Study on stability and bifurcation of electromagnet-track ......cycle caused by Hopf bifurcation,...
Transcript of Study on stability and bifurcation of electromagnet-track ......cycle caused by Hopf bifurcation,...
ORIGINAL PAPER
Study on stability and bifurcation of electromagnet-trackbeam coupling system for EMS maglev vehicle
Xiaohao Chen . Weihua Ma . Shihui Luo
Received: 20 October 2019 /Accepted: 25 August 2020 / Published online: 7 September 2020
� Springer Nature B.V. 2020
Abstract The stability and bifurcation behavior of
the electromagnet-track beam coupling system of the
electromagnetic suspension maglev vehicle are stud-
ied by the theoretical and numerical analyses. The
stability domain of three key dynamics parameters of
the track beam as well as the Lyapunov coefficient at
the degenerated equilibrium point is calculated. The
topological structure of the solution to the coupling
system near the Bautin bifurcation point is determined.
The results show that in the engineering practice, the
intermediate frequency should be avoided in the
natural frequency of the track beam; when a low
frequency is taken, it should be reduced as much as
possible; when a high frequency is taken, it should be
increased as much as possible and the damping ratio
should also be increased, so that the system can remain
stable while a lighter-weight track beam is used,
thereby reducing the engineering costs. Besides, when
the parameters are near the Bautin bifurcation point,
the system will have complex dynamics behaviors.
Within a certain parameter range, multiple stable and
unstable limit cycles exist simultaneously, so that the
system tends to have different stable solutions under
different initial disturbances.
Keywords Maglev vehicle � Stability analysis �Hopf bifurcation � Bautin bifurcation � Limit cycle
1 Introduction
EMS (electromagnetic suspension) maglev vehicles
rely on electromagnetic levitation to avoid mechanical
contact in traditional wheel-rail trains, thus enjoying
the advantages of being smooth and comfortable, with
low noise, small turning radius and strong climbing
ability [6, 14]. It also has broad application and
development prospects in the intercity highway and
low-speed urban traffic system. With electromagnet
suspended under the track beam by electromagnetic
attraction, the stable levitation of the vehicle is
realized by the active control of the electromagnetic
coil current [21]. However, the variation of the
electromagnetic force acting simultaneously on the
flexible track beam will cause the vibration, which
then changes the size of the air gap between the
electromagnet and the track, thus in turn affecting the
variation of the electromagnetic force, thereby form-
ing the electromagnet-track beam coupled vibration
system. If the parameters of the vehicle system do not
match the track beam system, coupled self-excited
vibrations may occur between the electromagnet and
the track beam, affecting the stable levitation of the
vehicle. The EMS maglev transportation system often
X. Chen � W. Ma (&) � S. LuoTraction Power State Key Laboratory, Southwest Jiaotong
University, Chengdu, People’s Republic of China
e-mail: [email protected]
X. Chen
e-mail: [email protected]
123
Nonlinear Dyn (2020) 101:2181–2193
https://doi.org/10.1007/s11071-020-05917-8(0123456789().,-volV)( 0123456789().,-volV)
uses an elevated mode for the track. The flexibility of
the track beam, which has a significant impact on the
levitation stability [13, 18], is the main cause of the
self-excited vibration of the electromagnet-track beam
coupling, especially during stationary or low-speed
operation [1, 8].
In view of the vibration problem of the electro-
magnet-track beam coupling system of maglev vehi-
cles, some scholars studied from the perspective of
levitation controller. Liang [9] expounded the charac-
teristics of electromagnetic interaction between the
electromagnet and the track beam, and studied the
influence of the control system parameters on the
performance of levitation system. Zou [23] concluded
that the system would have the same homoclinic
bifurcation, Hopf bifurcation and chaos under differ-
ent parameters, which was the root cause of the self-
excited vibration of the coupling system. Wang
[11, 12] focused on the time delay of the control
system, pointing out that the time delay parameter can
not only suppress the resonance response, but also
control the generation of chaos. Zhang [17] studied the
amplitude variation of the periodic solution to the
coupling system when self-excited vibration occurred,
using the perturbation method to calculate the limit
cycle caused by Hopf bifurcation, and verified its
effectiveness by numerical simulation. Li [7] used the
Nyquist stability criterion, the Routh table and the root
locus diagram to obtain the stability conditions of the
electromagnet-track beam coupling system under
static levitation. Other scholars have concentrated on
the impact of the mechanical structure of the vehicle
on the coupled vibration. Zhao [19, 20] equated the
electromagnetic force with linear spring-damping
force, established a high-speed maglev vehicle-track
beam coupled vibration model, and simulated its
vibration characteristics in high-speed operation. Kim,
Han et al. [3, 4] established a three-dimensional
vehicle-track beam model for coupling numerical
calculation, and compared the calculation results with
the experimental results to prove the validity of the
model.
As the coupled self-excited vibration problem has a
great impact on engineering applications, it is of great
practical significance to study the stability of electro-
magnet-track beam coupling system of EMS maglev
vehicles. In the field of the traditional railway, many
researchers used the Hurwitz determinant to determine
the Hopf bifurcation point of the railway wheelset
[10, 22] and the central manifold reduction method to
reduce the dimensionality of the system, then giving
the stability or bifurcation form of the system [16], but
few researchers used this method to study the maglev
system.
Due to the diversified structural modes of the EMS
maglev vehicle, the parameters of the track beam and
the vehicle vary a lot in different modes. In order to
obtain general conclusions, this paper establishes the
minimum coupling system model by only considering
the flexible track beam and the vertical motion of the
single electromagnet levitated underneath. The non-
linearity of the model is mainly derived from the
inverse square characteristics of the electromagnetic
interaction. This paper is aimed to figure out the
calculation method of the stability domain of the
electromagnet-track beam coupling system, and deter-
mine the stability of the degenerated equilibrium point
and the form of Hopf bifurcation by using the
Lyapunov coefficient when it has valid critical
parameters in the system.
2 System model and differential equation
of motion
The electromagnet-track beam coupling system is the
basic unit of the maglev vehicle levitation system. The
dynamics model is shown in Fig. 1. The electromag-
net of mass m is levitated below the track beam.
It is proved by engineering practice that almost all
the coupled vibration problems are caused by the first-
order vibration instability of the track beam [15].
Therefore, when examining the coupled vibration
problem between the electromagnet and the track
beam, it is reasonable to consider that the track beam is
Fig. 1 System dynamics model
123
2182 X. Chen et al.
equivalent to a single-degree-of-freedom vibrating
body with mass M, support stiffness k and damping c.
The coupling system has two mechanical degrees of
freedom, namely the vertical displacement of the
electromagnet z and the deflection of the track beam at
the levitation position w. The electromagnetic force
between the electromagnet and the track beam fm is a
function of the electromagnet coil current i and the
levitation gap d. The coil current i is driven by a
voltage u acting across the coil, which has a total
resistance of R. The levitation gap is d ¼ d0 þ z� w,
where d0 is the rated levitation gap. The electromag-
netic force between the electromagnet and the track
beam fm can be expressed as follows
fm i; dð Þ ¼ l0N2A
4
i
d
� �2
; ð2:1Þ
where l0 is the permeability in vacuum, N is the
number of turns of the coil, and A is the area of the
magnetic pole of the electromagnet.
According to Kirchhoff’s law, the relationship
between coil current and control voltage can be
written as
u ¼ iRþ L0 _i� kL _d; ð2:2Þ
where coefficients L0 ¼ l0N2A
2d and kL ¼ l0N2Ai
2d2.
The dynamics equation of the mechanical system
can be written in the following form by Newton’s law
of motion
m z:: ¼ mg� fm i; dð Þ;
€w ¼ r fm i; dð Þ � mg½ � � x2w� 2nx _w;
�ð2:3Þ
where g is the acceleration of gravity, r ¼ 1M is the
dynamic exchange coefficient of the electromagnetic
force on the track beam, x ¼ffiffiffikM
qis the natural
frequency of the track beam, n ¼ c2ffiffiffiffiffiMk
p is the damping
ratio of the track beam, and the zero points of z and
w are located in the static equilibrium position.
Due to the inverse square characteristic of the
electromagnetic force, the open-loop electromagnetic
levitation system is unstable, thus feedback control of
the control voltage u is required. The levitation system
of EMS maglev vehicle mostly uses double-loop
feedback control to adjust the levitation gap. The
levitation electromagnet is equipped with gap sensor,
acceleration sensor and current sensor, which can
monitor the levitation gap d, the vertical acceleration
of the electromagnet €z and the coil current I, respec-
tively. By the feedback on the levitation gap d and theintegrated vertical acceleration of the electromagnet €z,
that is, the vertical velocity of the electromagnet _z, thetarget levitation current is obtained
ie ¼ kP d� d0ð Þ þ kD _zþ i0; ð2:4Þ
where kP and kD are the state feedback coefficients, i0
is the stable levitation current, and i0 ¼ 2d0N
ffiffiffiffiffiffimgl0A
q.
To speed up the current tracking speed, current loop
is used to have feedback control on the voltage across
the coil
u ¼ ke ie � ið Þ þ iR; ð2:5Þ
where ke is the current feedback coefficient.
Taking Eqs. (2.4) and (2.5) into Eq. (2.2), the
differential equation of the circuit can be obtained
L0 _i� kL _d ¼ ke kP d� d0ð Þ þ kD _zþ i0 � i½ �: ð2:6Þ
In conclusion, the motions of the electromagnet and
the track beam are represented by Eq. (2.3). Current in
the coil is represented by Eq. (2.6). The ordinary
differential equations describing electromagnet-track
beam coupling system are obtained. To convert
equations into standard forms, by transforming
x1; x2; x3; x4; x5½ � ¼ z; _z;w; _w; i� i0½ � and expanding
d, L0 and kL, the system equations can be given by
_x1 ¼ x2;
_x2 ¼ � l0N2A
4m
x5 þ i0x1 � x3 þ d0
� �2
þg;
_x3 ¼ x4;
_x4 ¼ rl0N
2A
4
x5 þ i0x1 � x3 þ d0
� �2
�mg
" #� x2x3 � 2nxx4;
_x5 ¼2keðx1 � x3 þ d0Þ
l0N2AkP x1 � x3ð Þ þ kDx2 � x5½ � þ x5 þ i0
x1 � x3 þ d0x2 � x4ð Þ:
8>>>>>>>>>>><>>>>>>>>>>>:
ð2:7Þ
3 Analysis on stability and bifurcation
To study the influence of the track beam parameters r,x and n on the stability of the coupling system, let a ¼,
then (2.6) can be written into the following state space
for
_x ¼ J að Þxþ f x; að Þ; ð3:1Þ
where J að Þ is the Jacobian matrix of the system,
_x ¼ _x1; _x2; _x3; _x4; _x5½ �T, x ¼ x1; x2; x3; x4; x5½ �T and
123
Study on stability and bifurcation of electromagnet-track 2183
f x; að Þ ¼ 0; f2 x; að Þ; 0; f4 x; að Þ; f5 x; að Þ½ �T.Then the stability of the system at the equilibrium
point is examined. Obviously, x ¼ 0 is a steady-state
solution of the system, and the Jacobian matrix at the
equilibrium point is given by
J að Þ ¼
0 1 0 0 02g
d00 �2g
d00 �N
d0
ffiffiffiffiffiffiffiffiffiffil0Agm
r
0 0 0 1 0
�2mgrd0
02mgrd0
�x2 �2nxNrd0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffil0Amg
p2d0kekPl0N2A
2d0kekDl0N2A
þ 2
N
ffiffiffiffiffiffiffiffimg
l0A
r�2d0kekP
l0N2A� 2
N
ffiffiffiffiffiffiffiffimg
l0A
r� 2d0kel0N2A
26666666664
37777777775:
ð3:2Þ
The characteristic equation of (3.2) can be written
in the following alternative form
k5 þ a1 að Þk4 þ a2 að Þk3 þ a3 að Þk2
þ a4 að Þkþ a5 að Þ ¼ 0;ð3:3Þ
Where
a1 að Þ ¼ 2d0kel0N2A
þ 2nx;
a2 að Þ ¼ 2kekDN
ffiffiffiffiffiffiffiffiffiffiffig
ml0A
rþ 4ked0nx
l0N2Aþ x2;
a3 að Þ ¼ 2kekPN
ffiffiffiffiffiffiffiffiffiffiffig
ml0A
r� 4gkel0N2A
þ 2kekPN
ffiffiffiffiffiffiffiffimg
l0A
r� 4kemg
l0N2A
� �r
þ 4kekDnxN
ffiffiffiffiffiffiffiffiffiffiffig
ml0A
rþ 2ked0x2
l0N2A
a4 að Þ ¼ 4kekPN
ffiffiffiffiffiffiffiffiffiffiffig
ml0A
r� 8gkel0N2A
� �nxþ 4kekDx2
N
ffiffiffiffiffiffiffiffiffiffiffig
ml0A
r
a5 að Þ ¼ 2kekPN
ffiffiffiffiffiffiffiffiffiffiffig
ml0A
r� 4gkel0N2A
� �x2
Let Ta ¼ span r;x; nf g be the space of the track
beam parameters r;x and n, and set S has all the entireelements in the parameter space Ta that allow all the
eigenvalues of Eq. (3.3) to have negative real parts.
According to the first method of Lyapunov, if all the
eigenvalues of Eq. (3.3) have negative real parts, the
equilibrium point of the system (3.1) is stable. The
elements in the set S make the system stable at the
equilibrium point.
In order to study the Hopf bifurcation of the system
(3.1), it is necessary to find a condition that the
Eq. (3.3) has a pair of pure imaginary eigenvalue and
the other eigenvalues have negative real parts. If the
transversality condition is not zero at the same time,
the Hopf bifurcation occurs at this point. Since there is
not only one parameter being discussed, whether the
transversality condition is satisfied is determined by
the changing directions of the parameters, which will
be discussed by the numerical analysis in Sect. 4.
Here, the following theorem is applied to Eq. (3.3).
Theorem 3.1 [15]. Let the real coefficient algebraic
equation
kn þ a1kn�1 þ � � � þ an�1kþ an ¼ 0; ð3:4Þ
the necessary and sufficient condition for having a
pair of pure imaginary eigenvalue and the remaining
n - 2 eigenvalues have negative real parts is
ai [ 0 i ¼ 1; 2; . . .nð Þ;Di [ 0 i ¼ n� 3; n� 5; . . .ð Þ;Dn�1 ¼ 0;
where Di is the ith order principal minor of the
Hurwitz determinant of Eq. (3.4).
By applying Theorem (3.1) to Eq. (3.3), the con-
dition that there is a pair of pure imaginary eigenvalue
and the remaining eigenvalues have negative real parts
is
ai [ 0 i ¼ 1; 2; 3; 4; 5ð Þ;
D2 ¼a1 1
a3 a2
�������� ¼ a1a2 � a3 [ 0;
D4 ¼
a1 1
a3 a1
0 0
a1 1
a5 a4
0 0
a3 a2
a5 a4
���������
���������¼ a1a2a3a4 � a23a4 � a21a
24
� a1a22a5 þ a2a3a5 þ 2a1a4a5 � a25 ¼ 0:
ð3:5Þ
Let the set C be the entire elements satisfying the
condition (3.5) in the parameter space Ta. The
elements in set C put the system in a critical state
with a degenerated equilibrium point. In order to
determine the stability of this point, it is necessary to
analyze the nonlinear part of the system. System (3.1)
is a 5-dimensional system. In order to examine its
stability and bifurcation, the central flow reduction
method is usually used to simplify complex high-
dimensional system into corresponding planar system,
and then the paradigm theory and Poincare canonical
theory are applied to determine the form of bifurcation
that may occur in a degenerated system [2]. However,
these methods require multiple transformations of the
base of the space, which are very complicated and
difficult to program. Therefore, the method based on
Fredholm’s theorem is used to ‘‘project’’ the system
123
2184 X. Chen et al.
into the critical feature space and its complement
space to obtain its restricted version of the general
normalization equation. According to this method, the
simplification and normalization of the central man-
ifold are performed simultaneously. The obtained
formula contains only the Jacobian matrix and its
transposed critical eigenvectors, as well as the Taylor
series expansion according to the original base at the
degenerated equilibrium point, greatly reducing the
complexity of symbol calculation and numerical
calculation [5].
Let 8ac 2 C and J acð Þ have a pair of pure
imaginary eigenvalues �ixc while the remaining
eigenvalues have negative real parts, then the nonlin-
ear part of the system fc xð Þ ¼ f x; acð Þ ¼ Oðjjxjj2Þ is
smooth enough near the equilibrium point. Taylor
series expansion at the equilibrium point is
f c xð Þ ¼ 1
2B x; xð Þ þ 1
6C x; x; xð Þ þ Oðjjxjj4Þ; ð3:6Þ
where B n; gð Þ and C w; g; fð Þ are symmetric multi-
linear vector functions of w; g; f 2 C5, which can be
given by
Bi w; gð Þ ¼X5j;k¼1
o2fi x; acð Þoxjoxk
jx¼0wjgk; i ¼ 1; . . .; 5;
ð3:7Þ
Ci w; g; fð Þ ¼X5j;k;l¼1
o3fi x; acð Þoxjoxkoxl
jx¼0wjgkfl; i ¼ 1; . . .; 5:
ð3:8Þ
Let q 2 C5 be the complex eigenvector correspond-
ing to the pure imaginary eigenvalue �ixc,
J acð Þq ¼ ixcq; J acð Þ�q ¼ �ixc�q; ð3:9Þ
and introduce adjoint eigenvector p 2 C5,which not
only has the following properties
J acð ÞTp ¼ �ixcp; J acð ÞT �p ¼ ixc�p; ð3:10Þ
but also satisfy the normalization condition p; q ¼ 1,
where p; q ¼P5i¼1
�piqi 2 C is the standard inner product
in C5.
The eigenvalue subspace of J acð Þ corresponding toq and �q is Tc ¼ span Re qcð Þ; Im qcð Þf g. Ts is the
characteristic subspace corresponding to other
eigenvalues of J acð Þ, then Ts ¼ fyjy 2 R5; p; y ¼ 0g,and the state space x 2 R5 can be decomposed into
x ¼ zqþ �z�qþ y; zqþ �z�q 2 Tc; y 2 Ts; ð3:11Þ
and
z ¼ p; x;y ¼ x� p; xq� �p; xq:
�ð3:12Þ
In the (z, y) coordinates, system (3.1) becomes
_z ¼ ixczþ hp; f c zqþ �z�qþ yð Þi;_y ¼ J acð Þyþ f c zqþ �z�qþ yð Þ � hp; f c zqþ �z�qþ yð Þiq��p; f c zqþ �z�qþ yð Þ�q:
8<:
ð3:13Þ
Integrating (3.6) into (3.13), the system’s restricted
version of the general normalization equation is obtained
based on the central manifold reduction method and
Poincare canonical theory, and then the expression of the
first Lyapunov coefficient of the system is obtained. There
is a detailed derivation process in Ref. [5]. The first
Lyapunov coefficient is expressed as
l1 acð Þ ¼ 1
xcRec1; ð3:14Þ
where
c1 ¼1
2p;C q; q; �qð Þ þ B �q; 2ixcI � J acð Þð Þ�1B q; qð Þ
� �D
�2B q; J acð Þ�1B q; �qð Þ� �E
If l1(ac)\ 0, the degenerated equilibrium point is
stable. If the changes of the system parameters satisfy
the transversality condition at the same time, super-
critical Hopf bifurcation occurs.When the equilibrium
point loses stability as the parameter value changes, a
stable limit cycle is split from the equilibrium point. If
l1(ac)[ 0, the degenerated equilibrium point is
unstable, and if the transversality condition is satisfied,
subcritical Hopf bifurcation occurs in the system.
When the equilibrium point loses stability as the
parameter value changes, an unstable limit cycle
merges with the equilibrium point. However, if
l1(ac) = 0, Bautin bifurcation may occur. To investi-
gate the stability of the equilibrium point and the
topological structure near the bifurcation point, further
analysis on the higher-order parts of the nonlinear part
of the system is required.
Extend Eq. (3.6) into
123
Study on stability and bifurcation of electromagnet-track 2185
f c xð Þ ¼ 1
2!B x; xð Þ þ 1
3!C x; x; xð Þ þ 1
4!D x; x; x; xð Þ
þ 1
5!E x; x; x; x; xð Þ þ Oðjjxjj6Þ
;
ð3:15Þ
where
Di w; g; f; mð Þ ¼X5
j;k;l;s¼1
o4fi x; acð Þoxjoxkoxloxs
jx¼0wjgkflms;
i ¼ 1; . . .; 5;
ð3:16Þ
Ei w; g; f; m; tð Þ ¼X5
j;k;l;s;r¼1
o5f i x; acð Þoxjoxkoxloxsoxr
jx¼0wjgkflmstr
i ¼ 1; . . .; 5:
ð3:17Þ
According to the derivation process of the first
Lyapunov coefficient l1 acð Þ, the expression of the
second Lyapunov coefficient l2 acð Þ can be obtained.
Similarly, the detailed derivation process is given in
Ref. [5].
l2 acð Þ ¼ 1
12xcRe½ p;E q; q; q; �q; �qð Þ þ D q; q; q; �h20
�þ 3D q; �q; �q; h20ð Þ þ 6D q; q; �q; h11ð Þ þ C �q; �q; h30ð Þþ 3C q; q; �h21
þ 6C q; �q; h21ð Þ þ 3C q; �h20; h20
þ 6C q; h11; h11ð Þ þ 6C �q; h20; h11ð Þ þ 2B �q; h31ð Þþ 3B q; h22ð Þ þ B �h20; h30
þ 3B �h21; h20
þ 6B h11; h21ð Þi�;
ð3:18Þ
where
h20 ¼ 2ixcI � J acð Þð Þ�1B q; qð Þ;h11 ¼ �J acð Þ�1B q; �qð Þ;h30 ¼ 3ixcI � J acð Þð Þ�1 C q; q; qð Þ þ 3B q; h20ð Þ½ �;h31 ¼ 2ixcI � J acð Þð Þ�1½D q; q; q; �qð Þ þ 3C q; q; h11ð Þ
þ 3C q; �q; h20ð Þ þ 3B h20; h11ð Þ þ B �q; h30ð Þþ 3B q; h21ð Þ � 6c1h20�;
h22 ¼ �J acð Þ�1½D q; q; �q; �qð Þ þ 4C q; �q; h11ð Þþ C �q; �q; h20ð Þ þ C q; q; �h20
� þ 2B h11; h11ð Þ
þ 2B q; �h21
þ 2B �q; h21ð Þ þ B �h20;h20
�:
At this time, the stability of the system at the
equilibrium point and the local topological structure
near the bifurcation point can be determined by the
plus/minus sign of the second Lyapunov coefficient
l2 acð Þ.Let the set U ¼ faja 2 Ta&a 62 S
SCg and let it be
the complement of the parameter space Ta for the
union of sets S and C. Every element in the set
U makes at least one eigenvalue of Eq. (3.3) with a
positive real part, making the system unstable at the
equilibrium point.
The parameter space Ta is divided into three
subsets. If the track beam parameter a 2 S, the system
is stable at the equilibrium point. If the track beam
parameter a 2 U, the system is unstable at the
equilibrium point. However, if a 2 C, it is compli-
cated: if the first Lyapunov coefficient l1 að Þ\0, the
system is stable at the equilibrium point and super-
critical Hopf bifurcation occurs if the transversality
condition is satisfied; if l1 að Þ[ 0, the system is
unstable at the equilibrium point and subcritical Hopf
bifurcation occurs if the transversality condition is
satisfied; if l1 að Þ ¼ 0, Bautin bifurcation occurs, and
the stability at the equilibrium point and the topolog-
ical structure near the bifurcation point are determined
by the plus/minus sign of the second Lyapunov
coefficient l2 að Þ.
4 Numerical analysis
In this section, analysis on an electromagnet-track
beam coupling system is taken as an example to
demonstrate the conclusions obtained in Sect. 3, and
the stability domain and the bifurcation diagrams of
the equilibrium point under different track beam
parameters are drawn by using the parameters
described in the appendix.
4.1 Stability domain of the equilibrium point
Since the values of all parameters are known, it is easy
to calculate the distribution of the sets S,C andU in the
parameter space Ta according to Theorem (3.1), as
shown in Fig. 2. In this figure, the blue area is the set S,
the uncolored area is the set U, and the curved yellow
surface is the set C.
Figure 3 shows the curves composed of the
elements of the set C in the space span r;xf g when
n takes different values. The elements on the curve all
belong to the set C, so that Eq. (3.3) has a pair of pure
123
2186 X. Chen et al.
imaginary eigenvalue and the remaining eigenvalues
have negative real parts. At this time, the system has a
degenerated equilibrium point, and the first Lyapunov
coefficient corresponding to the elements on the curve
calculated according to Eq. (3.14) is indicated by the
solid line when l1\0 and indicated by dotted lines
when l1 [ 0, respectively. The elements of the solid
line make the degenerated equilibrium point stable,
and the supercritical Hopf bifurcation occurs when the
parameters cross the curve. The elements of the dotted
line make the degenerated equilibrium point unstable,
and the subcritical Hopf bifurcation occurs when the
parameters cross the curve. If the parameters does not
cross the curve but is tangent to the curve, the Hopf
bifurcation does not occur at this time because the
transversality condition is not satisfied.
The elements below the curve belong to the set S, so
that all eigenvalues of Eq. (3.3) have negative real
parts and the system is stable at the equilibrium point.
The elements above the curve belong to the set U, so
that there is at least one eigenvalue of the Eq. (3.3)
with a positive real part, and the system is unstable at
the equilibrium point.
It can be seen from Fig. 3 that the curve formed by
the elements of the set C has a significant inflection
point around x ¼ 150 rad/s. In the low frequency part
before the inflection point, the critical value rdecreases with the increase of x, that is, before the
inflection point, as the natural frequency of the track
beam increases, a larger equivalent mass of track beam
is required to stabilize the system. When the natural
frequency is smaller than that of the inflection point,
the stability domain varies little with the damping ratio
of the track beam n. In the high frequency part after theinflection point, the critical value r increases with the
increase of x, that is, after the inflection point, as the
natural frequency increases, a lower equivalent mass
of track beam can also stabilize the system. When the
natural frequency is larger than that of the inflection
point, the stability domain changes significantly with
the change of n. Therefore, increasing n can signifi-
cantly increase the stability domain of the system.
When the natural frequency x of the track beam is
close to that of the inflection point, the value range of rfor the system to be stabilized is very small, so that a
considerably large equivalent mass of the track beam
is required for the stability.
It can be seen from the analysis above that the range
of the natural frequency of the track beam can be
roughly divided into three parts: low frequency
(before the inflection point), intermediate frequency
(near the inflection point), and high frequency (after
the inflection point). In engineering practice, the
intermediate frequency (near the inflection point)
should be avoided in the natural frequency of the
track beam; because in this range, the stability of the
equilibrium point can be ensured only when the track
beam is considerably heavy, which greatly increases
the engineering cost. When a low frequency (before
the inflection point) is taken as the natural frequency of
the track beam, it should be reduced as much as
possible, which can help keep the system stable while
a lighter-weight track beam is chosen. Similarly, when
a high frequency (after the inflection point) is taken,
both the damping ratio of the track beam and the
Fig. 2 Distribution of the sets S,C andU in the parameter space
Ta
Fig. 3 Curves composed of the elements of the set C in the
space span r;xf g when n takes different values
123
Study on stability and bifurcation of electromagnet-track 2187
natural frequency should be increased as much as
possible, since the damping ratio of the track beam nwill greatly affect the performance of the system. In
this case, choosing a lighter-weight track beam can
also keep the system stable, which reduces engineer-
ing costs.
4.2 Bifurcation analysis
The bifurcation analysis of the system is conducted by
selecting the comparatively more typical track beam
damping ratio: let n ¼ 0:5%. Figure 4 is the stability
domain diagram of the system in the parameter space
span r;xf g, and the first Lyapunov coefficient at the
edge of the stability domain.
The points in the shaded part of the figure are the
elements in the set S. When the parameters are in the
shaded part, the system is stable at the equilibrium
point. The points of the blue curve are the elements in
the set C. When the parameters are on the blue curve,
the equilibrium point is degenerated. The first Lya-
punov coefficient indicated by the red curve shows
that when x\64:3 rad=s or x[ 147:7 rad=s, the first
Lyapunov coefficient of the degenerated equilibrium
point is l1\0, the equilibrium point is stable, and
supercritical Hopf bifurcation occurs when the param-
eters cross the curve; when 64:3 rad=s\x\147:7 rad=s, the first Lyapunov coefficient l1 [ 0,
the equilibrium point is unstable, and the subcritical
Hopf bifurcation occurs when the parameters cross the
curve; when x ¼ 64:3 rad=s or x ¼ 147:7 rad=s, the
first Lyapunov coefficient l1 ¼ 0, and Bautin bifurca-
tion occurs when the parameters cross the curve from
this degenerated equilibrium point.
In order to better understand the topological
structure near the bifurcation point, the dynamics
behaviors of the system near the Bautin bifurcation
point at x ¼ 64:3 rad=s are investigated. Four differ-
ent groups of parameters around the point are used for
numerical simulation and shown in Fig. 5, where
a1 � a4 are the parameter combinations: x1; r1½ � ¼62 rad/s; 5:5� 10�3 kg�1�
, x2; r2½ � ¼ 62 rad/s;½6:3� 10�3 kg�1�, x3; r3½ � ¼ 66 rad/s; 5:5�½10�3 kg�1� and x4; r4½ � ¼ 66 rad/s; 6:1� 10�3½kg�1�. A non-stiff differential equation solver (the
standard NDSolve function in MATHEMATICA) is
used for the numerical integration. Under the initial
disturbance x ¼ 0; 0; 0:003 m; 0; 0½ �T, the projections
of the phase diagram on the plane d� _d under the fourparameters combination are presented. The blue curve
is the phase trajectory, with the black point and the
gray point representing the stable and unstable equi-
librium points, respectively, and the black solid curve
and the imaginary curve representing the stable and
unstable limit cycles, respectively. What needs to be
pointed out is that the system dynamics behavior is
complicated when the parameter combination a3 is
taken, therefore, an additional set of initial distur-
bances x ¼ 0; 0; 0:0033 m; 0; 0½ �T is added for simu-
lation, with its phase trajectory represented by a red
curve.
It is worth noting that due to the physical limitations
of electromagnets, the completely smooth Eq. (2.6)
does not describe the global behavior of the system. It
is necessary to modify Eq. (2.6) as follows:
Fig. 4 Stability domain diagram of the system in the parameter
space span r;xf g and the first Lyapunov coefficient at its edge
when n ¼ 0:5%Fig. 5 Simulation parameter points selected near the Bautin
bifurcation point
123
2188 X. Chen et al.
1. Since the coil current is not reversible, the
minimum value is 0A, that is, when x5 þ i0\0,
the value of x5 in the numerical integration is �i0;
2. Since the levitation gap cannot be negative, when
x1 � x5 þ c0\0, it is considered to have a colli-
sion and the calculation is stopped accordingly.
As can be seen from Fig. 6, the results of numerical
analysis and theoretical analysis are consistent. When
the parameter takes a1, the system can return to the
equilibrium position under the initial disturbance.
When the parameters take a2 and a4, under the initialdisturbance, the system will tend to maintain a
constant vibration amplitude and finally be stabilized
in the periodic solution, and at that time, the equilib-
rium point is unstable. When the parameters take a3,the situation is much more complicated. The
stable equilibrium point, the stable limit cycle and
the unstable limit cycle coexist in the figure. When the
initial disturbance is small, the system can return to the
equilibrium position, yet when the initial disturbance
is larger, the system is inclined to the stable limit cycle
for periodic motion.
Figure 5 can be used to determine the direction of
the Hopf bifurcation transition. Obviously, as the
bifurcation parameter moves from left to right relative
to the Bautin bifurcation point, the supercritical Hopf
(a) (c)
(b) (d)
Fig. 6 The phase diagram of the system with different
parameters a The phase diagram when the parameter takes a1b The phase diagram when the parameter takes a2 c The phase
diagramwhen the parameter takes a3 d The phase diagramwhen
the parameter takes a4
123
Study on stability and bifurcation of electromagnet-track 2189
bifurcation point becomes a subcritical Hopf bifurca-
tion point. To observe this transition process, the
Newton–Raphson iterative method is applied to track
the evolution of periodic solutions. At the same time,
the Floqurt multiplier of the periodic solution and the
eigenvalues of the Jacobian matrix of the static
solution are monitored, as they provide the stability
information of the limit cycle and the equilibrium
point, respectively. Figure 7 is the two Hopf bifurca-
tion diagrams near the Bautin bifurcation point. In
Fig. 7, the dotted line indicates the unstable equilib-
rium point or the limit cycle, while the solid line
indicates the stable equilibrium point or the limit
cycle, with the symbols * and o indicating the Hopf
bifurcation point and the fold bifurcation point,
respectively. Figure 8 is a blowup of Fig. 7 near the
Hopf bifurcation point. It can be clearly seen that as xincreases and passes the Bautin bifurcation point, a
supercritical Hopf bifurcation transforms into a sub-
critical Hopf bifurcation.
As shown in Fig. 7a, when damping ratio of the
track beam n ¼ 0:5% and the natural frequency
x ¼ 62 rad=s, if r is smaller than that of the fold
bifurcation point C1, the system can always return to
the equilibrium position under the initial disturbance;
if r is larger than that of the fold bifurcation point C1
and smaller than that of the Hopf bifurcation point A1,
the system will return to the equilibrium position
under a small initial disturbance, yet tend to have a
periodic motion with a large amplitude under a large
initial disturbance; if r is larger than that of the Hopf
bifurcation point A1 and less than that of the fold
bifurcation point B1, the system tends to have a
periodic motion with a smaller amplitude under a
smaller initial disturbance, and have a periodic motion
with a larger amplitude under a larger initial distur-
bance, with equilibrium position being unstable; if r is
larger than that of the fold bifurcation point B1, the
system will always tend to have a periodic motion with
a large amplitude under the initial disturbance, and the
equilibrium position is unstable. As shown in Fig. 7b,
when damping ratio of the track beam n ¼ 0:5% and
the natural frequency x ¼ 65 rad=s, if r is smaller
than that of the fold bifurcation point B2, the system
can always return to the equilibrium position under the
initial disturbance; if r is greater than that of the fold
bifurcation point B2 and less than that of the Hopf
bifurcation point A2, the system will return to the
equilibrium position under a small initial disturbance
and tend to have a periodic motion with a large
amplitude under a large initial disturbance; if r is
greater than that of the Hopf bifurcation point A2, the
system will always tend to have a periodic motion with
a large amplitude under the initial disturbance, and the
equilibrium position is unstable.
5 Conclusion
In this paper, stability and bifurcation analyses are
studying based on the mathematical model of the
*
(a)
(b)
*
Fig. 7 Hopf bifurcation diagram near Bautin bifurcation point
a Bifurcation diagram of the system as r changes when x ¼62 rad=s bBifurcation diagram of the system as r changes whenx ¼ 65 rad=s
123
2190 X. Chen et al.
electromagnet-track beam coupling system of EMS
maglev vehicles. A simplified dynamics model of a
single electromagnet levitated below the flexible track
beam is established under some certain assumptions,
which reduces the degree of freedom of the whole
system without losing the basic characteristics, and
some general conclusions applicable to different
structural modes of EMS maglev vehicles are drawn.
Based on the Lyapunov’s first method, the space
formed by the three key parameters of the track beam
r, x and n is decomposed into a stable set S, an
unstable set U and a degenerated set C. In order to
examine the stability of the degradation condition and
study the bifurcation behavior of the system near the
degenerated equilibrium point, the first Lyapunov
coefficient at the degenerated equilibrium point is
calculated by using codimension one and codimen-
sion two bifurcation theory of the nonlinear dynamics
system. The first Lyapunov coefficient is used to
determine which kind of Hopf bifurcation occurs in
the system under some certain bifurcation parameters.
When the first Lyapunov coefficient is 0, the second
Lyapunov coefficient obtained by the higher-order
Taylor expansion of the nonlinear part of the model is
needed to understand the local topology of the
coupling system better.
The distribution of the set S, U and C in the
parameter space is obtained by numerical analysis.
The shape of the stable domain set S in the parameter
space indicates that the range of the natural frequency
of the track beam x can be roughly divided into three
parts: low frequency (before the inflection point),
intermediate frequency (near the inflection point), and
high frequency (after the inflection point). In engi-
neering practice, the intermediate frequency (near the
inflection point) should be avoided in the natural
frequencyx of the track beam. If x takes a value from
the low frequency part (before the inflection point), it
should be reduced as much as possible, which can help
keep the system stable while a lighter-weight track
beam is used. At this time, the damping ratio of the
track beam n has an insignificant influence on the
stability domain. If x takes a value from the high
frequency (after the inflection point) part, the damping
ratio n will greatly affect the performance of the
system. In this case, both n and x should be increased
as much as possible, so as to ensure that the system can
keep stable with a lighter-weight track beam, thus
reducing the engineering costs.
When the parameter n has a fixed value, the Bautinbifurcation point of the system is determined by
calculating the first Lyapunov coefficient correspond-
ing to the elements in the set C. Numerical simulations
are then carried out by selecting the combination of
parameters near the Bautin bifurcation point, and it is
found that the system has complex dynamics behav-
iors near the Bautin bifurcation point.
In order to better observe the Hopf bifurcation
transition near the Bautin bifurcation point, the
Newton–Raphson iterative method is used to track
the evolution of the periodic solution. From the change
of the Hopf bifurcation diagram, it can be found that
the supercritical Hopf bifurcation is transformed into
subcritical Hopf bifurcation as the bifurcation
(a)
(b)
Fig. 8 Blowup of Fig. 7 near the Hopf Bifurcation points
123
Study on stability and bifurcation of electromagnet-track 2191
parameter moves from left to right relative to the
Bautin bifurcation point. In the case of supercritical
Hopf bifurcation, there are two fold bifurcations,
while in the case of subcritical Hopf bifurcation, there
is one fold bifurcation. As the bifurcation parameters
change, there may be multiple limit cycles coexisting,
so that the system tends to have different stable solu-
tions under different initial disturbances.
Acknowledgements This study was funded by the National
Natural Science Foundation of China (Grant Number
51875483), the National Key R&D Program of China (Grant
Number 2016YFB1200601-A03) and Independent Topics of
National Key Laboratories (Grant Number 2020TPL-T04).
Funding This study was funded by the National Natural
Science Foundation of China (Grant Number 51875483).
Compliance with ethical standards
Conflict of interest The authors declare that they have no
conflict of interest.
Appendix: Parameters of electromagnet-track
beam coupling system in Sect. 4
Parameter Description Value
m Mass of electromagnet 550 kg
g Gravity acceleration 9.8 m/s2
l0 Permeability in vacuum 4p� 10�7 T m/A
N Coil turns of electromagnet
core
800
A Area of the magnetic pole of
the electromagnet
0.014 m2
d0 Rated levitation gap 8 mm
i0 Stable levitation current 11.07 A
kP Gap feedback coefficient 3500
kD Velocity feedback coefficient 300
ke Current feedback coefficient 25
References
1. Alberts, T.E., Oleszczuk, G., Hanasoge, A.M.: Stable levi-
tation control of magnetically suspended vehicles with
structural flexibility. In: American Control Conference
(2008)
2. Carr, J.: Applications of Centre Manifold Theory. Springer,
New York (1981)
3. Han, H.S., Yim, B.H., Lee, N.J., et al.: Effects of the
guideway’s vibrational characteristics on the dynamics of a
maglev vehicle. Veh. Syst. Dyn. 47(3), 309–324 (2009)
4. Kim, K.J., Han, J.B., Han, H.S., Yang, S.J.: Coupled
vibration analysis of maglev vehicle-guideway while
standing still or moving at low speeds. Veh. Syst. Dyn.
53(4), 587–601 (2015)
5. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory.
Springer, New York (1995)
6. Lee, H.W., Kim, K.C., Ju, L.: Review of maglev train
technologies. IEEE Trans. Magn. 42(7), 1917–1925 (2006)
7. Li, J.: The Vibration Control Technology of EMS Maglev
Vehicle-Bridge Coupled System. Doctor, National Univer-
sity of Defense Technology (2015)
8. Li, J.H., Li, J., Zhou, D.F., et al.: Self-excited vibration
problems of maglev vehicle-bridge interaction system.
J. Cent. South Univ. 21(11), 4184–4192 (2014)
9. Liang, X.: Study on MAGLEV Vehicle/Guideway Coupled
Vibration and Experiment on Test Rig for A Levitation
Stock. Doctor, Southwest Jiaotong University (2014)
10. Sedighi, H., Shirazi, K.: Bifurcation analysis in hunting
dynamical behavior in a railway bogie: using novel exact
equivalent functions for discontinuous nonlinearities. Sci.
Iran. 19, 1493–1501 (2012)
11. Wang, H., Li, J.: Sub-harmonic resonances of the non-au-
tonomous system with delayed position feedback control.
Acta Phys. Sin. 56(5), 2504–2516 (2007)
12. Wang, H., Li, J., Zhang, K.: Stability and Hopf bifurcation
of the maglev system with delayed speed feedback control.
Acta Autom. Sin. 33(8), 829–834 (2007)
13. Wang, K., Luo, S., Ma, W., et al.: Dynamic characteristics
analysis for a new-type maglev vehicle. Adv. Mech. Eng.
9(12), 1–10 (2017)
14. Yan, L.: Suggestion for selection of Maglev option for
Beijing-Shanghai high-speed line. IEEE Trans. Appl.
Supercond. 14(2), 936–939 (2004)
15. Zhang, J., Yang, Y., Zeng, J.: An algorithm criterion for
Hopf bifurcation and its application in vehicle dynamics.
Acta. Mech. Sin. 32(5), 596–605 (2000)
16. Zhang, T., Dai, H.: Bifurcation analysis of high-speed
railway wheel-set. Nonlinear Dyn. 83(3), 1511–1528 (2016)17. Zhang, L.: Research on Hopf Bifurcation and Sliding Mode
Control for Suspension System of Maglev Train. Doctor,
Hunan University (2010)
18. Zhang, M., Luo, S., Gao, C., et al.: Research on the mech-
anism of a newly developed levitation frame with mid-set
air spring. Veh. Syst. Dyn. 56(12), 1797–1816 (2018)
19. Zhao, C., Zhai, W.: Dynamic characteristics of electro-
magnetic levitation systems. J. Southwest Jiaotong Univ.
39(4), 464–468 (2004)
20. Zhao, C., Zhai, W.: Guidance mode and dynamic lateral
characteristics of low-speed maglev vehicle. China Railway
Sci. 26(6), 28–32 (2005)
21. Zhou, D., Hansen, C.H., Li, J., et al.: Review of coupled
vibration problems in EMS maglev vehicles. Int. J. Acoust.
Vib. 15(1), 10–23 (2010)
22. Zobory, I., Nhung, T.: A mathematical investigation of the
dynamics of drive-systems of railway traction vehicles
under stochastic track excitation. Period. Polytech. Transp.
Eng. 20, 101–107 (1992)
123
2192 X. Chen et al.
23. Zou, D., She, L., Zhang, Z., et al.: Maglev vehicle and
guideway coupling vibration analysis. Acta Electron. Sin.
38(9), 2071–2075 (2010)
Publisher’s Note Springer Nature remains neutral with
regard to jurisdictional claims in published maps and
institutional affiliations.
123
Study on stability and bifurcation of electromagnet-track 2193