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.

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: 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 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.

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.

Application of Air Spring in Controlling Line Spectra

2005 ◽  
Author(s):  
Qi-Wei He ◽  
Shi-Jian Zhu ◽  
Jing-Jun Lou ◽  
Lin He
Keyword(s):  
Power Spectrum ◽  
Vibration Isolation ◽  
Period Doubling ◽  
Performance Indices ◽  
Air Spring ◽  
Radiated Noise ◽  
Chaotic Vibration ◽  
Route To Chaos ◽  
Marine Vessels ◽  
Period Doubling Bifurcations

Application of air spring in controlling line spectra in the radiated noise of marine vessels was studied. Starting with the route to chaos and the scaling property of the power spectrum in the cascade of period-doubling bifurcations, the method of chaotic vibration isolation was advanced. The performance indices for the presented method were also given. The method was experimentally verified.

scholarly journals Chaotic dynamics of a pulse modulated control system for a heating unit

2018 ◽  
Vol 224 ◽  
pp. 02055
Author(s):  
Yuriy A. Gol’tsov ◽  
Alexander S. Kizhuk ◽  
Vasiliy G. Rubanov
Keyword(s):  
Control System ◽  
Differential Equations ◽  
Chaotic Dynamics ◽  
Period Doubling ◽  
Classical Period ◽  
Nonlinear Phenomena ◽  
Pulse Control ◽  
Heating Unit ◽  
Border Collision Bifurcation ◽  
Period Doubling Bifurcations

The dynamic modes and bifurcations in a pulse control system of a heating unit, the condition of which is described through differential equations with discontinuous right–hand sides, have been studied. It has been shown that the system under research can demonstrate a great variety of nonlinear phenomena and bifurcation transitions, such as quasiperiodicity, multistable behaviour, chaotization of oscillations through a classical period–doubling bifurcations cascade and border–collision bifurcation.

Period-Doubling Bifurcation and Non-Typical Route to Chaos of a Two-Degree-Of-Freedom Vibro-Impact System

2000 ◽  
Vol 68 (4) ◽  
pp. 670-674 ◽  
Author(s):  
G. L. Wen and ◽  
J. H. Xie
Keyword(s):  
Finite Number ◽  
Period Doubling ◽  
Degree Of Freedom ◽  
Period Doubling Bifurcation ◽  
Periodic Response ◽  
Impact System ◽  
Route To Chaos ◽  
Two Degree Of Freedom ◽  
Impact Motion ◽  
Period Doubling Bifurcations

A nontypical route to chaos of a two-degree-of-freedom vibro-impact system is investigated. That is, the period-doubling bifurcations, and then the system turns out to the stable quasi-periodic response while the period 4-4 impact motion fails to be stable. Finally, the system converts into chaos through phrase locking of the corresponding four Hopf circles or through a finite number of times of torus-doubling.

Routes to Chaos in Squeeze-Film Dampers

2004 ◽  
Author(s):  
Jawaid I. Inayat-Hussain ◽  
Njuki W. Mureithi
Keyword(s):  
Period Doubling ◽  
Squeeze Film ◽  
Saddle Node Bifurcation ◽  
Chaotic Vibrations ◽  
Saddle Node ◽  
Highly Nonlinear ◽  
Squeeze Film Dampers ◽  
Route To Chaos ◽  
Type 3 ◽  
Period Doubling Bifurcations

This work reports on a numerical study undertaken to investigate the imbalance response of a rigid rotor supported by squeeze-film dampers. Two types of damper configurations were considered, namely, dampers without centering springs, and eccentrically operated dampers with centering springs. For a rotor fitted with squeeze-film dampers without centering springs, the study revealed the existence of three regimes of chaotic motion. The route to chaos in the first regime was attributed to a sequence of period-doubling bifurcations of the period-1 (synchronous) rotor response. A period-3 (one-third subharmonic) rotor whirl orbit, which was born from a saddle-node bifurcation, was found to co-exist with the chaotic attractor. The period-3 orbit was also observed to undergo a sequence of period-doubling bifurcations resulting in chaotic vibrations of the rotor. The route to chaos in the third regime of chaotic rotor response, which occurred immediately after the disappearance of the period-3 orbit due to a saddle-node bifurcation, was attributed to a possible boundary crisis. The transitions to chaotic vibrations in the rotor supported by eccentric squeeze-film dampers with centering springs were via the period-doubling cascade and type 3 intermittency routes. The type 3 intermittency transition to chaos was due to an inverse period-doubling bifurcation of the period-2 (one-half subharmonic) rotor response. The unbalance response of the squeeze-film-damper supported rotor presented in this work leads to unique non-synchronous and chaotic vibration signatures. The latter provide some useful insights into the design and development of fault diagnostic tools for rotating machinery that operate in highly nonlinear regimes.

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