high-Tc Superconducting Levitation System Research Articles

Chaotic Motions in Dynamic High-Tc Superconducting Levitation System with Thermal Effects

The chaotic motions were shown by the vibrations of a bulk permanent magnet (PM) supported by a cylindrical high-Tc superconductor (HTSC) under external disturbance in very early experiments. There are few theoretical studies on the nonlinear behaviors of the HTSC levitation system as the complex interaction between the HTSC and PM. In this paper, we developed a numerical program to analyze the nonlinear vibrations of the levitated PM coupled with the electromagnetic force, taking into account the heat diffusion on HTSC. When the levitation system displayed chaotic oscillations by vertical excitation, the electromagnetic and thermal characteristics of the HTSC will change. The dynamic responses are shown by comparing the calculation results in which the thermal effects are taken into account and not. The numerical results display the displacements of PM, magnetic levitation forces, the acceleration, and the maximum Lyapunov exponent. The results show that the thermal effect will influence the stability of the levitation system.

Influence of the physical parameters on the limits of magnetic stiffness of high-Tc superconducting levitation systems

Magnetic stiffness is one of the important stability parameters of high temperature superconducting levitation systems. Till to now, great efforts have been made to understand levitation properties including flux penetration, magnetization curves, levitation force, ac susceptibilities, etc. In this paper we present a quadratic approximation method for the limit of magnetic stiffness in a high temperature superconducting levitation system based on Kim’s critical state model and Ampère law. The system is composed with a cylindrical permanent magnet (PM) and a coaxial high temperature superconductor (HTS). It is found that the limit of magnetic stiffness depends upon both the penetration history of shielding currents distribution in HTS and applied magnetic field gradients. Furthermore, the influence of the physical parameters, such as critical current density in HTS and applied magnetic field, on the limits of magnetic stiffness is investigated in detail. The obtained results display that magnetic stiffness decreases with the increasing of critical current density, since shielding currents have not penetrated into the large portion of the HTS. With the increase of applied magnetic field, the magnetic stiffness obtain a larger magnification factor. It is related to the increase of the shielding current penetration volume and the internal magnetic field in HTS.

Vibration reduction using autoparametric resonance in a high-Tc superconducting levitation system

High-Tc superconducting levitation systems have very small damping and enable stable levitation without control. Therefore, they can be applied to various kinds of application. However, there are some problems that small damping produces large vibration and nonlinearity of magnetic force can generate complicated phenomena. Accordingly, analysis of these phenomena and reduction of vibration occurring in the system are important. In this study, we examined reduction of vibration without using any absorbers, but utilizing autoparametric resonance caused by nonlinear coupling between vertical oscillation and horizontal oscillation. We conducted numerical analysis and experiments in order to investigate motions of a rigid bar levitated by the electromagnetic force from high-Tc superconductors. As a result, if the ratio of the natural frequency of vertical oscillation and that of horizontal oscillation is two to one, the vertical oscillation decreases while the horizontal oscillation is excited. Thus, it was confirmed that the amplitude of a primary resonance can be reduced by occurrence of autoparametric resonance without using any absorbers.

Experimental Study of Suppressing a Parametric Resonance in a HTSC Levitation System by a Horizontal Pendulum

This research deals with the experimental investigation of reduction in the amplitude of nonlinear oscillation in a high-Tc superconducting magnetic levitation system by using a vibration absorber. High-Tc superconducting levitation systems have very low damping and stable levitation without control. In such low-damped systems, complicated phenomena of dynamics can be easily generated by nonlinearity of the magnetic force. Our previous research showed that if the frequency of vertical excitation is in the neighborhood of double the natural frequency in the horizontal direction, horizontal oscillation can occur parametrically through nonlinear coupling between vertical motion and horizontal motion. This research investigates an effect of a vibration absorber on the parametrically excited horizontal oscillation of the levitated body. We performed experiments on a symmetric rigid bar, levitated by the electromagnetic force between superconductors fixed on the base, and magnets fixed at both ends of the bar. A simple damped horizontal pendulum is attached to the main system as a vibration absorber, so that it is linearly coupled with the main system. Through these experiments, it has been confirmed that this kind of a vibration absorber is useful for reducing the amplitude of a parametric resonance in a high-Tc superconducting levitation system.

Reduction in the Amplitude of the Primary Resonance Combined with the Parametrically Excited Horizontal Oscillation in a High-Tc Superconducting Levitation System

This research proposes reduction in the amplitude of the primary resonance combined with the parametrically excited horizontal oscillation in a High-Tc superconducting levitation system. High-Tc superconducting levitation systems have very low damping and stable levitation without control. However, in such low-damped systems, complicated phenomena of dynamics can be easily generated by nonlinearity of the magnetic force. Our previous research showed that, when the frequency of vertical excitation is in the neighborhood of double the natural frequency in the horizontal direction, horizontal oscillation can occur parametrically through nonlinear coupling between vertical motion and horizontal motion. This research discusses on effects of the parametrically excited horizontal resonance on the primary resonance. We discussed possible nonlinear interaction called internal resonance between the vertical motion and the horizontal motion whose natural frequency is chosen as half that of the vertical motion. The primary resonance reduction based on this mechanism was successfully verified numerically. This kind of nonlinear oscillation can be useful for reducing resonance of a high-Tc superconducting levitation system.

208 高温超電導磁気浮上系における係数励振的振動の線形動吸振器による低減

This research deals with numerical analysis of effects of a vibration absorber on the amplitude of nonlinear oscillation in a high-Tc superconducting magnetic levitation system. High-Tc superconducting levitation systems have very low damping and stable levitation without control. In such low-damped systems, complicated phenomena of dynamics can be easily generated by nonlinearity of the magnetic force. Our previous research showed that, if the frequency of vertical excitation is in the neighborhood of double the natural frequency in the horizontal direction, horizontal oscillation can occur parametrically through nonlinear coupling between vertical motion and horizontal motion. This research investigates an effect of a vibration absorber on the parametrically excited horizontal oscillation of the levitated body. We calculated numerically the dynamics of the rigid body which is levitated by the magnetic force acting between superconductors, which are fixed on shake table, and magnets, which are fixed to both the ends of rigid body. A simple damped horizontal pendulum is attached to the main system as a vibration absorber, so that it is linearly coupled with horizontal motion of the main system. Numerical results proved that this kind of vibration absorber is effective in reducing the amplitude of a parametric resonance in a high-Tc superconducting levitation system.

546 高温超電導磁気浮上系におけるオートパラメトリック共振を利用した振動低減

546 高温超電導磁気浮上系におけるオートパラメトリック共振を利用した振動低減

Subharmonic Resonance of a Rotor Supported by a High-Tc Superconductor

This paper investigates subharmonic resonance, one of typical nonlinear phenomena, appearing in a high-Tc superconducting bearing system. In such low-damping systems having noncontact support by magnetic force, complex dynamical behaviors can be caused by nonlinearity of the magnetic force. In this study, the magnetic rotor, supported by a high-Tc superconducting bulk, is bound to a D.C. motor with a universal joint. According to the equation of motion, caused by the quadratic term of the levitation force, subharmonic resonance of the rotor can occur when the rotor rotates, not at its critical speed, but at twice its critical speed. This analytical prediction was confirmed by our numerical and experimental results. Additionally, our study also found occurrence of internal resonance induced by this subharmonic resonance. These findings beyond linear regimes can be important for the design of high-Tc superconducting levitation systems.

Modal bifurcation in a high-Tc superconducting levitation system

This paper deals with modal bifurcation of a multi-degree-of-freedom high-Tc superconducting levitation system. As modeling of large-scale high-Tc superconducting levitation applications, where plural superconductingbulks are often used, it can be helpful to consider a system constitutingof multiple oscillators magnetically coupled with each other. This paperinvestigates nonlinear dynamics of two permanent magnets levitated above high-Tc superconducting bulks and placed between two fixed permanent magnets without contact.First, the nonlinear equations of motion of the levitated magnets were derived.Then the method of averaging was applied to them. It can be found from theobtained solutions that this nonlinear two degree-of-freedom system can have twoasymmetric modes, in addition to a symmetric mode and an antisymmetric mode bothof which also exist in the linearized system. One of the backbone curves in thefrequency response shows a modal bifurcation where the two stable asymmetric modesmentioned above appear with destabilization of the antisymmetric mode, thus leadingto modal localization. These analytical predictions have been confirmed in ournumerical analysis and experiments of free vibration and forced vibration. Theseresults, never predicted by linear analysis, can be important for application of high-Tc superconducting levitation systems.

Internal resonance of an elastic body levitated above high-Tc superconducting bulks

In high-Tc superconducting magnetic levitation systems, levitated bodies can keep stable levitation with no contact and no control and thus their damping is very small. Thanks to these features, their applications to various apparatus are expected. However, on account of their small damping, the nonlinearity of electromagnetic levitation force can give notable effects upon motion of the levitated bodies. Therefore this nonlinearity must be taken into account to accurately analyze the dynamical behavior of the levitated bodies. Structures of such a levitated body can show elastic deformation if the large electromagnetic force acts on it. Therefore, we need to deal with the model as an elastic body. As mentioned above, nonlinear characteristics easily appear in this elastic vibration on account of the small damping. Especially when the ratio of the natural frequencies of the eigenmodes is integer, internal resonance can occur. This nonlinear resonance is derived from nonlinear interactions among the eigenmodes of the elastic levitated body. This kind of internal resonance of an elastic body appearing in high-Tc superconducting levitation systems has not been studied so far. This research especially deals with internal resonance of a beam supported at both its ends by electromagnetic forces acting on permanent magnets. The governing equation with the nonlinear boundary conditions for the dynamics of a levitated beam has been derived. Numerical results show internal resonance of the 1st mode and the 3rd mode. Experimental results are qualitatively in good agreement with numerical ones.

Drift of Levitated/Suspended Body in High-$T_{c}$ Superconducting Levitation Systems Under Vibration—Part I: A Criterion Based on Magnetic Force-Gap Relation for Gap Varying With Time

Levitation drift in the high-Tc superconducting levitation systems is directly related to the safe operation of the systems. In these levitation systems, the gap between a superconductor and a permanent magnet may decrease, increase, or keep unvarying with time. Based on the numerical simulations of the magnetic force-gap hysteresis relations of two different physical processes in field cooling, and the dynamic features at some given positions on major magnetic force-gap loops, a criterion described by the slopes of the given position on minor and major loops is proposed in this study. According to the suggested criterion, the drift phenomenon can be characterized by judging gap varying with time for a given levitation system. In addition, the characteristic of continuous space range of the equilibrium position of levitated/suspended body has been further exhibited from the numerical results.

Drift of Levitated/Suspended Body in High-$T_{c}$ Superconducting Levitation Systems Under Vibration—Part II: Drift Velocity for Gap Varying With Time

To study levitation drift further, i.e., the gap between a superconductor and a permanent magnet varying with time in high-Tc superconducting levitation systems, drift velocity is introduced. Based on the numerical simulations of the dynamic response of a levitated body, and according to the essential reasons for drift, the drift velocity is first divided into two regimes: Vff [related to flux flow (FF)] and Vtc [related to flux creep (FC)]. The drift velocity is shown to be mainly dependent upon properties of superconductivity (such as the critical current density of superconductors), initial disturbances, and applied excitations (such as the amplitude and the frequency of external excitations). Furthermore, the corresponding influences of the drift velocities Vff and Vfc have been investigated quantitatively in this paper.

Parametrically excited motion of a levitated rigid bar over high-Tc superconducting bulks

High-Tc superconducting levitation systems achieve, under no contact support, stable levitation without control. This feature can be applied to flywheels, magnetically levitated trains, and so on. But no contact support has small damping. So these mechanisms can show complicated phenomena of dynamics due to nonlinearity in their magnetic force. For application to large-scale machines, we need to analyze dynamics of a large levitated body supported at multiple points. This research deals with nonlinearly coupled oscillation of a homogeneous and symmetric rigid bar supported at its both ends by equal electromagnetic forces between superconductors and permanent magnets. In our past study, using a rigid bar, we found combination resonance. Combination resonance happens owing to the asymmetry of the system. But, even if support forces are symmetric, parametric resonance can happen. With a simple symmetric model, this research focuses on especially the parametric resonance, and evaluates nonlinear effect of the symmetric support forces by experiment and numerical analysis. Obtained results show that two modes, caused by coupling of horizontal translation and roll motion, can be excited nonlinearly when the superconductor is excited vertically in the neighborhood of twice the natural frequencies of those modes. We confirmed these resonances have nonlinear characteristics of soft-spring, hysteresis and so on.

Numerical method to the excited high-Tc superconducting levitation system above the NdFeB guideway

A numerical method based on the flux flow creep model is introduced to simulate the properties of the high-Tc superconductor (HTS) levitated above the NdFeB permanent magnetic guideway. Compared with the former method based on the critical state model, this method is more effective because the flux flow creep model considers the time factor and the thermal activation of the electromagnetic phenomenon of the HTS. Both dynamic magnetic forces and motions of the levitating HTS bulk can be solved by this numerical method

Magnetic stiffness of a coupled high-Tc superconducting levitation system

High-Tc superconducting levitation systems, such as flywheels and linear-drives, have very small damping, because of their no contact support. Thus, they can show complicated phenomena of dynamics due to nonlinearity in their magnetic force or torque. In mechanical design of such systems, it is important to evaluate their nonlinear dynamics, though it has not been well studied so far. As one modeling of such an application system, we consider a magnetically coupled system of several permanent magnets and high-Tc superconducting bulks. It is a multi-degree-of-freedom dynamical system showing complicated motion caused by nonlinear interaction between levitated magnets and superconducting bulks. This research deals with magnetic stiffness of a coupled superconducting levitation system. As a fundamental dynamical property of the system, we evaluate the stiffness by doing experiment and analysis, and examine nonlinear effect on this property.

Control of flywheel system with high-Tc superconducting bearings

The high-Tc superconducting levitation system is stable in both radial and axial directions without contacts, so it has been applied as bearings of an energy storage flywheel. To have high efficiency, the rotating speed of the wheel is significantly high, and it is necessary to pass through critical speeds. Rotating tail stocks are often attached at both ends of the shaft, and those are removed after passing through the critical speeds. But the method is complex in practical use. The present article discusses a method for passing through the critical speeds by using magnetic actuators. The analytical results for the system are first obtained, then a control method for the system is presented. Numerical calculations are carried out for a typical problem, and it is ascertained that the present system and control method are applicable to high-Tc superconducting bearings.

VIBRATION CONTROL FOR A HIGH-Tc SUPERCONDUCTING NON-LINEAR LEVITATION SYSTEM

In the present paper the modelling and control of a high-Tc superconducting levitation system are discussed. First, the analytical expressions for obtaining the non-linear levitation force, given by the present author, are clarified; and then a vibration control method is presented in which feedback currents involving frequency weights are used. In the method, the square of the displacement in the frequency domain is taken as a cost function. Optimal coefficients of the transfer function of the controller are obtained by minimizing the cost function. Hence, the spillover instability due to the non-linear vibrations will be reduced. Numerical calculations have been carried out for some typical problems. To validate the present method, experimental tests have also been carried out.

Approximate boundary conditions in a circular conductor and their application to nonlinear vibration analyses of high-Tc superconducting levitation system

When a levitated superconductor vibrates, the levitation force has nonlinear relationship among an air gap, amplitude, and frequency, so the usual static analysis for levitation forces is invalid. There are two phenomena of a flux creep and flux flow when the conductor vibrates in a magnetic field. The present article provides an analytical result for the levitation forces of a thick superconducting disc with consideration of the phenomena and effects of flux variations on the critical current. It is significantly difficult to have the analytical result of levitation force by using the exact boundary conditions. This paper presents approximate boundary conditions which give an appropriate result. To validate the proposed boundary conditions, eddy currents in a conductor are first discussed, then the superconductor is discussed. Numerical results for the levitation forces were obtained and compared with the previously published experimental data. The levitation force becomes a restoring force having the nonlinear relationship, so it is difficult to solve vibrations. The present article gives a simplified method for solving nonlinear vibration problems for the levitated conductor. Numerical calculations were carried out for some typical examples.

Modeling in Nonlinear Vibrations of a High-Tc Superconducting Levitation System.

Three dimensional analytical results for the levitation force of a vibrating high-Tc superconducting levitation system were presented. When the levitated superconductor vibrates, the levitation force shows nonlinear relationships with the air gap, amplitude and vibration frequency, so that the convensional models which do not consider dynamic effects cannot be applied. In the model proposed by(Uesaka et al.), dynamic effects are considered, but the critical current is constant. We propose an improved Kim-model which is applicable to vibration problems of conductors. Numerical results for the levitation force were obtained and compared with previously published experimental data. A modeling method for the superconducting levitation system was presented, and the nonlinear spring constants obtained in an analysis which included the effects of air gap, amplitude and frequencies were given. Numerical calculations were carried out for nonlinear vibrations of the system based on the modeling results.