(Received 3 June 2017; accepted 10 August 2017; published online 24 August 2017). For a maglev system, vertical and lateral displacements of the levitation ...
Modeling dynamic behavior of superconducting maglev systems under external disturbances Chen-Guang Huang, Cun Xue, Hua-Dong Yong, and You-He Zhou
Citation: Journal of Applied Physics 122, 083904 (2017); doi: 10.1063/1.4986295 View online: http://dx.doi.org/10.1063/1.4986295 View Table of Contents: http://aip.scitation.org/toc/jap/122/8 Published by the American Institute of Physics
JOURNAL OF APPLIED PHYSICS 122, 083904 (2017)
Modeling dynamic behavior of superconducting maglev systems under external disturbances Chen-Guang Huang,1,a) Cun Xue,1 Hua-Dong Yong,2 and You-He Zhou2 1
School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, People’s Republic of China 2 Key Laboratory of Mechanics on Environment and Disaster in Western China, The Ministry of Education of China, Department of Mechanics and Engineering Sciences, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China
(Received 3 June 2017; accepted 10 August 2017; published online 24 August 2017) For a maglev system, vertical and lateral displacements of the levitation body may simultaneously occur under external disturbances, which often results in changes in the levitation and guidance forces and even causes some serious malfunctions. To fully understand the effect of external disturbances on the levitation performance, in this work, we build a two-dimensional numerical model on the basis of Newton’s second law of motion and a mathematical formulation derived from magnetoquasistatic Maxwell’s equations together with a nonlinear constitutive relation between the electric field and the current density. By using this model, we present an analysis of dynamic behavior for two typical maglev systems consisting of an infinitely long superconductor and a guideway of different arrangements of infinitely long parallel permanent magnets. The results show that during the vertical movement, the levitation force is closely associated with the flux motion and the moving velocity of the superconductor. After being disturbed at the working position, the superconductor has a disturbance-induced initial velocity and then starts to periodically vibrate in both lateral and vertical directions. Meanwhile, the lateral and vertical vibration centers gradually drift along their vibration directions. The larger the initial velocity, the faster their vibration centers drift. However, the vertical drift of the vertical vibration center seems to be independent of the direction of the initial velocity. In addition, due to the lateral and vertical drifts, the equilibrium position of the superconductor in the maglev systems is not a space point but a continuous range. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4986295] I. INTRODUCTION
Due to the achieved high critical current density and improved mechanical properties for high-temperature superconducting materials in recent years, several prototypes of maglev based on bulk superconductors (SCs) levitating over a guideway of permanent magnets (PMs) have been developed, with tremendous potential in uses such as magnetic bearings, spaceship propulsion systems, and transportation systems.1–5 Since the maglev realizes stable and passive levitation without any active control, these properties can effectively reduce contact noise and energy consumption.6 In the practical application of the maglev, the levitation force and stability are two key parameters especially needed to be analyzed and understood. Many experiments have shown that when the levitation body is subjected to an external disturbance, the vertical levitation force and the lateral guidance force will change simultaneously, which may induce an undesirable modification of the levitation point.7–10 Therefore, it is necessary to thoroughly study the levitation behavior of superconducting maglev systems under external disturbances. For a maglev system, the magnetic forces on the SC result from the interaction of the induced current in the SC with the inhomogeneous magnetic field created by the a)
Author to whom correspondence should be addressed: huangcg@nwpu. edu.cn
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guideway. Because of external disturbances, the SC moves away from the initial position and then undergoes a complex magnetic field variation, which will lead to the redistribution of the induced current. To understand the levitation performance change induced by external disturbances, many theoretical works have been carried out to analyze the maglev system.11–17 In these efforts, both the SC and the PM are assumed to be infinitely long and the SC is usually vertically or laterally moved on the guideway. Some studies adopt the frozen image model in terms of dipole approximation.11,12 Other authors use minimization of a related functional based on the critical state model.13–16 These models allow a realistic calculation of the levitation and guidance forces considering the important effects arising from the configuration of the guideway and hysteresis in SC. From experiments, it is well known that the magnetic forces in the maglev systems are time-dependent.18,19 Thus, the thermally activated flux motion of SC should be fully taken into account. So far, by selecting different electromagnetic potentials as state variables, such as the magnetic vector potential A, the electric scalar potential V, the current vector potential T, and the magnetic scalar potential x, a set of mathematical formulations have been derived from Maxwell’s equations to study the timedependent levitation of the bulk SC under external disturbances with the assumption that the induced current only flows in the ab-plane of the SC.20–25 This assumption is valid when the studied system has an axisymmetric geometry and the
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movement of the SC is restricted to the vertical direction. While for other situations, e.g., a bulk SC traverses above a rectangular guideway, such assumption is no longer satisfied. Recently, Ueda et al. proposed a three-dimensional model based on the A V formulation to analyze the vertical oscillation and lateral stability of a levitating transporter using the bulk SC.26,27 The levitation and guidance forces were calculated by taking the nonlinear current-voltage characteristic of the SC into consideration. Dias et al. developed a similar method to study the lateral force behavior in a field cooled superconducting maglev system.28 In terms of the T x and A V formulations, Ma et al. presented a series of numerical studies on the dynamic behavior of the magnetic forces on the SC under the vertical or horizontal movement in a maglev system.29–31 It is obvious that all the works mentioned above mainly focus on the disturbance-related levitation of the maglev by assuming the SC moving along a horizontal or vertical straight path. However, from the perspective of dynamics, the motion trajectory of the SC under external disturbances should generally be expected to be a complex spatial curve. Based on Newton’s second law and the A V formulation, in this work, we build a two-dimensional numerical model to simulate the dynamic behavior of superconducting maglev systems. In this model, the boundary conditions are implicitly incorporated, and thus, the simulation domain is only restricted to the superconducting material. By using such a model, we present a systematic study of the effect of external disturbances on two typical maglev systems consisting of an infinitely long SC and a guideway of different arrangements of infinitely long parallel PMs. The non-linear characteristic of the SC is described by a power law approximation. The levitation height, lateral displacement, and vertical and lateral drift velocities of the SC are calculated by changing the magnitude and direction of the disturbance-induced initial velocity. In addition, the influence of flux motion and moving velocity of the SC on the levitation is also discussed. II. MODEL DESCRIPTION A. Dynamic equations
We consider two typical maglev systems popularly used in experiments and actual devices, as shown in Fig. 1. These systems are composed of a bulk high-temperature SC and a permanent-magnet guideway and the main difference
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between them is the configuration of the guideway. Here, we use the labels PM2L and PM3L to distinguish the guideways, where PM2L stands for two magnets with antiparallel horizontal magnetization and PM3L denotes three magnets with vertical magnetization in the central magnet and antiparallel horizontal magnetization in the lateral ones. In such systems, the SC and PM are infinitely long in the z direction and have rectangular cross-sections of aSC bSC and aPM bPM in the x and y directions, respectively. The PMs are assumed to have a uniform magnetization MPM and the horizontal interval between them is denoted as a in the guideways. In the practical operation of maglev systems, the vertical and lateral displacements of the SC may simultaneously occur around its equilibrium position if it is subjected to an external disturbance. To simulate such a disturbance, we consider the following process. First, the SC is zero-field cooled at a large distance h00 from the guideway. Second, the SC slowly moves down with a fixed velocity v00 to a static equilibrium height h0 , i.e., the working height. Finally, the SC suffers from a disturbance, and then, it has an instantaneous velocity, which is defined as v0 ¼ v0x x þ v0y y;
(1)
and the angle between the velocity and the negative direction of the y axis is expressed as h, as shown in Fig. 1(a). Denote the lateral displacement by w and height by h. From Newton’s second law, the dynamic equations of lateral and vertical movements of the SC can be expressed as € Fx ¼ 0; mw
(2)
mh€ þ mg Fy ¼ 0;
(3)
where m is the mass per unit length of the SC, g is the gravitational acceleration (commonly 9:8 N=kg), and Fx and Fy are the horizontal (guidance) and vertical (levitation) magnetic forces per unit length in the SC, respectively. Notice that in Eqs. (2) and (3) the rotation of the SC and the damping effect are ignored for simplicity. B. Electromagnetic equations
To describe the electromagnetic properties of the maglev system, the A V formulation of magnetoquasistatic
FIG. 1. Schematic view of the two studied maglev systems, where the arrows marked inside the PMs indicate their magnetized directions.
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Maxwell’s equations and a discretization procedure that previously appeared in Refs. 32 and 33 are employed. This numerical approach has an important advantage. Since the boundary conditions are implicitly incorporated into the model, one needs to discretize only the superconducting region and not the full space. Here, a brief overview of such approach is given as follows. For infinitely long geometry, the electric field E and the vector potential A in the SC are oriented along the z direction. Thus, we can regard them as scalar. From the A V formulation, the electric field at any point is expressed as @A @V ; (4) E¼ @t @z where V is the electric scalar potential. For the 2D problem, @V @z is a time-dependent constant on every SC cross-section and then it can be written as v ¼ @V @z . In the maglev system, the vector potential A can be expressed as A ¼ ASC þ APM ;
(5)
where ASC is the vector potential due to the shielding current J in the SC, and APM denotes the vector potential from the PMs. According to the 2D Poisson equation DASC ¼ l0 J, one obtains ð h 2 i l 2 ASC ðrÞ ¼ 0 J ðr0 Þln ðx x0 Þ þ y y0 dS0 ; (6) 4p S where l0 is the magnetic permeability in vacuum, and S is the cross section of the SC. By substituting (5) and (6) in (4), the following equation is finally obtained ð h 2 i @APM l0 @J ðr0 Þ 2 ln ðxx0 Þ þ yy0 dS0 v: (7) E¼ @t 4p S @t
M
N X dIj dAPM i þ þ v ¼ 0; (9) Mij dt dt j¼1 l0 Ð Ð 0 2 0 2 0 where Mij ¼ 4pDS If i 2 S S ln½ðx x Þ þ ðy y Þ dSdS . i j ranges from 1 to N, a final form of the discrete equation can be obtained
Ei þ
(10)
where 1 is a unit vector. Combining Eq. (8) and the constraint condition 1T I ¼ 0 (zero net current) for the SC, Eq. (10) can be rewritten as33 M
dI E0 n dAPM ¼ 0; þ n TI þ T dt dt Ic
(11)
1
T
M where Ic ¼ DSJc , T ¼ IN 11 and IN is the identity 1T M1 1 matrix. Once the time evolution of the vector potential produced by the guideway in the superconducting region is known, the current profile in the SC at any instant can be obtained by solving Eq. (11) with some popular solvers, such as the solver ode15s in MATLAB.37 It has been proven in Ref. 33 that these solvers can reliably deal with Eq. (11). After the current profile is obtained, the guidance and levitation forces per unit length in the SC can be calculated as
Fx ¼
N X
Fy ¼
Ii By;i ;
i¼1
N X
Ii Bx;i ;
(12)
i¼1
where Bx;i and By;i are the x and y components of the magnetic induction intensity produced by the guideway. C. Numerical procedure
Using the established dynamic and electromagnetic models, the dynamic behavior of the maglev system under external disturbances can be analyzed by taking the following two steps (see their flow diagrams in Fig. 2): (i)
For the SC, it has a highly nonlinear current-voltage characteristic and the constitutive relation between the electric field and the current density can be expressed with a power-law as follows:21,34–36 n J E ¼ E0 ; (8) Jc where E0 is the critical electric field (commonly 104 V=m), Jc is the critical current density, and n is the creep exponent related to the depinning barrier of the material. In order to numerically solve the problem, we divide the SC into N ¼ aSC =Dx bSC =Dy elements of infinite length in the z direction with a rectangular cross-sectional area DS ¼ Dx Dy, as shown in Fig. 1(b). In terms of a set of appropriate basis functions, the following weak form at element i is obtained from Eq. (7) by means of the usual weighted residual approach32,33
dI dAPM þ v1 ¼ 0; þEþ dt dt
In the process of the SC moving with a fixed velocity v00 from the height h00 to h0 , the time evolution of the vector potential produced by the PMs in the SC can be expressed as dAPM dAPM dAPM dAPM ¼ vx þ vy ¼ v00 ; dt dx dy dy
(13)
where vx and vy are the x and y components of the velocity for the SC. By substituting (13) in (11), the current profile in the SC can be obtained by solving the following equation, i.e., M (ii)
dI E0 n dAPM ¼ 0: þ n TI þ v00 T dy dt Ic
(14)
After the SC reaches the working height h0 ðFy mg ¼ 0Þ, it suffers from a disturbance and then moves in the xy plane. Combining the Eqs. (2), (3), (11) and (12), the dynamic behavior of the SC can be determined by the following coupled equations, i.e.,
€þ mw
N X
Ii By;i ¼ 0;
(15)
i¼1
mh€ þ mg
N X i¼1
Ii Bx;i ¼ 0;
(16)
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FIG. 2. Flow diagrams of the present numerical simulation.
dI E0 dAPM dAPM þ vy M þ n TIn þ T vx dx dy dt Ic
! ¼ 0;
(17)
and the initial conditions are taken into account by t ¼ 0 : h ¼ h0 ; w ¼ 0; vx ¼ v0x ; vy ¼ v0y ; I ¼ I0 ;
(18)
where I0 is the current vector of the SC obtained in step (i). ^ velocity ð^ ^ hÞ, For a given position ðw; v x ; v^y Þ and current ^I corresponding to the time ^t , the next values of ðw; hÞ, ðvx ; vy Þ and I at the new time t ¼ Dt þ ^t can be calculated by performing the numerical procedures as follows: ^ v^x , v^y , ^I, and calculate the ^ h, Step 1: input the values of w, coefficient matrixes M and T. ^ vk ¼ v^x , vk ¼ v^y and ^ hk ¼ h, Step 2: set k ¼ 0, wk ¼ w, x y k ^ I ¼ I, where the superscript k represents the number of iteration steps.
Step 3: calculate the magnetic forces with the parameters wk , hk , and Ik , and solve the Eqs. (15) and (16) with the ini^ v^x , and v^y . The values of wkþ1 , hkþ1 , vkþ1 ^ h, tial values w, x kþ1 and vy at t ¼ Dt þ ^t are obtained. PM PM Step 4: calculate ðvx dAdx þ vy dAdy Þ with the parameters and vykþ1 . Solve the Eq. (17) with the initial wkþ1 , hkþ1 , vkþ1 x value ^I and then obtain the value Ikþ1 at t ¼ Dt þ ^t . Step 5: judge the convergence condition of the iteration by: jwk wkþ1 j < d; jhk hkþ1 j < d; kIk Ikþ1 k2 < d; (19) where d is a given value to control the precision of iteration. If the Eq. (19) is not satisfied, then set wk ¼ wkþ1 , hk ¼ hkþ1 , Ik ¼ Ikþ1 and k ¼ k þ 1, and go back to step 3. Otherwise, go to step 6. Step 6: obtain w ¼ wk , h ¼ hk , vx ¼ vkx , vy ¼ vky and I ¼ Ik at t ¼ Dt þ ^t .
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FIG. 3. Calculated levitation force Fy versus the height h for the SC descending from h00 to h0 with different n and v00 . In each image, the inset is a comparison of numerical values of the power law model at n ¼ 201 with theoretical results of the Bean model.
III. RESULTS AND DISCUSSION
We use the numerical model proposed in Sec. II to simulate the two studied maglev systems. In both systems, the dimensions are bPM ¼ aPM ¼ 0:05 m, bSC ¼ aSC =3 ¼ bPM . The magnetization for the PMs is M ¼ 7:95 105 A=m, the critical current density in the SC is Jc ¼ 3:7 106 A=m2 , the initial height h00 ¼ 5bPM and the working height h0 ¼ 0:5bPM . These parameters are chosen as the typical values usually used in experiments and other theoretical works.38–42 A. Vertical movement of the SC
It has been well known that the levitation force is determined by the current density in the SC and the magnetic field generated by the guideway. The current density profile in our calculation is strongly affected by two essential parameters. One is the creep exponent related to the depinning barrier of the superconducting material; another parameter is the SC moving velocity representing the
relative movement of the SC and the guideway. To analyze the role of these parameters, we first consider the SC descending process from h00 to h0 in the two studied maglev systems and show the calculated levitation force Fy as a function of the height h for different n and v00 in Fig. 3. It can be seen that the obtained Fy curves for PM2L and PM3L have some differences in their values, but their trends are completely consistent with each other. These curves are monotonically increased as h decreases and the slopes of them increase in magnitude with increasing v00 . In at h ¼ relation to this, Fy reaches a greater maximum Fmax y h0 for larger v00 . It is interesting that with the increase of n, value increases for small v00 , whereas it decreases the Fmax y with n for large v00 . That is to say, the variation of Fmax y depends on the moving velocity of the SC. However, such dependence no longer exists for sufficiently large n. From our calculations, we find that when n ¼ 201, the obtained Fy values for different v00 always coincide with the theoretical curve based on the Bean model, as shown in the insets of Figs. 3(a) and 3(b).
FIG. 4. Calculated field lines and current profiles for the two studied maglev systems with different n and v00 values at h ¼ h0 .
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To fully understand the physical origin of the effect of these parameters on levitation, in Fig. 4 we display the field lines and current profiles for the two studied maglev systems with different n and v00 at h ¼ h0 . From this figure, one can observe some common features in the configurations PM2L and PM3L. When the SC is placed above the guideway, flux vortices will practically only penetrate into the superconducting regions near to the guideway, where the interaction of the magnetic flux density with the shielding currents leads to a levitation force. Because of thermal activation, the trapped flux in the SC has a possibility of escaping the pinning site. For small v00 , since the SC moves slowly, the flux vortices have enough time to penetrate deeper into the SC. This causes the shielding currents in the SC to be reduced and the levitation force to be decreased. Supposing n becomes smaller, larger penetration regions of the flux vortices will be formed to result in a further decrease in the levitation force [see Figs. 4(a), 4(b), 4(f), and 4(g)]. However, for large v00 , the situation is just the opposite. Because the flux vortices have little time to relax, their penetration will be restrained in some small regions close to the bottom of the SC. Meanwhile, the shielding currents in these regions are significantly increased to maintain the magnetic shielding in the interior, which in turn enhances the levitation force. With the decrease of n, the
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restriction of flux penetration becomes more serious and finally, a larger levitation force is obtained [see Figs. 4(c), 4(d), 4(h) and 4(i)]. Furthermore, for the case of n ¼ 201, the obtained current and field profiles for different v00 [see Figs. 4(e) and 4(j)] are almost the same as ones for the Bean model, as shown in Refs. 40 and 42. This is coincident with the fact that in the limit of infinite n, the power-law model will consistently reduce to the Bean model. B. Dynamic levitation of the SC under external disturbances
In this sub-section, we select v00 ¼ 104 m=s and n ¼ 21 for the SC to discuss its dynamic behavior under external disturbances. Once the SC slowly reaches the static equilibrium height h0 , it suffers from an external disturbance and then has an instantaneous velocity v0 to move in the xy plane. We first discuss a special case of h ¼ 0 and v0 ¼ 0:01 m=s in the two studied maglev systems and present the calculated height and levitation force for the SC in Fig. 5. From Fig. 5(a), one sees that the SC is freely vibrated in the y direction after being disturbed. With the increase of time, the vertical vibration center gradually drifts downwards, i.e., the levitation height decreases. This feature is consistent with the previous experimental results shown in Refs. 43–45.
FIG. 5. Simulations of the two studied maglev systems at h ¼ 0 and v0 ¼ 0:01 m=s. (a) and (b) Time responses of the height and levitation force. (c) and (d) Levitation force-height relations.
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FIG. 6. Simulations of the two studied maglev systems with different h at v0 ¼ 0:01 m=s. (a) and (b) Movement trajectories of the SC. (c) and (d) Time responses of the levitation height and lateral displacement. (e) and (f) Time responses of the vertical and horizontal drift velocities of the SC.
Interestingly, the drift velocity of the vertical vibration center seems to be related to the amplitude of dynamic response and, thus, the PM3L exhibits a larger drift velocity relative to the PM2L. Moreover, flux vortices will move in and out as the SC oscillates, which results in the energy losses. As a result, the amplitude of dynamic response decreases with time little by little. From Fig. 5(b), it is found that in the process of free vibrating of the SC, the calculated Fy value varies always around a certain value, i.e., the value of gravity
of the SC. Due to the magnetic hysteresis, the oscillation amplitude of Fy decays over time. This decay characteristic can also be directly observed from the levitation force-height curves, as shown in Figs. 5(c) and 5(d). It is obvious that only when h is less than a certain value will the hysteresis have an important effect on Fy . However, if h is larger than such certain value, the hysteresis effect becomes negligible and Fy starts to present an approximately linear relationship with h.
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FIG. 7. Simulations of the two studied maglev systems with different v0 at h ¼ 90 . (a) and (b) Movement trajectories of the SC. (c) and (d) Time responses of the levitation height and lateral displacement. (e) and (f) Time responses of the vertical and horizontal drift velocities of the SC.
We further consider the general case of h 6¼ 0 and show some levitation characteristic curves for the two studied maglev systems with different h at v0 ¼ 0:01 m=s in Fig. 6. From Figs. 6(a) and 6(b), it is found that due to the mutual coupling between the dynamic equations of lateral and vertical movements [see Eqs. (15)–(17)], the movement trajectory of the SC is very complicated and exhibits a typical characteristic of nonlinearity. To see more clearly its movement process, the time evolutions of the levitation height and
lateral displacement are, respectively, shown in Figs. 6(c) and 6(d). One can see that as time increases, the SC is simultaneously vibrated in the x and y directions, and the lateral and vertical vibration centers drift along their vibration directions. The lateral offset of the horizontal vibration center shows a monotonic increase with increasing h. However, the vertical offset of the vertical vibration center seems to be independent of h. In other words, at a certain moment, the vertical vibration center will drop down to the same height
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no matter what value of h is taken. To quantitatively describe such two drifts, the vertical and lateral drift velocities are defined as Vyy ¼
ht þT ht ; T
Vxx ¼
wt þT wt ; T
(20)
where ht is the levitation height at time t ¼ t when the SC moves down just through the vertical equilibrium position ðFy mg ¼ 0Þ, wt is the lateral displacement of time t at which the SC moves rightward just through the horizontal equilibrium position ðFx ¼ 0Þ, and T is a time-dependent vibration cycle of the SC. In Figs. 6(e) and 6(f), we show the calculated vertical and lateral drift velocities for PM2L and PM3L. Obviously, the configuration PM3L yields larger Vxx and Vyy values relative to the configuration PM2L. For PM2L, the Vxx or Vyy value decreases rapidly in the initial period, and after the initial period, it gradually tends to a stable value. While for PM2L, with the increase of time, its Vxx or Vyy value first rapidly decreases to a minimum and then begins to slowly increase. In Fig. 7, we display the calculated movement trajectory, levitation height, lateral displacement, and vertical and horizontal drift velocities of the SC for the two studied maglev systems with different v0 at h ¼ 90 . From Figs. 7(a) and 7(b), one sees that after being disturbed, the levitation point of the SC is changed and it gradually moves away from the initial working position under the combined action of gravity and magnetic forces. That is to say,, the equilibrium position of the SC is not a space point but a continuous range. By separating the lateral and vertical movements, it can be seen from Figs. 7(c) and 7(d) that the SC is periodically vibrated in both x and y directions. For the y direction vibration, the vertical vibration center drifts downwards no matter what value of v0 is achieved, and bigger v0 will lead to a larger vertical offset for the vertical vibration center. However, for the x direction vibration, only when v0 is larger than a certain value will the horizontal vibration center start to drift laterally. Namely, the appearance of the lateral offset of the horizontal vibration center depends on the v0 value. These features can be also found indirectly from Figs. 7(e) and 7(f). In the process of SC free vibrating, the calculated Vxx or Vyy value for PM2L and PM3L is increased with the increase of v0 in the initial period, whereas in other periods such monotonous relation may not exist. In addition, we notice that in the initial period, the Vxx value stays near zero for small v0 . However, if v0 is larger than a certain value, Vxx starts to increase rapidly exceeding zero. IV. CONCLUSIONS
In terms of Newton’s second law and the A V formulation together with the E J power law, in this work, we build a two-dimensional numerical model to study the effect of external disturbances on two typical maglev systems composed of a superconductor and a guideway formed by different arrangements of permanent magnets. The results show that in the process of the superconductor vertically moving from a large distance to the working position, the levitation force is related to the exponent n of the power law model
and the moving velocity of the superconductor. With the increase of n, the levitation force is reduced for large velocity, whereas it becomes enhanced for small velocity. After the superconductor reaches the working position, it suffers from an external disturbance and then has a disturbanceinduced initial velocity. Under the combined action of gravity and magnetic forces, the superconductor is periodically vibrated in the lateral and vertical directions and, simultaneously, its levitation point gradually drifts along the two vibration directions. The drift velocity of the levitation point is closely dependent on the magnitude and direction of the initial velocity. In addition, by comparing the configurations PM2L and PM3L, we find that PM3L yields a larger levitation force relative to PM2L, but the capacity of resisting external disturbance for the latter is significantly superior to that for the former. ACKNOWLEDGMENTS
We acknowledge the support from the National Natural Science Foundation of China (Grant Nos. 11602195, 11472120, and 11421062), the National Key Project of Magneto-Constrained Fusion Energy Development Program (Grant No. 2013GB110002), Program for New Century Excellent Talents in University of Ministry of Education of China (No. NCET-13-0266), the National Key Project of Scientific Instrument and Equipment Development (No. 11327802), and the Fundamental Research Funds for the Central Universities (Grant No. 3102017zy042). 1
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