Internal Resonance Phenomena of the Jeffcott Rotor With Nonlinear Spring Characteristics

2004 ◽  
Vol 126 (4) ◽  
pp. 476-484 ◽  
Author(s):  
Yukio Ishida ◽  
Tsuyoshi Inoue
Keyword(s):  
Internal Resonance ◽  
Critical Speed ◽  
Nonlinear Phenomena ◽  
Nonlinear Analyses ◽  
Rotor Systems ◽  
Gyroscopic Moment ◽  
Resonance Phenomena ◽  
Jeffcott Rotor ◽  
The One ◽  
Two Degree Of Freedom

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 pf>0 and that of a backward-whirling mode pb<0 have the relation of internal resonance pf:pb=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.

Internal Resonance of the Jeffcott Rotor With Nonlinear Spring Characteristics

2001 ◽  
Author(s):  
Yukio Ishida ◽  
Tsuyoshi Inoue
Keyword(s):  
Internal Resonance ◽  
Critical Speed ◽  
Nonlinear Phenomena ◽  
Almost Periodic ◽  
Periodic Motions ◽  
Nonlinear Analyses ◽  
Rotor Systems ◽  
Gyroscopic Moment ◽  
Jeffcott Rotor ◽  
Two Degree Of Freedom

Abstract 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 pf and that of a backward whirling mode pb have the relation of internal resonance pf : pb = 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. While, in many cases of the practical rotating machinery, such a relationship holds apprximately due to small gyroscopic moment. 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 small gyroscopic moment. Especially, 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 the usage of the Jeffcott rotor in nonlinear analyses of rotor systems may induce incrrect results.

Chaotic Vibration and Internal Resonance Phenomena in Rotor Systems: Part I — Theoretical Analysis

2003 ◽  
Author(s):  
Tsuyoshi Inoue ◽  
Yukio Ishida
Keyword(s):  
Internal Resonance ◽  
Critical Speed ◽  
Rotor System ◽  
Nonlinear Phenomena ◽  
Combination Resonance ◽  
Critical Speeds ◽  
Rotor Systems ◽  
Gyroscopic Moment ◽  
Chaotic Vibration ◽  
Resonance Phenomena

Naturally, the gyroscopic moment is small for the many practical rotating machineries. In addition, some mechanical elements of a rotor system make various types of nonlinearity such as clearance in a ball bearing (Yamamoto, 1955)(Yamamoto, 1977), oil film in a journal bearing (Tondl, 1965), geometrical nonlinearity due to the shaft elongation (Shaw, 1988),(Ishida, 1996), etc. In such rotor systems, the natural frequencies of a forward whirling mode pf and a backward whirling mode pb almost satisfy the relation of internal resonance pf : pb = 1 : (−1). And then, the critical speeds of a backward harmonic oscillation and a supercombination oscillation are near from the major critical speed. Similarly, in the vicinity of two times of the major critical speed, the critical speeds of the forward and the backward subharmonic resonances of order 1/2 and the combination resonance are close to each other. Therefore, the internal resonance phenomena may occur at the major critical speed and two times of the major critical speed. However there are few studies on the nonlinear phenomena of the rotor systems due to the influence of internal resonance. In this study, we use a 2DOF rotor model and investigate the dynamic characteristics of nonlinear phenomena, especially the chaotic vibration, due to the internal resonance at the major critical speed and the critical speed of two times of the major critical speed. The following are clarified theoretically: (a) the Hopf bifurcation and consecutive period doubling bifurcations possible route to chaos occur at the major critical speed and at two times of the major critical speed, (b) another chaotic vibration from the combination resonance occur at two times of the major critical speed. The results demonstrate that the chaotic vibration is common nonlinear phenomena in the nonlinear rotor system when the effect of the gyroscopic moment is small.

Internal Resonance Phenomena of an Asymmetrical Rotating Shaft: Critical Speed of Twice the Major Critical Speed

1999 ◽  
Author(s):  
Yukio Ishida ◽  
Tsuyoshi Inoue
Keyword(s):  
Internal Resonance ◽  
Critical Speed ◽  
Nonlinear Phenomena ◽  
Rotating Shaft ◽  
Disk System ◽  
Rotor Systems ◽  
Resonance Phenomena ◽  
Resonance Relation ◽  
Asymmetric Rotor ◽  
Asymmetrical Rotating Shaft

Abstract Unstable vibrations appear in the vicinities of several critical speeds in asymmetric rotor systems with nonlinear spring characteristics. However, when the natural frequencies satisfy internal resonance relation exactly or approximately, these phenomena may change remarkably. In this paper, such internal resonance phenomena of an asymmetric shaft-disk system are studied theoretically and experimentally. The changes in nonlinear phenomena during the transition from the system with internal resonance to the system with no internal resonance are also investigated.

Internal Resonance Phenomena of an Asymmetrical Rotating Shaft

2005 ◽  
Vol 11 (9) ◽  
pp. 1173-1193 ◽  
Author(s):  
Yukio Ishida ◽  
Tsuyoshi Inoue
Keyword(s):  
Internal Resonance ◽  
Critical Speed ◽  
Nonlinear Phenomena ◽  
Resonance Case ◽  
Rotating Shaft ◽  
Disk System ◽  
Resonance Phenomena ◽  
Previously Treated ◽  
Unstable Vibration ◽  
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.

Chaotic Vibration and Internal Resonance Phenomena in Rotor Systems: Part II — Experimental Analysis

2003 ◽  
Author(s):  
Tsuyoshi Inoue ◽  
Yukio Ishida
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Internal Resonance ◽  
Critical Speed ◽  
Nonlinear Phenomena ◽  
Rotor Systems ◽  
Dynamical Characteristics ◽  
Chaotic Vibration ◽  
Mechanical Elements ◽  
Theoretical Results

In the practical rotating machinery, the gyroscopic moment is often small. In addition, some mechanical elements of a rotor system make various types of nonlinearity. In such rotor systems, the natural frequency of a forward whirling mode pf and that of a backward whirling mode pb almost satisfy the relation of internal resonance pf : pb = 1 : (−1). However there are few studies on the nonlinear phenomena of the rotor systems due to the influence of internal resonance, especially the experimental investigation have not been reported. In this study, we investigate experimentally the dynamical characteristics of nonlinear phenomena due to the internal resonance at the major critical speed and at two times of the major critical speed. The following are clarified experimentally: (a) the amplitude modulated vibration due to Hopf bifurcation from the steady state vibration occur and then the chaotic vibration occur at the major critical speed, (b) the amplitude modulated vibration due to Hopf bifurcation from the steady state subharmonic resonance of order 1/2 occur and then the chaotic vibration occur at two times of the major critical speed, (c) another chaotic vibration from the combination resonance occur at two times of the major critical speed. These experimental results match to the theoretical results (Part I).

Chaotic Vibration and Internal Resonance Phenomena in Rotor Systems

2005 ◽  
Vol 128 (2) ◽  
pp. 156-169 ◽  
Author(s):  
Tsuyoshi Inoue ◽  
Yukio Ishida
Keyword(s):  
Internal Resonance ◽  
Critical Speed ◽  
Period Doubling ◽  
Nonlinear Phenomena ◽  
Rotor Systems ◽  
Chaotic Vibration ◽  
Consecutive Period ◽  
Route To Chaos ◽  
Harmonic Resonance ◽  
Period Doubling Bifurcations

Rotating machinery has effects of gyroscopic moments, but most of them are small. Then, many kinds of rotor systems satisfy the relation of 1 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.

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