The Development of the Spherical Aerostatic Bearings for 3DRW · The Development of the Spherical Aerostatic Bearings for 3DRW ... Science / Japan Aerospace ... the spherical-flywheel

The Development of the Spherical Aerostatic Bearings for 3DRW

By Keita Tanaka1), Yuichi Tsuda2), Takanao Saiki2), Yoji Shirasawa3), Ryo Jifuku1)

1)Department of Aeronautics and Astronautics, The University of Tokyo, Tokyo, Japan 2)The Institute of Space and Astronautical Science / Japan Aerospace Exploration Agency, Sagamihara, Japan

3)JAXA Space Exploration Center / Japan Aerospace Exploration Agency, Sagamihara, Japan

Abstract: The three-dimensional reaction wheel (3DRW) is a newly developed attitude control system of a satellite, which is characterized by a single levitated-wheel rotating around any one of the axis. The authors have developed the test models of 3DRW and analyzed their performance by experiments and calculations. This paper is about the development of 3DRW, especially the development of the bearings which realize the stable support of the fly-rotor without any contact using air pressure.

1. Introduction

The three-dimensional wheel (3DRW) is a altitude control device for satellites, characterized by a single spherical wheel levitating and rotating about any one of the axes. The principal idea of 3DRW was originally proposed by Muller et al1) suggesting the possibility of the spherical-flywheel system which would achieve a light-weight and powerful attitude control system of space vehicles, but its possibility remains unsettled.

3DRW requires two principal architectures: a rotor-floating system and a rotor-rotating system. In this paper, we propose a gas bearing as a rotor-floating system and show the characteristic of it. A gas bearing has no interference with electronic, magnetic and other force fields and the bearing system can be completely separated from other components and that makes the design and analysis of it easy.

To realize the stable positional control of the rotor, it is important to understand the characteristics of spherical gas bearings such as pressure distribution, load capacity, stiffness and flow rate. Some previous works clarified them but few researches are there relating the use of gas bearings in the space. What is required for the space gas bearing is having the minimum robust characteristic against various small disturbances.

In this study, the authors have developed a spherical gas bearing with a circular slot restrictor and analyzed its performance by experiments and numerical calculations. The result shows that this type of bearing can generate necessary floating force with low supply pressure and low flow rate, and that leaves the possibility to realize 3DRW system operated by a compact pump such as piezoelectric type.

2. The static property of a spherical gas bearing

2.1. Air-film thickness Of a spherical gas bearing, the air-film thickness varies depending on the position on the bearing surface. The bearing is

assumed to have has the same radius of curvature as the rotor and it is displaced in the center when the rotor is levitated. The air-film thickness h* can be written as a function of the angle measured from the load line θ. Applying the law of cosines, h* can be obtained as,

!!! 222* sincos)( hRRhh ""+= (1)

where h is the thickness of the air-film at the bottom of the bearing, that is, rotor-levitation width. When h is sufficiently small compared to the radius of the rotor R, Eq. (1) becomes,

!! cos)(* hh = (2)

This approximation works well in the bearing performance calculation.

Fig. 1 Air-film thickness

2.2. Pressure distribution, load capacity, flow rate

The load capacity of the bearing W is obtained by integrating the pressure over the bearing surface.

!!"!!

!

!RdRpp

dsppW

a

sa

##$=

$=

%

%2

0sin2cos))((

))(( (3)

The flow is considered as laminar, viscous and incompressible and the velocity of the flow normal to the direction of the flow is neglected. Thus, the pressure is assumed to be constant over that direction. The Inertia force is neglected and only the force caused by the pressure gradient and the viscous shearing is considered. The Reynolds equation of the time-invariant flow in a narrow parallel gap is given by

xph

Bxq !

!"=

µ12

3 (4)

where qx is the flow in the x direction, h is the thickness of the gap and B is the width.

2.2.1. A spherical slot-inlet bearing Fig. 2 shows the aerostatic bearing with a slot-type restrictor. The Reynolds equation in the spherical coordinate can be

obtained by replacing B with 2πRsinθ, h with hcosθ and x with Rθ in Eq. (4),

!µ

!!"! #

#$=

pR

hRq 1

12

)cos(sin2

3

(5)

where qθ is the flow in the θ direction.

Fig. 2 Slot-inlet bearing

The boundary conditions relating to the pressure are

apppp

==

==

: :

2

11

!!

!! (6)

This means the pressure within the slot-inlet is equal to inlet pressure p1. Through integrating Eq. (5) under the boundary conditions, Eq. (6), the flow rate out of the bearing qout is derived.

!!"

#$$%

&+'

'=

1

21

22

2

13

tantan

log2tantan3

((

((µ

) aout

pphq

(7)

In the same way, the pressure distribution p can be written as follows.

)(

tantan

log2tantan

tantan

log2tantan

1

1

21

22

2

222

2

aa pppp !

""#

$%%&

'+!

"#

$%&

'+!

=!

((

((

((

(( (8)

The pressure at the inlet of the bearing p1 is calculated by considering the condition that the load capacity of the bearing calculated in Eq. (3) is equal to the rotor weight, Mg.

!

Mg= (p1 " pa )cos# $ 2%Rsin#0

#1& $Rd#+ (p (# )" pa )cos# $ 2%Rsin##1

# 2& $Rd#

= %R 2 (p1 " p2 )' sin 2#1 +

"1+cos2#1cos2#2

" 2sin 2#1 logtan#2tan#1

(

) *

+

, -

tan 2#2 " tan2#1 + 2log tan#2

tan#1

(

) *

+

, -

(

)

* * * * *

+

,

- - - - -

(9)

After redeploying Eq. (9), the inlet pressure p1 is derived as follows.

MgR

pp)tantan(

tantan

log2tantan

12

222

1

21

22

2

21!!"

!!

!!

#

$$%

&''(

)+#

=# (10)

This equation expresses that the inlet pressure p1 depends on the configuration of the bearing and the rotor weight. Next, the flow from the chamber to the slot-inlet on the bearing surface is considered. It is considered to the flow in a narrow

circular cylindrical gap. The flow rate into the bearing qin can be obtained by replacing B with 2πRsinθ, h with hsl, and integrating it.

sl

sslin l

pphRq 13

1 12sin2 !

!=µ

"# (11)

where hsl is the slot width and lsl is the slot length. According to the law of mass conservation, the flow rate into the bearing and out of the bearing should be equal if there is no

leak. Supplying pressure ps can be calculated using Eq. (7), Eq. (10) and Eq. (11). With that, the pressure distribution on the bearing surface is completely obtained.

2.2.2. A spherical slot-inlet ring-type bearing A spherical ring-type bearing is the bearing which has a hole/outlet at the bottom of it (Fig.3). The flow from the slot-inlet is

divided into an upward and a downward flow and then released to outside. The pressure at the outlet at the bottom of the bearing is equal to the air pressure. The load capacity of this kind of bearings is generally worse than that of non-ring type bearings which have the same configuration, but the performance analysis of them is necessary because it is inevitable to create a hole in the bearing surface to install sensors to detect the motion of the rotor in practical case.

To calculate the pressure distribution on the surface, the same procedures are conducted as non-ring spherical bearings, that is, Eq .(5), except that calculating a downward flow requires to multiply the right side of the equation by the negative number to express the backward flow. As for boundary conditions, there is little difference. They become

a

a

pppppp

==

==

==

: : :

2

11

0

!!

!!

!!

(12)

θ 1 is the angle of the slot inlet, and θ 0 and θ 2 are the edge angles. Integrating the Reynolds equation under the boundary conditions Eq. (12) gives the flow rate out of the bearing.

!

qout ="h 3 (p1 # p2 )

3µ1

tan2$2 # tan2$1 + 2log tan$2

tan$1

%

& '

(

) *

%

&

' ' ' ' '

#1

tan 2$0 # tan2$1 + 2log tan$0

tan$1

%

& '

(

) *

(

)

* * * * *

(13)

The load capacity can be written by.

!

Mg= (pd (" )# pa )cos" $ 2%Rsin"0

"1& $Rd + (pu (" )# pa )cos" $ 2%Rsin""1

" 2& $Rd"

= %R 2 (p1 # p2 )' #

#1+cos2"1cos2"0

# 2sin 2"1 logtan"0tan"1

(

) *

+

, -

tan 2"0 # tan2"1 + 2log tan"0

tan"1

(

) *

+

, -

(

)

* * * * *

+

#1+cos2"1cos2"2

# 2sin 2"1 logtan"2tan"1

(

) *

+

, -

tan 2"2 # tan2"1 + 2log tan"2

tan"1

(

) *

+

, -

+

,

- - - - -

(14)

where pd and pu respectively represent the pressure distribution of the downward flow and the upward flow. The inflow rate is given by Eq. (11).

In summary, the inlet pressure p1 can be calculated from Eq. (14), the flow rate q from Eq. (13) and the supply pressure ps from Eq. (11).

Fig. 3 Slot-inlet ring-type bearing

3. Experiment of levitation

Some spherical gas bearing experiments are conducted for verification of the theory developed above. 3.1. Experimental set-up

Fig. 4 and 5 shows experimental systems, the rotor, the gas bearings and other instruments. Two types of bearings are used for the test: The bearing No.1 is a perfect spherical bearing without any hole. The bearing No.2 is a spherical bearing with a hole at the bottom of it, but it can be sealed by an attachment not to leak the air. Table 1. summarizes the specifications of each bearing. Notice that these two types of bearings has the difference in the inlet and edge angle.

Table. 1 Specification of bearings

No.1 No.2a No.2b Curvature Radius R [mm] 17.46 Inlet angle θ 1 [deg] 27 51 51 Outer edge angle θ 2 [deg] 66 76 76 Inner edge angle θ 0 [deg] - - 10 Slot width hsl [µm] 5 Slot length lsl [mm] 10

Fig. 4 Rotor and Bearing Fig. 5 Experimental set-up

3.2. Experimental results 3.2.1. Bearing No.1

Fig. 6 and Fig. 7 indicate respectively the air-thickness and flow rate with regard to the supplying pressure for the bearing No.1. The plots are the theoretical and experimental values. As these graphs show the calculation can predict almost accurately the levitation height but contains large error concerning the flow rate.

Fig. 6 Air-film thickness h for Bearing No.1 Fig. 1 Flow rate q for Bearing No.1

3.2.2. Bearing No.2 The experimental and theoretical results for the bearing No.2 appear in Fig. 8 and Fig. 9. The bearing No.2 can change its

performance by attaching/detaching a seal-like component to/from the hole at the bottom of it. The following graphs include both results from the with-seal-version and the without-seal-version (which is represented as ring-type in the graphs).

We see from these graph that there are much error between the theoretical values and the experimental values compared with the result of the experiment using the bearing No.1. The experiment results in achieving less levitation height with less air flow rate than expected.

In addition, these results indicate that making a hole at the bottom of the bearing can much affect its load capacity but not its flow rate.

Fig. 8 Air-film thickness h for Bearing No.2 Fig. 9 Flow rate q for Bearing No.2

4. Conclusion

The authors have developed a spherical gas bearing for 3DRW and evaluated it by experiments and numerical calculations. The result shows that the gas bearing which used in this study can serve as a rotor-floating system of 3DRW. The following is its reasoning.

The first experiment indicates the calculation can almost exactly predict the bearing performance, the levitation height and the flow rate when the supply pressure is given (Fig. 6 and Fig. 7). It also suggests that the small supplying pressure, only 0.04 MPa, can make the rotor levitated. This fact shows that it would be able to be used as a sustainment system for 3DRW.

The second experiment shown in Fig. 8 and Fig. 9 insists that the bearing No. 2 cannot get as good performance as No.1. It requires 0.2 MPa to levitate the flywheel. It is likely that this performance inferior is caused by the manufacture error. Because the experiment system requires very high degree of accuracy as much as 1.0 µm, it is not easy to eliminate all the manufacture error.

In addition, these experiments confirm that it is possible to levitate the ball with ring-type bearings. When developing 3DRW, it is very important to monitor the movement of the rotor while controlling it. The problem of how to install sensors always accompanies the development of the system. The result indicates that making the bearing a ring-type does not necessarily disturb the development of 3DRW.

The future work of this research is to estimate the stability of bearings such as the stiffness and the damping. To theoretically calculate the values of them is not difficult, but to measure is another story. Verifying the data integrity of them between theories and experiments is demanding, because they are exactly what are required to check to create the practical 3DRW which can be operated in the space of gravityless. In this study, we have done the tests on the ground where the gravity exists and we have developed the bearing system which can resist the gravity of the ball. In fact, however, it is not necessary for the bearing to have such a big power in space. What is required for the space gas bearing is having the minimum robust characteristic against various small disturbances. Thus how to simulate the space condition and how to execute experiments are problems to be solved.

Acknowledgments

We would like to express our thanks to Ono Denki Co. for cooperating in developing the gas bearings.

References

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