Sd 1

1

Wear, 52 (1979) 1 - 11 @ Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands

DESIGNOFEXTERNALLYPRESSURIZED~ROUSGASREAR~GS WITHJOURNALROTATION N. S. RAO

~e~a~~e~~of~ech~nical

Engineering, Indian Institute of Tech~olo~,

(Received

in final form February 27,1978)

June 13,1977;

Khamgpur (India)

Summary The steady state characteristics of externally pressurized porous gas bearings with journal rotation are obtained from a numerical solution of the modified Reynolds equation using the Unite difference technique. Design charts for the load capacity and the attitude angle for various design parameters are presented. The dimensionless mass rate of flow is given in a short table. A numerical example is given to illustrate the design procedure.

1. Introduction Gas-lubricated porous journal bearings have been used for a considerable time [ lf.Initial experimental work [ 21 showed that such bearings, supported on rubber O-rings, could be rotated up to a speed of about 4200 Hz with a shaft diameter of about 6 mm. There are many theoretical analyses of journal bearings but very few for a rotating journal bearing [ 1,3]. Majumdar 14f presented a theoretical analysis for the design of non-rotating journal bearings considering a three-Dimensions compressible flow through the porous bush and Sun [ 51 presented a steady state solution for a rotating journal bearing using a linearized ph method and considering one-dimensional flow through the porous bush. The present work estimates theoretically the steady state characteristics of a rotating journal bearing using pressure perturbation theory. 2. Formulation

and analysis

Figure 1 shows a gas journal bearing with a porous surface. The porous bushing would normally be attached to an impermeable housing and a closed cavity would be formed between the two components. The compressed gas is fed from behind the bushing at a constant pressure pg. The coordinate system (x, y, z) is fixed with respect to the stationary bushing (Fig. 1). The thickness of the porous bushing is small compared with the radius of the

Fig. 1. A schematic

diagram of a porous gas journal bearing.

bearing, and as a result the gas may be considered as flowing only in the radial direction. With the usual assumptions for gas lubrication of porous bearings, the pressure distribution in the lubricating film is governed by the modified Reynolds equation [ 51 :

The boundary

conditions

p=1 ap -= 37

of the bearing are aty=klandO
0

(2)

P(d,Y) = is(fl + Sn,Y) For small eccentricity ratios the pressure may be perturbed with the eccentricity ratio E and a first-order pe~urbation solution is valid: p=p*

+e& (3)

E = 1 +ecose where P = is@,?)

PO =&I(Y)

PI =P1(G)

Substituting eqn. (3) in eqn. (1) and collecting order powers of e gives

e1:

a; +($“E$

+ 1.5

(g,‘g

the zeroth-order

cos 0

and first-

3

(4b)

--

where Q = popI. j,

The boundary

aty=OandO
aty=kland0<0<2n atj=OandO<e

0

Q(YJ ) = QW

< 2n

__; = (1 -P%osh (L/DA;‘2Y) cash (L/DA;‘2) By substituting +

ae2

(5b)

+ 277)

Solving eqn. (4a) with the boundary

_ a2Q

accordingly: W

0

Q=O -= aQ a7

are modified

at~=+landO<0<2n

= 1

a& -= a?

conditions

conditions

given by eqn. (5a) gives

+ _-2 s

(6)

for a2$j lay2 from eqn. (6), eqn. (4b) can be written

(_D )2a2Q -ay2 +i.5qp: L

= A ;

5

p.

-

-_p:)cose

&sine

+A,Q

(7)

ae

Equation (7) was solved for Q numerically (by iteration) in a finite difference form using an EC 1030 digital computer. A successive over-relaxation scheme was used with the central difference method to represent the partial derivatives. The load capacity and attitude angle were then computed: 1

W,

&

LRpar

r

Lf

0 0 1

2

LRp,c WE _

W

J-P,

,,ss o =

E

(jq

case

de dy

VW

PQ

2n

WtI

@,=-=

Q

2%

_=-

+ ,p

PO

sine de dy

(8b)

4

The mass rate of flow of gas through the bearing can be written G = 2 j’2j* 0 0

-

;r;),_,

RdBdyia

~a

(94

After simplification eqn. (9a) can be written G

G=

=A -22~B

(KRLP,P .hH)

(9b)

where A=

n(j,2 - l)tanh(L/DA;‘2) P,(LIDM,

@cl

After obtaining Q from eqn. (7), the integrations in eqns. (8) and (9) were performed numerically by computer. The static stiffness of the bearing can be written S = dry

(10)

For the linear first-order perturbation method, the stiffness S can be written

Therefore the stiffness (at any radial clearance) can be calculated from the dimensionless load capacity W.

3. Results and discussion The analysis shows that there are four independent parameters which determine the characteristics of this type of bearing, namely L/D ratio, supply pressure j& , feeding parameter AP and bearing number A. The load capacity W and attitude angle 9 for a bearing with L/D = 1.0 are given in Figs. 2 - 5 for different feeding parameters (AP = 0.5, l.O,z.6 and 4.0). Figure 6 shows the effect of the feeding parameter AP on W and 4. T&e effect of the L/D ratio on W and 9 is given in Fig. 7. The results for W and 4~ for bearings with L/D = 0.5, 2.0 and_3.0 are given in Figs. 8, 9 and 10 respectively. The mass rate of flow G for these bearings can be calculated from the d&a presented for A and B in Table 1. Table 2 shows the comparison of W and G with previous results [ 51.

24

I

Fig. 2. Variation of w and i$ with h and Fs for L/D = 1.0, A, = 2.0: --,W;---,(b* Fig. 3. Variation of w and $Jwith A and ss for L/D = 1.0, &, = 0.6 : -,

w; - - -, @.

24

I2

24

i

0 ------A

I

3

5

7

9

II

----A

Fig. 4. Variation of w and r# with A and & for L/D = 1.0, A, = 4.0: -,F;---,c$. Fig. 5. Variation of w and 4 with A and & for L/D = 1.0, A, = 1.0: -,%;---,

3.1. Effect of supply pressure W increases as j?, increases. Qtdecreases with increasing p, for all values of A for L/D = 0.5 and 1.0, but increases with j& for bearings with &/II = 2.0 and 3.0 for higher values of bearing number A (Figs. 2 - 10). G increases with increasing j&, (Table 1).

#.

6

;‘\::, ,

1 ‘t ,!

6,

,f20

,--ih7-_,6 ’

/’

/ 8 5

A-II-0

1' /:

I2

AX2'0 8

4 I-'JI 3

I'\c____--I

---*:;,7j4

Fig. 6. Variation -, W;---,$.

of w and 4 with feeding

parameter

/\p for L/D = 1.0 and Fs = 5.0:

Fig. 7. Variation --, 0.

of w and 4 with L/D ratio and A for Fs = 3.0, A, = 2.0; -,w;

Y

4

0 0

1

3

5

7

9

II

0

I

-A (a)

Fig. 8. Variation (b) AP = 2.0.

3

5

7

9

II

h @I

of w and # with A and Fs for L/D = 0.5: -,

@; - - -, #. (a) &, = 1.0;

3.2. Effect of bearing number w increases as A increases. rpincreases with A to attain a maximum value and then decreases as A is further increased. ??-is little affected as is indicated by the values of A and B presented in Table 1. B is less than A

r 12

48

IO

40

8 136

32 I 248

4

lb

2

8

0 0

_L_~_l._.J I 3

t 5

L

17

9

0 II ----WA

-A

tb)

(a) Fig. 9. Variation (b) Ap = 2.0.

of w and Qtwith A and & for L/D = 2.0: -J?;---,

----an

-A

(4 Fig. 10. Variation (b) &, = 2.0.

4. (a) Ap = 1.0;

(b) of w and @ with I\ and & for L/D = 3.0: _,~;_-_

, 4. (a) hp = 1.0;

and A is independent of A, and therefore for all practical purposes ?? can be taken as being equal to A which is independent of A and E. Thus g can be obtained in the closed form given by eqn. (SC).

8 TABLE1 ValuesofA and B (eqn.(9b))forvarious operational L/D

0.5 1.0

2.0 3.0

A*

A

1.0 2.0 0.5 1.0 2.0 4.0 1.0 2.0 1.0 2.0

conditions

A

p, = 3.0

p," 5.0

p,= 8.0

7,742 7.213 7.213 6.380 5.262 4.038 4.038 2.941 2.778 1.973

13,937 12.984 12.984 11.484 9.473 7.268 7.269 5.294 5.001 3.552

22.865 21.302 21.302 18.841 15.541 11.925 11.925 8.626 8.206 5.828

B

&= 0.5

1.0

2.0

1.0

0.5

1.0

2.0

4.0

2.0

1.0

2.0

3.0

1.0

2.0

G=A-2265

1.0 5.0 10.0 1.0 5.0 10.0 1.0 5.0 10.0 1.0 5.0 10.0 1.0 5.0 10.0 1.0 5.0 10.0 1.0 5.0 10.0 1.0 5.0 10.0 1.0 5.0 10.0 1.0 5.0 10.0

3.0

0.014 0.014 0.011 0.020 0.019 0.014 0.083 0.098 0.012 0.103 0.107 0.027 0.087 0.078 0.027 0.037 0.027 0.009 0.215 0.397 0.167 0.099 0.137 0.041 0.177 0.569 0.296 0.061 0.147 0.048

p, = 5.0 0.043 0.040 0.033 0.062 0.058 0.051 0.244 0.262 0.120 0.308 0.309 0.241 0.263 0.246 0.197 0.112 0.095 0.071 0.618 0.879 0.983 0.296 0.343 0.361 0.481 1.011 1.612 0.177 0.293 0.405

-

p, = 8.0 0.113 0.109 0.099 0.163 0.157 0.145 0.637 0.658 0.596 0.806 0.805 0.762 0.697 0.665 0.616 0.301 0.270 0.231 1.601 1.886 2.450 0.779 0.811 0.942 1.220 1.792 2.915 0.461 0.570 0.886

9

TABLE 2 Comparison of present results with the results of ref. 5 (Table 1 and Fig. 10) A

1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 L/D = l.O,p,

w

F

This work

ref. 5

This work

ref. 5

5.25 5.30 5.36 5.44 5.54 5.65 5.78 5.82 6.05

5.14 5.22 5.30 5.40 5.50 5.60 5.70 5.80 5.92

7.220 7.228 7.226 7.232 7.235 7.238 7.238 7.240 7.240

7.280 7.280 7.279 7.277 7.275 7.273 7.270 7.268 7.264

= 5.0, AP = 4.0, E = 0.4.

3.3. Effect of feedingparameter Figure 6 shows that w increases with AP to a maximum value around AP = 2.0 and then decreases as A,, is further increased for all values of A. I#Jdecreases rapidly and then increases slightly as A, increases for low values of bearing number (A = 2.0), whereas it increases continuously with increasing A, for high values of A (A = 11.0). c decreases with AP for all supply pressures and L/D ratios (values of A from Table 1).

3.4. Effect of L/D ratio For small bearing numbers (i.e. at low speeds of operation) w increases with increasing L/D ratio to a maximum around L/D = 1.0 and then_ decreases with a further increase in L/D ratio. For higher values of A, W increases continuously with increasing L/D ratio (Fig. 7). It is also observed that the variation of w with A for higher L/D ratios is more pronounced even for small values of A (Figs. 9 and 10) which indicates that the L/D ratio enhances the effect of journal rotation. 4 increases as the L/D ratio increases for all values of A (Fig. 7). c decreases with increasing L/D ratio (Table 1).

3.5. Comparison with previous results Table 2 shows a comparison of w and ?? with the values obtained by Sun [ 51; agreement between the two methods is good for small eccentricity ratios (E 4 0.5). Although the present analysis gives the steady state characteristics for small eccentricity ratios, there is no difficulty in obtaining convergence of the numerical process for any arbitrary value of bearing number A as reported in ref. 5.

10

3.6. Design procedure

The use of the design data presented can be illustrated by a numerical example. A journal bearing operating at a speed of 20 000 rev mine1 has the following specifications at room temperature (293 K): L = 5 X 10d2 m

D = 5 X 10e2 m

3, = 5.065 X lo5 Pa

Pa = 1.013 X lo5 Pa

K=lX

rl = 18.3 X 1Ou5 N s rnd2

10-15m2

H=5X

10e3m

Figure 6 shows that the maximum load occurs when A, = 2.0. Using A, = 2.0 gives = 1.68 x 10-s

Therefore = 8.0

Now from Fig. 2 the corresponding load can be obtained as w = 6.62 and from Table 1

c: = 9.473 - 0.442~ W and G can be obtained by substituting variables. The stiffness is given by s=

LRP, -

c

W

=2X105

Fore=0.4,W=330NandG=

the dimensional values of the

Nm-l 7.5X10”

kgs-‘.

4. Conclusions (1) The method presented gives the steady state characteristics of a bearing with journal rotation operating at small eccentricity ratios (E < 0.5). (2) The results are expressed in dimensionless form and are presented for a wide range of de&n conditions which will be helpful for bearing design. (3) Thcmass flow rate z is almost independent of the speed of rotation A and thus G can be taken as equal to A which is expressed in closed form (eqn. (90 (4) the static stiffness S can be obtained from W and C.

11

Acknowledgments The author thanks Dr. B. C. Majumdar for stimulating discussions and valuable suggestions. The author is grateful to Prof. R. Mishra for his kind interest and encouragement.

Nomenclature A, B C D e G, G Hh, h K L P, F P’,P pa _ p4, ps pot Pl

Q R S w, w W,, w,

We,We x, Y, 2 ;,y,; 77 A 4 Pa Q 0

defined in eon. (9c) radial clearance journal diameter eccentricity G = G/(KRLpg,/r/H), mass rate of flow (dimensionless) t_ickness of porous bushing. h = h/C, film thickness permeability coefficient of porous material length of the bearing 5 = p/p,, pressure (absolute) in the bearing clearance p’ = p’/pa, pressure (absolute) in the porous media ambient pressure (absolute) Ps = PslPar SuPPlY pre=ure defined in eon. (3) -POP1

journal radius stiffness of the bearing W = WILRp,e, load capacity of the bearing W, = WJLRp,e, load capacity parallel to the line of centres We = We /LRp,e, load capacity perpendicular to the line of centres coordinates e/C, eccentricity ratio 6 = x/R, y = y/(L/2), z= z/H, nondimensional coordinates coefficient of absolute viscosity of gas (~Qu/P,)(R/C)~, bearing compressibility number (dimensionless) 12KR2/C3H, feeding parameter (dimensionless) ambient density attitude angle journal rotation speed

References 1 H. J. Sneck, A survey of gas-lubricated porous bearings, J. Lubr. Technol., 90 (4) (1968) 804 - 809. 2 A. G. Montgomy and F. Sterry, A simple air bearing motor for very high rotational speed, AERE Rep. no. ED/R 19671, Atomic Energy Research Establishment, Harwell, Oxon., Gt. Britain, 1955. 3 B. C. Majumdar, Gas lubricated porous bearings: a bibliography, Wear, 36 (1977) 269 273. 4 B. C. Majumdar, Design of externally pressurized gas-lubricated porous journal bearings, Tribol. Int. (April 1976) 71 - 74. 5 D. C. Sun, Analysis of the steady state characteristics of gas-lubricated porous journal bearings, J. Lubr. Technol., 97 (1) (1975) 44 - 51.