Tilting effects on the performance of hydrostatic and hybrid gas thrust bearings

Tilting effects on the performance of hydrostatic and hybrid gas thrust bearings Z.S. Safar*

A hydrostatic gas thrust bearing is analysed to determine the effect of tilting on its performance characteristics. The governing Reynolds Equation is solved simultaneously with the mass flow rate equation by numerical methods to determine the inlet pressure and the pressure distribution throughout the pad. It is shown that the load, friction, and lubricant mass flow rate are strong functions of the bearing number X, restrictor coefficient Xo, supply pressure Ps and tilt parameter e Externally pressurized gas lubricated bearings are commonly used in industrial practice due to their potential advantages, which include zero starting friction and small viscous running friction, negligible heat generation and freedom from vibration. The principal disadvantage is the need for continuous supply of pressurized gas. Pressurized air bearings have been applied in many fields I , for example machine tools, measuring and inspection instruments and process, manufacturing and medical equipment. Most hydrostatic bearings are usually designed to operate under parallel conditions, but manufacturing tolerances, structural and thermal deformations result in the bearing surfaces being relatively inclined as shown in Fig 1. Several investigators have studied the problem of misalignment in hydrostatic incompressible bearings 2-6 and significant data are available for the bearing designer. In comparison the aerodynamic gas bearing has received much less attention. Some analytical and experimental results are available for the effect of misalignment in externally gas lubricated journal bearings 7 . The effect of misalignment was studied by Andrisano and Maggiore s for a porous thrust bearing using the lumped parameters method ;'it was concluded that misalignment has a great effect on the load capacity of the bearing. The purpose of this work is to analyse the effect of misalignment on the important performance characteristics of an externally pressurized orifice compensated air thrust bearing. The fluid film thickness is taken as a function of the radial and circumferential directions, the Reynolds Equation is solved by numerical techniques and the load capacity friction and rate of mass flow are presented as functions of bearing number, orifice coefficient, supply pressure and tilt parameter.

Analysis

A

ca F

f

g ho h

Mo Mp P Pa Ps Po

p

R r, 0, z r, Z

ri, ro T

T~ H, 1), W

W 7 C

?, Xo /a p £Z

Area of orifice Coefficient of discharge Friction force onthe bearing Coefficient of friction Gravitational acceleration Film thickness at the centre of the bearing Lubricant film thickness Mass flow through the orifice Mass flow thrmlgh the pad Lubricant pressure Ambient pressure Supply pressure Static pressure downstream of inlet Dimensionless pressure Gas constant Polar coordinates Dimensionless coordinates Inner and outer radii of the bearing Gas temperature Supply temperature Radial, circumferential and axial velocity components Load carrying capacity Pressure ratio (= Po/Ps) Angle of tilt (Fig 1) Film thickness variation (= 7ro/ho) Bearing number Orifice coefficient Fluid viscosity Density of lubricating gas Rotational speed

I n solving Eq (1) the following dimensionless variables are introduced

The pressure distribution is described by the Reynolds Equation

3 aP ~ a 3P ~(hP) r or(Oh r~r)+~- ~ (ph -~-~)=6ug2r 2 30

Nomenclature

p___e Pa (1)

It is assumed that the gas film behaves as an ideal gas with constant specific heat, also the flow is considered isothothermal with Pip = constant. *Mechanical Engineering Department, Florida Atlantic University, Boca Raton, Florida 33431, USA

0301-679X/81/040199-04 $02.00 © 1981 IPC Business Press

f

(2)

ro

h h-= To = 1 +eicosO ?, = 6/~g2ro=

hg l"a TRIBOLOGY international AugtJCt 1981

199

Safar - Tilting effects on the performance o f hydrostatic and hybrid gas thrust bearings

Thus the dimensionless Reynolds is

o (fi~~ o~ ~. + ~ (~3 oP ~ OPho7- ~ - ) 0o ~ - ) = 2 x : 0-7-

(3)

with the boundary conditions

P(~, 0) Po =

fi(1,0) = 1

(4)

P(,~, 0) =P(~, 0 + 2~) In order to determine the inlet pressure Po, the rate of mass flow through the orifice is equalized to that through the pad thus mo = mp The mass flow through the pad is calculated as

(5)

1

n-1

_CdAPsI2g n )12[Po]n[ X/Ts R ( n : l L~J

1

Po t~--]2

LI-(~) J

~

F [~=qdO = XoPs~

(7)

_(

)

2

(8)

24 Cd/aro2

hgPaPs

[

2gnRT x/2 n_l ]

AO

[ (Ph)i'f+l-(Ph)i']-l]

The finite difference equation is solved by the Gauss-Seidel method. The boundary conditions are given by Eq (4). To determine fro, the expression 0ff/07 is assumed and the inlet pressure fro is calculated using Eq (7), the procedure is repeated until the inlet pressure converges to a desired tolerance. The load capacity is calculated using the expression

~_

where Xo is the orifice coefficient and given by ao

1

c i j - (A0): (Ph3)i'J+'/2

a i / - ( 52~" _p~h3 ) ( ) ~_,/~,j ei/= ( A2rr)i (p~3)i- V2,j

fij -

From continuity considerations the following equation is obtained 1 n-1 1

o

1

b i j - (AO) 2 (ph3)i,j-1/2

(6)

The mass flow rate at the throat of the orifice

Mo

bijPi,j-1 + c ijPi,j+ l + aijff i-l,j + dijff i+ l,/+ eijP)j = fij where

dij = - (bij +cff + ai/+ ei] )

Mp = f2~ Ph 3 ~P o 12-~ ~-~ r [r=ridO

1

The dimensionless lubrication equation is solved in finite difference form. The finite difference grid has n subdivisions of length AF= (1 -f)/n on the radial width, while the circumferential arc is subdivided into m increments of length A0 = 2n/m. With grid coordinates i andj in the F and 0 directions, Eq (3) transforms in the finite difference form

W

Paro2

_f2.fl(fi_l)FdYdO o 7i

+(~o_l)~r~_i2

(10)

The friction force is given by (9)

if,_

F _ f2~rfl 1 av Pahoro o r i ~ - z fdFd0

(11)

where

vitro - Paho 2--

1 8P

X

2~- ovZ--;(22 - ~) + ~-(1 - 2 )

(12)

The mass flow rate is

~ I f2.M p aP M = PaPah~ - 12 o 3r- ~- I~-=~d0

!

°uI i

Fig 1 Configuration of tilted thrust bearing

200

TRIBOLOGY international August 1981

(13)

Results The pressure distribution and the performance characteristics of a hydraulic thrust bearing are calculated for different values of bearing number X, film thickness variation e, restrictor coefficient ko and the values of supply pressure ffs = 2 and 5 respectively. Figs 2 and 3 show the variation of the load capacity with bearing number for e varying from 0-0.8. At e=O (parallel surface operation) the load is independent of the bearing number. The dependence of load on the bearing number is more sensitive at higher values of tilt parameter. The load is sensitive at higher values of tilt parameter. The load is increased as the supply pressure Ps is increased. The dependence of the friction force P on the bearing number is shown in Figs 4 and 5. For parallel surface operation the relation between the friction and the bearing number is linear. It is shown in the figures that the friction force is increased with the tilt parameter and decreased as the supply pressure is increased. The variation of the load with X and e is more pronounced than that for the friction force.

Safar - Tilting effects on the performance o f hydrostatic and hybrid gas thrust bearings

1.3 XO= I

•

=0.8

XO= I 1.2

--

I.I

--

• =0.8

0.6

1.0

g 8

o~

o 0.9 0.6

0.4

0.8 --

2

i

i

I

1

0

2

4

6

Bearing

I 8

0.7 -Z

o

I0

number, ),

0.6 I 0

1

I

I

I

z

4

6

8

Fig 2 Distribution o f load capacity with bearing number for Ps = 2, and Xo = 1

10

Bearing number, X

Fig 3 Distribution o f load capacity with bearing number for Ps = 5, and Xo = 1

+I

~=2 k0= I

XO= I

/~

• =0.8

]

=0.8

.

g

3

._o

o

,,=

9.2

i_g LL

1

I

I

I

2

4

6

8

I0

0

Z

Bearing number, X

Fig 4 Distribution or friction force with bearing number for Ps = 2, and Xo = 1

4

6

8

I0

Beorir~ number, X

Fig 5 Distribution or friction force with bearing number for Ps = 5, and Xo = 1

TRIBOLOGY

i n t e r n a t i o n a l A u g u s t 1981

201

Safar - Tilting effects on the performance o f hydrostatic and hybrid gas thrust bearings

~--5 X 3

-

=6

X =6 >.-

o

,t = 0 8 ~ ~

3

0.6

70

0.4

8

0.6 o E

g

0.4

Z

2

o J

I

I I 0.2

I 0.4

I 0.6

I

I

0.3 Pressure ra~io, a

0.6

[ 0.8

I

0.5 Pressureroti0,a Fig 7 Distribution o f lubricant mass f l o w rate with restrictor coefficient for Ps = 5 and X = 6

Fig 6 Distribution o f load capacity with restrictor coefficient for ~ = 5 and X = 6

It was f o u n d from the calculations that the dependence o f the lubricant mass flow rate on the bearing n u m b e r is very small and m a y be neglected. Therefore for a given

Air Dry * Solid

Film

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202

[ I I 1 0.4 0.6 Restrictor coefficient, k o

LO

I

i

70

I

0.8

Restrictor coefficient,X o 0

I 02

T R I B O L O G Y international August 1981

supply pressure and restrictor coefficient the supply o f gas is constant and i n d e p e n d e n t o f the bearing number. For a given bearing n u m b e r X = 6 and supply pressure F s = 5 the effect o f restrictor coefficient Xo on the load capacity is shown in Fig 6. It is concluded that the load is increased w i t h Xo, and that d e p e n d e n c e is more p r o n o u n c e d at higher values o f tilt parameter. It was f o u n d from the calculations that the friction is i n d e p e n d e n t o f the restrictor coefficient, while the lubricant mass flow rate is greatly d e p e n d e n t on Xo as shown in Fig 7.

References 1. Wunsch M.L. and NimroD W.M. Industrial Applications of Gas Bearings in the UK. I. Mech. E. and I, Prod. E., Proceeding Conf. Externally Pressurized Bearings, 1972, Paper C18, 114-132 2. Bennet T.P. Resistance to Tilt of Hydrostatic Slipper Pads. BHRA Report, December 1961, RR 712 3. Fisher M.J. A Theoretical Determination of Some Characteristics of a Tilted Hydrostatic Slipper Bearing. BHRA Report, April 1962, RR 72 4. Royle J.K. and R a i z a d a R.S. Numerical Analysis of Effects of Tilt, Sliding and Squeeze Action on Externally Pressurized Oil-Film Bearing. L Mech. E., 1966, 4th Lub. and Wear Cony. 5. Howarth R.B. and N e w t o n M.J. Investigation of the Effects of Tilt and Sliding on the Performance of Hydrostatic Thrust Bearings. 1. Mech. E. and L Prod. E., Proceedings Conf. Externally Pressurized Bearings, 1972, Paper C20 6. Safar Z.S. Frictional Behaviour of a Hydrostatic Thrust Bearing Under Dynamic Loading Conditions. Presented at the 7th Leeds-Lyon Symposium, Leeds, September 1980. Proceedings of the 7th Leeds-Lyon Symposium, IPC Science and Technology Press Ltd, Guildford, UK, 1981, 249-254 7• Markho P.H., Grewai S.S. and Stowell T.B. An Experimental Investigation of the Effect of Misalignment and Directionality on the Performance of an Externally Pressurized Orifice Compensated Air Journal Bearing. ASME TRANS, Series F, Jan 1979, 101(1) 28-32 8• Andrisano A. and Maggiore A. Theoretical and Experimental Analysis of an Externally Pressurized Porous Gas Thrust Bearing. Tribology International, 1978, 11(5) 285-288

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