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.

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.

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.

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.

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.

Period-1 to Period-8 Motions in a Nonlinear Jeffcott Rotor System

2020 ◽  
Author(s):  
Yeyin Xu ◽  
Albert C.J. Luo
Keyword(s):  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Mapping Method ◽  
Rotor System ◽  
Period Doubling Bifurcation ◽  
Periodic Motions ◽  
Bifurcation Tree ◽  
Rotor Systems ◽  
Period 2 ◽  
Jeffcott Rotor

Abstract In this paper, a bifurcation tree of period-1 to period-8 motions in a nonlinear Jeffcott rotor system is obtained through the discrete mapping method. The bifurcations and stability of periodic motions on the bifurcation tree are discussed. The quasi-periodic motions on the bifurcation tree are caused by two (2) Neimark bifurcations of period-1 motions, one (1) Neimark bifurcation of period-2 motions and four (4) Neimark bifurcations of period-4 motions. The specific quasi-periodic motions are mainly based on the skeleton of the corresponding periodic motions. One stable and one unstable period-doubling bifurcations exist for the period-1, period-2 and period-4 motions. The unstable period-doubling bifurcation is from an unstable period-m motion to an unstable period-2m motion, and the unstable period-m motion becomes stable. Such an unstable period-doubling bifurcation is the 3rd source pitchfork bifurcation. Periodic motions on the bifurcation tree are simulated numerically, and the corresponding harmonic amplitudes and phases are presented for harmonic effects on periodic motions in the nonlinear Jeffcott rotor system. Such a study gives a complete picture of periodic and quasi-periodic motions in the nonlinear Jeffcott rotor system in the specific parameter range. One can follow the similar procedure to work out the other bifurcation trees in the nonlinear Jeffcott rotor systems.

Period-1 Motion to Chaos in a Nonlinear Flexible Rotor System

2020 ◽  
Vol 30 (05) ◽  
pp. 2050077 ◽  
Author(s):  
Yeyin Xu ◽  
Zhaobo Chen ◽  
Albert C. J. Luo
Keyword(s):  
Period Doubling ◽  
Rotor System ◽  
Harmonic Amplitude ◽  
Flexible Rotor ◽  
Periodic Motions ◽  
Bifurcation Tree ◽  
Rotor Systems ◽  
Saddle Node ◽  
Jeffcott Rotor ◽  
And Control

In this paper, a bifurcation tree of period-1 motion to chaos in a flexible nonlinear rotor system is presented through period-1 to period-8 motions. Stable and unstable periodic motions on the bifurcation tree in the flexible rotor system are achieved semi-analytically, and the corresponding stability and bifurcation of the periodic motions are analyzed by eigenvalue analysis. On the bifurcation tree, the appearance and vanishing of jumping phenomena of periodic motions are generated by saddle-node bifurcations, and quasi-periodic motions are induced by Neimark bifurcations. Period-doubling bifurcations of periodic motions are for developing cascaded bifurcation trees, however, the birth of new periodic motions are based on the saddle-node bifurcation. For a better understanding of periodic motions on the bifurcation tree, nonlinear harmonic amplitude characteristics of periodic motions are presented. Numerical simulations of periodic motions are performed for the verification of semi-analytical predictions. From such a study, nonlinear Jeffcott rotor possesses complex periodic motions. Such results can help one detect and control complex motions in rotor systems for industry.

Analytical Solutions for Period-1 Motions in a Nonlinear Jeffcott Rotor System

2014 ◽  
Author(s):  
Jianzhe Huang ◽  
Albert C. J. Luo
Keyword(s):  
Analytical Solutions ◽  
Periodic Motion ◽  
Rotor System ◽  
Eigenvalue Analysis ◽  
Hopf Bifurcations ◽  
Harmonic Amplitude ◽  
Periodic Motions ◽  
Bifurcation Tree ◽  
Rotor Systems ◽  
Jeffcott Rotor

In this paper, the analytical solutions of period-1 solutions are developed, and the corresponding stability and bifurcation are also analyzed by eigenvalue analysis. The Hopf bifurcations of periodic motions cause not only the bifurcation tree but quasi-periodic motions. The quasi-periodic motion can be stable or unstable. Displacement orbits of periodic motions in the nonlinear Jeffcott rotor systems are illustrated, and harmonic amplitude spectrums are presented for harmonic effects on periodic motions of the nonlinear rotor.

Vibration Suppression of Rotating Machinery Utilizing an Automatic Ball Balancer and Discontinuous Spring Characteristics

2009 ◽  
Vol 131 (4) ◽  
Author(s):  
Jun Liu ◽  
Yukio Ishida
Keyword(s):  
Rotational Speed ◽  
Large Amplitude ◽  
Small Amplitude ◽  
Critical Speed ◽  
Vibration Suppression ◽  
Rotating Machinery ◽  
Almost Periodic ◽  
Periodic Motions ◽  
Speed Range ◽  
Suppression Method

Automatic ball balancer is a balancing device where two balls inside a hollow rotor move to optimal rest positions automatically to eliminate unbalance. As a result, vibrations are suppressed to the zero amplitude in the rotational speed range higher than the major critical speed. However, it has the following defects. The amplitude of vibration increases in the rotational speed range lower than the major critical speed. In addition, almost periodic motions with large amplitude occur in the vicinity of the major critical speed due to the rolling of balls inside the rotor. Because of these defects, an automatic ball balancer has not been used widely. This paper proposes the vibration suppression method utilizing the discontinuous spring characteristics together with an automatic ball balancer to overcome these defects and to suppress vibration. The validity of the proposed method is confirmed theoretically, numerically, and experimentally. The results show that amplitude of vibration can be suppressed to a small amplitude in the vicinity of the major critical speed and the zero amplitude in the range higher than the major critical speed.

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