Forced oscillations in the vicinities of both the major and the secondary critical speeds of a continuous asymmetrical rotor with the geometric nonlinearity are discussed. When the self-aligning double-row ball bearings support the slender flexible rotor at both ends, the geometric nonlinearity appears due to the stiffening effect in elongation of the shaft if the movements of the bearings in the longitudinal direction are restricted. The nonlinearity is symmetric when the rotor is supported vertically, and is asymmetric when it is supported horizontally. Because the rotor is slender, the natural frequency $pfn$ of a forward whirling mode and $pbn$ of a backward whirling mode have the relation of internal resonance $pfn:pbn=1:(−1)$. Due to the influence of the internal resonance, various phenomena occur, such as Hopf bifurcation, an almost periodic motion, the appearance of new branches, and the diminish of unstable region. These phenomena were clarified theoretically and experimentally. Moreover, this paper focuses on the influences of the nonlinearity, the unbalance, the damping, and the lateral force on the vibration characteristics.

1.
Yamamoto
,
T.
,
Ishida
,
Y.
, and
Aizawa
,
K.
, 1979, “
On the Subharmonic Oscillations of Unsymmetrical Shafts
,”
Bull. JSME
0021-3764
22
(
164
), pp.
164
173
.
2.
Yamamoto
,
T.
,
Ishida
,
Y.
, and
Ikeda
,
T.
, 1981, “
Summed-and-Differential Harmonic Oscillations of an Asymmetrical Shaft
,”
Bull. JSME
0021-3764
24
(
187
), pp.
183
191
.
3.
Ishida
,
Y.
,
Nagasaka
,
I.
,
Inoue
,
T.
, and
Lee
,
S.
, 1996, “
Forced Oscillations of a Vertical Continuous Rotor With Geometric Nonlinearity
,”
Nonlinear Dyn.
0924-090X
11
(
2
), pp.
107
120
.
4.
Ishida
,
Y.
,
Nagasaka
,
I.
, and
Lee
,
S.
, 1997, “
Forced Oscillations of a Continuous Rotor With Geometric Nonlinearity (Internal Resonance Phenomena at Harmonic and Subharmonic Resonances)
,”
Proceedings of ASME DETC1997
, Paper No. VIB-4097.
5.
Shabaneh
,
N.
, and
Zu
,
J. W.
, 2003, “
Nonlinear Dynamic Analysis of a Rotor Shaft System With Viscoelasticlly Supported Bearings
,”
ASME J. Vibr. Acoust.
0739-3717
125
, pp.
290
298
.
6.
Yamamoto
,
T.
, and
Ishida
,
Y.
, 2001
Linear and Nonlinear Rotordynamics
,
Wiley
,
New York
.