scholarly journals Chaotic Vibrations and Internal Resonance Phenomena in Rotor Systems. Case that the Critical Speeds of a Combination Resonance and Subharmonic Resonances of Order 1/2 are close.

2001 ◽  
Vol 67 (660) ◽  
pp. 2434-2441
Author(s):  
Tsuyoshi INOUE ◽  
Yukio ISHIDA ◽  
Takuji MURAYAMA
Keyword(s):  
Internal Resonance ◽  
Combination Resonance ◽  
Critical Speeds ◽  
Rotor Systems ◽  
Chaotic Vibrations ◽  
Resonance Phenomena ◽  
Subharmonic Resonances

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 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.

Forced Oscillations of a Continuous Rotor With Geometric Nonlinearity: Internal Resonance Phenomena at Harmonic and Subharmonic Resonances

1997 ◽  
Author(s):  
Yukio Ishida ◽  
Imao Nagasaka ◽  
Seongwoo Lee
Keyword(s):  
Geometric Nonlinearity ◽  
Internal Resonance ◽  
Forced Oscillations ◽  
Center Line ◽  
Rotating Shaft ◽  
Restoring Force ◽  
Resonance Phenomena ◽  
Geometric Stiffening ◽  
Resonance Curves ◽  
Subharmonic Resonances

Abstract Harmonic resonances and subharmonic resonances of order 1/2 and order 1/3 in a continuous rotating shaft with distributed mass are discussed. The restoring force of the shaft has geometric stiffening nonlinearity due to the extension of the shaft center line. It is supposed that a distributed bias force, such as the gravity, works. The possibility of their occurrences, the shapes of resonance curves, and internal resonance phenomena are investigated.

scholarly journals Normal form analysis of stator rub in rotating machinery

2018 ◽  
Vol 148 ◽  
pp. 04004
Author(s):  
Alexander Shaw ◽  
Alan Champneys ◽  
Michael Friswell
Keyword(s):  
Normal Form ◽  
Internal Resonance ◽  
Normal Forms ◽  
Rotating Machinery ◽  
Form Analysis ◽  
Rotor Systems ◽  
Resonance Phenomena ◽  
Ideal System ◽  
Rotating Shafts

This work considers analysis of sustained impacting cycles of rotating shafts with potentially many disks. The insight that this is an internal resonance phenomena makes this an ideal system to be studied with the method of normal forms. However, the presence of arbitrary non smooth nonlinearities due to impact and rub mean that the method must be extended by incorporating an Alternating Frequency/Time (AFT) step to capture nonlinear forces. The process results in an elegant formulation that can model a very wide variety of rotor systems and is demonstrated by comparing against simulation of a contacting overhung rotor.

The Analytical Imbalance Response of Jeffcott Rotor During Acceleration

2000 ◽  
Vol 123 (2) ◽  
pp. 299-302 ◽  
Author(s):  
Shiyu Zhou ◽  
Jianjun Shi
Keyword(s):  
Natural Frequency ◽  
Initial Condition ◽  
Constant Acceleration ◽  
Transient Vibration ◽  
Critical Speeds ◽  
Rotor Systems ◽  
Jeffcott Rotor ◽  
Physical Insight

Since many rotor systems normally operate above their critical speeds, the problem of accelerating the machine through its critical speeds without excessive vibration draws increasing attention. This paper provides an analytical imbalance response of the Jeffcott rotor under constant acceleration. The response consists of three parts: transient vibration due to the initial condition of the rotor, “synchronous” vibration, and suddenly occurring vibration at the damped natural frequency. This solution provides physical insight to the vibration of the rotor during acceleration.

scholarly journals Dynamics of a rotor system coupled with water-lubricated rubber bearings

2018 ◽  
Vol 232 (23) ◽  
pp. 4263-4277 ◽  
Author(s):  
YF Shi ◽  
M Li ◽  
GH Zhu ◽  
Y Yu
Keyword(s):  
Elastic Deformation ◽  
Dynamic Characteristics ◽  
Hydrodynamic Lubrication ◽  
Dynamic Performance ◽  
Elastohydrodynamic Lubrication ◽  
Rotor System ◽  
Critical Speeds ◽  
Rotor Systems ◽  
Mass Unbalance ◽  
Rubber Bearings

Dynamic behaviour is significantly important in the design of large rotor systems supported on water-lubricated rubber bearings. In this study, the mathematical model of elastohydrodynamic lubrication of the bearing is established based on the theory of hydrodynamic lubrication after considering the elastic deformation of rubber, and the dynamic characteristics of water-lubricated rubber bearings are analysed under small perturbation conditions according to the load increment method and the finite difference method. Next, the differential equation of rotor systems coupled with the water-lubricated rubber bearing is deduced using Lagrange’s approach, and its critical speeds, stability, and unbalanced responses are analysed in detail. The numerical results show that several parameters, such as the eccentricity, length–diameter ratio, and clearance of bearing and the rotating speed of the rotor, have a great impact on the dynamic performance of water-lubricated rubber bearings, and this influence cannot be ignored, especially in the case of large eccentricity ratios. The dynamic characteristics of rotor systems guided by water-lubricated rubber bearings reveal that the critical speeds are much lower than the ones under the rigid supports because of the elastic deformation, and they also indicate that the rotor system supported on water-lubricated rubber bearings has a weaker stability. In addition, the steady-state responses of the rotor system are analysed when the mass unbalance of the propeller exists, and the effect of the thickness of the rubber liner is also considered.

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