In this paper, the effects of forward and backward whirl mechanism on the existence and the stability of multiple nonlinear normal modes (NNMs) in a four degree-of-freedom (DOF) rotor/stator rubbing system with cross-coupling stiffness and dry friction are investigated analytically. The NNMs may possess either positive or negative modal frequencies, corresponding, respectively, to the inherent motions of forward or backward whirl, and can be either stable or unstable. The relationship between the NNMs, regarding to their stability, and the forced system responses of the system is of great interest. It is found that a stable NNM corresponds to a forced harmonic response with the same whirl direction and frequency as the NNM, and an unstable NNM may still influence some forced system responses by contributing a frequency component equal to the modal frequency to the response spectrum.

References

References
1.
Rosenberg
,
R. M.
,
1962
, “
On Normal Vibrations of a General Class of Nonlinear Dual-Mode Systems
,”
ASME J. Appl. Mech.
,
29
(
1
), pp.
7
14
.10.1115/1.3636501
2.
Shaw
,
S. W.
, and
Pierre
,
C.
,
1991
, “
Nonlinear Normal Modes and Invariant Manifolds
,”
J. Sound Vib.
,
150
(
1
), pp.
170
173
.10.1016/0022-460X(91)90412-D
3.
Chati
,
M.
,
Rand
,
R.
, and
Mukherjee
,
S.
,
1997
, “
Modal Analysis of a Cracked Beam
,”
J. Sound Vib.
,
207
(
2
), pp.
249
270
.10.1006/jsvi.1997.1099
4.
Vestroni
,
F.
,
Luongo
,
A.
, and
Paolone
,
A.
,
2008
, “
A Perturbation Method for Evaluating Nonlinear Normal Modes of a Piecewise Linear 2-DOF System
,”
Nonlinear Dyn.
,
54
(
4
), pp.
379
393
.10.1007/s11071-008-9337-3
5.
Kerschen
,
G.
,
Peeters
,
M.
,
Golinval
,
J. C.
, and
Vakakis
,
A. F.
,
2009
, “
Nonlinear Normal Modes—Part I: A Useful Framework for the Structural Dynamicist
,”
Mech. Syst. Signal Process.
,
23
(
1
), pp.
170
194
.10.1016/j.ymssp.2008.04.002
6.
Peeters
,
M.
,
Viguie
,
R.
,
Serandour
,
G.
,
Kerschen
,
G.
, and
Golinval
,
J. C.
,
2009
, “
Nonlinear Normal Modes—Part II: Toward a Practical Computation Using Numerical Continuation Techniques
,”
Mech. Syst. Signal Process.
,
23
(
1
), pp.
195
216
.10.1016/j.ymssp.2008.04.003
7.
Jiang
,
D.
,
Pierre
,
C.
, and
Shaw
,
S. W.
,
2005
, “
The Construction of Non-Linear Normal Modes for Systems in Internal Resonance
,”
Int. J. Non-Linear Mech.
,
40
(
5
), pp.
729
746
.10.1016/j.ijnonlinmec.2004.08.010
8.
Casini
,
P.
, and
Vestroni
,
F.
,
2011
, “
Characterization of Bifurcating Non-Linear Normal Modes in Piecewise Linear Mechanical Systems
,”
Int. J. Non-Linear Mech.
,
46
(
1
), pp.
142
150
.10.1016/j.ijnonlinmec.2010.08.002
9.
Stoykov
,
S.
, and
Ribeiro
,
P.
,
2011
, “
Nonlinear Free Vibrations of Beams in Space Due to Internal Resonance
,”
J. Sound Vib.
,
330
(
18–19
), pp.
4574
4595
.10.1016/j.jsv.2011.04.023
10.
Szemplinska-Slupnicka
,
W.
,
1979
, “
The Modified Single Mode Method in the Investigations of the Resonant Vibration of Non-Linear Systems
,”
J. Sound Vib.
,
63
(
4
), pp.
475
489
.10.1016/0022-460X(79)90823-X
11.
Benamar
,
R.
,
Bennouna
,
M.
, and
White
,
R. G.
,
1991
, “
The Effects of Large Vibration Amplitudes on the Mode Shapes and Natural Frequencies of Elastic Structures—Part I: Simply Supported and Clamped–Clamped Beams
,”
J. Sound Vib.
,
149
(
2
), pp.
179
195
.10.1016/0022-460X(91)90630-3
12.
Qaisi
,
M. I.
,
1998
, “
Non-Linear Normal Modes of a Continuous System
,”
J. Sound Vib.
,
209
(
4
), pp.
561
569
.10.1006/jsvi.1997.1246
13.
Jiang
,
D.
,
Pierre
,
C.
, and
Shaw
,
S. W.
,
2005
, “
Nonlinear Normal Modes for Vibratory Systems Under Harmonic Excitation
,”
J. Sound Vib.
,
288
(
4–5
), pp.
791
812
.10.1016/j.jsv.2005.01.009
14.
Jiang
,
J.
, and
Ulbrich
,
H.
,
2005
, “
The Physical Reason and the Analytical Condition for the Onset of Dry Whip in Rotor-to-Stator Contact Systems
,”
ASME J. Vib. Acoust.
,
127
(
6
), pp.
594
603
.10.1115/1.1888592
15.
Chen
,
Y. H.
, and
Jiang
,
J.
,
2013
, “
Determination of Nonlinear Normal Modes of a Planar Nonlinear System With a Constraint Condition
,”
J. Sound Vib.
,
332
(
20
), pp.
5151
5161
.10.1016/j.jsv.2013.04.040
16.
Jiang
,
J.
, and
Wu
,
Z. Q.
,
2010
, “
Determine the Characteristics of a Self-Excited Oscillation in Rotor/Stator Systems From the Interaction of Linear and Nonlinear Normal Modes
,”
Int. J. Bifurcation Chaos
,
20
(
12
), pp.
4137
4150
.10.1142/S0218127410028252
17.
Jiang
,
J.
,
Ulbrich
,
H.
, and
Chavez
,
A.
,
2006
, “
Improvement of Rotor Performance Under Rubbing Conditions Through Active Auxiliary Bearings
,”
Int. J. Nonlinear Mech.
,
41
(
8
), pp.
949
957
.10.1016/j.ijnonlinmec.2006.08.004
18.
Jiang
,
J.
, and
Ulbrich
,
H.
,
2001
, “
Stability Analysis of Sliding Whirl in a Nonlinear Jeffcott Rotor With Cross-Coupling Stiffness Coefficients
,”
Nonlinear Dyn.
,
24
(
3
), pp.
269
283
.10.1023/A:1008376412944
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