In many cases of rotating systems, such as jet engines, two or more coaxial shafts are used for power transmission between a high/low-pressure turbine and a compressor. The major purpose of this study is to predict the nonlinear dynamic behavior of a coaxial rotor system supported by two active magnetic bearings (AMBs) and contact with two auxiliary bearings. The model of the system is formulated by ten degrees-of-freedom in two different planes. This model includes gyroscopic moments of disks and geometric coupling of the magnetic actuators. The nonlinear equations of motion are developed by the Lagrange's equations and solved using the Runge–Kutta method. The effects of speed parameter, speed ratio of shafts, and gravity parameter on the dynamic behavior of the coaxial rotor–AMB system are investigated by the dynamic trajectories, power spectra analysis, Poincaré maps, bifurcation diagrams, and the maximum Lyapunov exponent. Also, the contact forces between the inner shaft and auxiliary bearings are studied. The results indicate that the speed parameter, speed ratio of shafts, and gravity parameter have significant effects on the dynamic responses and can be used as effective control parameters for the coaxial rotor–AMB system. Also, the results of analysis reveal a variety of nonlinear dynamical behaviors such as periodic, quasi-periodic, period-4, and chaotic vibrations, as well as jump phenomena. The obtained results of this research can give some insight to engineers and researchers in designing and studying the coaxial rotor–AMB systems or some turbomachinery in the future.

References

References
1.
Schweitzer
,
G.
,
Maslen
,
E.
,
Bleuler
,
H.
,
Traxler
,
A.
,
Cole
,
M.
, and
Keogh
,
P.
,
2009
,
Magnetic Bearings: Theory, Design, and Application to Rotating Machinery
,
Springer
, Berlin.
2.
Chiba
,
A.
,
Fukao
,
T.
,
Ichikawa
,
O.
,
Oshima
,
M.
,
Takemoto
,
M.
, and
Dorrell
,
D. G.
,
2005
,
Magnetic Bearings and Bearingless Drives
,
Elsevier
,
London
.
3.
Nayfeh
,
A. H.
, and
Balachandran
,
B.
,
1995
,
Applied Non-Linear Dynamics: Analytical, Computational and Experimental Methods
,
Wiley
,
New York
.
4.
Ji
,
J. C.
,
Yu
,
L.
, and
Leung
,
A. Y. T.
,
2000
, “
Bifurcation Behavior of a Rotor Supported by Active Magnetic Bearings
,”
J. Sound Vib.
,
235
(
1
), pp.
133
151
.
5.
Yang
,
X. D.
,
An
,
H. Z.
,
Qian
,
Y. J.
,
Zhang
,
W.
, and
Yao
,
M. H.
,
2016
, “
Elliptic Motions and Control of Rotors Suspending in Active Magnetic Bearings
,”
ASME J. Comput. Nonlinear Dyn.
,
11
(
5
), p.
054503
.
6.
Ji
,
J. C.
, and
Hansen
,
C. H.
,
2001
, “
Non Linear Oscillations of a Rotor in Active Magnetic Bearings
,”
J. Sound Vib.
,
240
(
4
), pp.
599
612
.
7.
Zhang
,
W.
,
Yao
,
M. H.
, and
Zhan
,
X. P.
,
2006
, “
Multi-Pulse Chaotic Motions of a Rotor-Active Magnetic Bearing System With Time-Varying Stiffness
,”
Chaos, Solitons Fractals
,
27
(
1
), pp.
175
186
.
8.
Zhang
,
W.
, and
Zu
,
J. W.
,
2008
, “
Transient and Steady Nonlinear Responses for a Rotor-Active Magnetic Bearings System With Time-Varying Stiffness
,”
Chaos, Solitons Fractals
,
38
(
4
), pp.
1152
1167
.
9.
Zhang
,
W.
, and
Zhan
,
X. P.
,
2005
, “
Periodic and Chaotic Motions of a Rotor-Active Magnetic Bearing With Quadratic and Cubic Terms and Time-Varying Stiffness
,”
Nonlinear Dyn.
,
41
(
4
), pp.
331
359
.
10.
Kamel
,
M.
, and
Bauomy
,
H. S.
,
2010
, “
Nonlinear Study of a Rotor–AMB System Under Simultaneous Primary-Internal Resonance
,”
Appl. Math. Model.
,
34
(
10
), pp.
2763
2777
.
11.
Jang
,
M. J.
, and
Chen
,
C. K.
,
2001
, “
Bifurcation Analysis in Flexible Rotor Supported by Active Magnetic Bearings
,”
Int. J. Bifurcation Chaos
,
11
(
8
), pp.
2163
2178
.
12.
Inayat-Hussain
,
J. I.
,
2010
, “
Nonlinear Dynamics of a Magnetically Supported Rigid Rotor in Auxiliary Bearings
,”
Mech. Mach. Theory
,
45
(
11
), pp.
1651
1667
.
13.
Inayat-Hussain
,
J. I.
,
2009
, “
Geometric Coupling Effects on the Bifurcations of a Flexible Rotor Response in Active Magnetic Bearings
,”
Chaos, Solitons Fractals
,
41
(
5
), pp.
2664
2671
.
14.
Inayat-Hussain
,
J. I.
,
2010
, “
Nonlinear Dynamics of a Statically Misaligned Flexible Rotor in Active Magnetic Bearings
,”
Commun. Nonlinear Sci. Numer. Simul.
,
15
(3), pp.
764
777
.
15.
Xie
,
H.
,
Flowers
,
G. T.
,
Feng
,
L.
, and
Lawrence
,
C.
,
1999
, “
Steady-State Dynamic Behavior of a Flexible Rotor With Auxiliary Support From a Clearance Bearing
,”
ASME J. Vib. Acoust.
,
121
(
1
), pp.
78
83
.
16.
Payyoor
,
N.
,
Tiwari
,
M.
, and
Gupta
,
K.
,
2013
, “
Non-Linear Dynamics, Instability and Chaos of Two Spool Aero Gas Turbine Rotor System
,”
ASME
Paper No. GTINDIA2013-3582.
17.
Chiang
,
H. W. D.
,
Hsu
,
C. N.
, and
Tu
,
S. H.
,
2004
, “
Rotor-Bearing Analysis for Turbomachinery Single- and Dual-Rotor Systems
,”
J. Propul. Power
,
20
(
6
), pp.
1096
1104
.
18.
Ferraris
,
G.
,
Maisonneuve
,
V.
, and
Lalanne
,
M.
,
1996
, “
Prediction of the Dynamic Behavior of Non-Symmetrical Coaxial Co- or Counter-Rotating Rotors
,”
J. Sound Vib.
,
195
(
4
), pp.
649
666
.
19.
Maurice
,
L.
, and
Adams
,
J. R.
,
2010
,
Rotating Machinery Vibration From Analysis to Troubleshooting
,
CRC Press
, Boca Raton, FL.
20.
Baruh
,
H.
,
1999
,
Analytical Dynamics
,
WCB/McGraw-Hill
, New York.
21.
Nasar
,
S. A.
, and
Unnewehr
,
L. E.
,
1983
,
Electromechanics and Electric Machines
,
Wiley
,
New York
.
22.
Jianxue
,
X.
,
2009
, “
Some Advances on Global Analysis of Nonlinear Systems
,”
Chaos, Solitons Fractals
,
39
(4), pp.
1839
1848
.
23.
Wolf
,
A.
,
Swift
,
J. B.
,
Swinney
,
H. L.
, and
Vastano
,
J. A.
,
1985
, “
Determining Lyapunov Exponents From a Time Series
,”
Phys. D
,
16
(
3
), pp.
285
317
.
You do not currently have access to this content.