Yukio Ishida

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.

Forced oscillations of a vertical continuous rotor with geometric nonlinearity

Nonlinear forced oscillations of a vertical continuous rotor with distributed mass are discussed. The restoring force of the rotor has geometric stiffening nonlinearity due to the extension of the rotor center line. The possibility of the occurrence of nonlinear forced oscillations at various subcritical speeds and the shapes of resonance curves at the major critical speeds and at some subcritical speeds are investigated theoretically. Consequently, the following is clarified: (a) the shape of resonance curves at the major critical speed becomes a hard spring type, and (b) among various kinds of nonlinear forced oscillations, only some special kinds of combination resonances have possibility of occurrence.

Forced oscillations of a rotating shaft with nonlinear spring characteristics and internal damping (1/2 order subharmonic oscillations and entrainment)

Nonlinear forced oscillations of a rotating shaft with nonlinear spring characteristics and internal damping are studied. In particular, entrainment phenomena at the critical speeds of 1/2 order subharmonic oscillations of forward and backward whirling modes are investigated. A self-excited oscillation appears in the wide range above the major critical speed. The amplitude of this oscillation reaches a limit value and then a self-sustained oscillation occurs. In the vicinity of a 1/2 order subharmonic oscillation of a forward whirling mode, a self-excited oscillation is entrained by a subharmonic oscillation. In the vicinity of a 1/2 order subharmonic oscillation of a backward whirling mode, either a self-excited oscillation or a subharmonic oscillation occurs.Experiments were made by an elastic rotating shaft with a disc. Nonlinearity in its restoring force was due to an angular clearance of a bearing and internal damping was due to friction between the shaft and an inner ring of the bearing. A self-excited oscillation was observed in the range above the major critical speed and this self-excited oscillation was entrained by a 1/2 order subharmonic oscillation of a forward whirling mode.

Nonstationary Oscillation of a Rotating Shaft With Nonlinear Spring Characteristics During Acceleration Through a Major Critical Speed (A Discussion by the Asymptotic Method and the Complex-FFT Method)

Nonstationary oscillations during acceleration through a major critical speed of a rotating shaft with nonlinear spring characteristics are discussed. First, the first approximate solutions of steady-state and nonstationary oscillations are obtained by the asymptotic method. Second, the amplitude variation curves of each oscillation component are obtained by the complex-FFT method. It is clarified that the first approximation of the asymptotic method has comparatively large quantitative error in the case of nonstationary solutions. In addition, the influences of each nonlinear component in polar coordinate expression on nonstationary oscillations are investigated.

Nonlinear Forced Oscillations Caused by Quartic Nonlinearity in a Rotating Shaft System

This paper deals with nonlinear forced oscillations in a rotating shaft system which are caused by quartic nonlinearity in a restoring force. These oscillations are theoretically analyzed by paying attention to the nonlinear components represented by the polar coordinates. It is clarified which kind of nonlinear component has an influence on each oscillation. In experiments it was shown that, when the shaft was supported by double-row angular contact ball bearings, the restoring force had nonlinear spring characteristics involving quartic nonlinearity in addition to quadratic and cubic ones. Experimental results were compared with the theoretical results regarding the probability of occurrence and the shapes of the resonance curves.

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.

Vibration Suppression of Nonlinear Rotor Systems Using a Dynamic Damper

Due to the inevitable imbalance of rotating machinery, resonance occurs when the rotational speed is in the vicinity of the critical speed. A rotor system supported by a single-row deep groove ball bearing shows nonlinear spring characteristics due to the clearance of bearings. In this article, passive vibration control of nonlinear rotor systems using a dynamic damper is studied. Theoretical analysis is performed to investigate the influence of nonlinearity on the vibration characteristics of controlled rotor systems, and the theoretical results obtained are confirmed by experiments. An example shows that the fixed-point theorem for optimization of the dynamic damper cannot be used when the rotor system has an isotropic symmetrical nonlinearity. The Newton-Raphson method is used to determine the optimal parameters of the dynamic damper for the nonlinear rotor, and passive vibration control utilizing the dynamic damper is achieved in the nonlinear rotor system.

Vibration suppression of the rotor system using both a ball balancer and axial control of the repulsive magnetic bearing

In this paper we describe a strategy to eliminate the disadvantages of both a ball balancer and a repulsive magnetic bearing through the simultaneous application of both to a rotor system. The resultant force of the magnetic synchronous force caused by the repulsive magnetic bearing and the unbalance force is cancelled by the ball balancer, and the vibration during the passage through the critical speed caused by the ball balancer is also reduced by the axial control of the repulsive magnetic bearing. As a result, both the vibration suppression during the passage through the critical speed and the vibration reduction to zero at the rotational speed range above the critical speed are achieved simultaneously. The vibration suppression effect of the proposed method is also validated experimentally.