Roles of superconducting magnetic bearings and ... - CyberLeninka

Physica C 483 (2012) 178–185

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Roles of superconducting magnetic bearings and active magnetic bearings in attitude control and energy storage flywheel Jiqiang Tang a,b,⇑, Jiancheng Fang a, Shuzhi Sam Ge b a b

School of Instrument Science and Opto-Electronics Engineering, Beihang University, Beijing 100191, China Department of Electrical & Computer Engineering, National University of Singapore, Singapore 117576, Singapore

a r t i c l e

i n f o

Article history: Received 20 April 2012 Received in revised form 10 July 2012 Accepted 12 July 2012 Available online 23 July 2012 Keywords: Superconducting magnetic bearing (SMB) Active magnetic bearing (AMB) Field cooling Gyroscopic effect Attitude control and energy storage flywheel (ACESF)

a b s t r a c t Compared with conventional energy storage flywheel, the rotor of attitude control and energy storage flywheel (ACESF) used in space not only has high speed, but also is required to have precise and stable direction. For the presented superconducting magnetic bearing (SMB) and active magnetic bearing (AMB) suspended ACESF, the rotor model including gyroscopic couples is established originally by taking the properties of SMB and AMB into account, the forces of SMB and AMB are simplified by linearization within their own neighbors of equilibrium points. For the high-speed rigid discal rotor with large inertia, the negative effect of gyroscopic effect of rotor is prominent, the radial translation and tilting movement of rotor suspended by only SMB, SMB with equivalent PMB, or SMB together with PD controlled AMB are researched individually. These analysis results proved originally that SMB together with AMB can make the rotor be stable and make the radial amplitude of the vibration of rotor be small while the translation of rotor suspended by only SMB or SMB and PM is not stable and the amplitude of this vibration is large. For the stability of the high-speed rotor in superconducting ACESF, the AMB can suppress the nutation and precession of rotor effectively by cross-feedback control based on the separated PD type control or by other modern control methods. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction Having a very high energy density, flywheels could be used as one single energy storage and attitude control device, forming a combined energy and attitude control system. This combination could lead to different improvements on mission design, e.g. mass saving, performance enhancement, reliability increase and so on [1,2]. Looking at the performance, the combined energy and attitude control system is much more promising than the traditional system (batteries with reaction wheels) [3–5]. Energy storage flywheel can have energy fed in the rotational mass of a flywheel, store it as kinetic energy, and release out upon demand [6–8]. But for the attitude control and energy storage flywheel (ACESF), not only the speed of the rotor is high, but also the position of the rotor must be controlled accurately to ensure that the vector of angular momentum have precise and stable direction, so the vibration of the high-speed rotor must be as little as possible to avoid disturbing the stability of spacecraft. Active magnetic bearings (AMBs) can accommodate very high spin speeds and have theoretically unlimited imbalance induced vibrations. Recently, the ⇑ Corresponding author at: School of Instrument Science and Opto-Electronics Engineering, Beihang University, Beijing 100191, China. Tel.: +86 01082317396; fax: +86 01082316813. E-mail address: [email protected] (J. Tang). 0921-4534/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physc.2012.07.007

frictional coefficients of high temperature superconducting magnetic bearings (SMBs) has achieved in the order of 106 or even smaller, what could contribute significantly to lower loss in energy storage flywheel and more precise in attitude control for space application [7]. Low stiffness of SMB is one major problem for industrial scale flywheel system. Mostly, a hybrid support of a flywheel rotor using conventional AMB together with SMB seems to provide a solution to this problem [9–11]. The SMB is used either to bear the weight of rotor or to stabilize the rotor by the auto-stability of the high temperature superconductor, but the relationship of magnetic suspension force or stiffness between the SMBs and AMBs and the behaviors of radial translation and tilting movement of the high-speed rotor suspended by SMB or SMB and AMB are still unclear yet. Fig. 1 is a prototype superconducting ACESF incorporating SMB and AMB we constructed to investigate the roles of SMB and AMB. The rotor suspension system consists of an axial high temperature superconducting magnetic bearing (HTSMB) at the lower end of the hollow composite fiber rotor, one axial AMB at the upper end of the rotor and two radial AMBs in the middle. The stators of the axial AMB and the axial HTSMB are mounted on the upper side and lower side of the vacuum enclosure, respectively. In this paper, all AMB are permanent magnet biased hybrid magnetic bearings. The axial AMB does most to reduce the load of rotor over axial

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J. Tang et al. / Physica C 483 (2012) 178–185

Housing Axial AMB

Sensors A Multi-ring Carbon fiber rotor

Radial AMB-Up Motor/ Generator

Shaft

Radial AMB-Lo Axial HTSMB

Sensors B

Fig. 1. Sketch of superconducting ACESF.

HTSMB and to control the axial translation of the hollow composite fiber rotor. The HTSMB, which is one alternative to axial MB, is entirely passive and used to support the rotor weight partially and to stabilize the rotor by the self-stability of high temperature superconductor. The displacements and tilting angles of rotor are measured by two position sensors (A and B), which are integrated axial/radial eddy current sensors. Radial AMBs including upper radial AMB (radial AMB-Up) and lower radial AMB (radial AMB-Lo) are used to control the radial movement and tilting of rotor. The stators of the radial AMBs and the motor/generator are fixed on a titanic shaft, which is mounted on the vacuum enclosure by bolts. Four touchdown bearings as auxiliary equipments are used to support the rotor when the magnetic bearings are turned off or fail. The axial protective gap is 1.5 mm and the radial protective gap is 0.12 mm. The rotor is driven by a DC motor. In order to reduce the aerodynamic drag and to avoid the frost of air within the gap of the HTSMB, the whole system is enclosed by a vacuum enclosure with pressure of about 1–5 Pa. With such a rotor suspension system, the rotor can be levitated stably and rotate at a very high speed. Based on the presented superconducting ACESF, this paper will research the relationship between SMB and AMB theoretically and originally for the purpose of attitude control and energy storage. By constructing the rotor model and force linearization of SMB and AMB respectively, the radial translation and tilting movement of rotor suspended by only SMB, SMB with equivalent PMB, SMB and AMB with PD control type are researched. These analysis results will give us some helpful hints to research the superconducting ACESF.

repulsive, so we use Fsupx, Fsupy and Fsupz to present the x component, y component and z component of it for the SMB. Taking the effects of motor into account, the x component, y component and z component of force of motor acting on rotor is presented by Fmotx, Fmoty and Fmotz, respectively. Tmot presents the driving moment of motor, and T the load moment on rotor. When the rotor rotates at a constant speed of X, the rotor system for the superconducting ACESF can be depicted as shown in Fig. 2. On the assumption that the axial and radial directions are separated and the dynamics of the rotor may be investigated independently, a complete flywheel rotor model including gyroscopic couples has been derived to analyze the relationship between SMB and AMB based on the motion equation of rotor derived according to the movement of the rotor’s center of mass.

m€xc ¼ F upx þ F lox þ F supx F motx €c ¼ F upy þ F loy þ F supy F moty my € þ J Xu _ ¼ F loy Llo þ F supy Lsup F upy Lup F moty Lmot Jw z

€ J z Xw_ ¼ F upx Lup F lox Llo F supx Lsup Ju

2. Flywheel rotor model Because the magnetic suspension force is attractive, we use Fupxp and Fupyp to present the x component and y component of force in the positive direction for the upper magnetic bearing respectively, Fupxn and Fupyn present the x component and y component of force in the negative direction for the upper AMB. Similarly, Floxp and Floyp present the corresponding force for lower AMB. Floxn and Floyn are used to present the x component and y component of force in the negative direction for the lower AMB. FA-AMB presents the magnetic suspension force of axial AMB in the positive direction of z axle. Due to the pinning effect, a lateral restoring force occurs in the radial direction. On the other hand, the superconducting magnetic suspension force may be attractive or

Fig. 2. Flywheel rotor model for the superconducting ACESF.

ð1Þ

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J. Tang et al. / Physica C 483 (2012) 178–185 LN 2 Container

Discal bulk HTS LN 2 Thermal insulator

(a)

(b)

(c)

Fig. 3. Arrangements of superconductors for SMB and superconducting suspension force measurement apparatus. (a) Schematic diagram of the arrangement of superconductors. (b) Six ring-shaped superconductors. (c) Suspension force measurement apparatus.

where w and u are the tilt angle around x axle and y axle, respectively, xc and yc are the displacements of rotor’s center of mass. Fupx = Fupxp Fupxn, Flox = Floxp Floxn, Fupy = Fupyp Fupyn, and Floy = Floyp Floyn. J is the transversal inertia moment of the rigid rotor around its x axle or its y axle, and Jz is the polar inertia moment of rotor around its z axle. Llo is the distance between the center of rotor and the center of lower magnetic bearing, Lsup the distance between the center of rotor and the center of SMB, Lup the distance between the center of rotor and the center of upper AMB, Lmot the distance between the center of rotor and the center of motor. In practice, we make the center of motor to coincide with the center of the rotor to ensure Lmot = 0, the force Fmotx, Fmoty and Fmotz of the motor are considered individually, so (1) become

m€xc ¼ F upx þ F lox þ F supx €c ¼ F upy þ F loy þ F supy my € þ J Xu _ ¼ F loy Llo þ F supy Lsup F upy Lup Jw

ð2Þ

z

€ J z Xw_ ¼ F upx Lup F lox Llo F supx Lsup Ju

Suspension force (N)

120 100 80 60 40 20 0 0

5

10

15

20

25

Distance (mm) Fig. 4. Static axial magnetic force of SMB with 5 mm cooling gap.

3. Axial superconducting magnetic bearings The HTS YBaCuO has the properties of flux exclusion and flux pinning when it is cooled by liquid nitrogen. Based on the repulsion and restoration between superconducting bulks and permanent magnets, the HTS can be applied to construct SMB which can provide an auto-stable repulsion force in the axial direction and restoring forces in the radial directions. The magnetic forces of SMB in every direction can be approximately described as follows:

FðiÞ ¼

Z Z Z

l0 MðHÞ

dH dH dV ¼ l0 MðHÞ V di di

ð3Þ

where M(H) is the magnetization of the superconducting block, H magnetic field vector, V the volume of superconductor, and i = x, y, z, where x and y presents the radial degrees and z presents the axial degrees, respectively. SMB has zero field cooling (ZFC) type and field cooling (FC) type with respect to the HTS cooling mode. ZFC bearings produce a large levitation force but are unstable in all directions, hence the additional stable control is required. FC bearings produce less level of levitation force, but they provide passive stability in all directions. With respect to a rotor passively supported by the SMB without any control mechanisms for flywheel energy storage, Mochimitsu Komori and Taku Hamasaki [12], Mochimitsu Komori [8, 9, 12, 13] found that the stiffness and the damping coefficient of the SMB system can be controlled. Rastogi and Alonso [10] studied the effects of superconductor’s thickness, diameter, and cooling height on the super-conducting magnetic suspension properties and pointed out that the maximal force is 44.3 N when the superconductor with 15 mm thickness and 6 mm cooling gap, the maximal force is 64.7 N when the superconductor with 15 mm thickness and 5 mm cooling gap. In the superconducting ACESF, the axial SMB consists of superconductors arranged as Fig. 3 shown. For permanent magnet NdFe-30, Br is 1.1 T, Hc is 8.424 105 A/m, the diameter of the melt-processed YBaCuO bulk superconductors is 28.5 mm and

Fig. 5. Schematic diagram and its configuration of radial AMB. (a) Sectional view. (b) End view. (c) Configuration.

181


x 10

Displancement Stiff (N/m)

100

Force (N)

80 60 40 20 0 -0.1

-0.1 -0.05

-0.05

1.4 1.3 1.2 1.1 1 0.9 0.8 -0.1

-0.1

0.1

0.1

0

0

0.05

0.05

Displacement (mm)

-0.05

-0.05

0

0

6

0.05

0.05

Displacement (mm)

Displacement (mm)

0.1

0.1

Displacement (mm)

(b)

(a)

Fig. 6. Simulative results of relationship among radial suspension force, displacement stiffness and displacement of radial AMB. (a) Relationship between radial suspension force and displacement. (b) Relationship between displacement stiffness and displacement.

236.8643

Current Stiff (N/A)

250

Force (N)

200 150 100 50 0 -1

-1 -0.5

-0.5

0

0

0.5

0.5

Current (A)

1

1

236.8643 236.8643 236.8643 236.8643 236.8643 -1

-1 -0.5

-0.5

0

0

0.5

0.5

Current (A)

(a)

Current (A)

1

1

Current (A)

(b)

Fig. 7. Simulative results of relationship among radial suspension force, displacement stiffness and current of radial AMB. (a) Relationship between radial suspension force and current. (b) Relationship between current stiffness and current.

(a)

(b)

Fig. 8. Experimental results of relationship among stiffness, displacement and current. (a) Relationship between displacement stiffness and displacement. (b) Relationship between current stiffness and displacement.

the thickness of them is 17 mm, six same YBaCuO bulk superconductors are arranged in a circle ring.

Before the rotor suspension system begins to work, the vacuum enclosure is vacuumized. When the axial AMB is not powered and


F supz ¼ ksupz da

ð4Þ

For the presented SMB, ksupz is 23 N/mm. In general, the radial stiffness ksupr is nearly about one thirds of axial stiffness, that is to say the radial stiffness ksupr is about 7.6 N/mm when the gap is 4 mm.

degree of the rotor is controlled by electromagnets. The electromagnets at opposite sides attract the rotor, and the total force acting on the rotor is the sum of the forces of the electromagnets. Because the interaction between ferromagnetic rotor and electromagnets is unstable, the position of the rotor must be measured by sensors, and the current in the coils is controlled to maintain this suspension. Although the AMB is a nonlinear actuator, the actuator will be approximately linear when the nonlinear current-force dependence is linearized by the biased permanent magnets or by supplying a bias current into both coils at opposite sides of the rotor. Step Response

-4

3

x 10

2.5 2

Amplitude (m)

the liquid nitrogen is not inputted into the cryogenic cooling system, the rotor is supported by the upper touchdown bearings, the minimum distance between superconductors and permanents is 4 mm. When the axial AMB is powered, the rotor is lifted upwards until it is oriented by the lower touchdown bearings, then the achieved maximum distance between the permanent magnets and superconductors, namely the cooling height, is 5.5 mm. At this moment, the liquid nitrogen is inputted and the HTSMB is field cooled. When the bulk superconductors are frozen absolutely, the power supplied to the axial AMB can be reduced gradually and the distance between the permanent magnets and superconductors reduces consequently. The designed maximum reduction is 1 mm and the operation cooling height is at least 4.25 mm. When the rotor rotates, the rotation speed degradation and the rotation power loss of the SMB were caused mainly by the braking magnetic force between the inhomogeneous magnetic field of the YBCO bulks and the induced eddy current in the PM rotor [16], but the HTS losses are fairly small to be neglected in this paper. When the operation cooling height changes due to the flux creep and flux flow, relaxation, losses and so on for SMB, the integrated axial/radial eddy current sensors can measure this change and the axial AMB then adjust it automatically to make the rotor be levitated stably. The static axial magnetic force of SMB in FC condition with 5 mm cooling gap is shown in Fig. 4. It is clear that there is hysteresis in the curves, but when the PM is moved away and back again by a small displacement, the minor force loop is reversible and it can be assumed that variations of magnetic forces are linear for small displacements [14,15]. The experimental results have indicated that the average axial stiffness of SMB is about 12 N/mm within the neighborhood of 5 mm gap while it increases to about 23 N/mm within the neighborhood of 4 mm gap. If the cooling gap of the axial SMB is given 5.25 mm and stable working position of rotor is 4.25 mm by decreasing the rotor downwards 1 mm, then the axial magnetic suspension force within the neighborhood of 4.25 mm can been simplified as average axial stiffness ksupz multiplying the drift da of rotor (mm)

1.5 1 0.5 0 -0.5

0

2

4

6

8

10

12

Time (s)

(a) Step Response

-7

20

x 10

15

Amplitude (m)

182

10

5

0

0

0.2

4. Active magnetic bearings

0.4

0.6

0.8

1

Time (s)

(b)

With the capability of high rotational speeds and good accuracy, AMB are applied in rotating machinery. In an AMB system, every

Step Response

-6

1.6

Table 1 Parameters for the ACESF.

x 10

1.4

Symbol

Quantity

Designed values

X J Jz m ksupz ksupr kR-AMBd kR-AMBc kA-AMBd kA-AMBc Lup/Llo

Speed of rotor Equatorial inertia moment of rotor Polar inertia moment of rotor Mass of rotor Axial stiffness of SMB Radial stiffness of SMB Displacement stiffness of radial AMB Current stiffness of radial AMB Displacement stiffness of axial AMB Current stiffness of axial AMB Distance between the center of rotor and the center of upper/lower AMB Distance between the center of rotor and the center of SMB Sensitivity of position sensor Proportional coefficient of the control circuit Differential coefficient of the control circuit

5233.33 rad/s 0.19995 kg m2 0.291762 kg m2 27.66 kg 23 kN/m 7.5 kN/m 630 kN/m 220 N/A 250–580 kN/m 520 N/A 0.048 m

Lsup ks kp kd

0.0945 m 0.075 V/um 6.03 0.6621

Amplitude (m)

1.2 1 0.8 0.6 0.4 0.2 0

0

0.2

0.4

0.6

0.8

1

Time (s)

(c) Fig. 9. Simulation results of radial translation of rotor suspended by different methods. (a) With SMB only. (b) SMB with PMB. (c) SMB with AMB.

183


where dr is the radial drift of rotor, dr is xc in x direction and dr is yc in y direction. ir is the control current in the inductance coils of radial AMB, and the subscript r represents x or y respectively. For examples, when the radial AMB is inputted 0 A, 0.1 A and 0.1 A respectively, the experimental results of relationship between displacement stiffness and displacement are depicted in Fig. 8a and relationship between current stiffness and displacement are depicted in Fig. 8b when rotor is in the center, drifted upward 0.05 mm or downwards 0.05 mm. From Fig. 8, we proved that the displacement of the radial AMB is close related with its position and it be constant when the rotor is in the center of the AMB, we also proved that the current stiffness is constant and does not change with the variety of current when the rotor is in the center of the radial AMB. When the rotor drifts, the current stiffness will change with its current proportionally. So the suspension force of radial AMB described by (5) is reasonable and feasible when the rotor is in its balanced position. The tested displacement stiffness kR-AMBd is 0.63 N/um and the tested current stiffness kR-AMBc is 220 N/A at the balanced position.

Fig. 10. Block diagram of tilting movement of rotor.

In this ACESF, there are two radial AMB: upper AMB and lower AMB. The radial AMB shown in Fig. 5 consists of installing sleeve, static laminated magnetic conductor with magnetic pole, static flux sleeve, permanent magnet, inductance coil, rotational laminated magnetic conductor and rotational flux sleeve. Between the static laminated magnetic conductor and the rotational laminated magnetic conductor, the magnetic working gap is formed. Because the radial AMB is permanent magnet biased hybrid magnetic bearings, the function of AMB is equivalent to the function of permanent magnetic bearing (PMB) plus the function of pure active magnetic bearing when a control current is supplied to it. If the control current is turned off, the function of AMB is equivalent to the function of PMB. Because the radial magnetic bearing is permanent magnetic biased style, the magnetic suspension force enlarges with the reducing of displacement, so the rotor in the balanced position tends to be unstable. In order to investigate suspension characteristics of the radial AMB, the simulative results of relationship among radial suspension force, displacement stiffness and displacement are depicted in Fig. 6. We can see from Fig. 6 that the displacement of the radial AMB is close related with its position and the displacement stiffness of radial AMB will increase greatly when the rotor is drifted to one of its magnetic poles. On the other hand, the displacement stiffness of radial AMB will be constant when the rotor is in the center of the AMB. At the same time, the simulative results of relationship among radial suspension force, displacement stiffness and current of radial AMB are depicted in Fig. 7. We can see from Fig. 7 that the suspension force of radial AMB changes proportionally with current while its current stiffness is constant and does not change with the variety of current when the rotor is in the center of the radial AMB. The suspension force FR-AMB of radial AMB in x direction or y direction is characterized clearly by displacement stiffness kRAMBd and current stiffness kR-AMBc at its balanced position.

F RAMB ¼ kRAMBd dr þ kRAMBc ir

5. Simulation and experimental results For the ACESF, the radial AMB-Up and radial AMB-Lo have same configuration parameters and are used to control the rotor’s radial movement and tilting. The displacement stiffness, current stiffness of radial AMB-Up is same as that of the radial AMB-Lo, respectively. The displacement stiffness, current stiffness in x direction is same as that in y direction for radial AMB-Up or AMB-Lo, respectively. And Llo = Lup = L. According to (4) and (5), we can obtain the equations of radial translation of rotor based on (2)

€xc ¼ ½ð2kRAMBd ksupr Þ xc þ 2kRAMBc ix =m €c ¼ ½ð2kRAMBd ksupr Þ yc þ 2kRAMBc iy =m y And the equations of tilting movement of rotor are

€ þ J Xu _ ¼ Lsup ksupr yc þ 2L kRAMBc iy Jw z € J Xw_ ¼ Lsup ksupr xc þ 2L kRAMBc ix Ju

If AMBs have PD control type, then the control current ir has the forms of kpdr + kdddr/dt. 5.1. Behavior of radial translation of rotor Generally, the rotor is well balanced, but the residual unbalance of rotor is unavoidable. For given 5 mg of unbalance mass, the unbalanced force acting on the rotor will exceed 1 N when the rotor rotates at a speed of 50,000 rpm. With these parameters

ð5Þ

0.1

0.2

0.08 0.06 0.04

0.05

y (mm)

y (mm)

0.1

0 -0.05

0.02 0 -0.02 -0.04

0.1

-0.06

-0.15 -0.1 -0.05

0

0.05

x (mm)

(a)

0.1

0.15

ð7Þ

z

0.15

-0.2 -0.15

ð6Þ

0.2

-0.08 -0.1 -0.1

-0.05

0

x (mm)

(b)

Fig. 11. Simulative traces of whirling of rotor (a) precession (b) nutation.

0.05

0.1

184


is still unstable even though the amplitude of vibration has decreased enormously due to the larger stiffness caused by PM in AMB. When the rotor is suspended by SMB and AMB with PD control type, the damping is introduced into the modeled system, the rotor becomes stable and its amplitude is small due to the larger stiffness and the introduced damping of the AMB.

1400

Frequencies of vibration (Hz)

1200

Nutation Nutation JzzΩ J Ω / JJ / Precession Precession

1000 800 600

5.2. Behavior of tilting movement of rotor 400

Set Mx and My as the moments acting on the rotor around x axis or y axis, respectively, we can draw the block diagram of the tilting movement of rotor suspended by AMB and SMB as shown in Fig. 10 based on (7) and its Laplace transform. Both Fig. 11 and (7) indicate that the tilting movement of rotor around x axis couples that around y axis through the angular momentum JzX, which indicates the gyroscopic effect of highspeed rotor. The moment around only x axis or y axis can make the rotor tilting around both x axis and y axis. For a slim rotor, the gyroscopic effect is slight, but for a discal rotor with large inertia, the gyroscopic effect is enormous. Based on the parameters of ACESF listed in Table 1, the simulative traces of the whirling of rotor are shown in Fig. 11, and their frequencies are shown in Fig. 12 when Jz/J = 1.459, X = 50,000 rpm and the angular stiffness of the rotor suspension system is 12,000 N m/rad. From Fig. 11, we can find that the simulative maximum amplitude of the nutation of rotor is about 0.05 mm while the simulative maximum amplitude of the precession of rotor is up to 0.15 mm, which exceeds the radial protective gap of the touchdown bearing. The simulation results prove that the rotor suspended by only SMB will be unstable at high speed. Based on prototype of superconducting ACESF shown in Fig. 13, the tested traces of nutation and precession are shown in Fig. 14 when the rotor suspended by only SMB. We can find that the maximum amplitude of the precession of rotor is up to 0.1 mm while the maximum amplitude of the nutation of rotor is about 0.018 mm, both of them are smaller than their corresponding simulative results respectively due to the large stiffness of PM introduced by AMB, but the amplitudes of the nutation and precession of rotor are too large to ensure the vector of angular momentum have precise and stable direction. To suppress the nutation and precession of the high-speed rotor for the ACESF, Mx and My are produced by radial AMB and SMB, Mx is Lsupksupr yc + 2L kR-AMBc iy and My is Lsupksupr xc + 2L kRAMBc ix in this case. There are two ways to suppress the negative effects of gyroscopic effect of rotor, one is robust control method, and the other is cross-feedback control based on the separated PD type control. But these two methods both depend on the control current iy in Mx and ix in My to realize the suppression of the

200 0 -200 0

100

200

300

400

500

600

700

800

900

Speed of rotor (Hz) Fig. 12. Frequencies of whirling of high-speed rotor.

Cryogenic system

Control system

Superconducting ACESF

Vacuum pump

Fig. 13. Prototype of superconducting ACESF.

listed in Table 1, the step responses of rotor suspended by only SMB, SMB with equivalent PMB or SMB with AMB are shown in Fig. 9 when the given external step disturbance is 1 N. We can find that the drift of rotor suspended by only SMB will not become zero since there is no damping in the modeled system and its amplitude is up to 0.27 mm, which exceeds the protective gap of the touchdown bearings and the rotor is unstable. Even if the protective gap of the touchdown bearings can be enlarged to avoid this collision, but the error of angular momentum of rotor is up to 31.8 N m s and will produce a large disturbance on the spacecraft when the rotor rotates at the speed of 50,000 rpm. When the AMB is not powered, the AMB is equivalent to a PMB, then the rotor suspended equivalently by SMB and PMB, the rotor 0.12

0.03

0.08 0.015

y (mm)

y (mm)

0.04 0 -0.04

0

-0.015

-0.08 -0.12 -0.12 -0.08 -0.04

-0.03 -0.015 0

x (mm)

(a)

0.04

0.08

0.12

0

0.015

x (mm)

(b)

Fig. 14. Tested traces of whirling of rotor (a) precession (b) nutation.

0.03

185

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

y (mm)

y (mm)


0

-0.05

0 -0.05

-0.1

-0.1

-0.15

-0.15

-0.2 -0.2 -0.15 -0.1 -0.05 0

0.05 0.1 0.15 0.2

-0.2 -0.2 -0.15 -0.1 -0.05 0

0.05 0.1 0.15 0.2

x (mm)

x (mm)

(a)

(b)

Fig. 15. Tested traces of rotor suspended by SMB together with AMB using cross-feedback based on separated PD control (a) the movement of the upper end of rotor (measured by sensor A) (b) the movement of the lower end of rotor (measured by sensor B).

nutation and precession of rotor. The tested traces of rotor suspended by SMB together with AMB using cross-feedback based on separated PD control are shown in Fig. 15, the dotted line circle with radius of 0.12 mm is the trace of the rotor determined by touchdown bearings, the amplitude of the vibration of rotor is 0.025 mm, which is only about one fifth of the value of the radial protective gap, all these prove that the rotor is suspended stably and the AMB is very important to improve the stability of the high-speed rotor for the ACESF. 6. Summary In the presented superconducting ACESF, the axial SMB is entirely passive and is used to stabilize the rotor and to support the rotor weight partially, the axial AMB does most to reduce the load of rotor over the axial SMB and to control its axial translation, a set of radial AMBs are used to control the rotor’s radial translation and tilting. By linearization, the force of SMB and AMB are simplified as stiffness multiplying the drift of rotor within the neighborhood of its balanced position. The displacement stiffness of the radial AMB is 630,000 N/m, which is about 82 times the radial stiffness of the axial SMB, and the current stiffness of radial AMB is 220 N/A. The translation of rotor suspended by only SMB, the displacement of rotor will not become zero and the amplitude of the vibration of rotor is large. The additional PMB can decrease the amplitude of the vibration of rotor due to its large stiffness but the rotor is not stable yet. When the rotor is suspended by axial SMB and AMB with PD control type, the rotor becomes stable and the amplitude of the vibration of rotor is small. For the highspeed rigid discal rotor with large inertia, the negative effect of gyroscopic effect of rotor is prominent, the AMB is very important to suppress the nutation and precession of rotor by robust control method or cross-feedback control based on the separated PD type control for the stability of the high-speed rotor in ACESF. Acknowledgment This work was supported in part by the National Natural Science Foundation of China under Grant 61174003.

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