Design data for externally pressurised spherical gas bearings

Design data for externally pressurised spherical gas bearings K.J. Stout and M. Tawfik*

Radial and thrust loading characteristics are given for both single row entry and double row entry bearings. The design data presented gives a basis for comparison of the two bearing configurations. The paper shows that the double entry configuration reduces the negative stiffness regions which occur in single entry bearings. Finally a procedure is given to enable the designer to select a bearing to meet his particular requirements.

Nomenclature

as e

/~ Btop Bbottom D

R

Flow shape factor Flow shape factor of top Flow shape factor of bottom Diameter of sphere edo --Pa Design pressure ratio, - Po --ea Supply pressure Flow rate Qr/ Dimensionless flow rate, a Pah Resultant load

/~

Dimensionless resultar~t load,

T

Axial bearing thrust load

Kg o Po Q

Dimensionless thrust load, W h/

h n Y z or*

7 e eA eR rl 0 R

(eo -Pa) 02 T

(eo --Pa)D2

Radial bearing load Dimensionless radial load,

Slot width Eccentricity Film thickness Number of slot entries per row Slot length Slot thickness 2ntho 3 Slot factor, - asnz 3 Direction of shaft displacement along a meridian Eccentricity ratio, e/h o Axial eccentricity ratio Radial eccentricity ratio Dynamic viscosity Angle on a parallel Angle on a meridian Bearing land angle Land angle ratio, ¢ 1 / ~

Suffices a

W (Po--Pa)D2

Partial spherical gas bearings have attracted interest in recent years for applications requiring low friction and three degrees of freedom in rotation. Such applications include rotary joints for pneumatic power transmission, instrument tables, gimbal bearirigs for stabilized intertial platforms, and space motion simulators) In this paper two forms of these bearings are considered: single and double row entry. The single row configuration offers greater accuracy in manufacture, but exhibits regions of negative stiffness which limits the loading which can be applied.2 The double entry configuration is less prone to negative stiffness.

o s, d

Ambient Design condition Value at slot

manufacture. Additional advantages include superior loadcarrying capacity and improved predictability. The results presented in this paper have been derived from an analysis which employs the finite difference technique. This technique 2 is useful because it predicts the loads accurately for any combination of eccentricities in both the radial and axial directions. The analysis assumes twelve inlet slots per row. In practice, this number of inlet slots has been found to be a suitable compromise between manufacturing complexity and performance of the bearings.

The control device used in this work was of slot feed configuration. The merits of various control devices have been discussed in an earlier publication a in which it was concluded that slot entry bearings are suitable for quantity

Slot entry design

* School of Mechanical & Production Engineering, Leicester Polytechnic, P.O. Box 143, Leicester LE2 9BH, UK.

The analysis assumes that the pressure at the outlet from all entry slots is equal at e =0.0 for both Single or double

The geometric configuration of partial spherical beatings is shown in Fig 1. The position of the entry slots, and the relationship of these entry slots to the total geometric shape of the bearings can be clearly seen.

TRIBOLOGY international June 1977 163

and hence the slot dimensions to be established. These charts will be discussed briefly later in the text, and their use explained in the design example.

Design parameters Double entry

Design parameters employed in this work are in a form which is convenient to the designer. They are:

Single entry

Radial load (IV)

= (Po -- Pa) D2 I¢

Axial load (T)

= (Po - Pa) D2 f

Flow rate (Q)

= Paho a Q/n

Fig I Geometric parameters of spherical bearings

Design characteristics of single entry bearings Single entry bearings have been previously discussed 2 and will be only briefly described here to enable a comparison between single and double entry geometries. Fig 2(a) shows how radial and thrust loading vary when the design pressure ratio Kgo is varied. It is shown that when Kg o = 0.5 a good range of combined radial and thrust loadings may be applied. When high thrust loads with small or negligible radial loads are required a high.Kgo value is preferred.

row entry. It is necessary to adjust the dimensions of the inlet slot to balance the flow resistance through the control device to that of the flow through the bearing clearance. For single row entry this does not present any problems. In the double entry case, however, the slot dimensions of the top row differ from those of the bottom row. To assist the designer, charts are provided which enables slot factors

Single entry spherical gas bearings P* / P * = 5

o

~L= 2 0 .

//-60"

~Z = 4 0 .

P,/Po= 5

Kg = 0.:5

0.05 II..-

0.(

",, \.~'-.1.~'--30 * "~o* .75 \ \ _ (I, 50 °

~.5 0.10 +

/

Kgo

= 0.5

~2

= O*

4~t

= 35*

~2

= 70"

~bI = O*

04

?( 7"=90 *),=0.5 IO'

3C

I0

00.5

0.3 ~ ( × - - 0 " ),=0.5

"60*

"60°

~z = 70* 0.5

4C

\

0.15 -

~ = 35*

Kg,= 0.5 5C

Kg = 0.5

f

13

~*),:-o.5

,¢~02

W(7=O*lt=o"

Ol

,JoO

0o

C

oJo

II--

P*/Pa

o-

0.1.5

~

30 o

-60* / /

P* / P, = 5

~bL = 35*

~, = O*

9 /

/

J

30

K9=0.7

~b2 = 70* 0.5

40

"60*

÷ ~,

50

IO

O. I 0 I ~ - - - - - - - - - - ~ / 20

a

o'., ~

~ 0.115

t. . . .

b

0.05

0.15

w

0

d

3 3 I~I~= 02

0.1

I0

o2og.,.o

04

) 0.2

,

=90,),=o.% 0.4

0.6

08

,!8

Kgo

Fig 2 Variation o f load parameters (a) with pressure ratio and (b) with eccentricity. Flow parameters and the product of dimensionless radial and thrust loads vary (c) with supply pressure and (d) with pressure ratio

164 TRIBOLOGY international June 1977

Alternatively, if low thrust loadings are to be experienced with high radial loads then low values of Kg o are applicable. A feature of single entry bearings is that they exhibit the characteristic of negative stiffness. This feature is only marginal in the case of loW values of Kgo, but becomes considerably more significant when Kg o increases. When the designer employs a bearing which exhibits such characteristics he is forced to operate well within the combination of loading which leads to an eccentricity ratio e = 1.0. This is the case, even when higher loads can be achieved at eccentricity ratios less than e = 1.0. Fig 2(b) is a further load map which shows how this negative stiffness region becomes more widespread in cases where the bearing half angle (¢2) is increased. In this example it is clearly shown that the bearing collapse is most prone in the direction of "/= 30 °. At the combination of radial and thrust loadings, which leads to 7 = 30 °, the maximum safe loadings must not yield an eccentricity ratio which exceeds e = 0.4. Figs 2(c) and (d) are design charts which show the effect of Po/Pa and Kg o on load-carrying capacity and flow rate. Some variation of a load parameter is expected with increasing supply pressure becuase flow rate varies with Po 2 not Po. It is clearly shown that the variations are minor

and are consistent with previous work concerning short journal bearings 4,5. It is therefore not necessary to be particularly concerned with the value of supply pressure at which the results are obtained. Fig 2(d) presents simplified data for a range ofKg o values for a typical supply pressure. This diagram gives further information in support of many of the comments made concerning Fig 2(b). Here the designer can find concise data relating to radial and thrust load parameters' variations with Kg o. Fig 2(d) shows that although thrust load capacity is increased with increasing Kgo, radial load is maximized at Kg o = 0.5.

Double

entry

bearings

Fig 3(a) presents load maps for double entry bearings. By comparing Fig 3(a) w i t h Fig 2(a), it becomes obvious that the negative stiffness regions are almost eliminated. The

general shape of the load maps indicates that these bearings have greater stiffness in both radial and thrust directions than the single entry configuration. A second feature of hnportance is that some improvement in the axial load capacity is gained. Unfortunately, the improvements in radial load parameter is not sustained to the same extent when the beating half angle is larger (Fig 3(b)). In this

Double entry sphericol gos beorings ~I P/P =5 0.05

~L:IO° (ib2=40 °

~/P=5 i

~ - 6 0 0

Kgo=03

~.-

IF-

Po/Po = 5

Kg o

= 0.5

¢~

Kgo

= O*

¢t ~z

= 20* = 70*

~ ~L ~z

= 0.7

= O*

= 20* = 70*

QIO

-~---'-.. /

O.I

o

0.15

__

\60 °

0.25

__

/ -60"

o.io - ~ J ~ ' <

K%: o.5

0.2 035

o.15

0.20 i~= 1.0

0.3

\60 °

04~ ,~.

/

-60 °

Kg o = 0 7

0.15

0.4

II0.20

÷

O25 a

0.5.=

E (=I.0

I 0.05

"-60* OI

i

0.15

005

b

0.15

~

005

0.15

c

Fig 3 Variation o f load parameters (a) with pressure ratio and (b & c) with eccentricity ratio and load angle at different values o f the design pressure ratio

T R I B O L O G Y international June 1977

165

example it may be seen that the negative stiffness region although reduced is still present. The maximum radial load parameter (in the line of 3' = 0 U) is only marginally higher than the single entry counterpart. By careful examination of flow paths within the bearing, and the resultant pressure profile, it may be concluded that the failure to gain further load capacity is due to increased circumferential flow which occurs largely between the two rows of inlet slots. The pressures, rising in the unloaded region of the bearing, assist in reducing load capacity. The designer can still take comfort from the fact that the double entry bearing can be loaded to a higher level than the equivalent single entry bearing because the region of negative stiffness is almost eliminated. Fig 3(c) shows that static instability increases with higher values of Kgo and again suggests that the designer must attempt to produce bearings which operate within an optimum range. It is suggested that for spherical bearings requiring combined radial and thrust loadings, the most suitable range for operation is 0.4 < Kg o < 0.6. Fig 4 shows how the pressure profile changes within the bearing clearance under different conditions of loading. On each diagram, the pressure profile attributed to five differing eccentricities are given. The four groups of curves represent different angles of loading. Therefore, by considering Fig 4 in conjunction with Fig 3(b), the reader can obtain an appreciation of the reasons why negative stiffness occurs.

Considering first Fig 4(a), in the concentric condition, (e = 0), it is seen that the pressure profile is approximately

Kg,=0.5

~L=O*

Oi LO

3

,~']L~--O

2/I I

5 4 3 a_* 2

I

I

I

)'=90* " ~ ~ m

to.5

' 7"- 60° b

tO.2 s

I t =0

o1.0

¢

i

i

Figs 5(a-d) are design charts of a form presented previously, and therefore requires little discussion as most of the comments previously made concerning the single entry type apply. These charts will assist the designer in obtaining sufficient information to aid him in designing spherical bearings. Fig 5(e) shows load and flow variations against land angle grL. It can be seen that as q~L increases the load paramenters and flow rate reduce. If design requirements include minimal flow rate, then single entry configurations are most suited. However, if a double entry configuration is chosen, due to its superior load carrying characteristics, it is suggested that a land angle ratio ~b in the range 0.25-+0.35 is normally chosen. This gives both good load carrying characteristics without incurring unacceptability high mass flow rates.

Design of entry slots

I

1.0

r

symmetrical. As the bearing eccentricity is increased there is greater film variation in the plane of applied load. As the rate of change of gap is far more severe in the lower regions of the bearing than the upper regions, the pressure gradient is more severe. The load capacity of the bearing is dependent upon the pressure profile and hence it is important to have high pressures at large values of ~ if high axial loads are to be achieved. This occurs when the angle ~L is reasonably small, and for bearing configurations shown ~L -- 20°. Fig 4(b) and (c) presents curves for angles of eccentricity of 60 ° and 30 ° and correspond to the angle at which the bearing clearance are approaching symmetry about the ~ plane. At high eccentricity the flow from the inlet slot is towards regions of increasing gap so that the pressure falls away rapidly. This is indicated by the 'sagging' effect of the profile, particularly between the boundaries and the inlet slots, and is the feature which has been found previously in spherical bearings:. The reader can clearly see that the area under the curve is reduced. This leads to a reduction in load capacity and hence negative stiffness. Fig 4(d) completes the picture, and is a case where the greatest rate of change of gap occurs at low values of ~.

i

~

!

),=0 ° - - ~ ~ y

Fig 6 and 7 assist the designer in dimensioning the feed slots. To obtain equal pressures in both rows of entry slots in the concentric position, which gives the most satisfactory loading conditions, it is necessary for the dimensions of the top row to be different from those of the bottom row. Fig 6 presents curves for the slot entry constant, asza/y (Top), which are directly related to parameters such as supply pressure, pressure ratio, and beating half angle. Fig 7 presents analytically calculated shape factors which apply to both the top and lower parts of the bearing. These shape factors may be considered as being closely related in a dimensionless form to concentric flow which takes place in the bearing. By employing the relationship given in the diagram, the slot entry constants for both entries can be calculated. The application of these constants to determine entry dimensions will be demonstrated in the following design example.

d

Design example Fig 4 Pressure profile in a double entry spherical gas bearing changes with eccentricity

166

TRIBOLOGY international June 1977

A hemispherical air bearing with a 70 ° half angle is required for a test rig. Space limitations require that the bearing diameter should not exceed 60mm. The bearing must

Double entry spherical gas beorincJs

I001

0.6

60

Kgo=0.5 @1=0° @1.=20 ° @2=70 ° 50

0.5

90

0.4

80

@1= O*

@t= I0" @Z= 40 °

T(T =00)~ =o. 5

,P"

Io 30

4

l

0.9

..o~

~(y =90")( =0.5

40

o

Kg° = 0.5

Po/Po=5 @1=0° @,=20= ~

o3 I; ,o

0.7

60

0.6

0.1 IO 50

,05

0

04

o'%b

T(y = 90°). =_0.5 20

_/

oj

I0

/

q

2

a

0.2

Qo

W(Y =0")~ =o.5 ~1y:90.).=o,5 6 8 I0

4

W(y = 900).= o.5

04

012

b

Po/Po

I

0:8

o6

Kgo

6O

0.6 P/P=5

Kgo= 0.5

@t/@2=0.25

0.5

50

Po/Po=5 Kgo=0.5 @2=70* @~=0

50

90°)* =0.5 0.5

2O

04

i0 t

@2=0 °

m

'

40 ..0.2

_

T(y =Oo],f o ,-..=.-.,.~ = ,5

I o 30

-o3 I;

20

I0

w~T-w ~ = 0 5

30

40

50

@2

d

~

o2

~

01

_. - 90 an*~ W(y)~=05

Tii.

W(y = 90")~ = 0.5

0

6'0

70

,5

2'o

Design Procedure

The following paramters must be determined: Single or double entry configuration. Suitable pressure ratio K g o . Suitable bearing diameter D. Design radial clearance h o. Concentric flow Q. Slot dimensions as, y, z. Calculations To determine whether a single or double entry bearings is most suitable for the design, and at the same time operates at a safe load capacity: 1. Calculate load ratio T/W

W

-

360 70

= 5 . 1 4 ~ 5.

2. On Figs 3 and 7, plot a line with a constant load ratio = 5 (~sing W= 0.05 with T= 0.25, and I47=0.1 with

r=0.5).

I

z5

37

C

T(y : 9 0 " ) , =-o.5

WlT=O*).=o.5 4

6

8

102

I IO I

I0

Po / Po

F ~ 5 In double entry bearings, load and flow parameters vary with supply pressure, pressure ratio, beating angle, and entry angle

e

support an axial load T = 360N in combination with a radial load W = 7ON. The flow rate must not exceed 1 litre/s and the supply pressure ratio Po/Pa = 5.

T

o 0

T.~_(Y= 9 0 0 )

( 7 ". 0 )o, f =-o.5 I

This line has been plotted on Figs 2(b) and 3(b). The designer Can select any point along these lines to satisfy the loading requirements. The actual point selected should be at an acceptable eccentricity ratio, at the same time avoiding negative stiffness regions. In this design example the lines drawn give two values of load parameters for a given eccentricity ratio. The point selected should be at the upper value to minimise bearing diameter and/or supply pressure. This point is marked with an asterisk on both figures. It gives 5 = 0 . 3 4 and I47=0.068 at e = 0 . 5 and 7 = 15 ° for the double entry bearings. For the single entry bearing 5 = 0 . 2 5 and W=0.05 at e =0.04 and 7 = 10 °. From Figs 2(b) and 3(b), the designer can clearly see that the double entry bearing can be loaded safely to a considerable higher eccentricity ratio. A disadvatage of employing the single entry bearing, in this case, is that when they are loaded to cause an eccentricity which is near to the negative stiffness region, any small shock loading may cause bearing collapse. To avoid this situation, the applied load must be designed below the value which corresponds to maximum collapse load parameter at an eccentricity ratio e = 1.0. For this example, (i) to minirnise the bearings diameter, (ii) to minimise supply pressure, (iii) to reduce the likelihood of bearings collapse, it is seen that the double entry configuration is preferable.

TRIBOLOGY international June 1977

167

2.2

From Fig 3(b), at this value of 7=, e is less than 0.5, d i s directly related to 7=.

2.0

~/=

1.8

70 0.4X

= 0.064

106 (0.052) 2

The combination of radial and thrust load parameters leads to an eccentricity ratio e=0.49 which is an acceptable solution.

1.6

Radial clearance 1.4

By assuming

flow rate = 1 litre/s:

ho = ~/ t'~0.o

1.2

le}@~

theconcentric

I:M o

where ~air = 1.8 X 10-2 cP (2.62 X 10-9 Reyn) 0.8

From figure 5(b) the flow parameter ~}o = 8.78

0.6

Theref°re h° = ~ /

0.4

109 X 1.82X 10- 5 X 1X 10 -3 1 - ~ ~ 1--~ X 8.78

= 0.0274mm

Thus by choosing the radial clearance h o = 0 . 0 2 5 m m , we satisfy the design requirements that the flow rate does not exceed 1 litre/sec.

0.2

Concentric flow 0

2

4

6 Po/ Pa

8

I0

Paho3 Q.o

The concentric flow rate Qo = - ~7

Fig 6 Slot entry constant is a function of pressure ratio Qo

-

1.03 X 9.8 X 104 X (0.025) 3 X 8.78 109 X 1.82X 10 -s

= 0.76

litres/s

Design of slots entries Pressure ratio

From Fi_$s 2(b), 3, and 5(b), it is seen that for a bearing having T/I#= 5, the most suitable pressure ratio is

Kgo =0.5. Diameter

1 To dimension the feed slots, it is necessary to consider the number of inlet slots around the bearing when defining their geometry. As previously discussed, it has been found that the number of slots, n = 12, around the circumference is a good compromise between load capacity and manufacturing cost.

By choosing double entry bearings for Po/Pa = 5, Kg o = 0.5, e=0.5

Double entry sphericol gos beoring

T=0.34 and W=0.068

Co-Ca Thus - = 0.4 ea

i

1

MN/m 2

~2 = 7 0 *

@l~ _ o =

_20~

\\\,,4o.

J(eo - rP a ) T

~

~

0.5

04

It

I0 °

I

=J

360 0.4X0.34× 106

Selecting D = 52mm: 360

~=

0.4X 10~(0.052) 2

168

= 51.449 m m

,,~l

I

0'3

0'2

I I

DO

I I

I

o,

BTopl n a'(Bottom) = at(Top) t~T°P BBottom

= 0.33

T R I B O L O G Y international June 1977

Fig 7 Flow shape factors

I

I

0.2

0.5

B Bottom/n

I --

0.4

2 It is usually convenient to select the slot thickness, z = 0.0125 m m for gas bearing applications. 3 The slot width a is a function of the bearing diameter, such that a small land occurs between each feed slot. A convenient dimension for the slot width (as) is given by the following expression:

Po Qo = W= T = ho = =

Dimensions of feed slots

0.757rDcos$ s as =

where Ss is the slot entry angle from the horizontal plane. 0 . 7 5 X rrX 5 2 X cos20 °

as(tOp) =

12 0.75 X rrX 52X cos50 °

as(bottom ) =

12

=

o~*asmZ3 2nho a

From Fig 6 when Po/Pa = 5, Kg o = 0.5, $ = 70 °, and for double entry bearing (4 = 0.25). a* (top) = 0.7 a* (top)as(tOp)z 3 27rh° 3 0.7X9.6X 12X (0.0125) a 2X 7rX (0.025) a

= 1.6mm

It has been shown that the double entry bearing reduces, and in some cases eliminates the negative stiffness region which has been exhibited in the single entry type. The improved static stability allows the designer to operate the double entry bearing at higher loadings and at greater eccentricities more safely. A penalty is paid in that the concentric flow rate is approximately doubled. Increasing the bearing included angle ($2-¢~) offers limited improvements in radial load capacity, although large increases in axial load capacity is achieved. It may be seen from the charts provided that radial load capacity reaches a maximum at approximately ~2 = 50°. Experience has shown that the ratio o f radial to axial load (W/T) is limited to 0.3 for most applications. When rotational freedom is required in more than one axis, the designer is probably best served by employing spherical bearings. In such cases the designer must ensure that the radial loadings falls within the capability of the bearing which is predominately suited for axial loadings.

Acknowledgements

y ( b o t t o m ) is determined from Fig 7.

The authors would like to acknowledge the sponsorship o f the British Council, the Science Research Council, and Bentley Engineering who are supporting the research into externally pressurized gas bearings. In addition, the authors would like to thank Dr D.J. Picken and Dr J.W.L. Warren whose help and encouragement have been greatly appreciated.

~*(bottom) = a*(top) Bt°p B bottom For ¢i = 0 ° and SL = 20° Bto p = 0.122 (from the left-hand diagram) ¢2 = 70 ° and ¢L = 20°

References

J~bottom = 0.06 (obtained from the right-hand diagram)

2 X rr X (0.025) 3

1

2

1 . 4 2 × 6 . 6 × 12×(0.0125) 3 Thus y (bottom) =

= 2.2mm

Finalised Bearing Design

3

The final design parameters for the bearing required are as follows:

4

Double entry bearing n = D =

6.6mm 2.2 mm 0.0125mm

= 6.6 mm

~

Therefore y(top) =

9.6mm 1.6mm 0.0125mm

Design data has been presented for the prediction o f combined axial and radial loading o f double entry partial spherical bearings. The performance predictions o f the double entry bearing has been compared with the single entry configuration published previously. 2

asnZ3

y (top)

Bottom

Conclusions

2rrYho 3

Therefore y =

as = y = z =

Top

= 9.6mm

4 The slot length y is calculated from the expression: ~*

0.5KN/m 2 0.76 litres/sec 70 N a t e = 0 . 5 3 6 0 N a t e =0.5 0.025mm

12 feed slots per row 52mm

5

Anon. Nasa Contributions to Fluid-film lubrication NASA SP-5058, 1969 Stout K. J. and Rowe W.B. Analysis of Externally Pressurized Spherical Gas Bearings Employing Slot Restrictors, Tribology June 1972 pp 121-127 Stout K.J. and Rowe W.B. Externally Pressurized Bearings Design for Manufacture. Tribology International June 1974 pp 98-106 Shires G.L. and Dee C.W. Spherical Pressurized Fluid Bearings with Feed Slots. Paper 10 5th gas bearing symposium University of Southampton (April 1971) Shires G.L. and Dee C.W. Pressurized Fluid Bearings with Inlet Slot paper 7 3rd gas bearing symposium, Southampton (April 1967)

T R I B O L O G Y i n t e r n a t i o n a l June 1977

169