Tsuyoshi Inoue

Detection of a Rotor Crack Using a Harmonic Excitation and Nonlinear Vibration Analysis

Detection of a rotor crack based on the nonlinear vibration diagnosis using harmonic excitation force is investigated. The open-close mechanism of crack is firstly modeled by a piecewise linear function. In addition, another approximation crack model using a power series function that is convenient for the theoretical analysis is used. When the power series function crack model is used, the equations of motion of a cracked rotor have linear and nonlinear parametric terms. In this paper, a harmonic excitation force is applied to the cracked rotor and its excitation frequency is swept, and the nonlinear resonances due to crack are investigated. The occurrence of various types of nonlinear resonances due to crack are clarified, and types of these resonances, their resonance points, and dominant frequency component of these resonances are clarified numerically and experimentally. Furthermore, nonlinear theoretical analyses are performed for these nonlinear resonances, and it is clarified that the amplitudes of these nonlinear resonances depend on the nonlinear parametric characteristics of rotor crack. These results enable us to detect a rotor crack without stopping the system during on-line operation.

Internal Resonance Phenomena of the Jeffcott Rotor With Nonlinear Spring Characteristics

The Jeffcott rotor is a two-degree-of-freedom linear model with a disk at the midspan of a massless elastic shaft. This model, executing lateral whirling motions, has been widely used in the linear analyses of rotor vibrations. In the Jeffcott rotor, the natural frequency of a forward-whirling mode p f (>0) and that of a backward-whirling mode p b (<0) have the relation of internal resonance p f :p b = 1: (-1). Recently, many researchers analyzed nonlinear phenomena by using the Jeffcott rotor with nonlinear elements. However, they did not take this internal resonance relationship into account. Furthermore in many practical rotating machines, the effect of gyroscopic moments are relatively small. Therefore, the one-to-one internal resonance relationship holds approximately between forward and backward natural frequencies in such machinery. In this paper, nonlinear phenomena in the vicinity of the major critical speed and the rotational speeds of twice and three times the major critical speed are investigated in the Jeffcott rotor and rotor systems with a small gyroscopic moment. The influences of internal resonance on the nonlinear resonances are studied in detail. The following were clarified theoretically and experimentally: (a) the shape of resonance curves becomes far more complex than that of a single resonance; (b) almost periodic motions occur; (c) these phenomena are influenced remarkably by the asymmetrical nonlinearity and gyroscopic moment; and (d) the internal resonance phenomena are strongly influenced by the degree of the discrepancies among critical speeds. The results teach us that the usage of the Jeffcott rotor in nonlinear analyses of rotor systems may induce incorrect results.

Vibration Suppression Using Electromagnetic Resonant Shunt Damper

An electromagnetic actuator has the property to convert mechanical energy to electrical energy and vice versa. In this study, an electromagnetic resonant shunt damper, consisting of a voice coil motor with an electric resonant shunt circuit, is proposed. The optimal design of the shunt circuit is obtained theoretically for this electromagnetic resonant shunt damper. Furthermore, the effects of parameter errors of the elements of the electromagnetic resonant shunt damper are also investigated. The applicability of the theoretical findings for the proposed damper is justified by the experimental analysis.

Chaotic Vibration and Internal Resonance Phenomena in Rotor Systems

Rotating machinery has effects of gyroscopic moments, but most of them are small. Then, many kinds of rotor systems satisfy the relation of I to (-1) type internal resonance approximately. In this paper, the dynamic characteristics of nonlinear phenomena, especially chaotic vibration, due to the 1 to (-1) type internal resonance at the major critical speed and twice the major critical speed are investigated. The following are clarified theoretically and experimentally: (a) the Hopf bifurcation and consecutive period doubling bifurcations possible route to chaos occur from harmonic resonance at the major critical speed and from subharmonic resonance at twice the major critical speed, (b) another chaotic vibration from the combination resonance occurs at twice the major critical speed. The results demonstrate that chaotic vibration may occur even in the rotor system with weak nonlinearity when the effect of the gyroscopic moment is small.

Internal Resonance Phenomena of an Asymmetrical Rotating Shaft

In general, asymmetrical shaft-disk systems have been investigated where unstable vibrations may occur. Most studies have treated a single resonance case for the linear system, and we have previously treated a single resonance case for the nonlinear system. However, when natural frequencies have a simple integer ratio relation in a nonlinear asymmetrical shaft-disk system, an internal resonance may occur and the vibration phenomena change remarkably compared to the characteristics of a single resonance case (the case without internal resonance). In this study, the internal resonance phenomena of an asymmetrical shaft are investigated theoretically and experimentally in the vicinities of the major critical speed, and twice and three times the major critical speed. We clarify that the shape of the resonance curves changes, almost periodic motions occur, and, especially, the occurrence of unstable vibration at the rotational speed of twice the major critical speed is extremely affected by the internal resonance. Further, we show the change of nonlinear phenomena between the systems with and without internal resonance.

Nonlinear Vibration Analysis of the Wind Turbine Blade (Occurrence of the Superharmonic Resonance in the Out of Plane Vibration of the Elastic Blade)

The use of wind turbine generator has rapidly spread as a one of the foremost clean energy sources. Recently, as the size of the wind turbine generator has become larger, its maintenance has become more difficult. However, there are few studies on the vibration analysis and its suppression in the conventional researches. The wind turbine is a special type of rotating machinery which has a long heavy blade rotating in the vertical plane under the action of the gravitational force. The wind power acting on the wind turbine blade varies periodically because of the height-dependent characteristics of the wind. Therefore, the dynamical design and analysis of the wind turbine blade requires a more thorough study. This paper investigates the fundamental vibration characteristic of an elastic blade of the wind turbine. The nonlinear vibration analysis of the superharmonic resonance is performed, and its characteristics are explained. Furthermore, the effect of the interaction of both the gravitational force and the wind force on the superharmonic resonance is clarified.

Nonlinear Analysis of Rotor Dynamics by Using the Method of Multiple Scales

The method of multiple scales is modified to nonlinear analysis in rotor systems. Amplitude equations for forward and backward whirling modes are directly derived and the method makes it easier to understand resonance mechanism. As an example, we analyze near the major critical speed the nonlinear dynamics of a horizontally supported Jeffcott rotor and show that nonlinear and gravity effects cause the backward whirling mode in addition to the forward one. Some experiments are performed and the validity of the theoretical results is confirmed.

Vibrations of an Asymmetrical Shaft With Gravity and Nonlinear Spring Characteristics (Isolated Resonances and Internal Resonances)

An asymmetrical shaft supported by single-row deep groove ball bearings with clearance has nonlinear spring characteristics, rotating stiffness difference in shaft, and static stiffness difference in support at the shaft ends. When an asymmetrical rotor is supported horizontally, a harmonic excitation due to the unbalance and a double frequency excitation due to the coexistence of gravity and shaft asymmetry work simultaneously. As a result, this system becomes a nonlinear parametrically excited system with multiple periodic excitation forces. In this paper, nonlinear forced vibrations and parametrically excited vibrations are investigated theoretically and experimentally. Vibration characteristics of both isolated resonances and internal resonances are studied.

Vibration of the Rigid Rotor Supported by a Repulsive Magnetic Bearing (Influences of Magnetic Anisotropies of Magnets)

This study focuses on the vibration of the rotor supported by a repulsive-type passive magnetic bearing. It evaluates the restoring force of a repulsive magnetic bearing numerically by considering the effects of magnetic anisotropies of the inner and the outer magnets. The study mainly investigates the occurrences of linear and nonlinear parametric characteristics, and the excitation forces for each pattern of magnetic anisotropies. Moreover, it clarifies the effects of each magnetic anisotropy pattern on the occurrences of various resonance phenomena theoretically and experimentally.

Suppression of the Forward Rub in Rotating Machinery by an Asymmetrically Supported Guide

In rotating machinery, rubbing occurs between the rotor and the stator, at the seal, between the rotor and the guide and between the rotor and the backup bearing. The backward rub or the partial impact vibration can be avoided by lubricating the contact surface sufficiently in order to decrease the friction. However, forward rub may still occur in such a case with a lubricated contact surface. Once such a forward rub occurs, it remains even if the rotational speed increases to much larger than the first bending critical speed and it is difficult to escape from this forward rubbing condition automatically. This paper proposes the suppression method of this forward rub by introducing the directional difference in the support stiffness of the guide or the backup bearing. The nonlinear theoretical analysis clarifies and explains the usefulness of the proposed method and it is also validated experimentally.