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Prepared under

Contract No. Nonr-3 8 15 (00) (X) T a s k NR 062-297

Supported jointly by

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S p a c e Adminis 'ora tion

A d m i n i s t e r e d by

S. Doroff Fluid D y n a m i c s Branch O f f i c e of N a v a l Research D e p a r t m e n t of the N a v y

O N FLUID INERTIA EFFECTS IN INFINITELY WIDE FOIL BEARINGS

Prepared by : A. Eshel M e m b e r pf #he Research Staff

Approved by:

M a n a g e r , p e c h a n i c s Section

Approved by: Ee2&< -J--

W. A. Gross Vice P r e s i d e n t , Research & A d v a n c e d Technology, D i r e c t o r of Research

REPRODUCTION I N PART OR I N WHOLE IS PERMITTED FOR ANY PUROSE O F THE UNITED STATES GOVERNMENT.

Ampex Corporation Research and Advanced Technology Division

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4

ACKNOWLEDGEMENTS

This work was sponsored by the Office of Naval Research,

Fluid Dynamics Branch, under Contract Nonr 38 15 (00) (X) administered by

Mr. S. Doroff.

Thanks are due to M i s s T. Woo for a s s i s t ance in programming.

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ABSTRACT

Equations for a foil over a lubricating f i l m in which the effects

of fluid inertia a r e taken into account a re derived. Approximate solutions

showing the effect of inertia and fluid compressibility a re obtained.

effect of inertia is to increase the fluid f i l m thickness.

The

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NOMENCLATURE

Compresibility Parameter 'are T

C I

C* Normalized Compressibility Parameter (1 + C)H*

f

h

H - H

I

I*

k, m, n L

P

pa

r

r

R 0

t

ta

T

V 0

V r h V

Shape Factor for Velocity Profile

Radial Clearance h Dimensionless Clearance = - r

Normalized Clearance = H/H*

Inertia Parameter 1/ZpaU2/(T/U 0 )

Normalized Inertia Parameter IH*

Exponential Measures

Distance of Guide from Point of Tangency

Pressure under the Foil

Atmospheric Pressure

- 2/3

0

Polar Angle

Spindle Radius

Local Radius of Curvature

Temperature

Ambient Temperature

Tension per Unit Width of Foil

Velocity Component in the Angular Direction

Velocity Component in the Radial Direction

Dimensionless Radial Component of Velocity

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NOMENCLATURE (Cont)

A U

U

cp

P

P

Pa

n

Dimensionless Angular Component of Velocity

Speed of Foil or Shaft

Dimensionless Parameter 6rJ.u T

Small Perturbation of Clearance

Air Viscosity

Air Density at (p, t a Air Density a t Ambient Conditions (p t ) a’ a

Dimensionless Pressure = ( p - pa)/(T/ro)

Polar Angle

Wrap Angle

Dimens ionle s s Polar Angle

Normalized Polar Angle

Dimensionless Radial Distance from Spindle

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1. 0 INTRODUCTION

2. 0 ANALYSIS

3. 0 ASYMPTOTIC BEHAVIOR

REFERENCES

P a g e

1

1

10

17

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LIST OF FIGURES

1. Schema tic View of Problem under Considera tion

2. Roots of Characteristic Equation

3. Bounds of Uncertainty in the Evaluation of H* (C* =

4. Nominal Clearance H* as a Function of Compressibility (C)

and Inertia (I) Parameters

5. Nominal Clearance H* as a Function of Compressibility (C) and

the Normalized Inertia (I*) Parameters

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ON FLUID INERTIA EFFECTS IN

INFINITELY WIDE FOIL BEARINGS

1. 0 INTRODUCTION

Recent applications of foil bearings (Fig. 1) in high speed tape trans-

ports and in high speed rotor supports necessitate the study of fluid inertia

effects. This report is devoted to an approximate treatment of these effects.

2. 0 ANALYSIS

We star t with the steady, two-dimensional, compressible Navier-

Stokes equations in polar coordinates:

1

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U

Fig. 1 Schematic View of Problem under Consideration

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The continuity equation is

The perfectly flexible foil equation is

1 c - p-Pfi - q

The isothermal equation of state is

P g-, The curvature in polar coordinat.es is

and the clearance

(4)

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In previous analyses of foil bearings [ 11, a s m a l l parameter

expansion was used to formally discard terms of higher order.

lowing, the same expansion is used for the Navier-Stokes equations.

will yield a consistent set of equations, including the effect of inertia.

This reduction may not be correct in the large gap inlet and exit regions

of the flow since the expansion is incomplete in terms of boundary con-

ditions. I t is expected, however, that for small inertia parameters, the

effect of the boundary conditions will not penetrate into the small gap

regions.

resul ts must be viewed as estimates only.

In the fol-

This

Due to this rather heuristic character of the derivation, the

We introduce the dimensionless parameters

and t h e dimensionless variables

A - k ( Y = v

foil bearing number

compressibility parameter

inertia parameter

(9)

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where k, m, n a re constants to be determined.

In accordance with conventional foil bearing theory, the following

requirements are imposed on the equations for the purpose of the determination

of k, m, n: (a) The expansion of equation (6) has to retain the capability

of satisfying the condition t h a t the tape becomes flat at infinity. (b) The

purpose of this work is to study cases in which inertia effects are signi-

ficant. It will be required, therefore, that at least s o m e inertia term will

be of the order of s o m e viscous term. (c) Finally, it is required that the

iateral and longitudinal terms in the continuity equations will be of the same

order. These three requirements resul t in the conditions

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With these results, the equations can be shown to degenerate, to the zeroth

approximation, into Prandtl's boundary layer equations.

If w = /-u

We assume that the velocity profile is parabolic. The profile

which sat isf ies the boundary conditions

is

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where f is to be determined.

The three unknown functions V , H, f will be determined from

the momentum and continuity integrals between q = 0 and

from the elastic equation.

= H and

The momentum integral becomes

The continuity integral is

Equations (ZO), (24), and (25) constitute the approximate formulation of the

problem. The first integral of Eq. (25) is

where ( 1 + C ) H* denotes the integration constant.

tation of H* is the value of H in the central or uniformity region of the

bearing.

The physical interpre-

It is also a measure of the flow rate through the bearing.

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From previous ana lyses [ 11, the boundary conditions representing

the straight asymptotic behavior of the foil far from the spindle are:

The formulation is, thus, a function of three parameters: I, C, H*.

It is convenient to eliminate H* from Eq. (26) For this purpose, the

trans forma tion

is used. The equations become

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with

Following the procedure of[ 11, the solution is started from the central

region towards the exit with a matched H* a t t h e central region.

done by a linearized approach

This is

where

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The linearized continuity and momentum equations become

The roots of the characterist ic equation are graphically shown in Fig. 2,

3. ASYMPTOTIC BEHAVIOR

It can be observed that as

- f l Thus, the'boundary conditions cannot be satisfied s ince H

approach a constant as required by Eq. (32).

that in the region of interest H is fairly small. H is, therefore, nearly

parabolic, until finally the logrithmic contribution becomes overwhelming.

Thus, one may consider that the boundary condition of asymptotically straight

tape is approximately satisfied. *

does not

It must be observed, however, - - /If

*This interpretation of the resul ts is due to Prof. H. G. Elrod Jr.

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Due to the imperfect satisfaction of the boundary conditions, a

degree of uncertainty is involved which increases with the inertia parameter

I and vanishes for I+O. In the actual computation work the point a t which

the logrithmic behavior takes over was assumed to lie betweenH = 20 and

H = 1000. The value of H* taken as "correct" was chosen as that corres-

ponding to H = 100.

- -

Figure 3 shows the corresponding range of uncertainty in H*.

The main result of this work is given in Fig. 4. Here, the values of H*

as a function of the compressibility and inertia parameters is given.

similar chart, where H* is expressed in terms of C and I* which naturally

appear in the equations, is given in Fig. 5.

A

For the purpose of practical calculations, a n analytical formula

was fitted to the results of Fig. 4:

The formula approximates the results well in the region

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4 .c

3 .c

Hf

2.(

I .I

(

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1 I I f

/

2.0

/

Fig. 3. Bounds of Uncertainty in the Evaluation of H* (C*= *)

5

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I .6

I .2

H*

0.8

0.4

0

VALUES TAKEN AT 6 l s l O O

01 I 0.2 0.3 0.4 0. I

Fig. 4 Nominal Clearance H* as a Function of Compressibility (C) and Inertia (I) Para meters

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Fig. 5 Nominal Clearance H* as a Function of Com- pressibility (C) and the Normalized Inertia (I*) Parameters

16

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REFERENCES

1. A. Eshel and H. G. Elrod Jr., "The Theory of the Infinitely Wide,

Perfectly flexible Self-Acting Foil Bearing, 'I Journal of Basic

Engineering, Trans. ASME, Series D, Vol. 87, Dec. 1965,

pp. 831 - 836.

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1 ORIGINATIN G ACTIV ITY (Corporate author)

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6. REPORT DATE

T i i l v 15 1968 8 s . C O N T R A C T O R G R A N T NO.

Contract No. Nonr-3 8 15 (00) (X) b. P R O J E C T NO.

C .

d.

On Fluid Inertia Effects in Infinitely Wide Foil Bearings

7a. T O T A L NO. OF PAGIES 7 6 . NO. OF R E F S

25 1

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Technical Report

Eshel, A,

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12. SPONSORING MILITARY ACTIV ITY

Department of the Navy Office of Naval Research Washinston. D. C. 20360

I DD I FORM JAN 6 4 1473 Uncla s sif ied

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Security Classification 4.

KEY WORDS

Foil Bearings

Fluid Mechanics

Reynolds Equation

Fluid Films

Ma the ma tical Ana 1 ys is

Inertia Effects

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