position of shaft center to the phase of admission arc on the partial admission had some effect on the stability of the rotor system. (4) Theoretical analysis of destabilizing force considered partial admission condition gave qualitative description of the experimental results from the scale model.Copyright © 2005 by ASME