As is well known, zero and one nodal diameter ($k=0$ and $k=1$) modes of a blade system interact with the shaft system. The former couples with torsional and/or axial shaft vibrations, and the latter with bending shaft vibrations. This paper addresses the latter. With respect to $k=1$ modes, we discuss, from experimental and theoretical viewpoints, in-plane blades and out-of-plane blades attached radially to a rotating shaft. We found that when we excited the shaft at the rotational speed of $Ω=|ωb−ωs|$ (where $ωb$ is the blade natural frequency, $ωs$ the shaft natural frequency, and $Ω$ is the rotational speed), the exciting frequency $ν=ωs$ induced shaft-blade coupling resonance. In addition, in the case of the in-plane blade system, we encountered an additional resonance attributed to deformation caused by gravity. In the case of the out-of-plane blade system, we experienced two types of abnormal vibrations. One is the additional resonance generated at $Ω=ωb/2$ due to the unbalanced shaft and the anisotropy of bearing stiffness. The other is a flow-induced, self-excited vibration caused by galloping due to the cross-sectional shape of the blade tip because this instability disappeared in the rotation test inside a vacuum chamber. The two types of abnormal vibrations occurred at the same time, and both led to the entrainment phenomenon, as identified by our own frequency analysis technique.

1.
Matsushita
,
O.
,
Tanaka
,
M.
,
Kanki
,
H.
, and
Kobayashi
,
M.
, 2009,
Vibration of Rotating Machinery—Fundamentals of Practical Vibration Analysis
,
Corona
,
Tokyo, Japan
, Chaps. 10 and 11, p.
242
, p. 234, pp. 266–267.
2.
2009, Mechanical Vibration—Torsional Vibration of Rotating Machinery-Part 1: Land-Based Steam and Gas Turbine Generator Sets in Excess of 50MW, ISO 22266-1, International Organization for Standardization.
3.
Genta
,
G.
, 2005,
Dynamics of Rotating Systems
,
Springer
,
New York
, Chap. 13, pp.
465
469
, p. 512.
4.
Hagiwara
,
N.
,
Ono
,
Y.
, and
Iijima
,
H.
, 1985, “
Natural Frequencies of Forward and Backward Whirling of Blades (1st Report Analysis of Influences of Shaft and Rotating Effects)
,”
Trans. Jpn. Soc. Mech. Eng., Ser. C
0387-5024,
51
(
463
), pp.
555
562
.
5.
,
R.
, and
Drewczyński
,
M.
, 2004, “
Natural Frequencies and Modes Shapes of Two Mistuned Bladed Discs on the Shaft
,”
ASME
Paper No. GT2004-54265.
6.
,
R.
, and
Drewczyński
,
M.
, 2006, “
Forced Vibration of Several Bladed Discs on the Shaft
,”
ASME
Paper No. GT2006-90158.
7.
Santos
,
I. F.
,
Saracho
,
C. M.
,
Smith
,
J. T.
, and
Eiland
,
J.
, 2004, “
Contribution to Experimental Validation of Linear and Non-Linear Dynamic Models for Representing Rotor-Blades Parametric Coupled Vibrations
,”
J. Sound Vib.
0022-460X,
271
(
3–5
), pp.
883
904
.
8.
Christensen
,
R. H.
, and
Santos
,
I. F.
, 2006, “
Active Rotor-Blade Control Vibration Using Shaft-Based Electro-Magnetic Actuation
,”
ASME J. Eng. Gas Turbines Power
0742-4795,
128
, pp.
644
652
.
9.
Anegawa
,
N.
,
Fujiwara
,
H.
, and
Matsushita
,
O.
, 2008, “
Resonance and Instability of Blade-Shaft Coupled Bending Vibrations
,”
Turbomachinery (Turbomachinery Society of Japan)
,
36
(
4
), pp.
52
60
.
10.
Anegawa
,
N.
,
Fujiwara
,
H.
, and
Matsushita
,
O.
, 2008, “
Blade-Shaft Coupled Resonance Vibration by Using Active Magnetic Bearing Excitation
,”
ASME
Paper No. GT2008-50904.
11.
Anegawa
,
N.
,
Fujiwara
,
H.
,
Okabe
,
A.
, and
Matsushita
,
O.
, 2008, “
Resonance and Instability of Blade-Shaft Coupled Bending Vibrations With In-Plane Blade Vibration
,”
The International Journal of Fluid Machinery and Systems
,
1
(
1
), pp.
169
180
.
12.
Anegawa
,
N.
,
Fujiwara
,
H.
, and
Matsushita
,
O.
, 2008, “
Blade-Shaft Vibration Measurement in Rotating Machinery
,”
Dynamics and Design Conference
,
The Japan Society of Mechanical Engineers
,
Yokohama, Kanagawa, Japan
, No.
639
.
13.
Yamamoto
,
T.
, and
Ishida
,
Y.
, 2001,
Linear and Nonlinear Rotordynamics: Modern Treatment and Application
,
Corona
,
Tokyo, Japan
, Chap. 8, p.
216
.
14.
Iijima
,
H.
,
Goto
,
S.
, and
Hagiwara
,
N.
, 1985, “
Natural Frequencies of Forward and Backward Whirling of Blades (2nd Report Experiment by Models With Flat Blades Fixed to a Shaft in Its Middle)
,”
Trans. Jpn. Soc. Mech. Eng., Ser. C
0387-5024,
51
(
472
), pp.
3228
3234
.
15.
2008,
Flow Induced Vibrations Classification and Lessons From Practical Experiences
, 2nd ed.,
The Japan Society of Mechanical Engineers
,
Tokyo, Japan
, Chap. 2, p.
84
.
16.
Nakamura
,
T.
, and
Kaneko
,
S.
, 2008,
Flow Induced Vibrations Classifications and Lessons From Practical Experiences
,
Elsevier Science
,
New York
.
17.
Inoue
,
J.
, and
Sueoka
,
A.
, 2002,
Kikairikigaku II
,
Rikogakusha
,
Tokyo, Japan
, Chap. 4, p.
93
.