research article design and mathematical analysis of a...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013, Article ID 836058, 17 pageshttp://dx.doi.org/10.1155/2013/836058

Research ArticleDesign and Mathematical Analysis of a NovelReluctance Force-Type Hybrid Magnetic Bearing for Flywheelwith Gimballing Capability

Chun’e Wang and Jiqiang Tang

School of Instrument Science and Opto-electronics Engineering, Beihang University, Beijing 100191, China

Correspondence should be addressed to Jiqiang Tang; tjq [email protected]

Received 13 March 2013; Revised 22 April 2013; Accepted 22 April 2013

Academic Editor: Igor Andrianov

Copyright © 2013 C. Wang and J. Tang.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Magnetically suspended flywheel (MSFW) with gimballing capability fulfills requirements of precision and maneuvers for spaceapplications. A novel reluctance force-type hybridmagnetic bearing (RFHMB) is presented based on analysis of demerits of Lorentzforce-type magnetic bearing and common RFHMB. It features that radial and axial magnetic bearing units are integrated into acompact assemblywith four separate biased permanentmagnets and two conical stators; four radial poleswith shoes and rotormadeof iron-based amorphousness can reduce eddy loss. Equivalent magnetic circuits of permanent magnets and their control currentsare presented. Simulation results indicate flux density fluctuates from 0.272 T to 0.41 T; rotor tilting does not affect the suspensionforce when rotor only tilts around X-axis or Y-axis. When rotor drifts in X, Y, or Z direction and tilts around X-axis or Y-axissimultaneously, force in corresponding directions slightly increases with tilting angle’s enlargement, but the maximum change doesnot exceed 14%. Additional tilting torque mainly determined by uniformity of flux density in conical air gaps is 0.05Nm whichis far smaller than 11Nm in common RFHMB; magnetic suspension force is effectively decoupled among X, Y, and Z directions;results prove that MSFW with gimballing capability theoretically meets maneuvering requirement of spacecraft.

1. Introduction

Flywheel is an inertial actuator in a spacecraft attitudecontrol system to generate suitable attitude control torquesfor correcting spacecraft attitude deviation or adjusting to anassigned attitude [1–4]. Flywheel generates attitude controltorque by changing the rotational speed of rotor; this torquehas high torque precision but cannot meet the maneuver-ing requirement fully for some space applications [5, 6].Compared with conventional ball bearing flywheels, themagnetically suspended flywheel (MSFW) has advantagesof no friction, free lubrication, low losses, and theoreticallyunlimited lifetime [7–11]. As for the 5-degree-of-freedom(DOF) MSFW, the rotor of it can be actively controlled inall 5 DOFs, so the momentum vector of the flywheel canbe tilted actively with respect to the spacecraft body (namedas “vernier gimballing”) to generate control moment in theother two directions [12, 13]. For these space applications ofan MSFW with gimballing capability, its structure has been

studied in [12, 14]. Bichler and Eckardt [15] and Yabuuchiet al. [16] proposed an attitude control scheme for largeangle attitude maneuvers; Kim et al. [14] researched theinteraction dynamics between a satellite and on board gim-balling MSFWs. Being superior to other attitude actuators,this kind of MSFW promises to fulfill the requirements ofboth precision and maneuvers and it is possible to decreasethe number of the needed flywheels to control the attitude ofspacecraft.

With respect to the MSFW with gimballing capability,the magnetic bearings can be divided into Lorentz force-type magnetic bearings and reluctance force-type magneticbearings according to the physical cause of the generationof the force. For the Lorentz force-type magnetic bearings,the force per unit length of conductor generated by the inter-action of the current

𝑖 and the flux density 𝐵 is orthogonal

to the flux lines, independent of the air gap, and linearlydependent on the current [17]. Bichler and Eckardt [15]and Eckardt [18] presented a Lorentz force-type magnetic

2 Mathematical Problems in Engineering

bearing for a low noise momentum wheel with gimballingcapability. For the reluctance force-type magnetic bearings,the magnetic force per unit area is given by 𝐵2/𝜇

0, where 𝐵 is

the magnetic flux density in the air gap and 𝜇0is the vacuum

permeability. Fang et al. [19, 20] and Tang et al. [19, 21]paid attention to how to design the structure of a permanentmagnet biased axial hybridmagnetic bearing, how to improveits gimballing control ability, and how to reduce its eddy losswhen the rotor rotates at a high speed [22]. However, dueto the magnetic flux being inversely proportional to the gaplength, the tilting angle of rotor is very small and cannot bebeyond 0.1∘ generally; the generated control moment is onlyabout 0.1 Nms, which is too small to meet the maneuveringrequirement of spacecraft fully.

In the presented new MSFW with gimballing capabilityshown in Figure 1, there are one rotor, one novel reluctanceforce-type hybrid magnetic bearing (RFHMB), one tiltingmagnetic bearing, one integrated axial/radial displacementsensor, one motor, two touchdown bearings, one shaft, andone base. The presented novel RFHMB, which is located inthe center of the flywheel, is used to control both the axialand the radial translation of the rotor. The tilting magneticbearing shown in Figure 2 is arranged in the outer rim of theflywheel rotor and is used to make the rotor rotate around𝑋- and 𝑌-axes by Lorentz force to generate large controlmoment due to the gyroscope effect of the high-speed rotor.The displacements and tilting angles are measured by theintegrated axial/radial displacement sensors. Two touchdownbearings are used to support the rotor when the magneticbearings are turned off or in case of failure, and allow therotor spin down from its full operating speed to zero. Dueto the novelty and specialty of RFHMB, the rotor of thisMSFW with gimballing capability can be tilted beyond 1∘and its outputting control moment is up to 3.3Nm; bothof them are larger by far than that of the existing MSFWwith gimballing capability. So the presented newMSFWwithgimballing capability canmeet themaneuvering requirementof spacecraft fully.

This paper emphasizes the design of this novel RFHMB;it features that the radial magnetic bearing unit and axialmagnetic bearing unit are integrated as a compact assembly.Permanent magnets are used to generate the radial and axialbiased fluxes to support the rotor in order to reduce theconsumption of power when this MSFW is on the earth.To improve the suspension properties of this RFHMB, thenovel structure of this RFHMBwith conical contours for axialsuspension is presented firstly to avoid torque coupling whenrotor is tilted. In order to decouple the suspension forces notonly between 𝑋 and 𝑌 dimension but also between radialdirection and axial direction, suspension properties in radialand axial directions, torque characteristics, and couplingpropertieswhen rotor is shifted or tilted are researched deeplyby theoretical analysis and finite element method. Theseresearch results indicate that the additional tilting torque ofthe bearing can be greatly decreased and the magnetic fluxesamong 𝑋, 𝑌, and 𝑍 directions are almost decoupled fromeach other. The MSFW with gimballing capability can meetthe maneuvering requirement of spacecraft theoreticallybecause of the presented novel RFHMB.

2. Demerits of Existing ReluctanceForce-Type Magnetic Bearing for Flywheelwith Gimballing Capability

For the MSFW with gimballing capability, the reluctanceforce-type magnetic bearings used in it can be classifiedinto two types generally according to their arrangement andstructure. One type shown in Figure 3 includes an upper 2-DOF radial magnetic bearing (2-DOF RMB-UP), a lower 2-DOF radial magnetic bearing (2-DOF RMB-LO), an upper 1-DOF axial magnetic bearing (1-DOF AMB-UP), and a lower1-DOF axial magnetic bearing (1-DOF AMB-LO). The struc-tures and photos of 2-DOF RMB and 1-DOF AMB are shownin Figure 4, respectively. Another type including one 2-DOFRAM, an upper 3-DOF axial magnetic bearing (3-DOFAMB-UP ), and a lower 3-DOF axial magnetic bearing (3-DOF AMB-LO ) is shown in Figure 5, the structure of 2-DOFRMB is the same as that in Figure 4(b), and the structuresand photos of 3-DOF AMB are shown in Figure 6 [21]. Allof the 2-DOF RMB, 1-DOF AMB, and 3-DOF AMB arepermanentmagnet biasedmagnetic bearings; having positivedisplacement stiffness is one of their intrinsic characteristics.

For the MSFW shown in Figure 3, set 𝐽𝑝, 𝐽𝑒, andΩ as the

polar inertial moment, equatorial inertial moment, and therotational speed of rotor, respectively. If we want to tilt therotor around𝑋-axis (or 𝑌-axis) anticlockwise for angle 𝜑 perunit time by this pair of 2-DOFRMBs to generate the requiredgyroscope momentum𝑀req, then

𝑀req = 𝐽𝑝Ω × ��. (1)

The generation of additional tilting torque is shown inFigure 7, where𝐿

𝑟presents the distance between the center of

2-DOF RMB-UP or 2-DOF RMB-LO and𝑌-axis. If we denote𝐹ru-l and 𝐹ru-r as the left suction force and the right suctionforce of 2-DOF RMB-UP , respectively, their unbalanced forceis 𝐹ru-r − 𝐹ru-l. In the same manner, we denote 𝐹rl-l and 𝐹rl-ras the left suction force and right suction force of 2-DOFRMB-LO , their unbalanced force with being 𝐹rl-l − 𝐹rl-r, thenthe required driving momentum𝑀

𝑑is

𝑀𝑑= 𝐽𝑒⋅ �� = (𝐹ru-r − 𝐹ru-l + 𝐹rl-l − 𝐹rl-r) ⋅ 𝐿𝑟. (2)

Then (1) becomes

𝑀req = 𝐽𝑝Ω × [𝐹ru-r − 𝐹ru-l + 𝐹rl-l − 𝐹rl-r

𝐽𝑒

⋅ 𝐿𝑟] . (3)

However due to the tilting angle, the right suction force of1-DOF AMB-UP presented by 𝐹au-r becomes larger than itsleft suction force 𝐹au-l, and the left suction force of 1-DOFAMB-LO presented by 𝐹al-l becomes larger than its rightsuction force 𝐹al-r. If the distance between the force centerof 1-DOF AMB-UP or 1-DOF AMB-LO and𝑍-axis is 𝐿

𝑎, then

the additional torque𝑀𝑎acting on the rotor produced by 1-

DOF AMBs is

𝑀𝑎= (𝐹au-r − 𝐹au-l + 𝐹al-l − 𝐹al-r) ⋅ 𝐿𝑎. (4)

Mathematical Problems in Engineering 3

Novel RFHMB

Touchdown bearing Integrated axial/radial displacement sensor

Tilting magnetic bearing Rotor

Shaft Base Motor

S S

SSS

SS

S

S

S

N

N

N

NN

NN

N

N

N

Figure 1: Momentum flywheel with gimballing capability.

Windings Framework Windings Framework

Figure 2: Stator of Lorentz force-type tilting magnetic bearing.

Rotor

Shaft 1-DOF AMB-UP

2-DOF RMB-UP

1-DOF AMB-LO

2-DOF RMB-LO

Figure 3: MSFW with 2-DOF RMBs and 1-DOF AMBs.

So the actual outputting gyroscope momentum 𝑀act ofthe rotor is

𝑀act = 𝐽𝑝Ω × [(𝐹ru-r − 𝐹ru-l + 𝐹rl-l − 𝐹rl-r) ⋅ 𝐿𝑟

+ (𝐹au-r − 𝐹au-l + 𝐹al-l − 𝐹al-r) ⋅ 𝐿𝑎] × (𝐽𝑒)−1

.

(5)

The error between the required gyroscope momentum𝑀req and the actual outputting gyroscope momentum𝑀actis 𝐽𝑝Ω× [(𝐹au-r −𝐹au-l +𝐹al-l −𝐹al-r) ⋅ 𝐿𝑎]/𝐽𝑒, which is difficult

to be measured or be compensated effectively, so the attitudeof the spacecraft cannot be controlled accurately.

Similarly, if the rotor is tilted around 𝑋-axis (or 𝑌-axis)anticlockwise for angle 𝜑 per unit time by outer torque,which maybe produced by some special mechanism suchas the tilting magnetic bearings shown in Figures 1 and 2to generate the required gyroscope momentum, the errorbetween the required gyroscope momentum 𝑀req and theactual outputting gyroscope momentum 𝑀act is 𝐽

𝑝Ω ×

[(𝐹ru-r−𝐹ru-l+𝐹rl-l−𝐹rl-r)⋅𝐿𝑟+ (𝐹au-r−𝐹au-l+𝐹al-l−𝐹al-r)⋅𝐿𝑎]/𝐽𝑒,which is larger than that produced by a pair of 2-DOF RMBs.

For the MSFW shown in Figure 5, the generation ofadditional tilting torque is shown in Figure 8. Setting thedistance between the force center of 3-DOF AMB-UP or 3-DOF AMB-LO and 𝑍-axis as 𝐿

𝑎, when the rotor is tilted

around 𝑋-axis (or 𝑌-axis) anticlockwise for angle 𝜑 perunit time by this pair of 3-DOF AMBs to generate therequired gyroscope momentum 𝑀req, the required drivingmomentum𝑀

𝑑provided the left suction force and the right

suction force of 3-DOF AMB-UP presented by 𝐹au-l and 𝐹au-r;the left suction force and the right suction force of 3-DOFAMB-LO presented by 𝐹al-l and 𝐹al-r are

𝑀𝑑= 𝐽𝑒⋅ �� = (𝐹au-r − 𝐹au-l + 𝐹al-l − 𝐹al-r) ⋅ 𝐿𝑎. (6)

For the 2-DOF RMB in this MSFW, the left suction forcepresented by 𝐹r-l equals the right suction force presented by𝐹r-r, and both of them cross through the centroid of rotorO, and then the torque made by them is zero and the actualoutputting gyroscope momentum 𝑀act of the rotor is notaffected. But the tilting angle of rotor is very small and cannotbe beyond 0.1∘ generally because the air gaps of both 2-DOFRAM and 3-DOF AMB are about 0.3mm, which is too smallfor the rotor to tilt about a large angle.

If the rotor is tilted around 𝑋-axis (or 𝑌-axis) anticlock-wise for angle 𝜑 per unit time by outer torque produced bytiltingmagnetic bearings shown in Figures 1 and 2 to generatethe required gyroscope momentum, the torque presented by(6) becomes the additional torque acting on the rotor, so theerror between the required gyroscope momentum𝑀req andthe actual outputting gyroscope momentum 𝑀act is 𝐽𝑝Ω ×

(𝐹au-r − 𝐹au-l + 𝐹al-l − 𝐹al-r) ⋅ 𝐿𝑎/𝐽𝑒, which is also difficult tobe measured or be compensated effectively, so the attitude ofthe spacecraft cannot be controlled accurately either.


SS

NN

(a)

𝑌

𝑋

(b)

Figure 4: Structure and photos of magnetic bearings, (a) 1-DOF AMBs, (b) 2-DOF RMBs.

Rotor

3-DOF AMB-UP

2-DOF RMB

3-DOF AMB-LO

Figure 5: MSFW with one 2-DOF RAM and two 3-DOF AMBs.

However, no matter what the additional torque is, theadditional torque acting on the rotor is equal to the productof the unbalanced force and torque arm of the bearing.Even if the positive displacement stiffness is the intrinsiccharacteristic of AMB or RMB and it is difficult to reducethese unbalanced forces when the rotor tilts, it is preferableto reduce the torque arm of AMB or RMB by adopting thenovel RFHMB to make the additional torque to be zeroapproximately.

Stator pole

Windings

S SN N

𝑋

𝑌

Figure 6: Structure and photos of 3-DOF AMB.

3. Structure Design and Magnetic Analysis

3.1. Structure. Yabuuchi et al. [16], Sawada et al. [13], andHoriuchi and Inoue [23] presented a conical pure active reluc-tance force-type magnetic bearing for the attitude control ofsatellite. When these kinds of magnetic bearings are used inMSFW with gimballing capability, the air gap of them arelarger than those in common magnetic bearings to enlargethe tilt angle of the flywheel rotor, but they will consumeplenty of power to support the rotor on the earth. Since


𝐹au-l

𝐹ru-l 𝐹ru-r

𝐹au-r

𝐿a

𝐿r

𝐹rl-l𝐹rl-r

𝐹al-r

𝐹al-l

𝑋

𝑍

𝑌 𝜑

𝜑

Figure 7: Generation of additional tilting torque on rotor forMSFWwith 2-DOF RMBs and 1-DOF AMBs during tilting.

𝐹au-l

𝐹au-r

𝐿r

𝐿a

𝑋

𝑍

𝑌𝐹r-l 𝐹r-r

𝐹al-r

𝐹al-l

𝑂 𝜑

Figure 8: Generation of additional tilting torque on rotor forMSFWwith 2-DOF RMBs and 1-DOF AMBs during tilting.

the force generated by conical stator has component forcesin both radial and axial directions, it is crucial to solve theproblem of force decoupling to make its control easier. Whenthe flywheel rotor tilts with respect to the stator, the air gapis not uniform any more and the flux in one channel (e.g., 𝑋channel) is prone to leak into adjacent channel (𝑌 channel),which will make the coupling between different channelsmore serious. In order to solve the problem of couplingbetween radial and axial forces in conical magnetic bearing,Watkins et al. proposed a structure with pole piece separatedinto two parts tapering mirror images of one another [24],but this structure will result in large additional tilting torquewhen it is used in MSFW with gimballing capability.

For this presented novel RFHMB as shown in Figure 9,its main function is to ensure that the rotor of this MSFWbe suspended stably in the central position, so the specialstructure of RFHMB is our original novelty and how toanalyze its suspension properties is our main contributions.In this novel RFHMB, the radial magnetic bearing unitand axial magnetic bearing unit are integrated as a hybridreluctance force-type magnetic bearing assembly. Permanentmagnets are used to generate the radial and axial bias fluxesto suspend the rotor without consumption lots of powerwhen this MSFW is on the earth. The radial stators shownin Figure 10 include 4 stator poles and are located in thecenter of RFHMB, their relative permanent magnets are

also separated into four separate segments by nonmagneticmaterials among them. Two stator poles in the same channelare connected by the stator yoke to be a stator subassembly in𝑋 or 𝑌 channels. These radial stator subassemblies of 𝑋 and𝑌 channels are connected perpendicularly with nonmagneticmaterial between them, so the magnetic flux of 𝑋 and 𝑌channels are independent from each other. There are twoconical stators, one is located at the upper end and anotheris located at the lower end, and each axial conical stator hasone control winding.These conical stators are used to controlthe translation of rotor in𝑍 direction only and have no effectson these forces in radial direction. Because these two conicalworking air gaps formed between the stator and rotor of thebearing can make the normal direction of the midst conicalfaces direct to the centroid of the rotor, so the torque arm ofthe bearing is approximately to zero, the additional torqueacting on the rotor can be reduced approximately to zeroconsequently when the rotor is tilted.

3.2. Magnetic Analysis. The rotor of the MSFW with gim-balling capability is suspended in the central position by thenovel RFHMB. When the rotor is disturbed to move to 𝑍+direction, the upper conical air gap would become largerwhile the lower conical air gap would get smaller along 𝑍direction, so the flux density and force would get even smallerat the upper conical stator, and the rotor would subject to anadditional force in 𝑍+ direction. The integrated axial/radialdisplacement sensor can detect this axial translation of rotorand transmit related signals to the control system to preventthe rotor from translating further in 𝑍+ direction. Then thecontrol system sends switching signals to generate controlcurrent through these axial windings; this process results ina suitable corrective control flux. The novel RFHMB withflux paths in 𝑋 direction is shown in Figure 11, where thebiased fluxes generated by permanent magnets are denotedby the solid lines and the control flux generated by windingsare denoted by dash line. The biased flux generated from thepermanent magnet, which is radically magnetized, is dividedinto two parts after passing through the radial air gaps: oneis the biased flux flowing across the radial stator pole, airgap, the rotor, upper axial air gap, and its upper axial stator;the other is a flux flowing across the radial stator pole, theair gap, the rotor, the lower axial air gap, and its lower axialstator. The control flux generated by the current in axialcontrolling windings flows across the upper stator pole, theupper axial air gap, the rotor, the lower axial air gap, and thelower stator pole. By adding the biased flux at the large airgap side and subtracting the biased flux at the small air gapside, the bearing can produce a restoring force along the axialdirection.

When the rotor moves in 𝑋+ direction, the flux densityand force under the𝑋+ radial stator pole decreasewhile thoseunder the 𝑋− radial stator pole increase, so an additionalforce is generated in𝑋+ direction. Due to the conical statorsin axial magnetic bearing, their flux density and force wouldalso decrease under the𝑋+ part. Since the component forcesin 𝑍 direction generated by the upper and lower conicalstators can be counteracted with each other, the component


Upper axial conical stator pole

Lower axialconical stator pole

radial stator pole

4 separate permanent magnets

4 separate radial windings

Upper axial windings

Lower axial windings N

NS

S

Base

Conical surface

Conical surface

4 separate

Conical surface

Conical surface

Figure 9: Schematic diagram of novel RFHMB.

Radial windings Separate PM

Stator yoke

Radial stator pole

(a) (b) (c)

Figure 10: Radial stator of the novel RFHMB, (a) one stator subassembly in 𝑋 or 𝑌 channels, (b) combination of two stator assemblies, (c)radial stator.

force in 𝑋 direction is only left. The integrated axial/radialdisplacement sensor can also detect this state and transmitrelated signals to the control system to prevent the rotorfrom translating further in 𝑋+ direction. The control fluxgenerated by the control current in radial windings in 𝑋channel is represented by dash line in Figure 12; it flowsthrough the𝑋+ radial air gap, the𝑋 channel stator yoke, andthe X− radial air gap; then the bias flux in the 𝑋+ directionis enhanced and the biased fluxes in the 𝑋− direction isdecreased; therefore, a restoring force is produced and therotor is moved back toward the central position.

If we define𝐹pm and𝑅pm as themagneticmotive force andreluctance of each permanent magnet (A), 𝛿

𝑥, 𝛿𝑦, and 𝛿

𝑧as

the displacement in 𝑋, 𝑌, and 𝑍 directions (m), respectively,the reluctance of radial air gap in radial magnetic bearingunit in 𝑋+ direction is presented by 𝑅

𝑥+(1/H), that in 𝑋−

direction is presented by 𝑅𝑥−

(1/H). Similarly, 𝑅𝑦+

and 𝑅𝑦−

present reluctance of radial air gap in𝑌+ and𝑌− direction forthe radial magnetic bearing unit (1/H); 𝑅

𝑧+and 𝑅

𝑧−present

reluctance of air gap of axial magnetic bearing unit in 𝑍+and 𝑍− direction (1/H), respectively. At the same time, thecontrol currents in 𝑋 direction and 𝑌 direction for radialwindings and that for axial windings are presented by 𝐼

𝑥,

𝐼𝑦, and 𝐼

𝑧(A); their relative magnetic motive forces of each

control windings are presented by 𝐹𝑐𝑥(A), 𝐹

𝑐𝑦(A), and 𝐹

𝑐𝑧(A),

respectively. 𝐹𝑐𝑥= 𝑁𝑟𝐼𝑥, 𝐹𝑐𝑦= 𝑁𝑟𝐼𝑦, and 𝐹

𝑐𝑧= 𝑁𝑎𝐼𝑧, where

𝑁𝑟is the turn number of the radial control windings and𝑁

𝑎

is the turn number of the axial control windings.Considering the influence of the leaked flux, the equiv-

alent magnetic circuits of permanent magnets in this novelRFHMB are simplified as shown in Figure 13, and its equiv-alent magnetic circuits of control current are simplified asshown in Figure 14. From Figures 13 and 14, we can findout that not only the magnetic fluxes of the radial magneticbearing unit in𝑋 and 𝑌 channels are independent from eachother, but also the magnetic fluxes of these conical statorshave no effect on that of the radial magnetic bearing unit.Even if the air gap becomes uniform badly when the flywheelrotor tilts with respect to the stator, there are no couplebetween𝑋 and𝑌 channels in the radialmagnetic bearing unitor between radial magnetic bearing unit and axial magneticbearing unit at all. So the difficult problem of decouplingbetween different channels is solved by this presented specialstructure accordingly.

According to the equivalent magnetic circuit shown inFigures 13 and 14, the biased fluxes and control fluxes of each


𝑆𝑆 𝑁𝑁𝑍

𝑋

Figure 11: Novel RFHMB with paths of flux in𝑋 direction.

air gaps can be calculated out according to the Ohm’s Law ofthe magnetic circuit:

[

[

[

[

[

[

[

[

Φ𝑝𝑥+

Φ𝑝𝑥−

Φ𝑝𝑦+

Φ𝑝𝑦−

Φ𝑝𝑧+

Φ𝑝𝑧−

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

𝑅𝑥

𝑅𝑥+

0

𝑅𝑥

𝑅𝑥−

0

0

𝑅𝑦

𝑅𝑦+

0

𝑅𝑦

𝑅𝑦−

𝑅𝑧

𝑅𝑧+

𝑅𝑧

𝑅𝑧+

𝑅𝑧

𝑅𝑧−

𝑅𝑧

𝑅𝑧−

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

⋅

[

[

[

[

[

2𝐹pm

𝜎𝑝(𝑅pm + 𝑅𝑥 + 𝑅𝑧)

2𝐹pm

𝜎𝑝(𝑅pm + 𝑅𝑦 + 𝑅𝑧)

]

]

]

]

]

, (7)

where 𝜎𝑝is the leakage coefficient of the biased flux and 𝜎

𝑟

and 𝜎𝑧are the leakage coefficients of the radial and axial

control fluxes, respectively. 𝑅𝑥= 1/(1/𝑅

𝑥++ 1/𝑅

𝑥−), 𝑅𝑦=

1/(1/𝑅𝑦++ 1/𝑅

𝑦−), and 𝑅

𝑧= 1/(1/𝑅

𝑧++ 1/𝑅

𝑧−).

And the control flux in each channel can be expressed as

Φ𝑐𝑥=

2𝐹𝑐𝑥

[(𝑅𝑥++ 𝑅𝑥−) ⋅ 𝜎𝑟]

,

Φ𝑐𝑦=

2𝐹𝑐𝑦

[(𝑅𝑦++ 𝑅𝑦−) ⋅ 𝜎𝑟]

,

Φ𝑐𝑧=

2𝐹𝑐𝑧

[(𝑅𝑧++ 𝑅𝑧−) ⋅ 𝜎𝑧]

.

(8)

Hence the total flux of each air gap can be expressed as

Φ𝑥+= Φ𝑝𝑥+

+ Φ𝑐𝑥, Φ

𝑥−= Φ𝑝𝑥−

− Φ𝑐𝑥,

Φ𝑦+= Φ𝑝𝑦+

+ Φ𝑐𝑦, Φ

𝑦−= Φ𝑝𝑦−

− Φ𝑐𝑦,

Φ𝑧+= Φ𝑝𝑧+

+ Φ𝑐𝑧, Φ

𝑧−= Φ𝑝𝑧−

− Φ𝑐𝑧,

(9)

where Φ𝑥+

is the total flux of bias and control fluxes in X+direction, Φ

𝑥−is the total flux in X− direction, Φ

𝑦+and Φ

𝑦−

are the total fluxes in 𝑌+ and 𝑌− directions, respectively, andΦ𝑧+

and Φ𝑧−

are the total fluxes in 𝑍+ and 𝑍− directions,respectively.

In order to express the magnetic field distribution of thenovel RFHMB explicitly, the 3D FEM model of the designedRFHMB is set up in ANSYS 11.0. When the rotor locatesat the equilibrium position, the static biased magnetic fielddistribution of the bearing is obtained as shown in Figure 15.

4. Analysis of Eddy Loss

In the novel RFHMB, the segmented radial stator poleswill make the magnetic density in the radial rotor unevenunavoidably; what will result in the enormous eddy loss whenthe rotor rotates at high speed.The power consumption of theRFHMB includes the following: (1) copper loss 𝑃cu caused bythe windings in RFHMB, and𝑃cu = 𝑖

2𝑅, where i is the current

in the winding and R is the resistant of the winding; (2) ironloss is caused by the magnetic field, which is produced notonly by the alternative modulating current in windings butalso by the biased permanent magnets. Iron loss includesstatic suspension iron loss and rotational iron loss and canalso be divided into hysteresis loss 𝑃

ℎand eddy loss 𝑃

𝑒:

𝑃ℎ= 𝐾ℎ∗ 𝑓ℎ∗ 𝐵

1.6

𝑚𝑉𝑓𝑒, (10)

𝑃𝑒= 𝐾𝑒Δ

2𝑓

2

ℎ𝐵

2

𝑚𝑉𝑓𝑒, (11)

where 𝐾ℎis the material constant, 𝐾

𝑒is coefficient of the

material constant, 𝑓ℎis the alternative frequency of the

magnetic field produced by the alternative current, 𝐵𝑚

isthe magnitude of the changed magnetic density for thealternating magnetic field, Δ is the thickness of the material,and 𝑉

𝑓𝑒is the volume of the iron core for the magnetic pole

[21].With respect to the RFHMB, there is a biased magnetic

field and free of magnetic field alternation between positiveand negative, so the hysteresis loss 𝑃

ℎis small enough to

be neglected. The eddy loss 𝑃𝑒is an important issue to

be focused on. According to (11), we can reduce the eddyloss of the novel RFHMB by optimization of stator poles toreduce themagnitude of the changedmagnetic density for thealternating magnetic field and selection of the soft magneticmaterial with low loss properties such as laminated material,iron-based Nanocrystalline, and amorphousness.

When the stator pole of RFHMB is divided into foursegments, a qualitative conclusion can be drawn that themag-netic density in the radial rotor plate produced by these poleswould change when the rotor rotates. In order to make thedifference of the magnetic density affected by the permanent


𝑆𝑆 𝑁𝑁𝑜

Y

X

Figure 12: Paths of radial flux in𝑋 direction for the novel RFHMB.

𝐹pm 𝑅pm

Φpmy

Φpmx

Φpz+

Φpz−

𝐹pm 𝑅pm

𝐹pm

𝐹pm

𝑅pm

𝑅pm

𝑅x+

𝑅x−

𝑅y+

𝑅y−

𝑅z+

𝑅z−

Φpx+

Φpx−

Φpy+

Φpy−

Figure 13: Equivalent magnetic circuits of permanent magnets inthe RFHMB.

𝑅x+

𝑅x−

𝑅z+

𝑅z−

𝑅y+

𝑅y−

Φcx

Φcy

Φcz

𝐹cx

𝐹cx

𝐹cy

𝐹cy

𝐹cz

𝐹cz

Figure 14: Equivalent magnetic circuits of control current.

Nodal solutionStep = 2Sub = 1Time = 2BSUM (avg)RSYS = 14SMN = 0.716𝐸−05SMX = 2.051

1

0.716𝐸

−05

0.22

7882

0.45

5757

0.68

3632

0.91

1507

1.13

9

1.36

7

1.59

5

1.82

3

2.05

1

Figure 15: Magnetic field distribution of the RFHMB.

magnets and/or the control current in the air gap smaller, wedesigned a special structure of the stator pole with shoes forthe novel RFHMB.The original stator poles without shoes areshown in Figures 16(a); 16(b) is the diagram of stator withpoles shoes. FromFigure 16, we can findout that the clearancebetween poles has been reduced from 30

∘ to 5∘. In orderto satisfy the requirement of high magnet energy productand small temperature coefficient, these separate permanentmagnets in the RFHMB are samarium cobalt permanentmagnet and their hysteresis force 𝐻cb and remanence 𝐵

𝑟are

larger than 760KA/m and 1.05 T, respectively. When the airgap 𝛿 of the RFHMB is 0.25mm, the radial distributions ofmagnetic density affected by only the permanent magnets or


𝑋

𝑌

30∘

30 ∘

30∘

30 ∘

(a)

𝑋

𝑌

5 ∘

5∘

5 ∘

5∘

(b)

Figure 16: Radial stator of the novel RFHMB, (a) without pole shoes, (b) with pole shoes.

0 40 80 120 160 200 240 280 320 3600

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Radi

al m

agne

tic fl

ux d

ensit

y Br

(𝑇)

Circumferential angle (∘)

(a)

Nodal solutionStep = 2Sub = 1Time = 2BSUM (avg)RSYS = 14SMN = 0.106𝐸−03SMX

Path

= 0.478801

10.106𝐸

−03

0.05

3294

0.10

6483

0.15

9671

0.21

2859

0.26

6047

0.31

9236

0.37

2424

0.42

5612

0.47

8801

(b)

Figure 17: Radial flux density distribution affected only by the permanent magnets in the air gap of RFHMB without pole shoes, (a) radialflux density curves, (b) radial flux density distribution.

both permanentmagnets and control current in the air gap ofthe novel RFHMB without pole shoes are depicted in Figures17 and 18.The radial distributions ofmagnetic density affectedby only the permanent magnets or both permanent magnetsand control current in the air gap of the novel RFHMB withpole shoes are depicted in Figures 19 and 20.

From Figure 17 to Figure 20, we can find out that dueto the existence of pole shoes, the areas of the radial polesfor the novel RFHMB are enlarged; then the maximummagnetic density affected by only permanent magnets in theair gap is reduced from 0.47 T to 0.38 T while that affectedby permanent magnets and control current 0.1 A in the air

gap is reduced from 0.6 T to 0.52 T; the variations ofmagneticdensity are about 0.08 T; and the reduction of the maximummagnetic density will reduce the eddy loss of the novelRFHMB effectively. On the other hand, we also can findout that due to the existence of pole shoes, the clearancebetween poles has been reduced from 30

∘ to 5∘, althoughthe fluctuation of the maximum magnetic density affectedby only the permanent magnets or both permanent magnetsand control current in the air gap still exists but the dutyratio between the maximummagnetic density andminimummagnetic density increases greatly, so the eddy loss of thenovel RFHMB will be reduced accordingly.


0 40 80 120 160 200 240 280 320 3600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Radi

al m

agne

tic fl

ux d

ensit

y Br

(𝑇)


(a)

Nodal solutionStep = 2Sub = 1Time = 2BSUM (avg)RSYS = 14SMN = 0.256𝐸−03SMXPath

= 0.611012

1

0.256𝐸

−03

0.06

8118

0.13

598

0.20

3841

0.27

1703

0.33

9565

0.40

7427

0.47

5289

0.54

3151

0.61

1012

(b)

Figure 18: Radial flux density distribution affected by the permanent magnets and the control current 0.1 A in the air gap of RFHMBwithoutpole shoes, (a) radial flux density curves, (b) radial flux density distribution.

0 40 80 120 160 200 240 280 320 3600

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Radi

al m

agne

tic d

ensit

y Br

(𝑇)


(a)

Nodal solutionStep = 2Sub = 1Time = 2BSUM (avg)RSYS = 14SMN = 0.003448SMXPath

= 0.391381

1

0.00

3448

0.04

6552

0.08

9656

0.13

2759

0.17

5863

0.21

8967

0.26

207

0.30

5174

0.34

8278

0.39

1381

(b)

Figure 19: Radial flux density distribution affected by the permanent magnets in the air gap of RFHMBwith pole shoes, (a) radial flux densitycurves, (b) radial flux density distribution.

According to (11), we can find out that the soft magneticmaterials with low loss properties such as laminatedmaterial,iron-based Nanocrystalline, and amorphousness have greateffect on the eddy loss of the novel RFHMB. Some testedresults of eddy loss for different soft magnetic materialshave verified that the iron loss per unit mass of iron-basedNanocrystalline with 0.03mm films is the smallest and theeddy loss of the novel RFHMB is smaller when its rotor

made of Nanocrystalline with 0.03mm films. Because theiron-based Nanocrystalline with 0.03mm films is very crispand always in reeled state, it is difficult to handle themfurther, then the iron-based amorphous with 0.1mm films isemployed for radial rotor of the novel RFHMB.

For a MSFW with polar angular momentum is 68N⋅m⋅swhen its rotor rotates at the speed of 5000 r/min, the photoof the novel RFHMB is shown in Figure 21, and its main


0 40 80 120 160 200 240 280 320 360−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Radi

al m

agne

tic fl

ux d

ensit

y Br

(𝑇)


(a)


= 0.531598

1

0.00

3307

0.06

2006

0.12

0705

0.17

9404

0.23

8103

0.29

6802

0.35

5501

0.41

42

0.47

2899

0.53

1598

(b)

Figure 20: Radial flux density distribution affected by the permanent magnets and the control current 0.1 A in the air gap of RFHMB withpole shoes, (a) radial flux density curves, (b) radial flux density distribution.

Figure 21: Photo of the stator of the novel RFHMB.

parameters are listed in Table 1.The presented novel RFHMBcan make the rotor title around 𝑋- or 𝑌-axis about 1∘ andthe generated control moment is up to 3.3N⋅m. It is verifiedthat this novel RFHMB is very helpful to make MSFW withgimballing capability meet the maneuvering requirement ofspacecraft fully.

5. Suspension Properties

5.1. Noncoupling Suspension Forces. By the principle of virtualwork, the component force 𝐹

𝑟𝑥in 𝑋 direction and the com-

ponent force 𝐹𝑟𝑦in 𝑌 direction of the resultant forces acting

on the rotor generated by the radial stators are integrated as

𝐹𝑟𝑥= ∫

𝛼12

𝛼11

𝜇0𝑟𝑠ℎ𝑟𝑠(Φ𝑥+𝑅𝑥+)

2

⋅ cos 𝜃2𝛿

2

𝑔(𝛿𝑟, 𝛾, 𝜃)

𝑑𝜃

+ ∫

𝛼22

𝛼21

𝜇0𝑟𝑠ℎ𝑟𝑠(Φ𝑥−𝑅𝑥−)

2

⋅ cos 𝜃2𝛿

2

𝑔(𝛿𝑟, 𝛾, 𝜃)

𝑑𝜃,

𝐹𝑟𝑦= ∫

𝛼32

𝛼31

𝜇0𝑟𝑠ℎ𝑟𝑠(Φ𝑦+𝑅𝑦+)

2

⋅ sin 𝜃2𝛿

2

𝑔(𝜌𝑟, 𝛾, 𝜃)

𝑑𝜃

+ ∫

𝛼42

𝛼41

𝜇0𝑟𝑠ℎ𝑟𝑠(Φ𝑦−𝑅𝑦−)

2

⋅ sin 𝜃2𝛿

2

𝑔(𝜌𝑟, 𝛾, 𝜃)

𝑑𝜃,

(12)

where 𝛼11, 𝛼21, 𝛼31, and 𝛼

41are the starting angles of the

four radial stator poles; 𝛼12, 𝛼22, 𝛼32, and 𝛼

42are the ending

angles of the four radial stator poles; 𝑟𝑠is the outer radius of

the stator; ℎ𝑟𝑠is the height of radial stator iron and 𝛿

𝑔is the

length of the radial air gaps, which is variable with the angle𝜃 around the circumference. 𝛿

𝑟is the rotor displacement in

radial direction, 𝛿𝑟= √𝛿

2

𝑥+ 𝛿

2

𝑦, 𝛾 is the direction angle of the

rotor displacement vector, and 𝛾 = arctan(𝛿𝑥/𝛿𝑦).

As for the force generated by the axial conical stator, itincludes the component force not only in𝑍 direction but alsoin𝑋 and 𝑌 directions. All the component forces 𝐹

𝑎𝑥, 𝐹𝑎𝑦and

𝐹𝑎𝑧can be integrated as

𝐹𝑎𝑥= ∫

2𝜋

0

∫

𝐿

0

𝜇0(𝑟𝑐− 𝑙 cot𝜑) (Φau𝑅au)

2

⋅ cos 𝜃2𝛿

2

au (𝜌𝑟, 𝛾, 𝜃, 𝑧)𝑑𝑙 𝑑𝜃

+ ∫

2𝜋

0

∫

𝐿

0

𝜇0(𝑟𝑐− 𝑙 cot𝜑) (Φal𝑅al)

2

⋅ cos 𝜃2𝛿

2

al (𝜌𝑟, 𝛾, 𝜃, 𝑧)𝑑𝑙 𝑑𝜃,


−0.2−0.15−0.1−0.0500.050.10.150.2

−0.5−0.3

−0.10.1

0.30.5

−400

−200

0

200

400

𝐹𝑥

(N)

𝐼𝑥 (A)

𝛿𝑥 (mm)

(a)

−0.3−0.2−0.10 0.1 0.2 0.3

−0.5−0.3

−0.10.1

0.30.5

−300

−200

−100

0

100

200

300

𝐹𝑧

(N)

𝐼𝑧 (A) 𝛿𝑧 (mm)

(b)

Figure 22: Characteristics of suspension force, (a) 𝐹𝑥versus 𝛿

𝑥and 𝐼𝑥, (b) 𝐹

𝑧versus 𝛿

𝑧and 𝐼𝑧.

𝐹𝑎𝑦= ∫

2𝜋

0

∫

𝐿

0


2

⋅ sin 𝜃2𝛿

2

au (𝜌𝑟, 𝛾, 𝜃, 𝑧)𝑑𝑙 𝑑𝜃

+ ∫

2𝜋

0

∫

𝐿

0


2

⋅ sin 𝜃2𝛿

2

al (𝜌𝑟, 𝛾, 𝜃, 𝑧)𝑑𝑙 𝑑𝜃,

𝐹𝑎𝑧= ∫

2𝜋

0

∫

𝐿

0


2

⋅ cos𝜑2𝛿

2

au (𝜌𝑟, 𝛾, 𝜃, 𝑧) ⋅ sin𝜑𝑑𝑙 𝑑𝜃

− ∫

2𝜋

0

∫

𝐿

0


2

⋅ cos𝜑2𝛿

2

al (𝜌𝑟, 𝛾, 𝜃, 𝑧) sin𝜑𝑑𝑙 𝑑𝜃,

(13)

where 𝑟𝑐is the maximum outer radius of the axial conical

stator, 𝐿 is the axial length of the conical stator iron, and 𝛿auand 𝛿al are the length of the upper and lower conical air gaps,respectively, both varying with the rotor displacement.

The resultant forces generated by both the radial and theconical stators in𝑋, 𝑌, and 𝑍 directions can be expressed as

𝐹𝑥= 𝐹𝑟𝑥+ 𝐹𝑎𝑥,

𝐹𝑦= 𝐹𝑟𝑦+ 𝐹𝑎𝑦,

𝐹𝑧= 𝐹𝑎𝑧.

(14)

Based on the above equations and parameters of the novelRFHMB listed in Table 1, the obtained force-displacementand force-current characteristics are shown in Figure 22.From Figure 22, we can find out that both the radial magneticforce and the axial magnetic force are nearly proportional tothe displacement of rotor and their relative control current.We also calculated out that the force-displacement stiffnessand force-current stiffness of radial magnetic bearing unitare 0.65N/𝜇m and −380N/A, and the force-displacement

Table 1: Main parameters of the RFHMB.

Symbol Quantity Designedvalues

𝜌𝑐0

Length of conical gap 0.8mm𝜌𝑟0

Length of radial gap 0.5mm𝐷𝑠ra Outer diameter of radial stator 116mm𝐷𝑠co Maximum outer diameter of conical stator 102mm𝐿𝑠ra Height of radial stator iron 17mm𝐿𝑠co Height of conical stator 6.5mm𝐻𝑐

Coercive force of permanent magnet 760 kA/m𝐵𝑟

Remanence of permanent magnet 1.05 T𝑁𝑟

Turns of radial control windings 220𝑁𝑎

Turns of axial control windings 180𝜎𝑝

Leakage coefficient of bias flux 1.45𝜎𝑟

Leakage coefficient of radial control flux 1.25𝜎𝑎

Leakage coefficient of axial control flux 1.2

stiffness and force-current stiffness of axial magnetic bearingunit are 0.34N/𝜇m and −318N/A, respectively.

In order to analyze the magnetic coupling among 𝑋,𝑌, and 𝑍 directions, the changes of magnetic forces withvariation of current and displacement are analyzed when therotor is drifted or tilted. Because of the symmetry of theRFHMB structure, not all the force coupling analyses in everydirection are presented here. For example, when the rotor isdrifted by a certain displacement (𝛿

𝑥) in 𝑋 direction and its

corresponding control current 𝐼𝑥is given in these windings

in this direction, the force in 𝑌 direction can be obtained incase that the offset 𝛿

𝑦= 0.1mm and the current 𝐼

𝑦= 0; the

coupling analysis between𝑋 and𝑌 channels are illustrated inFigure 23. The relationship among 𝐹

𝑦, 𝛿𝑥, and 𝐼

𝑥is shown in

Figure 23(a); the relationship among 𝐹𝑧, 𝛿𝑥, and 𝐼

𝑥is shown


70

68

66

64

62

601

0.50

−0.5−1 −0.2

−0.10

0.10.2

𝐹𝑦

(N)

𝐼𝑥 (A)

𝛿𝑥 (mm)

(a)

10.5

0−0.5

−1 −0.2−0.1

00.1

0.2

40

38

36

34

32

30

𝐹𝑧

(N)

𝐼𝑥 (A)

𝛿𝑥 (mm)

(b)

10.5

0−0.5

−1 −0.3 −0.2 −0.1 0 0.10.30.2

75

70

65

60

𝐹𝑥

(N)

𝐼𝑧 (A)

𝛿𝑧 (mm)

(c)

Figure 23: Force coupling among different channels, (a) 𝐹𝑦versus 𝐼

𝑥and 𝛿

𝑥, (b) 𝐹

𝑧versus 𝐼

𝑥and 𝛿

𝑥, (c) 𝐹

𝑥versus 𝐼

𝑧and 𝛿

𝑧.

in Figure 23(b); and the relationship among 𝐹𝑥, 𝛿𝑧, and 𝐼

𝑧is

shown in Figure 23(c), respectively.We can find out that the offset and control current in

windings in one direction have little influence on the forcein other directions. When the displacement in 𝑋 directionreaches 0.25mm,which is the length of protection air gap, thevariation of force in𝑌 direction is only 1.6% and the variationof force in 𝑍 direction is 2.5%, which can be calculated fromFigures 23(a) and 23(b). When the position of the rotor isfixed, the forces in 𝑌 and 𝑍 directions remain unchanged nomatter how large the current in 𝑋 direction is. The force in𝑋 direction varies a little with the current and displacementin 𝑍 direction due to the radial component force generatedby the conical stator, and its maximum variation is not betterthan 6%.Therefore, magnetic fluxes of the novel RFHMB arealmost decoupled among different directions when the rotorshifts in some direction.

When the rotor tilts around 𝑋- or 𝑌-axis, the forcecoupling is also analyzed in case that the rotor drifts a certaindisplacement in one direction, that is, 𝑋, 𝑌, or 𝑍 direction.These main analysis results are illustrated in Figure 24 toexpress the relationships between 𝐹

𝑧and 𝛼, 𝐹

𝑥and 𝛼, and 𝐹

𝑦

and 𝛼, respectively.We can find out that when the centroid of the rotor

coincides with that of the novel RFHMB, that is, the rotordoes not drift in any direction (𝑋, 𝑌, or 𝑍 direction),the rotor tilting has no influence on its relative force. Butwhen the rotor drifts in some direction, the force in thisdirection will increase with the increasing of the tiltingangle whether the tilting angle is positive or negative. Whenthe rotor tilts around 𝑋-axis up to the designed maximumangle 1∘, the force change in 𝑍 direction is 9% in case of aaxial displacement 0.3mm while the force change will reach3% in 𝑋 directions or 14% in 𝑌 directions in case of a


−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−10

0

10

20

30

40

50

60𝐹𝑧

(N)

𝛼 (deg)

𝑍 = 0

𝑍 = 0.15mm𝑍 = 0.3mm

(a)

−100 0.2 0.4 0.6 0.8 1

0

10

20

30

40

50

60

70

80

90

𝛼 (deg)−1 −0.8 −0.6 −0.4 −0.2

𝐹𝑥

(N)

𝑋 = 0𝑋 = 0.1mm𝑋 = 0.2mm

(b)

−100 0.2 0.4 0.6 0.8 1

0

10

20

30

40

50

60

70

80

90

𝛼 (deg)−1 −0.8 −0.6 −0.4 −0.2

𝐹𝑦

(N)

𝑌 = 0

𝑌 =𝑌 =

0.1mm0.2mm

(c)

Figure 24: Force coupling when rotor tilts, (a) 𝐹𝑧versus 𝛼, (b) 𝐹

𝑥versus 𝛼, (c) 𝐹

𝑦versus 𝛼.

radial displacement 0.2mm, respectively.Therefore, the rotorshould be well suspended in the central position before therotor is tilted to generate the required control moment.

5.2. Low Additional Tilting Torque. When the rotor tiltsaround 𝑋- or 𝑌-axis, the additional tilting torque is causedmainly by the conical stators, especially when the normaldirection of the conical faces departs from the center of therotor. Assuming the rotor tilts around 𝑋- and 𝑌-axes by theangles of 𝛼 and 𝛽, respectively, the torque can be derived as

𝑀𝑥= ∫

2𝜋

0

∫

𝐿

0

𝜇0(𝑟𝑐− 𝑙 cot𝜑) (𝜙au𝑅au)

2

2𝛿

2

au (𝜃, 𝛼, 𝛽) sin𝜑⋅ 𝑙𝑥(𝜃, 𝑙) 𝑑𝜃 𝑑𝑙

− ∫

2𝜋

0

∫

𝐿

0

𝜇0(𝑟𝑐− 𝑙 cot𝜑) (𝜙au𝑅au)

2

2𝛿

2

al (𝜃, 𝛼, 𝛽) sin𝜑⋅ 𝑙𝑥(𝜃, 𝑙) 𝑑𝜃 𝑑𝑙,

𝑀𝑦= ∫

2𝜋

0

∫

𝐿

0


2

2𝛿

2

au (𝜃, 𝛼, 𝛽) sin𝜑⋅ 𝑙𝑦(𝜃, 𝑙) 𝑑𝜃 𝑑𝑙

− ∫

2𝜋

0

∫

𝐿

0


2

2𝛿

2

al (𝜃, 𝛼, 𝛽) sin𝜑⋅ 𝑙𝑦(𝜃, 𝑙) 𝑑𝜃 𝑑𝑙,

(15)

where 𝑙𝑥(𝜃, 𝑙) = (𝐿

𝑝+ 𝑙) sin𝜑 ⋅ sin 𝜃− (𝑟

𝑐− 𝑙 cot 𝜑) cos𝜑 ⋅ sin 𝜃

and 𝑙𝑦(𝜃, 𝑙) = (𝐿

𝑝+ 𝑙) sin𝜑 ⋅ cos 𝜃 − (𝑟

𝑐− 𝑙 cot 𝜑) cos𝜑 ⋅ cos 𝜃.



= 0.622976

Path 1

Path 2

1

0.00

1471

0.07

0527

0.13

9583

0.20

8639

0.27

7695

0.34

6751

0.41

5807

0.48

4863

0.55

3919

0.62

2976

(a)

0 40 80 120 160 200 240 280 320 3600.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44

Flux density of path 1Flux density of path 2


Flux

den

sity

BSU

M (𝑇

)(b)

Figure 25: Magnetic density fluctuation in the air gap of RFHMB (a) Magnetic field distribution (b) Magnetic flux density curves along path1 and 2.

When the rotor is tilted around 𝑋- or 𝑌-axes, theadditional tilting torque is mainly caused by the unbalancedforce and its related torque arm. The unbalanced force isdetermined by the uniform distribution of the flux densityaround the circumference. In case that the rotor tilts around𝑋-axis about 1∘, the distributions of flux density in the radialand axial air gaps derived by the finite elementmethod (FEM)are shown in Figure 25(a). The corresponding flux densitycurves along path 1 and path 2 are shown in Figure 25(b).

From Figure 25, we can find out that when the rotortilts around 𝑋- or 𝑌-axis, the flux density of the conical airgap is uniform around the circumference. The fluctuationof the flux density is from 0.272 T to 0.41 T. The variationtrend of the flux density in upper and lower conical airgaps is opposite at any circumferential angle. Therefore, therelated magnetic force is uniform around the circumferenceand the forces of upper and lower conical rotors cannot becounteracted at the same circumferential angle. With respectto the conical rotors, only the normal direction of the conicalmidst faces directs to the centroid of the rotor, so the torquearm still exists and an additional torque around 𝑋- or 𝑌-axis is generated unavoidably. For the conical axial magneticbearing in the presented novel RFHMB, the additional torque𝑀a-r can be calculated as

𝑀a-r = ∫𝑆1

𝑏

2

1(𝜃) ⋅ 𝑙

2𝜇0

𝑑𝐴 + ∫

𝑆2

𝑏

2

2(𝜃) ⋅ 𝑙

2𝜇0

𝑑𝐴, (16)

where 𝑏1(𝜃) and 𝑏

2(𝜃) are the flux densities that vary with the

circumferential angle 𝜃; 𝑆1 and 𝑆2 are the areas of upper andlower conical stators, respectively. 𝑙 is the normal distancebetween the vector of the unbalanced force generated by theconical axial magnetic bearing and the centroid of the rotor;it is very small and can be neglected in most cases.

On the other hand, if the rotor of the MSFW withgimballing capability is suspended by the common reluc-tance force-type magnetic bearing as shown in Figure 5, thegeneration of additional tilting torque on rotor is shown inFigure 8. Because the working surface of the 3-DOF MB isplane faces and the torque arm is comparatively large, itsadditional torque is expressed by (6). Considering the uni-formdistribution of the flux density and the additional torquegenerated by the radial rotors, the relationships between thetilting angle 𝛼 and the additional tilting torque 𝑀 of thenovel RFHMB and that of the common reluctance force-typemagnetic bearing with plane faces are analyzed by FEM andthese results are illustrated in Figure 26.

From Figure 26, we can find out that the additionaltilting torque for the presented novel RFHMB is 0.05Nm,which is very small, but the additional tilting torque for thecommon reluctance force-type magnetic bearing increaseswith the angle linearly and its maximum torque is up to11 Nm. We also find out that the angular stiffness calculatedfrom its torque-tilting angle curve for the presented novelRFHMB is 0.11Nm/∘ while the calculated angular stiffnessof the common reluctance force-type magnetic bearing isas much as 10.83Nm/∘. So the presented novel RFHMBwith low additional tilting torque will make the MSFW withgimballing capability have high-precision attitude controlability for spacecraft.

6. Conclusion

With respect to this MSFW with gimballing capability,magnetic bearings can be divided into Lorentz force-typemagnetic bearings and reluctance force-type magnetic bear-ings according to the physical cause of the generation of


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

0

2

4

6

8

10

12

Torque of the conical structureTorque of the plane structure

Torq

ue𝑀

(Nm

)

Tilting angle 𝛼 (∘)

Figure 26: Torque characteristics of the presented conical RFHMBand the comparative plane structure.

the force. The demerits of common reluctance force-typemagnetic bearing for flywheel with gimballing capability arethe existence of positive displacement stiffness, which willresult in unbalanced forces when the rotor tilts. For thenovel RFHMB, its novelties are the radial and axial magneticbearing units which are integrated as a compact assembly,permanent magnets are used to generate the radial and axialbiased fluxes, the radial stators include 4 stator poles, andtheir relative permanent magnets are also separated into fourseparate segments to make the magnetic flux of 𝑋 and 𝑌channels independent from each other. Two conical statorsare used to control the translation of rotor in only𝑍 directionwithout having effects on these forces in radial directionand to make the additional torque acting on the rotor zeroapproximately when the rotor is tilted.

The equivalent magnetic circuit of permanent magnetsand the equivalent magnetic circuits of control current areresearched and their main parameters obtained. To reducethe eddy loss caused unavoidably by the segmented radialstator poles, the radial stator poles are designed as fourseparate poles with shoes and their rotor is made of iron-based amorphous. When the rotor tilts around 𝑋- or 𝑌-axis, simulation results indicated that the fluctuation of fluxdensity is from 0.272 T to 0.41 T, the additional tilting torquesin the novel RFHMB is 0.05Nm while that in the commonreluctance force-type magnetic bearing increases with theangle linearly and its maximum torque is up to 11Nm. Byanalyzing the changes of magnetic forces with variation ofcurrent and displacement, we found out that magnetic fluxesof the novel RFHMB are almost decoupled among differentdirections when the rotor drifts in some direction. Whenthe rotor tilts around 𝑋- or 𝑌-axis, there are 2 conclusions:(1) the rotor tilting has no influence on its relative force ifthe centroid of the rotor coincides with that of the novelRFHMB; (2) the force in this direction will increase with

the enlargement of the tilting angle whether the tiltingangle is positive or negative when the rotor drifts in somedirection, but the maximum changes do not exceed 14%.Thisanalysis proved that not only the magnetic fluxes of the radialmagnetic bearing unit in 𝑋 and 𝑌 channels are independentfrom each other, but also the magnetic fluxes of these conicalstators have no effect on that of the radial magnetic bearingunit. All these results proved that the gimballingMSFWwiththe novel RFHMB can meet the maneuvering requirement ofspacecraft theoretically.

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China under Grant 61174003.

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