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IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 15, NO. 3, SEPTEMBER 2005
Influence of Flux Creep on Dynamic Behavior
of Magnetic Levitation Systems
With a High-Tc Superconductor
Xiao-Jing Zheng, Xiao-Fan Gou, and You-He Zhou
Abstract—Significant flux creep may be generated in some
superconductors with weak pinning, which could yield
highsuperconan influence on the dynamic behavior of a highductor-magnet levitation system. To investigate this influence, this
article presents a numerical analysis of dynamic features of the
levitation generated by an interaction between a high- superconductor (HTSC) and a permanent magnet (PM) after the flux creep
in the superconductor is taken into account in a macro-model of
superconductivity. The influence is comprehensively displayed by
comparing the predictions of dynamic responses of such systems
in which the flux creep in the superconductor is and is not taken
into account. The obtained results show that whether or not the
flux creep results in a noticeable influence to the levitation of
superconductor-magnet systems is mainly dependent upon properties of superconductivity and applied excitation, e.g., critical
current density of superconductors, and amplitude and frequency
of external excitations. When the critical current density is less
108 A/m2 , and the system is subjected to a periodic
than 4 5
excitation, the influence of flux creep should be taken into account
in the theoretical analysis.
Index Terms—Dynamic behavior, flux creep, highductor-magnet levitation system, numerical analysis.
supercon-
I. INTRODUCTION
H
superconductivity has been demonstrated
IGHtremendous potential for several fascinating applications
such as magnetic levitations, linear drives for transportation,
and magnetic flywheels for storing energy, etc. [1]–[3]. For
such applications, some experiments and simulations exhibited
that the magnetic levitation force had somewhat time-dependent relaxation, and the vibration center of levitated body had
drifted downward when a time-varying excitation was applied
to the system [4]–[11]. Such effects are directly related to the
safe operation and design of the systems.
When a permanent magnet (PM) moves above a superconductor, or/and vice versa, the magnetic field of the magnet penetrates the superconductor, creating shielding current or fluxiods
Manuscript received June 8, 2004; revised February 4, 2005. This paper was
recommended by Associate Editor J. Hull. This work was supported by the
Key Fund of the National Natural Science Foundation of China under Grant
10132010, by the Fund of the National Natural Science Foundation of China for
Outstanding Young Researchers under Grant 10025208, by the Fund of Pre-Research for Key Basic Researches of the Ministry of Science and Technology
of China, by the Fund of Natural Science Foundation of China under Grant
10472038, and by the Fund of Excellent Teachers in Universities of the Ministry of Education of China.
The authors are with the Department of Mechanics, School of Physical Sciences and Technology, Lanzhou University, Lanzhou, Gansu 730000, China
(e-mail: xjzheng@lzu.edu.cn; xfgou@163.com; zhouyh@lzu.edu.cn).
Digital Object Identifier 10.1109/TASC.2005.850537
in it. In this case, magnetic force is generated through interaction
between the shielding current and the applied magnetic field.
High- superconductors show flux flow and creep phenomena
that make the magnetic force time dependent. The flux flow and
creep phenomena are due to the motion of fluxiods which induce the electromotive force. According to the micro-view of
type-II superconductivity [12], [13], the fluxiods that are trapped
in the pinning potential barrier can escape from it by thermal activation. When the electric current flows in the superconductors,
the Lorentz force acts on the floxiods. Hence, the fluxiods can
moves more easily in the direction of the Lorentz force. This is
called the flux creep. When the Lorentz force is stronger than
the pinning force, the pinning effect on the fluxiods becomes
insignificant. In this case, the Lorentz force dominates the flux
flow. It has been found that the Joule loss caused by the flux flow
and creep makes the magnetic force decay in time.
In order to perform an analysis of macro-electromagnetism
of a high- superconductor (HTSC), the classical theory of
electromagnetic fields for normal conductors should be revised
such that the superconductivity phenomena can be described.
Especially, the Ohm’s law for a normal conductor has to be replaced by a nonlinear relation for a superconductor. Bean [14],
[23] proposed a macro-model of type-II superconductors, which
is referred to as the critical state model [15]. In this model,
the shielding current is generated only where a nonzero electric field exists, and the shielding current density is equal to its
critical value which is independent of magnitude of the electric
field. Otherwise, the shielding current is equal to zero where
there exists no electric field in the superconductor. Although
this model is simplest today, it is inherently nonlinear in the
macro-theory of electromagnetism of superconductors such that
the calculation of electromagnetic fields consumes much more
time to a superconductor than to a normal conductor. In order
to reflect those micro-phenomena (e.g., the flux creep and the
flux flow) of superconductivity as more as possible, some other
macro-models, such as the flux flow and creep model [4], [5]
and the flux flow model [6], [7], were proposed in literature.
Fig. 1 schematically shows these three constitutive relations,
from which one finds that both the flux flow and creep phenomena are considered only in the flux flow and creep model,
whereas the flux creep phenomenon is neglected in the flux flow
model.
Having measured the dynamic magnetic force of a HTSC,
Yoshida et al. [4], [5] gave some numerical analyzes based on
the flux flow and creep model. Their numerical results displayed
a good agreement with measurement data of magnetic force.
1051-8223/$20.00 © 2005 IEEE
ZHENG et al.: INFLUENCE OF FLUX CREEP ON DYNAMIC BEHAVIOR OF MAGNETIC LEVITATION SYSTEMS
3857
Fig. 2. Schematic drawing of a high-T superconductor-PM levitation system
subjected to an external excitation.
Fig. 1. Diagram of macro-models of high-T superconductors employed
recently. (a) Critical state model, (b) flux flow model, and (c) flux flow and
creep model.
Riise et al. [16] measured a time-varying magnetic force between a PM and a HTSC when the HTSC was subjected to an
impulse magnetic field, which shows that the magnetic force
relaxes logarithmically with time. They presumed that such relaxation could be generated from the thermally activated flux
creep in the superconductors. In addition, some other dynamic
features such as change of levitation height or drift of vibration
center of levitation body were also observed and measured in the
superconductor-PM levitation systems. For example, Terentiev
et al. [11] experimentally measured the vertical drift of levitated
body induced by a variable magnetic field and a vibration of
PM. They gave it an explanation that unpinning of flux lines by
the alternating field is a possible reason why the drift yields. In
HTSC magnetic bearings, the flux creep is considered as one
of the main reasons for the time decay of the levitation properties, namely, the magnetic forces and levitation heights [17].
Based on the flux flow model, an evaluation of dynamic characteristic of a PM freely levitated above an excited HTSC was conducted by Sugiura et al. [6]. Their numerical results confirmed
the downward drift of vibration center of the levitated PM. Hull
[3] gave a review of advanced development of the research in
this area. From it, one sees that almost all investigations of simulation of high- superconductor-PM levitation systems were
based either on the critical state model or on the flux flow model,
while little attention was paid on the influence of flux creep in
the levitation system [18]. As the superconductors with higher
critical or operating temperature are discovered, the phenomenon of “giant flux creep” [19] was observed. In such situation,
it is hence necessary to thoroughly understand the characteristic
of the systems as a whole when the flux creep in the high- superconductors generates a noticeable influence on dynamic features of the superconductor-PM levitation systems.
This paper will display some simulation results of dynamic
responses of the high- superconductor-PM levitation systems
to illustrate the influence of flux creep of superconductors on
dynamic behavior of the systems. After the numerical code of
solving the nonlinear and coupled problem is examined by some
existing experiments of free and forced vibrations of levitated
bodies moving in the vertically levitated direction, the effect is
quantitatively exhibited by comparing the numerical predictions
on the basis of the flux flow and creep model with those on the
flux flow model to a superconductor-PM levitation system.
II. BASIC EQUATIONS
In this section, we briefly introduce those essential equations
of the superconductor- PM levitation system what we are taken
into account. In such system, a cylindrical PM moves coaxially
over a cylindrical high- superconductor subjected to a vertical
excitation as shown in Fig. 2. Associated with the action of the
gravity force, a magnetic force arising from the interaction between the PM and the superconductor acts on the PM, which is
equal to the magnetic force exerted on the superconductor from
the PM, i.e.,
(1)
where represents the volume of the superconductor; is the
stands for the
shielding current density; and
is the induced part of due
magnetic induction, in which
to the shielding current in the superconductor, and
is the
excitation one of from the PM. Having introduced the magby [20]
netic vector potential to the PM, we can express
(2)
where
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IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 15, NO. 3, SEPTEMBER 2005
and are the radial and circumferential coordinates, respectively; is a longitudinal coordinate away from the center of the
(see Fig. 2); is the
magnet surface for a field point
radius of the cross-section of the PM; and is the nominal current strength usually determined by the residual magnetic field
of the permanent magnet.
In the macroscopic view, the shielding current density , the
magnetic induction , and the electric field intensity should
satisfy the Maxwell equations, i.e.,
(3)
, we
In order to determine the unknown function
have to study the vertical movement of the levitated body. Here,
we consider that the superconductor is cooled down below the
critical temperature for a sufficient critical current. Then the
to a static equiPM slowly moved from an initial position
. Denote
.
librium position
From the second Newton’s law, the dynamic equation of vertical movement of the PM can be expressed as
(8)
and the initial conditions are taken into account by
and the law of electric charge conservation
(9)
(4)
where is the gradient operator; and
indicates
is the
the magnetic field vector in the superconductor, and
magnetic permeability in vacuum.
To the high- superconductor, a general form of constitutive
relation linking the electric field intensity and the current density can be written by [6]
(5)
is a pre-specific function dependent on what superHere,
conductor features are taken into account in a macro-model of
high- superconductors. To the flux flow model of the highsuperconductors, for example, we have [6]
(6)
and to the flux flow and creep model, the constitutive relation is
formulated by [4], [5]
Here,
is the mass of the permanent magnet; stands for
the air damping coefficient; and is the gravitational acceler, and are respectively the displacement, velocity,
ation;
and acceleration of the PM relative to its static equilibrium porepresents the vertical component of magnetic force
sition;
;
indicates an external excitation force; and the overhead symbol dot represents the differentiation with respect to
of displacetime variable . When an external excitation
ment is applied to the superconductor, for example, we have
.
III. NUMERICAL PROCEDURE
Until today, almost no analytical solution has been found to a
problem of macro-electromagnetism of superconductors with finite geometric scale even if they have a simple geometric shape.
In this case, we pay our attention on establishment of a numerical code to solve the nonlinear and coupled problem of the
superconductor-PM levitation systems. Here, the -method of
eddy or shielding current and the finite element method for electromagnetic fields of superconductors are employed, and some
iteration approaches are chosen to solve the nonlinearity and
coupling. For example, a direct iteration is employed to the nonlinear differential equation (8) by the form
(10)
(7)
; represents
where
is the critical current density without
the creep resistivity;
stands for the pinning potential; indithermal activation;
cates the Boltzmann constant; is the absolute temperature; and
denotes the flow resistivity.
Due to nonlinearity of each constitutive relation between
electric field and current density , the electromagnetism of
HTSCs is nonlinear too. That is to say, the governing (2)–(5)
associated with either (6) or (7) are nonlinear. Meanwhile, it is
obvious that the electromagnetic fields in the superconductor
are dependent on spanning distance (or gap) and speed of
movement between the superconductor and the PM. Denote
. The magnetic force
the spanning distance by
changes with the spanning distance and its velocity except for
the distribution of magnetic field of PM and superconductor
parameters. Thus, we can formulate this dependency by a
mathematically.
function
Here, the superscript
iteration step, and
indicates the number of
(11)
The main steps of the numerical code are briefly introduced as
follows.
Step 1) Input the iterating values of
and
, i.e.,
and
. At the first step or
, they
are taken as their initial values of (9).
Step 2) Perform calculation of electromagnetic fields at th
step.
1) T-Method: From the anisotropy of in the MPMG-processed YBCO superconductor, we can assume that an YBCO
sample comprises thin plates parallel to - plane where supercurrents between the plates are negligibly small [21]. In this
paper, for simplicity, we focus our attention on this kind of superconductors that have only components of super-currents that
ZHENG et al.: INFLUENCE OF FLUX CREEP ON DYNAMIC BEHAVIOR OF MAGNETIC LEVITATION SYSTEMS
are parallel to the
coordinate plane (i.e., - plane). In this
case, the current vector potential that satisfies
may be simplified by
. Then, one can reduce the
governing (3) and (4) of electromagnetic fields of superconductors into the form
(12)
which is the governing equation for the unknown function of
current potential at the th step. In (12),
is an effective conductivity of the superconductor following the Ohm’s law
. Applying the Helmholtz’s formula to the induced magnetic induction in the superconductor, we get an integral equaas follows [21]:
tion of the induced magnetic induction
(13)
where is the unit vector normal to the surface of the superconductor, and indicates the distance between a source point and
a field one. Substituting (13) into (12), we obtain the governing
of the form
equation for
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3) Iteration Approach: To the nonlinear (16) of electromagnetic fields, another iteration is taken as
(17)
where the subscript
represents the number of it, and
. Here, and
eration step;
are the magnitudes of
and , respectively. In addition,
the initial conductivities
at the first step of iteration with
symbol are chosen by the sufficiently big values in the whole
superconductor.
4) Numerical Integration of (17): At each step , the CrankNicolson- method is employed to perform the numerical integration of (17) such that the response
varying with time is obtained numerically.
5) Precision Condition: Once the iterated solutions
and
are gained from the solution of (17), the iterated
is got at each element. Replacing
value
by their corresponding iterated values
the iterating values
at all elements, and repeating the iteration process of
(17), we may get the numerical solution of electromagnetic
fields of the superconductor until the following condition:
(18)
(14)
or
(15)
Here,
is the normal component of
.
2) Finite-Element Method: After the boundary conditions
is taken into account to
for unknown function
the differential equation (15), we use the FEM to get numerical
solution of the equation, to which the system of algebraic equations can be compactly expressed by the following matrix form
[22]:
(16)
and
are the square matrices of coefficients and
Here,
they are respectively relative to those coefficients of terms
and
in (15);
indicates the column matrix
;
means the column matrix of
of unknown function
with respect to time variable; and
differentiation of
represents the column matrix of excitation related to the last
term of (15). It is evident that the matrix
is dependent upon
. According
the equivalent conductivity, i.e.,
to the definition of conductivity, we know that the effective
to the superconductor is piecewise dependent
conductivity
on both electric field intensity and shielding current density
. Thus, the matrix equation of (16) is nonlinearly dependent
.
on the unknown column
is a pre-given
is satisfied at all elements. Here,
precision. After that, we can obtain the magnetic force
of
th iteration.
Step 3) Numerical integration of (10). After the Newmark- method is employed to take numerical
integration of (10), we get the numerical response
and
, or the iterated solutions of
and
.
Step 4) Replace the iterating solutions
and
by the iterated solutions
and
respectively, and repeat the calculations of steps 1–3
until the following precision condition:
(19)
is held. Here,
is a prescribed tolerance.
IV. NUMERICAL RESULTS AND DISCUSSION
Based on the numerical code introduced in the previous
section, we display some results of numerical tests and case
studies to the levitation systems considered. The geometric and
material parameters used in the calculations are listed in Table I
and II, some of which are the same as those given in Yoshida
et al. [4].
The first examination of the numerical code is to simulate the
measurement data of time-varying magnetic force of the superconductor-PM levitation system [4]. In this experiment, the levitated PM attached to a beam was driven to move to the fixed superconductor from 25 mm to 0.5 mm in gap between the magnet
and the superconductor at a specified constant speed of about
15 mm/s, then stops at the gap position of 0.5 mm. The magnetic
force is measured by means of a cantilevered beam with the PM
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TABLE I
GEOMETRICAL AND PHYSICAL PARAMETERS EMPLOYED IN SIMULATION
TABLE II
GEOMETRICAL AND PHYSICAL PARAMETERS EMPLOYED IN EXAMINATION 2
Fig. 4. Comparison of simulations with experimental data to drift
displacement of vibration center of the levitation system shown in Fig. 2 when
the system is applied by a harmonic excitation with amplitude A = 50 m and
frequency f = 50 Hz. The parameters of the system are listed in Table II.
Fig. 3. Comparison of numerical simulations with experimental measurement
data of magnetic force varying with time when the levitated PM moves to the
superconductor till to a position at which the PM movement stops. Here, the
parameters of J = 1:5 2 10 A/m ; U = 92 meV, = 7:62 2 10
1m
and others listed in Table I are employed.
fixed at its free end [4]. With time being, it is found in the experiments that the magnetic force varies with time due to deflection
of the beam. Fig. 3 displays time-variation of the magnetic force
obtained from both the experimental measurement and the simulation here. From this figure, one sees that after the magnetic
force approaches to a maximum value or peak, the predictions
on the basis of the flux flow and creep model are closer to the
experimental data than those based on the flux flow model. At
the stage before the peak, almost no difference between the predictions from these two models is found, but there is a difference
between the predictions and the experimental data. In practice
experiment, we know that the accelerating and decelerating periods are needed to realize the motion of PM. Once we consider the accelerating/decelerating period of 0.01 s and 0.05 s
in the theoretical simulation with the flux flow and creep model
respectively, the predictions exhibit the same as that shown in
Fig. 3 except for a small drop of the peak. When we increase
the moving velocity of PM in the simulation for example, it is
found that the peak in the prediction moves toward left notably
and the value of the peak has a slight change so the prediction
of magnetic force before the peak increases too. In this case, we
guess that one of possible reasons for generating this difference
is that some physical parameters, e.g., the critical current density
and the flow resistivity , of HTSC employed in simulation
are not accurate since they are not accurately measured.
The second examination of the numerical code is to predict
the experimental phenomenon of downward drift of vibration
center of levitated body in the superconductor-PM levitation
system [11]. In this experiment, a harmonic excitation of vertical
, is applied. Here, the amplitude and
displacement,
m and
Hz, respecfrequency are taken as
tively. At the same time, zero deviations of initial displacement
and velocity related to the equilibrium position of the system
were set in the experiment. In the literature [11], some parameters were given by their regions rather than fixed values. Here,
we take them in their regions with some determinant values that
are listed in Table II. Fig. 4 plots the vibration center of the
levitation body varying with time, which tells us that the predictions from the flux flow and creep model are agreement with
their measurement data well, but those from the flux flow model
are not.
From the above two examinations of the numerical code to
the high- superconducting levitation systems, we find that
this code enables us to simulate the superconductor-PM electromagnetic fields and the dynamic behavior of relevant levitation systems, and that the predictions to the systems on the basis
of the flux flow and creep model are more accurate than ones
superconductors. From
from the flux flow model of highTable II, one can also find that the experimental parameters including the scale of superconductor, critical current density, and
residual field of PM are too either small or low comparing to
practical applications. Since the dynamic behavior of the levitation system is dependent on the parameters, here, we give some
case studies when the parameters are taken in the range of practical applications.
At the following simulation of dynamic behavior of the magnetic levitation, the parameters in Table I are employed if they
are not specified separately. Here, the mass of PM is 93.8 g, and
following [6], the air damping coefficient is taken as 0.5 Ns/m
ZHENG et al.: INFLUENCE OF FLUX CREEP ON DYNAMIC BEHAVIOR OF MAGNETIC LEVITATION SYSTEMS
3861
Fig. 5. Simulations of response of free vibration of the levitated PM of the
levitation system when the superconductor is dealt with the flux creep model
and the flux flow and creep model, respectively.
Fig. 7. Simulations of restoring force exerted on the levitated PM for the cases
of free vibration of the levitation system when the superconductor is dealt with
the flux flow model and the flux flow and creep model, respectively.
Fig. 6. Simulations of response of forced vibration of the levitated PM of the
levitation system when the superconductor is dealt with the flux flow model and
the flux flow and creep model, respectively (A = 0:1 mm, f = 20 Hz).
Fig. 8. Absolute error of predictions of response displacement of free vibration
between two cases of the superconductor with and without consideration of flux
creep varying with the critical current density (z = 0; z_ = 15 mm/s).
for example. To display when the influence of flux creep in the
superconductor on the levitation behavior is notable, here, we
give some simulation results of difference between their displacement responses of the levitation system when the flux creep
is and is not considered, where the initial conditions are taken
and
mm/s. Figs. 5 and 6 display the dynamic
as
responses of free and forced vibrations of the levitated PM, respectively. For the case of free vibration in Fig. 5, it is found
that there is a noticeable downward drift of vibration center in
the response after the flux creep effect is taken into account but a
small drift only in an initial short time process is predicted when
the flux creep effect is neglected. When the levitation system is
subjected to a periodic excitation, the predictions on the basis
of these two models exhibit the downward drift of vibration
center, and the drift for the case when the flux creep is taken
into account is larger than one when it is neglected. Comparing
the simulation data of drift of vibration center with the experiment ones shown in Fig. 4, we find that the vibration center
mm for free vibrapositions change very fast (about to
tion and
mm for the forced vibration within 0.8 s), which
reveals that the drift phenomenon in practice is more severe if
the critical current density is not large enough. In addition, the
drift of vibration center for the forced vibration is different from
one for the free vibration in the levitation system. According to
the knowledge of dynamics, we know that the drift of vibration
center is mainly generated from the relaxation of magnetic force
at a gap position with time due to constant of gravity force of
levitated body. Fig. 7 displays the restoring force varying with
time for the case of free vibration of the levitation system. From
it, one sees that the restoring force when the flux creep is taken
into account decays faster than that when the flux creep is not
considered. Associated with the relaxation phenomenon of magnetic force as shown in Fig. 3, it is reasonable that the drift of
vibration center for the previous case is more severe than one for
the latter case. Due to that the flux creep in the high- superconductor leads to energy dissipation and decreasing of current
density [19], hence, the influence of flux creep in the superconductor model, which leads to larger drift of vibration center, has
to be considered in the dynamic simulations of a high- superconductor-PM levitation system in general case.
and
be the displacement of the levitated PM relaLet
tive to its static equilibrium position when the flux creep effect is
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IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 15, NO. 3, SEPTEMBER 2005
are dependent on the parameters of the system and the specific
A/m in this example, we
tolerance of the error, e.g.,
can neglect the effect of flux creep in the levitation system. For
the case of forced vibration of the system when it is applied by
,
a periodic excitation of displacement, or
the numerical results exhibit that the error is dependent on the
amplitude and frequency of excitation. For the levitation system
with a specific critical current density, the numerical results display that the error decreases with both amplitude and frequency
of the excitation, and increases with time. It is obvious that the
external excitation with larger amplitude and higher frequency
may lead to faster change of the magnetic field in the superconductor. According to the electromagnetic theory of the superconductor formulated as (3) and (5), it is obvious that the faster
change of magnetic field will result in a higher current density in
the superconductor such that the influence of flux creep become
weak. With time being, the vibration amplitude of the levitated
PM decreases due to both the air damping and one resulting
from the AC loss in the superconductor, thus, the influence of
flux creep increases with time. As an example, Fig. 9(a) and (b),
respectively, illustrate these characteristics of the absolute error
varying with either the excitation amplitude when the excitation
Hz or the excitation frequency
frequency is specified by
when the excitation amplitude is taken as
mm.
V. CONCLUSION
Fig. 9. Absolute error of predictions of response displacement of forced
vibration between two cases of the superconductor with and without
consideration of flux creep varying with either (a) excitation amplitude when
f = 5 Hz or (b) excitation frequency when A = 0:1 mm (z = 0; z
_ = 0).
and is not considered in the numerical simulation, respectively.
. It is evident that
Denote their absolute error by
the error characterizes the influence of flux creep in the superconductors on the superconducting levitation systems. The
numerical predictions exhibit that the absolute error is mainly
dependent on the critical current density of the superconductor,
the amplitude and frequency of the applied excitation, etc. except for other geometric and physical parameters. For the case
of free oscillation in this case study, some characteristic curves
of the absolute error varying with the critical current density
at different instants are plotted in Fig. 8. From it, we see that
the absolute error decreases with the critical current density in
the superconductor, and increases with time. When the critical
A/m (see Fig. 8), the
current density increases over
absolute error becomes small within the whole process of response and the influence of flux creep can be neglectable. According to the knowledge of high- superconductivity, in fact,
we know that the high critical current density implies strong pinning force existing in the superconductor. In this case, the influence of flux creep on the dynamic behavior of the levitation
system becomes weak as the critical current density increases.
Once the critical current density increases over a threshold that
A numerical code is employed in this paper to simulate
dynamic responses of a high- superconductor-PM levitation
system in which the shielding current and electromagnetic
fields in the superconductor are solved by the -method and
the finite element method after the flux creep in the superconductor is taken into account. From it, some experiments of
the levitation system are predicted out. The numerical results
exhibit that the drift of vibration center of the levitation systems
is mainly dependent upon the flux creep when the shielding
current density in the superconductor is not large enough. With
increasing of the critical current density and the amplitude and
frequency of the excitation, the influence becomes weak, while
the influence increases with time.
ACKNOWLEDGMENT
The authors sincerely appreciate the reviewers for their valuable suggestions in the revisions.
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3863
Xiao-Jing Zheng received the Master degree of
engineering science from Huazhong University of
Science and Technology, Wuhan, China, in 1984 and
the Ph.D. degree from Lanzhou University, Lanzhou,
China, in 1987.
Since 1991, she has been a Professor with the
Department of Mechanics of Lanzhou University,
Gansu, China. Her research interest includes analysis
of mechanics of multi-field coupling, electro-magneto-solid mechanics, and electrification mechanism
of aeolian sand grains, etc. She has published about
140 papers and two textbooks in her research areas.
Xiao-Fan Gou received the Ph.D. degree in superconductivity from Lanzhou University, Gansu,
China, in 2004.
At present, he is a Lecturer with the Department
of Engineering Mechanics of Hohai University, Nanjing, China. His research interests have been in the
numerical analysis of electromagnetic phenomena in
HTSC, including the application of superconductors
to levitation.
You-He Zhou received the Master degree of engineering science from Huazhong University of Science and Technologu, Wuhan, China, in 1984 and
the Ph.D. degree from Lanzhou University, Gansu,
China, in 1989.
He has been a Professor with the Department of
Mechanics, Lanzhou University, Lanzhou, China
since 1996. His research interest includes dynamic
analysis, electro-magneto-solid mechanics, and
dynamic control of smart structures, etc. He has
published about 160 papers and one textbook in his
research areas.