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.

1.
Yamamoto
,
T.
, 1955, “
On the Critical Speed of a Shaft of Sub-harmonic Oscillation
,”
Trans. Jpn. Soc. Mech. Eng.
0375-9466,
21-111
, pp.
853
858
(in Japanese).
2.
Yamamoto
,
T.
, 1957, “
On the Vibrations of a Rotating Shaft
,”
Mem. Fac. Eng. Nagoya Univ.
,
9-1
, pp.
19
115
.
3.
Yamamoto
,
T.
, and
Ishida
,
Y.
, 1977, “
Theoretical Discussions on Vibrations of a Rotating Shaft With Nonlinear Spring Characteristics
,”
Ing.-Arch.
0020-1154,
46
, pp.
125
135
.
4.
Tondl
,
A.
, 1965, “
Some Problems of Rotor Dynamics
,” Czechoslovak Academy of Sciences.
5.
Shaw
,
S.
, 1988, “
Chaotic Dynamics of a Slender Beam Rotating About Its Longitudinal Axis
,”
J. Sound Vib.
0022-460X,
124-2
, pp.
329
343
.
6.
Ishida
,
Y.
,
Nagasaka
,
I.
,
Inoue
,
T.
, and
Lee
,
S.
, 1996, “
Forced Oscillations of a Vertical Continuous Rotor With Geometrical Nonlinearity
,”
Nonlinear Dyn.
0924-090X,
11
(
2
), pp.
107
120
.
7.
Ehrich
,
F. F.
, 1991, “
Some Observations of Chaotic Vibration Phenomena in High Speed Rotordynamics
,”
J. Vibr. Acoust.
0739-3717,
113
, pp.
50
57
.
8.
Li
,
G. X.
, and
Païdoussis
,
M. P.
, 1994, “
Impact Phenomena of Rotor-Casing Dynamical Systems
,”
Nonlinear Dyn.
0924-090X,
5
, pp.
53
70
.
9.
Sethna
,
P. R.
, 1960, “
Steady-State Undamped Vibrations of a Class of Nonlinear Discrete Systems
,”
ASME J. Appl. Mech.
0021-8936,
27-1
, pp.
187
195
.
10.
Nayfeh
,
A. H.
, and
Balachandran
,
B.
, 1989, “
Modal Interactions in Dynamical and Structural Systems
,”
Appl. Mech. Rev.
0003-6900,
42-11-2
, pp.
175
201
.
11.
Tousi
,
S.
, and
Bajaj
,
A. K.
, 1985, “
Period-Doubling Bifurcations and Modulated Motions in Forced Mechanical Systems
,”
ASME J. Appl. Mech.
0021-8936,
52
, pp.
446
452
.
12.
Ishida
,
Y.
, and
Inoue
,
T.
, 2004, “
Internal Resonance Phenomena of the Jeffcott Rotor With Nonlinear Spring Characteristics
,”
ASME J. Vibr. Acoust.
0739-3717,
126
(
4
), pp.
476
484
.
13.
Yamamoto
,
T.
,
Ishida
,
Y.
, and
Ikeda
,
T.
, 1985, “
Super-Summed-and-Differential Harmonic Oscillations in a Symmetrical Rotating Shaft System
,”
Bull. JSME
0021-3764,
28
(
238
), pp.
679
686
.
14.
Stoker
,
J. J.
, 1950,
Nonlinear Vibrations in Mechanical and Electrical Systems
,
Wiley
,
New York
.
15.
Yamamoto
,
T.
, and
Yasuda
,
K.
, 1977, “
On the Internal Resonance in a Nonlinear Two-Degree-of-Freedom System (Forced Vibrations Near the Lower Resonance Point When the Natural Frequencies are in the Ratio 1:2)
,”
Bull. JSME
0021-3764,
20
(
140
), pp.
168
175
.
16.
Ehrich
,
F. F.
, 1992, “
Spontaneous Sidebanding in High Speed Rotordynamics
,”
Trans. ASME, J. Vib. Acoust.
1048-9002,
114
, pp.
498
505
.
17.
Wolf
,
A.
et al.
, 1985, “
Determining Lyapunov Exponents From a Time Series
,”
Physica D
0167-2789,
16
, pp.
285
317
.
18.
Yamamoto
,
T.
, 1961, “
On Sub-harmonic and Summed and Differential Harmonic Oscillations of Rotating Shaft
,”
Bull. JSME
0021-3764,
4
(
13
), pp.
51
58
.
19.
Moon
,
F. C.
, 1992,
Chaotic and Fractal Dynamics
,
Wiley
,
New York
.
20.
Baker
,
G. L.
, and
Gollub
,
J. P.
, 1996,
Chaotic Dynamics
,
2nd ed.
,
Cambridge University Press
, Cambridge.
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