Rotating Shaft System Research Articles

Nonlinear Torsional Vibrations of a Rotating Shaft System with a Magnet Coupling

Nonlinear Torsional Vibrations of a Rotating Shaft System with a Magnet Coupling

軸受台の動剛性を考慮した回転軸系の振動解析

The vibration characteristics of rotating shaft system should be computed considering the dynamic stiffness of bearing foundation, especially if this stiffness is rather low. But the accurate characteristics of such a system has not been computed, as far as known, so that the dynamic stiffness of the bearing foundation can have hardly been obtained.In this paper, a method to estimate the dynamic stiffness of bearing foundations by measuring their compliances dynamically is presented. On the other hand, the critical speeds and damping ratios of a model shafting system, consists of shaft, bearings and their foundations, was measured on the testing rig, and was compared to the estimated ones by using the dynamic stiffness obtained by the above mentioned method.There is good agreement between the measured and estimated ones, though the stiffness of bearing foundations of the testing rig is relatively low compared with actual machinaries. It is expected that the method described in this paper is useful to predict the vibration characteristics of rotating shaft system of actual machineries.

Vibration Control of Rotating Shaft with Self-Optimizing Support System

This paper proposes a new design concept of a self-optimizing support system for a rotating shaft. The purpose of this support system is to provide the self-tuning for the support stiffness in order that the vibration of a rotating shaft is usually situated on the near antiresonance with changes of a rotating speed. The optimal tuning values of a support stiffness are obtained by the on-line estimations of rotating angular velocities of a rotating shaft. The effect of the self-optimizing support system is proved by the tests of nonstationary responses for a fundamental rotor-shaft system.

Torsional vibration monitoring apparatus for rotating shaft system

Torsional vibration monitoring apparatus for rotating shaft system

Torsional vibration monitoring apparatus for rotating shaft system

Torsional vibration monitoring apparatus for rotating shaft system

Super-Summed-and-Differential Harmonic Oscillations of an Unsymmetrical Shaft and an Unsymmetrical Rotor

Super-summed-and-differential harmonic oscillations in rotating shaft systems are studied theoretically and experimentally. This kind of nonlinear forced oscillation occurred experimentally in the unsymmetrical shaft system carrying a disc and in the unsymmetrical rotor system where the rotor was mounted on a shaft with circular cross section (a round shaft). We have also compared these oscillations with those in the system where a disc is installed on a round shaft. It is concluded theoretically that large unsymmetry of a rotating shaft or a rotor makes it easier for super-summed-and-differential harmonic oscillations to occur in a rotating shaft system. Not only stable stationary oscillations but also unstable ones may occur. The experimental results are consistent qualitatively with those theoretical ones.

On the Shaft End Torque and the Unstable Vibrations of an Asymmetrical Shaft Carrying an Asymmetrical Rotor

In a rotating asymmetrical shaft carrying an asymmetrical rotor, there occur two types of unstable regions. These unstable regions change with the orientation angle ζ between the inequality of shaft stiffness and that of rotor inertia. The conditions under which unstable vibrations occur just as input energy into the rotating shaft system tends to increase the whirling amplitudes of the shaft, can be clearly ascertained. These conditions, which depend on the angle ζ, tend to show a good coincidence with those obtained by an analog computer. Moreover, when asymmetrical shaft stiffness Δij and asymmetrical rotor inertia Δ0 are chosen suitably, two types of unstable regions are eliminated.

A Variational Approach to the Analysis of Rotor Dynamics Problems

The differential equations for a rotating shaft system are derived variationally by the use of the Lagrange equations, where the kinetic and potential energy terms are obtained by integration of differential volume expressions which are in turn derived from system displacements and strains. Included, as a result, are the effects of initial and dynamic shaft curvature, gyroscopic moments, Coriolis forces, unbalances, rotatory inertia, static weight, and varying shaft cross-sections. Shown to be insignificant are the interacting effects of transverse shear, extensional displacements (and therefore axial constraint), torsion, product of inertia (for most shaft elements). Bearing forces are included as generalized forces on the shaft system. A sample solution of the equations uses the short bearing approximation to model bearing forces and derives expressions for shaft critical speed, equivalent bearing stiffness, and damping constants as functions of bearing eccentricity.

Torsional vibration monitoring apparatus for a rotating shaft system

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Subharmonic and Summed-and-Differential Harmonic Oscillations of an Unsymmetrical Rotor

Subharmonic and summed-and-differential harmonic oscillations of a nonlinear rotating shaft system where an unsymmetrical rotor is mounted on a shaft with circular cross section are treated. It becomes clear theoretically that the resonance curves of subharmonic oscillations of order 1/2 and summed-and-differential harmonic oscillations are classified in to four types. Therefore, similarly to those of an unsymmetrical shaft system reported previously, stable oscillation, unstable oscillation, or no oscillation appears. In an experimental apparatus in which single-row deep groove ball bearings were used, we observed the same types of the resonance curves as the theoretical results by changing the assembly of the apparatus and the size and location of the existing unbalance.

Internal Damping and Stability of Rotor-shaft Systems

Internal Damping and Stability of Rotor-shaft Systems

Experimental Investigation of Multi-Span Rotor Balancing Using Least Squares Method

The least-squares method which uses influence coefficients was applied to simultaneous balancing of a multibearing rotating shaft system, and an experiment was conducted to demonstrate its effectiveness. The rotor system used in the experiment was a 10-bearing, 5-span rotor system. Vibrations were detected simultaneously in horizontal and vertical directions. Using these data, correction weights were calculated with a computer program based on the above least-squares method. Balancing accuracy was considered with regard to two cases, horizontal and vertical vibrations. The balancing experiments were conducted for the cases where unbalance is distributed in one and in all rotors of the multispan rotor system. In both cases, good balance was achieved in one run.

Analysis of Coupled Vibration Response in a Rotating Flexible Shaft-Impeller System

This paper presents a way of analyzing the vibration of a rotor shaft system coupled with flexible impellers based on the transfer method. Each of the flexible impellers is modelled so that it comprises an inertia and an elastic hinge based on the assumption that any impeller’s vibration mode except its one-nodal diameter has no dynamical connection with a shaft. The stiffness of the hinge is based on the eigen frequency of its mode calculated by means of a finite element method, or obtained by experiment. The method developed allows the calculation of the vibration response of a rotor shaft system consisting of several flexible impellers and several bearings when it is subject to an out-of-balance force. Theoretical results are compared with experimental data and show very satisfactory agreement. This simple method appears therefore to be practical and efficient to predict the dynamical characteristics of a high speed rotor system with coupling of impeller flexibility.

Internal Damping and Stability of Rotor-Shaft System

Internal Damping and Stability of Rotor-Shaft System

Balancing method for use in multiple-span rotor shaft system and balancing system using same

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Torsional frequencies of multi-stepped shafts with rotors

An exact frequency determinant for natural frequencies of a multi-stepped shaft-rotor system, which includes the effect of shaft inertia, is developed. Frequency equations and modes of a heavy homogeneous engine, and those with one or two additional rotors, are derived using this result. Numerical results indicate that the effect of shaft inertia reduces with an increase in the mode number.

Theoretical discussions on vibrations of a rotating shaft with nonlinear spring characteristics

Various kinds of nonlinear forced oscillations may appear, when the restoring force of a rotating shaft has nonlinear spring characteristics. A rotating shaft system with gyroscopic moments acting does not experience rectilinear lateral vibrations of the shaft but a whirling type of motion. The nonlinear spring characteristics of the shaft are assumed to be distributed two-dimensionally and polar coordinates are used for their representation. Nonlinear spring characteristics expressed in polar coordinates may be characterized by a component with a constant value and other components whose magnitudes vary 1, 2, 3, 4, ... times, during a single whirl of the shaft around its equilibrium position. This type of representation gives a clear description of the phenomena of nonlinear forced oscillations and aids in the prediction of their occurrences. The present discussion centers on the subharmonic oscillation of order 1/3 of forward precession as a representative case. Other kinds of nonlinear oscillations are discussed briefly. Experimental results of previous reports may clearly be explained in the light of the results of this paper.

Stability Analysis of Rotating Shaft System with Many Bearings and Disks

Stability Analysis of Rotating Shaft System with Many Bearings and Disks

Stability Analysis of a Rotating Shaft System with Many Bearings and Disks

Stability Analysis of a Rotating Shaft System with Many Bearings and Disks

Effects of Flexible Mounting and Damping on the Synchronous Response of a Rotor-Shaft System

Synchronous response of a rotor supported by a damped, flexibly mounted bearing is similar to the well-known one of a rectilinear vibratory system having two degrees of freedom, or a dynamic vibration absorber. Designing the flexible support with an appropriate combination of equivalently concentrated mass of pedestal, bearing support stiffness, and optimum damping, the peak amplitude of a rotor-shaft system is found to be kept nearly to the value of rotor eccentricity. Experimental results of rotor unbalance resonance coincide well with the analytical ones.