Analysis of externally pressurized gas-lubricated conical bearings

Wear, 86 (1983)

201

201 - 212

ANALYSIS OF EXTERNALLY CONICAL BEARINGS

K. SRINIVASAN Department

PRESSURIZED

GAS-LUBRICATED

and B. S. PRABHU

of Applied Mechanics, Indian Institute of Technology, Madras 600036

(Received May 27,1982;

(India)

in revised form August 11,1982)

Summary An analysis is presented of externally pressurized conical gas bearings with discrete point pressure sources in the form of orifices located around the circumference at quarter stations from the end of the bearing. The governing Reynolds equation over the developed bearing area is solved by a direct numerical method to determine the pressure distribution. Numerical results are presented in the form of non-dimensional charts for load capacity, flow rate and stiffness for a conical bearing having I,/&, = 1 and semicone angles from 10” to 40”.

1. Introduction Many engineering applications involve rotating parts subjected to a combination of radial and axial loads. While in most cases this can be achieved by using separate journal and thrust bearings, there are advantages in supporting both of the loads by a single bearing. This study is an investigation of the feasibility of carrying both radial and thrust loads by using externally pressurized gas-lubricated conical bearings. Although gas films cannot support unit loads as heavy as liquid films can in self-acting bearings they have specific advantages over other types of lubricant. Because of the low viscosity internal shear stress and friction are low in gas films. Contamination problems can be solved by gas lubrication which is effective over an extremely wide range of temperatures. Gas bearings do not have a relatively high load-carrying capacity, but they are quite stiff so that they resist motion while the load changes. These advantages of gas lubrication have resulted in numerous applications such as bearings for jet engines, gyros, centrifuges, dental drills and machine tools. Gas bearings can operate aerodynamically or aerostatically or with a combination of aerostatic and aerodynamic lubrication. Gas bearings may have different shapes [l] with various methods of compensation [ 21. Many theoretical and experimental investigations on different types of externally pressurized gas bearings have been reported [3 - ES]. A theoretical study of 0043-1648/83/0000-0000/$03.00

@ Elsevier Sequoia/Printed

in The Netherlands

@ 3

202

Z

WT

LIC t

(a)

L--i

Fig. 1. (a) Conical bearing configuration;

(b) developed bearing surface.

externally pressurized orifice compensated conical gas bearings is dealt with in this paper by considering the bearing flow in two dimensions. An externally gas-lubricated conical bearing configuration is shown in Fig. l(a) and the bearing surface developed is shown in Fig. l(b). Gas is supplied to the bearing clearance through equally spaced orifices located around the circumference of the bearing bush. A theoretical estimate of the pressure distribution, radial and axial loads, flow requirements and stiffness of such a bearing is made by a numerical method.

2. Theoretical

analysis

The frustum of the cone which represents the conical bearing configuration may be developed as shown in Fig. l(b). The subtended angle On of the resulting sector is related to the cone angle p by en = 27r sin p

(1)

203

The mechanics of lubricating films are governed by Reynolds’ which may be written over the developed bearing as

With an isothermal written as 1 r

;(

“) rh 3Par

perfect

gas film having constant

l a i_laP) + ;;” ae h Pz

In non-dimensional

form this is

1 Z ?&P!E)+ R

;

For a steady stationary

(ph)

(2)

viscosity

eqn. (2) can be

=6pO;(Ph)+l2p$(Ph)

Z&~;)=*/;(R)

a __ + 2 #W

journal

to

eqn. (4) reduces

equation

(3)

(4)

(5) The non-dimensionalized

film thickness

is given by

- -

h(R,O)=l--ezsinfi-cos/3

(6)

where Ex(R) = Ex(&)

R-l + R Y _ 1 {ML) L

Ey(R) = e&)

R-1 + R Y _ 1 MRL)

- e&)1

- e&)1

L

In the present case results are presented for the bearing in which the taper angles of both bush and journal are equal to 0 and the journal moves parallel to the bush axis. Thus fi becomes a function of only the circumferential bearing arc. The pressure distribution over the developed area is obtained by writing eqn. (5) in finite difference form and solving it by iteration combined with successive overrelaxation satisfying the boundary conditions. The pressure at the ends of the bearing is generally atmospheric, but the pressures at the downstream of the orifices are not known. To calculate the pressure downstream of the orifices, the downstream pressure at one orifice (say orifice 3~) is assumed to be equal to the supply pressure and the downstream pressures at all other orifices are assumed to be equal to the ambient pressure. With these boundary conditions eqn. (5) is solved to obtain the pressure distribution over the bearing area. From this pressure distribution, dimensionless axial and circumferential flow factors are calculated. A similar procedure is repeated for all orifices. The mass flow rate in the axial direction from orifice x is given by

mass flow rate from orifice x to orifice m is given by

The circumferential

where

(9)

where &I = 0, - @D/h hn, = em + i3n/n, &,&%inj3 is the attitude angle at the mth orifice and &.r and & are the dimensionless smaller and larger boundary radii of the mth orifice over the developed bearing area. The dimensionless flow factors of orifice x are obtained from the mass rate of flow:

The total flow out of the xth orifice boundaries

The mass rate of flow through Qax = QA”P”IRgT;_ for k/
- 1)

is

orifice x is

1)/“2{&21k

-Fe

(k+1)lk)1’2

WW

205

and

(lab) For the steady state, outflow. The equations

Qol= QAI+

flow through the restrictor is equal to the bearing of mass flow for n orifices (n/2 orifices per row) are

2 (Qcl-)m - Qcm-I)

m=l n

Qo2 = Qa42+ z

(Qcz+m - Qcm+2)

(14)

m=l

n

Qon =

QA~ +

mgl (Qcn+m - Qcm-rn)

For the stationary steady journal, Reynolds’ equation is linear in P* and the principle and eqns. flow components depend on P *. Using the superposition (lo), (ll), (13) and (14) forF& > {2/(/z + l)}k’(k-l) we obtain l/2

2

t-1

&I

k-l

{P,l

anal

+ g

2/k _j&(k+l)/k)l/*

_

QAl

+

)(Fol$)

+

(Fom* - &)

= 0

(15) 1 i2

2

i-1 k-l

+

BD (PO, *lk - &, (k+l)lk}l’*-

mgLm+.(Fom*$) = 0

where BD =

12p(kR,T)“*CdA,Ps C3P a*

For-F& < {2/(k + 1)) k’(k- ‘) the first term in the above eqns. (15) will be

206 (k + 1)/2(k

-

1)

and other terms will remain the same. Equation (15) consists of n nonlinear simultaneous algebraic equations in ?jOi, PO,, . . . , PO,?and these equations are solved by Brown’s method [9] for various values of E, B,, ps and semicone angle /3. After obtaining the downstream pressure of the orifices, eqn. (5) is solved with these boundary conditions by an iteration technique to evaluate pressure distribution. The dimensionless load and flow components are evaluated as follows:

(16)

VT = sin /3 {“jD iis 0

(P - 1)R de dR

(17)

The above components are evaluated by numerical integration. sionless load components are expressed in the following form:

These dimen-

wR WR=

(18) (Ps -P,)LD&,j

cosp

WT w,=

(19) (Ps -P,)LD,

cosp

The non-dimensional

flow rate through

the bearing is given by

(20) The radial stiffness KR is given by K,

=

dW? de

When expressed

in non-dimensional

form this becomes

KRC

(21)

where

Ail computations

were carried out using an IBM 370 computer

system.

207

3. Results and discussion Results are presented for a conical bearing having L/D, = 1.0, semicone angles 0 = 10” - 40” and 16 orifice point pressure sources. Figures 2 - 9 show 0.6,

0

0

I, I

2

I 3

I ‘5

I

I 6

I 7

I 8

I 12

00

9

I

I 3

I Ia

I 5

I 6

I 70

I

00

80

Fig. -,

S.-Load w,;---,

cap_acity w,.

of the

conical

bearing

(I;/&

= 1.0; fl= 10”; PS = 3.0; n = 16):

Fig. -,

3. Load &;---,

capacity PT.

of the

conical

bearing

(L/DM = 1.0; p = 20”; ps = 3.0; n = 16):

13sip3 I-----

1.2 I'

,,-

4!

I'

/'/'--": IO//'

A'

a9//

OB iSO'-

,///

fr" OS-

;;:/

05-

//

A'

/

/

/

a4 t

Fig. -,

4. Load PB;---,

capacity iQ.

Fig. -,

5,Load cap_acity w,; - - -, Iv,.

of the

conical

bearing

(L/DM = 1.0; fi = 30”; Ps = 3.0; n = 16):

of the

conical

bearing

(LIDM = 1.0; p = 40”; ps = 3.0; n = 16):

9

Fig. 6. Load capacity -) IQ; ----, WT.

of the conical

bearing (L/&J

= 1.0; fl= 10”; PS = 5.0; n = 16):

Fig. 7. Load -, ivR;---,

capacity w,.

of the conical

bearing (L/I&

= 1.0; fl= 20”; H, = 5.0; n = 16):

Fig. 8. Load -, i&;---,

capacity WT.

of the conical

bearing (L/D111 = 1.0; p = 30”; ps = 5.0; R = 16):

Fig. 9. Load -,w'R;---,

capacity WT.

of the conical

bearing (L/DM = 1.0;p = 40"; P, = 5.0; R = 16):

the variation in the non-dimensional radial and thrust loads En and ET for various values of the bearing design parameter BD and eccentricity ratio e and for supply pressure ratios ps = 3 and Fs = 5. Figures 10 - 13 represent variation in ma with E for various values of BD and ps = 3. Figures 14 - 17 give the mass flow rate requirements. It can be seen from Figs. 2 - 9 that, for any semicone angle 0, Wa increases with decrease in thrust load @, as E increases. While WR increases

209

Fig. 10. Radial stiffness of the conical bearing (L/&J

= 1.0; fl= 10”; Ps = 3.0; n = 16).

Fig. 11. Radial stiffness of the conical bearing (L/DM = 1.0; p = 2O”;Ps = 3.0; n = 16).

0

0

03

06

09

0

0

&

06

03

09

6

Fig. 12. Radial stiffness of the conical bearing (L/l& = 1.0; p = 3O”;Fs = 3.0; n = 16). Fig. 13. Radial stiffness of the conical bearing (L/D,

= 1.0; p = 4O”;ps = 3.0; n = 16).

considerably from E = 0.6 to E = 0.9 forps = 3.0, there is little improvement in wR for the same eccentricity ratio range for Hs = 5.0 for a wide range of values of BD. This indicates that radial stiffness is greater at low e values than at high E values for Bs = 5.0. Figures 10 - 13 show that the radial stiffness is generally good for a wide range of values of e and BD for a supply pressure ratio Ps = 3. There is no particular Bn value which gives both maximum radial load and maximum stiffness. Thus to obtain maximum values of @n or Ra proper values of BD should be chosen. For a given L/r), value, a semicone angle fi is selected according to the order of thrust load to be taken by the bearing. BD is a function of the bearing design dimensions, lubricant properties and supply pressure ratio. For a bearing using com-

Fig. 14. Flow rate of the conical bearing (L/DM = 1.0; fl= 10’; n = 16). * -( ---,P,=5.0.

Ps = 3.0;

Fig. 15. Flow rate of the conical bearing (L/DM = 1.0; ~3= 20”; n = 16). * ----) ---,fis=5.0.

rj, = 3.0;

60

50

40 " '-30

20

10

0

1

3

5

7

9

6.3

Fig. 16. Flow rate of the conical bearing (L/~~ ---,P,=5.0.

= 1.0; fi = 30”; n = 16): -,

F.s = 3.0;

Fig. 11. Flow rate of the conical bearing (L/DM = 1.0; fl= 40”; n = 16). *-----,&=3.0; --,Ps”5.0.

pressed air as a lubricant if the supply pressure is selected on the basis of the external pressure source then the value of BD depends on the selection of the concentric radial clearance C and orifice dimensions. To achieve optimum values of Bn proper orifice d~ensjons need to be chosen. The optimum value of BD is 3 - 7 for Fs = 3.0 and 1 - 3 for Fs = 5.0 for all 0 values.

4. Conclusions The analysis shows that it is possible to use externally pressurized conical gas bearings for combined radial and axial loads. The results given in

211

non-dimensional form for various cone angles and a wide range of bearing design parameters can be used for conical bearing design. The analysis can be extended to bearings having any number of supply holes, ratio L/DM and orifice arrangements.

Acknowledgments The present work is a part of research sponsored by the Department Science and Technology, New Delhi. The authors thank Dr.-Ing. B. V. A. Rao for useful discussions.

of

References 1 W. A. Gross, Gas Film Lubrication, Wiley, New York, 1962. 2 W. A. Gross, Gas bearings: a survey, Wear, 6 (1963) 423 - 443. 3 0. Pinkus and B. Sternlicht, Theory of Hydrodynamic Lubrication, McGraw-Hill, New York, 1961. 4 R. Lemon, Analysis and experimental study of externally pressurized air lubricated journal bearings, J. Basic Eng., 84 (1962) 159 - 165. 5 G. L. Shires, The design of externally pressurized bearings. In N. S. Grassam and J. W. Powell (eds.), Gus Lubricated Bearings, Butterworths, London, 1964, Chap. IV. 6 V. N. Constantinescu, Gas Lubrication, American Society of Mechanical Engineers, New York, 1969. 7 B. C. Majumdar, On the general solution of externally pressurized gas journal bearings, J. Lubr. Technol., 94 (1972) 291 - 296. 8 S. M. Rohde and H. A. Ezzat, Computer-aided design of hybrid conical bearings. In S. M. Rohde (ed.), Fundamentals of Design of Fluid Film Bearings, American Society of Mechanical Engineers, New York, 1979, pp. 85 - 131. 9 G. D. Byrne and C. A. Hall, Numerical Solutions of Systems of Nonlinear Algebraic Equations, Academic Press, New York, 1973.

Appendix

A: nomenclature

A0

orifice area

BD C

12ErCdA,(kR,T)1’2Ps/C3P~2, bearing design parameter concentric radial clearance orifice discharge coefficient mean diameter of the conical journal displacement of the journal in the radial direction concentric position dimensional film thickness h/C, dimensionless film thickness ratio of specific heats of the gases stiffness and stiffness parameter length of the conical bearing

Cd DM e h h k KR,

L

gR

from

the

212

number of orifices (n/2 orifices per row) dimensional pressure P/P,, dimensionless pressure ambient pressure supply pressure P,/P,, supply pressure ratio downstream orifice pressure of xth orifice Pox/Pa, orifice pressure ratio of xth orifice mass rate of flow in axial direction from orifice point x (x = 1, 2, . ..) n) flow factor in the axial direction total mass flow rate 12pR,TQc/C3Pa2, dimensionless bearing mass flow rate mass rate of flow in the circumferential direction from orifice point 3c to orifice m (m = 1,2, . . . . n) flow factor in the circumferential direction mass rate of flow through orifice x radius represented over the developed bearing sector r/RM, dimensionless radius gas constant dimensional larger end radius of the developed bearing sector

%

RLIRM

RL

t T b WR

W _T WT x

y, 2

P E fx, EY, EZ 9 &I A 7 w c;

mean radius of the developed bearing sector dimensional smaller end radius of the developed bearing sector RsIRM time absolute temperature dimensional radial load dimensionless radial load dimensional axial load dimensionless axial load coordinates semicone angle of the conical bearing e JC, eccentricity ratio eccentricity ratios in X, Y, 2 directions (Fig. l(a)) attitude angle subtended angle of the developed bearing sector GpGR,*/P,C*, conical bearing number dimensionless time rotational speed of the conical journal w sin /3