Passive Micro Vibration Isolator Utilizing Flux Pinning Effect for Satellites

Information related to the origin of space and evolution of galaxy can be obtained using the observation satellites. In recent years, high pointing accuracy is demanded for getting more detailed data about distant stars and galaxies. As a result, vibration isolators that consist of a main structure and a TTM (Tip Tilt Mirror) have been adopted for observation satellites. However, cutting the low frequency vibrations off passively with the conventional methods is difficult. A vibration isolator that uses pinning effect is proposed for solving this problem. The pinning effect is acquired by cooling the type-II superconductor below the critical temperature and it generates a pinning force to maintain the relative distance and attitude between a type- II superconductor and a material that generates magnetic flux. The mission part and the bus part of the satellite are equipped with superconductors and permanent magnets and these parts perform short distance formation flight by applying the effect. This method can cut vibrations from low to high frequency bands off passively. In addition, Meissner effect can prevent collision of the mission and bus parts. In order to investigate the performance of this system, experiments and simulations are carried out and the results are discussed.


Background
Diverse information about galaxies and planets far from the Earth can be acquired using the observation satellites. High pointing accuracy has been demanded for obtaining information about further planets and galaxies in recent years. Disturbance is a problem when these satellites must observe with high pointing accuracy. The vibrations generated by the reaction wheels or the refrigerator in the bus part are some of these disturbances and the observed data blurs if these vibrations are transmitted to the mission part (Fig.1). Different methods that use the Tip Tilt Mirror (TTM) [1] and Sterwart platform [2] have been researched for restraining the influence of these vibrations. However, these methods hardly cut the vibrations of low frequency band off passively. Therefore, If an actuator, which is used for these conventional methods, is broken the observation satellite can not achieve high pointing accuracy. Besides, observation satellites that have both a primary and secondary mirror do not only experience the effect of vibrations but also the heat transfer. The heat is produced by equipments in the bus part and transmitted to the mission part through the conventional vibration isolator. This effect decreases the pointing accuracy as the heat strain causes vibration and deformation of the mirrors. In this paper, the micro vibration isolator utilizing the flux pinning effect is proposed.

Flux pinning effect
Flux pinning effect occurs between a type-II superconductor and a material that produces magnetic flux (e.g. permanent magnet). The state of the superconductor changes from the normal conducting state to the superconducting state by cooling below the critical temperature. In this case the superconductor gains two profperties that are called perfect conductivity and perfect diamagnetism. The perfect conductivity is the effect that the internal resistance in a superconductor can be approximated to zero. The perfect diamagnetism, which is also called the Meissner effect, is also generated by the cooled superconductor. The effect has property that the induced current prevents external magnetic flux intruding the superconductor. Currents that are named the shielding currents are induced on a superconductor by Lenz's law as a magnetic flux generating material approaches. In addition, the magnetic flux is repelled out of a superconductor by cooling below the critical temperature since Lorentz force is generated between the current and magnetic flux (Fig.2), even if the magnetic flux penetrates through a superconductor that is in normal conducting state. However, if a type-II superconductor has impurities and cracks, it does not repel out magnetic flux. These impurities and cracks in the type-II superconductor are not superconducting and hold the normal conducting state when the type-II superconductor is in superconducting state. Therefore, the flux is pinned in the impurities and cracks and pinning force competes with Lorentz force. Pinning force helps to maintain the relative distance and attitude between a type-II superconductor and magnetic flux generating material passively. The force can be approximated by a spring and damping force regarding the infinitesimal displacements.
Because of these reasons, many researches about devices with magnetic flux pinning effect (e.g. quake-free device [3] and transport mechanism [4]) have been conducted. In addition, this effect has been researched for formation flight technology where a number of satellites perform their mission by flying in formation. Electro-magnetic force [5], Coulomb force [6], Lorentz force [7] and pinning force [8] have been studied for controlling relative attitude and distance between the satellites since these formation flight methods have certain advantages compared with the formation flight using the thrusters. The proposed micro vibration isolator is composed of permanent magnets, electro-magnets and type-II superconductors (Fig.3). In this study, our aim is to develop a micro vibration isolator that can cut the vibrations from low to high frequency band off passively using the formation flight technology and flux pinning effect. Permanent magnets and electro-magnets are mounted on the bus part and the mission part is equipped with type-II superconductors. The mission and bus parts of the satellite that has the proposed micro vibration isolator performs a short distance formation flight. The relative distance and attitude are maintained by flux pinning effect. Moreover, collision of these parts is prevented by Meissner effect. The effect works when a superconductor cooled below the critical temperature, and the type-II superconductor repels out external magnetic flux. Thereby, the repulsive force is generated between the type-II superconductor and the permanent magnet which are getting closer. Pinning force can be approximated by the spring damping force and is easy to model. Due to the flux pinning effect, this isolator enables cutting off the low frequency vibrations without active control. Furthermore, the proposed micro vibration isolator can control the attitude of the mission part using the Meissner effect that we can control by the electro-magnets. Repulsive force is This proposed micro vibration isolator can resolve the problem of not only the vibration but also heat strain since the mission and bus parts are not connected with a structure as in a conventional vibration isolator. Besides since the method is passive any measures against actuator failure is not necessary.

Numerical model for the proposed micro vibration isolator
Frozen image model is used as a numerical model for calculating the pinning force that is generated between a type-II superconductor and a permanent magnet. This model proposed by Alexander Kordyuk [9] can be used assuming that the type-II superconductor is huge compared to the permanent magnet it is levitating on. In addition, a permanent magnet is approximated by magnetic dipole in this model. The pinning force is calculated using two images, which are called the frozen image and mobile image (Fig.4). These images are generated in the type-II superconductor when the superconductor is cooled below the critical temperature. Each image has magnetic moments that are same with the moment of the permanent magnet in magnitude but with different directions. Those magnetic moments is represented by following equation: ⃗ M mag , ⃗ M f and ⃗ M m is magnetic moment vector of a permanent magnet, a frozen image and a mobile image respectively. Magnetic dipole vector direction of mobile image is in line symmetry with respect to type-II superconductor's surface. Furthermore, this image moves with motion of the permanent magnet. In contrast, frozen image has magnetization vector whose direction is opposite of the mobile image and keeps the initial position even when the permanent magnet moves.
The pinning force as well as the torque can be calculated as the total force generated between the permanent magnet and each images. The method for calculating the force and torque are formulated in [10]. Similar expressions are used for calculating the pinning force and torque. The force and torque generated between each images and the permanent magnet can be determined with the following expressions: The pinning force F f,m and torque T f,m can be calculated using the magnetic moment M mag and magnetic field H f,m generated by frozen and mobile images. Here, subscript f or m means the value is generated between the frozen or mobile image and the permanent magnet. Magnetic field of each images at the position of the permanent magnet is given with the following expression: Here, ⃗ r f,m is the distance vector from images to the permanent magnet and the images have magnetic moment vector ⃗ M f and ⃗ M m . Finally, pinning force and torque between the permanent magnet and the type-II superconductor are obtained by

Equipment for measuring the pinning force
The sizes of the permanent magnet and the type-II superconductor (Fig.5) used in the experiment are given in Table.1. The measurement experiment using the equipment given in  aluminum and acrylic so as to magnetize the equipment. The permanent magnet is connected to the spring balance by string and attached to the aluminum circle plate. The displacement is measured by a laser rangefinder 'LK-G 405 and LK-G3000' by Keyence, and the pinning force is measured by the spring balance when the permanent magnet moves from the initial position. The pinning force was measured three times for which the initial distance between the permanent magnet and the type-II superconductor are 6.5[mm], 7.5[mm] and 8.5[mm], respectively.

Calculating the permanent magnet's magnetization
For understanding the validation of the numerical model, the magnetization of the permanent magnet was determined by measuring the magnetic field. The derived magnetization is substituted for frozen image model and the model is compared with the experimental results. A sheet that was marked with a line at every 5.0[cm] from initial point to 30[cm] was put on the table. The permanent magnet was put at the initial point and the magnetic field was measured at each points by the probe of 'Lake shore high cost performance gauss meter 421 type' (Fig.8). After measuring the magnetic field, the magnetization of the permanent magnet was estimated using the least-squares method and the measured magnetic field. Magnetic moment of the permanent magnet was estimated by the least-squares method. Magnetic field of only z direction(normal direction to the permanent magnet's plane) can be simplified using the following expression: Let 1 z approximates Z. Then Eq.7 is rewritten as the following expression: The sum of squares to be minimized is represented using the above equation (8) as follows: And, the partial derivation of E can be written as Then, magnetic moment M z is determined as 3.49[Am 2 ] , and the magnetization m z becomes 7.41 × 10 5 [A/m]. The comparison of the measured magnetic field with magnetic field calculated using the magnetic moment is given in Fig.9. Regarding this result we may say that the magnetic moment is almost correct. Hence, the magnetization can be represented with the numerical model.

Measurement results compared with the frozen image model
The results for the measured pinning force are given in Fig.10 Fig.11 and Fig.12. Initial distances are 6.5[mm], 7.5[mm] and 8.5[mm] respectively. The measured pinning force is similar in all three cases. The force that is averaged over three times is compared with the numerical results (Fig.13). In Fig.13, Sim and EX express a simulation and an experimental results respectively. The pinning force calculated by numerical model almost matches with the measured pinning force. Thus, this model can be used for the proposed micro vibration isolator.  Figure 13.
Comparison of calculated pinning force with the measured pinning force.

Required pinning force and torque for cutting off low frequency vibrations
It is assumed that the satellite has a mission and bus part each with a 500[kg] mass. Height of the mission part is 6.0[m] and the bus part is 2.6[m], Radius of both mission and bus part is 1.9[m] (Fig.3). The low spring coefficient is demanded for cutting the low frequency vibrations off. For cutting the vibrations at 0.1[Hz] band off passively, 0.2[N/m] spring coefficient is necessary. In addition, the satellite conducts a mission on halo orbit at L2 point of Earth-sun system. Solar radiation pressure affects the satellites predominantly on the halo orbit at L2 point. The solar radiation pressure can be evaluated using the following expression [11]: P s is a solar radiation coefficient and the value is 4.617 × 10 −6 [N/m 2 ]. A m is the effective area. H m is the length between the center of gravity and the point of sun pressure. q is the reflection coefficient and i is the angle between the effective area and the direction of sun pressure. In case of estimating the maximum torque and force by sun pressure, q is 1 and i is 0 • . The maximum force by solar radiation pressure can be calculated using difference of effective area between the mission and bus part. The maximum torque is calculated supposing that only the mission part rotates around the center of the bottom. In this case, the force and torque by solar radiation pressure are 1.19 × 10 −4 [N], 1.26 × 10 −3 [N m], respectively.
If the pinning force and torque are larger than the solar radiation pressure, the relative distance and attitude between the mission and bus parts can be maintained. In this research, the possibilities for the proposed micro vibration isolator, which can cut the vibrations at 0.1 [Hz] band off and also maintain the relative distance and attitude, are discussed

Analysis results by frozen image model
Pinning force and torque are simulated using the frozen image model for discussing the possibilities of the proposed micro vibration isolator. Low spring coefficient is needed for cutting the vibrations of low frequency band off between the mission and the bus part of the satellite. For cutting the vibrations at 0.1[Hz] band off, a spring coefficient of 0.2[N/m] must be achieved. In numerical analysis, the mission part moves along +z, -z and x axes in addition to the rotation around x and y axes (Fig.14). The proposed micro vibration isolator has one permanent magnet The results for the pinning force are shown in Fig.15, Fig.17 and Fig.19 when the spring coefficient is as given in Fig.16, Fig.18 and Fig.20, respectively. If the mission part moves along +z axis from the initial point, pinning force increases as in Fig.15. The force in +z direction gets the maximum value of -0.011[N] at 0.65[m] from the initial position. However, pinning force decreases as the mission part passes through 0.65[m] from the initial point. Hence, there is a possibility that the satellite separates if the external force is bigger than the pinning force of 0.011 [N]. However, it is considered that the distance can be maintained because the force of solar pressure is sufficiently smaller than the pinning force. Fig.17 is the force in case the mission part moves in -z direction. The force increases exponentially when the mission part approaches to the bus part. This repulsive force is caused by Meissner effect. The collision of the mission part and bus part can be prevented by this effect.
As to the force in x direction , the maximum value of the force is 0.023[N] at 0.4[m] from the initial position (Fig.19). In this case, not only the force in the x direction but also in the z direction is generated. The z direction force is caused by Meissner effect. The influence of the force in z direction must be considered for maintaining the relative distance and keeping high pointing accuracy. The x direction spring coefficient is 0.1[N/m] at initial point in Fig.20, therefore the low frequency vibrations can be cut off sufficiently.
When the mission part rotates around x and y axes, the same torque is generated (Fig.21,  Fig.22). However, the torque about z axis is not generated if the mission part rotates around z axis. The flux pinning effect allows the permanent magnet to rotate with low energy loss on the type-II superconductor if the change in the applying magnetic field is very small. Hence, when the proposed micro vibration isolator is mounted on the observation satellites, two of the permanent magnets and type-II superconductors are needed at minimum.
The calculated force and torque values as a result of the analyses are very small. However, the external force of solar pressure is in the order of -4 on halo orbit at L2 point, so these pinning force can maintain the relative distance between the bus and mission part. Regarding the torque, z axis torque is not generated. Therefore, more than two permanent magnets and type-II superconductors are necessary for preventing the mission part from rotating around z axis.

Conclusion
The passive micro vibration isolator using the flux pinning effect was proposed in this paper. The frozen image model was used for understanding the pinning force and torque generated between the bus and mission parts of the satellite. The pinning force can maintain the relative distance between the bus and mission parts on halo orbit at L2 point. However, regarding the relative attitude, torque around z axis can not be generated since the magnetic flux, which applies to the type-II superconductor, does not change. Since the pinning force is generated by the gradient of magnetic field, torque around z axis is not generated. It is needed that permanent magnets and superconductors are mounted on the mission and bus parts. The pinning force and torque must be calculated in case of using the permanent magnets and type-II superconductors. In addition, damping coefficient also must be calculated for cutting the low frequency vibrations off. However, in our case, we assume the damping force by the flux pinning effect is small. We need to have an eddy current damper [12] [13] or shunt damper [14] to acquire the damping force. Designing the attitude control system of the mission part is future work in this research.