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TECHNICAL PAPERS

# Chaotic Vibration and Internal Resonance Phenomena in Rotor Systems

[+] Author and Article Information
Tsuyoshi Inoue1

Department of Mechanical Science and Engineering, School of Engineering, Nagoya University, Nagoya, Aichi, 464-8603, Japaninoue@nuem.nagoya-u.ac.jp

Yukio Ishida

Department of Mechanical Science and Engineering, School of Engineering, Nagoya University, Nagoya, Aichi, 464-8603, Japanishida@nuem.nagoya-u.ac.jp

1

Corresponding author.

J. Vib. Acoust 128(2), 156-169 (Jul 13, 2005) (14 pages) doi:10.1115/1.2149395 History: Received June 22, 2004; Revised July 13, 2005

## Abstract

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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## Figures

Figure 1

Analytical model

Figure 2

Resonance curve for the 2DOF nonlinear rotor system which satisfies 1 to (−1) internal resonance approximately

Figure 3

Time histories, spectra, and Poincaré maps for the 2DOF nonlinear rotor system with ip=0.1 obtained by the numerical simulation of Eq. 1

Figure 4

Variations of amplitude P and orbits in phase planes Pcosβ−Psinβ obtained by numerical integration of Eq. 5 for various rotational speed ω

Figure 5

Bifurcation diagram of maximum values of amplitude P for the change of rotational speed

Figure 6

Influence of the magnitude of gyroscopic moment represented by parameter ip on the effect of 1 to (−1) internal resonance

Figure 7

Largest Lyapunov exponents for the vibration phenomena at the major critical speed calculated by using Wolf’s method for the data of numerical integration of Eq. 1

Figure 8

Natural frequencies pf and pb, and critical speeds of forward and backward subharmonic resonances of order 1∕2 and combination resonance pf−pb=ω for 2DOF rotor system with small ip

Figure 9

Resonance curves of forward subharmonic component Rf of subharmonic resonances of order 1∕2, and forward component Rf of combination resonance [pf−pb] for ε(1)=0.1

Figure 10

Time histories and Poincaré maps for the state points corresponding to the branch ab of the subharmonic resonance of order 1∕2

Figure 11

Largest Lyapunov exponents for the subharmonic resonance of order 1∕2

Figure 12

Resonance curves of forward subharmonic component Rf of subharmonic resonances of order 1∕2, and forward component Rf of combination resonance [pf−pb] for ε(1)=0.07

Figure 13

Time histories and Poincaré maps for the state points corresponding to the combination resonance [pf−pb]

Figure 14

Largest Lyapunov exponents for the almost periodic motion corresponding to the combination resonance [pf−pb]

Figure 15

Experimental system

Figure 16

Chaotic vibration at the major critical speed (experimental results)

Figure 17

Correlation dimension for m-dimensional pseudo-phase-space (m=2–20) (Experimental results: at ω=894rpm)

Figure 18

Largest Lyapunov exponents for the vibration around at the major critical speed corresponding to Fig. 1 (experimental results)

Figure 19

Natural frequencies p1,…,p4, and criticals speeds of forward and backward subharmonic resonances of order 1∕2, and combination resonance (p2−p3)=ω

Figure 20

Resonance curves of forward and backward components of subharmonic resonance of order 1∕2 and of combination resonance p2−p3=ω (Experiment: for a=80mm)

Figure 21

Time histories and Poincaré maps at twice the major critical speed (Experiment: for a=80mm)

Figure 22

Largest Lyapunov exponents for the subharmonic resonance of order 1∕2 corresponding to Fig. 2 (experimental results)

Figure 23

Resonance curves for forward component R2 of combination resonance p2−p3=ω (Experiment: for a=80mm)

Figure 24

Time histories and Poincaré maps of combination resonance p2−p3=ω at points g, h, and i (Experiment)

## Errata

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