Transi ation, Report T-A2049-26

",AL r A R I N 1 0 IH

AEROSTATIC [EXTERNALLY-PRESSURIZED [o- I' C\, A0 GAS LUBRICATED] THRUST BEARINGS.

41,u " 0

by J _ _ Q. I.

Erwin Loch

Translated from a doctoral thesis submitted ,to the Technical University of Graz, Austria

Prepared under

Contract Nonr-2342(O0)Task IR 062-316

Supported jointly byDEPARTMENT OF DEFENSE

ATOMIC ENERGY COMMISSIONNATIONAL AERONAUTICS AND SPACE ADMINISTRATION

Administered byOFFICE OF NAVAL RESEARCH

Department of the Navy

INJVAMIN MRANKLIN' PARKWAY AT 28)TH STREET, FIV! 4.. 1'1

Translation Report T-A2049-26

AEROSTATIC [EXTERNALLY-PRES SURIZEDGAS LUBRICATED] THRUST BEARINGS.

By

Erwin Loch

Translated from a doctoral thesis submittedto the Technical University of Graz, Austria.

Prepared underContract Nonr-2342(00)

Task NR 062-316

December 1965

Supported jointly byDEPARTMENT OF DEFENSE

ATOMIC ENERGY COMMISSIONNATIONAL AERONAUTICS AND SPACE ADMINISTRATION

Administered by

OFFICE OF NAVAL RESEARCHDepartment of the Navy

Reproduction in whole or in part; is permitted forany purpose of the U.S. Government

AEROSTATIC [EXTERNALLY-PRESSURIZEDGAS LUBRICATED] THRUST BEARINGS.

(Calculation and Experiments).

AEROSTATISCHE AXIALLAGER

(Berechnung und Versuche)

From the Faculty of Mechanical Engineering and Electrical Engineering

of the

Technical University of Graz, Austria

As a requirement for graduation to Doctor of Engineering

Thesis submitted by

Erwin Loch

A native of Innsbruck, Tirol (Austria)

Reviewer: Professor H. Winter

Co-Reviewer: Professor A. Steller

Summer 1962

&V -

The present research work was performed at the firmEscher Wyss, A. G., in their Research Laboratoriesat Zurich, Switzerland. This report is part of thedevelopment program for aerodynamic* and aerostatic*gas bearings.

The writer would like to thank especially Dr. C. Keller,Director of the Escher Wyss Research Division, and hisassistant, Dr. W. Spillmann, who graciously approved thepublication of this paper.

Zurich, Summer, 1962. Erwin Loch

* Editor's Note

The terms "aerodynamic" and "aerostatic" are used throughoutthis thesis. In the United States however, the generallyaccepted and preferred designations for these types of gas-lubricated bearings are "self-acting" and "externally-pressurized" respectively. This recommendation was made inabout 1958 by a Technical Coordination Committee formed toreview the research work in the field, sponsored by theOffice of Naval Research in conjunction with many otheragencies of the United States government.

Preface

This re,.port is a translation of the doctoral thesis by ErwinLoch, on the subject "Externally-Pressurized, Gas-Lubricated ThrustBearings", that was submitted to the Technische Hochschule, Graz,Austria in 1962.

It deals with the static and dynamic characteristics of single-and double-acting thrust bearings of many geometrical forms in such athorough manner that it seemed very desirable to translate thisimportant work into the English language to enhance its usefulness.The extensive theoretical analysis of these bearings included in thethesis is matched by the abundant experimental data taken from aseries of tests performed in the research laboratories of EscherWyss at Zurich, Switzerland.

We wish to expre--ss our thanks to Dr. Erwin Loch for authorizingthis translation and for his kind assistance in bringing it to com-pletion, and to Mr. Stanley Doroff, Fluid Dynamics Branch, Code 438,O.N.R., Washington for providing the necessary support.

The translation was done by Dr. Hans Figdor of the FranklinInstitute's Science Information Service. Technical editing wasdone by Prof. Dudley D. Fuller. Typing by Mrs. Anna H. Karle.

W. W. Shugarts, Jr., ManagerFriction & Lubrication Laboratory

• -ii-

Table of Contents

GermanChapter Text Translation

1 Principally used designations 5 4

2 Short review of the most important literature 6 6

SReview of the items covered in this study 9 8

4 Design and operation of aerostatic thrust bearings 11 10

5 Study of the problem of energy 14 15

6 Radial course of temperature 15 17

7 Laminar, isothermal clearance flow 20 23

1) Simple approximation - slow flow 20 23

2) Extended approximation 23 26

8 Discussion of the equations of flow 26 30

1) Influence of the kinetic energy 26 30

2) Influence of rotation 32 37

3) Gasdynamic influence 3L 38

4) Conditions in separation 37 42

5) Conditions at the transition from laminarinto turbulent flow 37 43

6) Influence of labyrinths in the bearing discs 38 45

9 Turbulent, isothermal clearance flow 40 46

10 Load capacity 45 52

1) Radial, isothermal laminar flow 45 53

2) Isothermal, turbulent radial flow 51 56

11 Optimization and economy 53 59

1) Optimum geometry for bearings withoutcompensation space 53 59

(A) Bearings for smallest flow 53 59

(B) Bearings for maximum load capacity -

flow ratio 54 65

~7 W ~ t

-2-

GermanChapter Text Translation

2) Optimum geometry for a bearing with

compensation space 57 66

3) Optimum height of the bearing clearance 61 71

4) Power input for different gases 62 72

5) Optimum nozzle pressure ratio in double-actingbearings 63 73

12 Properties and design of the throttle elements 66 78

1) Short cylindrical nozzles and apertures 67 79

2) Capillary nozzles 72 83

3) Venturi nozzles 72 84

4) Throttle elements with variable, pressure-dependent, cross section 78 87

5) Annular inlet slit 80 91

6) Gas Feed through porous inserts 80 92

13 Non-radial clearance flow 81 92

1) Potential flow 81 93

2) Load capacity 82 95

3) Amount of flow 83 95

4) Application of the potential theory 84 97

(A) Analytical method 84 97

(B) Numerical method 84 97

14 Bearing characteristics 99 109

1) Radial flow 101 114

2) Parallel flow 102 116

3) Potential flow 109 117

15 Results of the experiments 109 124

1) Heating 109 124

2) Pressure patterns 115 128

3) Bearing characteristics 115 129

p -

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GermanChapter Text Translation

4) Amount of flow 116 129

5) Load capacity 117 130

6) Nozzles with a counterbore at theside of the clearance 123 138

7) Measurement of the axial shift in a machine 124 138

16 Self-excited vibrations 127 141

1) Equation of vibration 127 141

2) Influences of stability 131 146

3) Frequency of vibration 135 149

4) Elimination of self-excited vibrations 136 152

(A) Damping by resonators 136 152

(B) Damping by interference 143 156

(C) Damping by friction 143 161

5) Vibrations of double-acting thrust bearings 144 161

Literature Cited 150 168

Illustrations of the experimental equipment, 153 171Figs 1 through 12.

000

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1. Principally used designations

A caloric working equivalent kcal/kpm

b width of the compensation channel m

d smallest nozzle diameter m

FD effective cross section 3urface of the inlet nozzles m2

FL bearing surface m2

g acceleration of gravity m/sec 2

G flow of bearing gas kg/sec

h height of the bearing slit m

H head by the compressor of bearing gas m

n number of the inlet nozzles

K absolute axial force kp

N output kpm/sec

p absolute pressure kp/m2

r radius of the bearing m

R gas constant m/0 K

s depth of the compensating channel m

t time sec

T temperature o K

v local velocity m/sec

w radial velocity averaged over h m/sec

Re Reynolds number

I! specific gravity kg/m3

Sdynamic viscosity kpsec/m2

2h adiabatic exponent

). heat conductivity of the gas kcal/msec 0 C

.j coefficient of friction

5 angular velocity 1/sec

- 1/se

-5-

Indices

a flow outwards in the direction + r

i flow inwards in the direction -r

B condition prior to exposure to the throttling elements

1 inlet to the bearing slit

2 outlet from the bearing slit

I carrying or loaded side of the bearing

II non-carrying or unloaded side of the bearing

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2. Short synopsis of the most important literature

Shires studies in a report (IX) the flow of air through long, narrowclearances with rectangular cross sections. Both laminar and turbulentflows were discussed. The results for different shapes of clearancesshowed a satisfactory agreement between theory and experiments as faras pressure profile, flow and temperature are concerned.Basic parameters in the comparison were the friction coefficientand the Reynold number; the results were also compared with thelaws found by Blasius for incompressible liquids. Shires concludedthat, in laminar flow, the resistance coefficients J,. are in goodagreement with the incompressible flow whereas, in turbulent flow,the _k , - values given by Blasius are, in general, not applicable.

The paper by Deuker and Wojtech (VII) describes the radial flow ofcompressible liquids between closely adjacent parallel circular discs.The kinetic energy term in the equation of flow was included inthe investigations. By several successive transformations of thevariable magnitudes, the equations are converted into an expressionwhich can be graphically integrated. From the plurality of calcu-,lated velocity curves, a solution for the pressure profile in thebearing is determined. This paper includes both compressible andnon-compressible cases. The author states that, in compressibleflow, the change of state of the gas can be assumed as p T =constant,the polytropic exponent m changing with the radius. A simplificationof the equations was achieved in the following way: For the flow inthe immediate vicinity of the inlet nozzle, i.e., with small radii,an isentropic change of state of the gas in the clearance was assumed;in the other areas of the bearing, an isothermal change. The investi-gation referred to various conditions at the inlet for Mach numbersbetween 0 and 1.

Comolet (VIII) compiles, in the form of a small textbook, the two-dimensional flows of viscous liquids and gases for the longitudinaland radial flow between two fixed parallel plates. It is based ona polytropic change of state of the gas. The investigations arewidely applicable, since also inertia forces, the effects of whichara sometimes considerable, have been considered. The equationsobtained allow the derivation of known results for the most simplecases. The experiments performed confirm the theoretical assumptionsand define the range of applications of the suggested equations. Forthe radial-divergent clearance (flow from a source in the center oftwo parallel circular discs), the flow is laminar and isothermalwhenever the Renumber at this location is smaller than 1100. Thiscritical Renumber for the sudden change from laminar to turbulentflow is approximately one half the number found for the parallelclearance. Comolet traces this back to the fact that the divergentflow in the immediate vicinity of the gas feed, i.e., with smallradii, has instability factors which do not occur in the paralleland convergent flow. In the case of thrust bearings where the flowis radially outward from a central chamber, Licht and Fuller (IV)derived the equations of laminar clearance flow for the pressureprofile, the height of the clearance, the load capacity and the

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flow. In doing this, they assumed that the change of stateof the gas was isothermal. In the general energy equation, however,the increase of kinetic energy of the radial velocity and the influ-ence of rotation were not taken into consideration. A comparisonbetween the theoretical results and the measured values showed agood agreement within the range of the experiments.

Laub (X) discusses in his paper the mathematical interrelationshipsfor similar bearings as (IV) and under the same conditions. Of greatinterest, in this connection, are the measuring technique used andthe experimental setup. This paper includes a synopsis of the mostimportant contributions in the field of aerostatic bearings.

Pigott and Macks (V) conducted experiments in a non-rotating thrustbearing which was fed by compressed air at temperatures up to 5400 C.Simplified expressions were used in the description of the flowthrough the capillary nozzles used and the bearing clearance. Theresults were shown in the form of curves and compared to the experi-mental values in a wide range of temperatures and load capacities.In this connection, they confirmed the fact that the load capacityof gas bearings is increased with increasing temperature of the gason account of the increased dynamic viscosity. In the range of veryhigh temperatures, however, the theoretical and experimental valuesfor the height of the clearance and throughput were no longer inagreement on account of distortion of the disc of the bearing.Besides, there is a short report on the occurrence of instabilitiesin the bearing, i.e., self-excited oscillations, in one of the exper-iments.

The authors Hughes and Osterle (VI) made a theoretical investigationof the influence of rotation in a revolving bearing disc on the loadcapacity of a static thrust bearing with laminar flow outward. Thisinvestigation had the following result: when the lubricant is aviscous liquid, the load capacity of the bearing changes considerablywith increasing angular velocity; if, however, a gas flows through thebearing clearance, the influence of rotation on the load capacity isvery small.

In another paper (XVI), Hughes and Osterle first defined the range ofapplicability for adiabatic and isothermal operating conditions withradial clearance flow, and then investigated the intermediary rangewith consideration of the heat transfer between gas and bearing discs.I8ased on a simplified model, it was found that the flow in the bearingclearance is in most cases, especially at high temperatures, predominantlyisothermal. Besides, equations for the calculation of the radial tempera-ture distribution were derived and the results if actual experiments werecompared with the isothermal and adiabatic flow.

The report by Licht, Fuller and Sternlicht (XVII) discusses the problemof the pneumatic instability of an aerostatic thrust bearing. Theappearance of self-excited vibrations is traced back to the compensationspaces which are in most oases arranged in the fixed bearing disc. Bymeans of a method which corrects nonlinear distortions, the problem isopened to an analytical study. Confirmed by the experimental results,the authors could show in a thrust bearing with a central compensatingspace that the greatest stability of self-excited vibrations (air-hammer)

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is reached with the lowest depths of the compensation space, thenarrowest bearing clearances and the highest input pressures.Further considerations and experiments showed the advantage ofnozzles with large cross sections and short boreholes (orifices)compared to nozzles with small cross section surfaces and longboreholes (capillaries).

Similar conditions and equations for the description of self-excitedvibrations in aerostatic thrust bearings are the basis of the paperby Roudebush (XVIII). By means of a digital computer, highly system-atical investigations on the influence of the various parameters(height of the clearance, pressure, compensating space, vibratingmass) on the stability of the bearing are performed. For some actualexamples, the amplitudes of vibration are determined and representedas a function of time. The author emphasizes that the volume of thecompensating space as well as the rigidity of the bearing are respon-sible for the pneumatic instabilities.

In the reference (XIX), the phenomenon of self-excited vibrations inaerostatic bearings is interpreted by means of a vectorial representa-tion. Richardson makes a distinction between a static and a dynamicrigidity of the bearing. The result of his studies is that the relationbetween dynamic force and deflection of the bearing behaves like a staticspring constant with a leading and trailing effect. While the restoringand deflecting force, respectively, of the sinusoidal motion of the bearingmass is trailing with the time, it is preceded by a negative damping force.The stability criterion was obtained from the condition of a positive damp-ing. In order to illustrate the application of the derived equations, acomparison was made between two customary aerostatic bearings with compen-sating channel and nozzles opening directly into the clearance, respectively.

I would like to mention the papers (XX), (XXI) and (XXII) by the authorsLicht and Elrod, without commenting on them; they relate, likewise, tothe problem of self-excited vibrations in aerostatic bearings.

Gottwald and Vieweg (XXIII) made calculations and model experimentsin static hydraulic and air bearings. In a conical bearing with annularclearance inlet and in a thrust bearing consisting of two circular discswith central inlet borehole, the mathematical interrelationships betweenthe course of the pressure in the clearance, the flow volume, theload capacity and the moment of friction were determined partly analyticallyand partly graphically. The influence of the size of the nozzle boreholediameters on the stability of the bearing was especially emphasized in onechapter.

3. Review of the items covered in this study.

Based on the energy phenomena, cn equation of general applicability forsmall disc clearances, for radial equilibrium of the compressibleclearance flow with consideration of rotation is derived.

Furthermore, it will be $hown that the heat transfer of a flowing gasbetween two discs in close vicinity to each other is very good and that,under certain conditions, the clearance flow is subject to an isothertaalchange of state.

-9-

All calculations and investigations have been extended to bearingsthrough which the flow is in two directions, i.e., outward and inward.For the laminar, isothermal radial flow, the pressure distribution inthe bearing clearance and the throughput volume are derived by meansof a simple approximation (w, dw = 0, 0. = 0) and an extended approx-imation (w.dw. + O, S2_+ 0).

A discussion of the flow equations obtained shows the influence ofkinetic energy, of rotation of a disc, of gas-dynamic effects andof labyrinth positions. This study is followed by a synopsis of theobservations on the conditions at the sudden transition from laminarto turbulent flow, which have been made by various authors.

Based on the friction values for the turbulent clearance flow givenin the literature, the course of the pressures and throughput volumesare calculated for the turbulent isothermal radial flow in a simplecase (w.dw= 0, _U= 0)

Although the solution of the load capacity integral for the simpleapproximation of the laminar isothermal radial flow in narrow clearancesfor the positive r - direction had already been known, the result couldbe evaluated in such a way that the carrying capacity of any bearingwhatsoever can be determined without calculations, by the introductionof a dimensionless factor Cka which depends only on the respectiveradii and pressure ratio. The solution of the load capacity integralof the laminar flow for the negative r - direction, which is, likewise,characterized by a dimensionless factor, Cki, as well as the integralsolution for the turbulent isothermal radial flow are added as novelitems.

Another chapter deals with the optimization and economical calculationsof the bearings. Various criteria lead to the establishment of theoptimum bearing geometry as far as the site of the gas inlet is concerned.The minimization of the total power input for an aerostatic bearing yieldsoptimum values for the compensating space and the height of the clearance.A comparison of the power losses illustrates the applicability of differentgases. For double-acting and symmetrically designed thrust bearings, an optimumnozzle pressure ratio is defined and calculated by means of minimization ofthe compressor output (bearing gas production).

The description of the individual throttle elements includes the propertiesand basic calculations for cylindrical nozzles and apertures, capillarynozzles, venturi nozzles and throttles with pressure-dependent, variablecross sections. With the aid of theoretical studies and the results ofmeasurements, the advantages of venturi nozzles over regular cylindricalnozzles will be shown (of. also German Patent 1,070,452). Whereas thepresent paper gives only a reference to the gas feed through surfaces ofporous material directly into the bearing clearance, the calculation anddesign of such bearings is the subject of another paper.

Another chapter is devoted to the non-radial isothermal clearance flow.If the influence of inertia of the gas and the disc rotation is disregarded,the flow distribution can be represented as a potential function where thecorrelation between * , 1 and p, w deviates from that of the incompressible

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flow. General relations between the pressure profile, the loadcapacity and the throughput volume are given. With the aid of abearing with several individual nozzles which open directly intothe bearing clearance, the influence of the number of nozzles onthe carrying capacity and throughput volume is determined and com-pared to the respective measuring results.

From the equilibrium condition for the amount of gas which flowsinto the bearing through the inlet nozzles and escapes again throughthe clearance, dimensionless bearing characteristics are derived andare represented in the form of curves, as functions of gas and pressureparameters for laminar and turbulent clearance flow. The characteristicsare applicable to feed systems with short cylindrical nozzles or aperturesand any arrangements whatsoever of bearings and flows.

Several experimental results of heating of the bearing, bearing character-istics, pressure profile in the clearance, volume of throughput andload capacity in specially designed bearing testing equipment confirm thecorrectness of the theoretical derivations. Besides, a simple method isdescribed Tor determination of axle shift actually occurring in a machine.

The derivation of the amplitude equation was taken from the literatureon self-excited vibrations in aerostatic bearings and was applied tobearings with ring-shaped supporting surface. With the aid of somecomparisons and measurements, the influences of feeding nozzles, thecompensation volume and other parameters on the bearing stability arepointed out. All previous developments and theoretical efforts con-cerning the design of stable aerostatic bearings relative to self-excited vibrations tend to dimension the compensating spaces in thebearing in such a way that the vibration described above does not occur.Conversely, in the present paper, methods for damping the vibrations aresuggested. The knowledge of the oscillating frequency of the bearingsystem allows the design of damping elements which can be installed inthe bearing discs in the form of Helmholtz resonators. Finally, axialvibrations can appear without the presence of compensating spaces.The possibilities of removing these instabilities are discussed.

4. Design and mode of operation of aerostatic thrust bearings.

The bearings described in this report consist of at least two paralleldiscs which are separated from each other by the bearing clearance;one of them is fixed on the rotating machine shaft, the other in thecasing either rigid or self-adjusting. The passage of the machineshaft through the fixed disc requires an annular supporting surface.In order to keep the bearing clearance permissible for safe operationas small as possible (low gas consumption), the bearing discs shouldbe plane, parallel to each other and should have finely ground surfaces.

The bearing gas is under increased pressure PB and is taken from asource located outside the bearing; it can be admitted to the bearing

p r

-11-

clearance in different ways. Upon entering into the clearance, thegas stream is divided in such a way that one part flows outward, theother towards the inside in the direction of the shaft center.

In order to attain a load capacity as high as possible in the bearing,it is desirable to utilize the highest pressure PB of the bearing gaswhich may be stored, e.g., in a container. In a bearing designedaccording to this requirement, the gas feeding duct should enter thebearing clearance with its whole cross section either directly oracross an annular compensating channel. If the volume of the gascontainer is large enough to keep the pressure in the reservoir PBconstant even at a high rate of withdrawal, (large bearing clearance)and if the pressure loss in the feeding duct is disregarded, thepressure distribution and the average loadability 1, i. e., thecarrying capacity of the bearing, is independent of the height ofthe clearance h. Such a bearing can be designed for one single loadcapacity only. Since any height of clearance can appear at the givenaxial force, the bearing is in an indifferent equilibrium. The through-put volume which is proportional to h' would assume undesirably highvalues at great heights of the clearance. Thus the bearing does nothave either a defined or a limited gas consumption.

If the bearing just described is fed directly from the compressor(for example, a multiple-stage piston compressor) without an inter-polated gas reservoir and this compressor aspirates permanently thesame amount of gas from the surroundings and pumps it into the bearing,a definite pressure PB adjusts itself corresponding to the height ofthe clearance and depending on h. Since, for example, the flowvolume is

G '- (PB 2 _ pý2) h3

for laminar isothermal clearance flow (equations 7.19 and 7.20), theaverage load capacity for G = constant is, therefore

1 2P -' PB P 2 + const

h3

Thus the load capacity of the bearing decreases with increasing heightof the clearance. Although such a bearing has a fixed equilibriumposition, it has a definite and limited gas consumption. The bearingsystem described here can, however, only be used where no changes ofload, or very slow ones, are to be expected. If there occurred suddenlyan appreciable additional load, the bearing clearance would drop to zeroon account of the compressibility of the medium in the feeding system,and only after a certain period of time, when the compressor has produceda higher pressure P'B in the duct system, a clearance corresponding to thenew load is formed. For reasons of design, the duct volume between com-pressor and site of inlet to the bearing cannot be brought to the desirable

!

RZ I

PP

jJI

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minimum value. Another disadvantage of this system appears when thebearing discs are in oblique position to each other. If the bearingis fed by only one single compressor, a major part of the gas wouldescape on the side of the wide bearing clearance and the pressure inthe duct system would be reduced considerably. The conditions areconsiderably better in bearings which are in principle similar, butfor incompressible liquids. The so-called supporting source bearingsin large machine tools (vertical boring mills) have proven very satis-factory and result in a higher efficiency and better level course thanhydrodynamic pivot bearings. They are designed in such a way thatindividual pockets which are separated from each other are arrangedin the fixed bearing disc. The lubricant is fed under pressure intoeach pocket by a separate feeding system (including an oil pump).Since the pumps are adjusted in a way that they supply the sameamount irrespective of the pressure, an oblique position of thefaceplate, for instance by unilateral loading, is avoided. Likewise,the height of the bearing clearance remains automatically the samewith all loads and r.p.m.

In aerostatic bearings, the supporting gas is led into the clearanceacross throttling elements. The throttle elements which remove thedisadvantages described above may consist of individual nozzles whichopen directly into the bearing clearance or into a compensating spaceor are designed as continuous annular clearance. The throttles limitthe volume of throughput at great heights of the clearance and thusachieve a maximum pressure Pl and average pressure f dependent on thebearing clearance. The volume of throughput G increases first withincreasing bearing clearance and reaches a constant value at the criticalflow out of the throttles. The limitation of quantity induces, however,also a reduction of pressure; thus the highest available feeding pressurePB will never be applied, but always only a lower pressure Pl.

r7FW;WY

744-Pl

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5. Study of the energy.

The gas is admitted into the bearing clearance under the pressure pl.Besides, a torsional moment Md, which is generally designated asbearing friction moment, is introduced. This torsional moment givesthe bearing gas a circumferential speed vT (z), i.e., an angularforce and overcomes in addition the tangential shearing stresscomponents. The actual course of vT (z) will,therefore, be adjusted in such a way that thebearing friction moment is greater than thecounter-moment on the fixed bearing disc.(Md> Mo). In order to facilitate the P2.analytical studies, a linear course,however, of vT across the bearing Lclearance h will be assumed.

v,,f . (5.1)

This is, of course, only an approximation, since P--8 - ""in this case Md = Mo; this means that no momentumflow, respectively. The error which occurs when

it is assumed that the disti'bution of v• is zlinear is small according to a'eference (1), p. 404and to (II) at very narrow disc distances. A greaterdeviation from this approximation occurs only at:

Retb km am h91aIe " q9 RT (5.1a)

Generally, however, Reh is less than 3 in thepractical design of gas bearings. The heightof the clearance in the bearing should be assmall as possible on account of the gas con-sumption. The lower limit of hmin for safeoperation must be selected more in considera-tion of oblique position of the discs by errorsin assembly and by deformation rather than of theroughness of the discs. Clearances of a widthbetween 15 and 30 x iO- 6 m having proven satisfactory, *r

h

O /,4,

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A centrifugal force which performs work when the flow is in radialdirection acts upon the displaced gas particles between the rotatingand the fixed disc. The exact connection between acceleration, thepressure and viscosity forces in radial direction can be derived fromthe Navier - Stokes equation for laminar flow. From

Vr = Vr (r,z)

v = v9 (r,z) and

vz =0

the following can be concluded:3Vr L F• B P a" Vr + Vr Vr aVr1 (5.2)

Vr - r y5- r + ÷ L cr a r r2- + -a j

If one turns from this equation which applied to the annular element2r Xdr.dz to average values of the width of the clearance (annular element2r3t.h.dr), this is transformed, if the following applies:

hW ._Svrd• (5.2a)

0

into

dw - d w +-4- + -dr9 0~~r~ ~Lrrrr ~

!1 daRReibungsarbeit der Wand-schubspannungen in radi-aler Richtung

wiLbbeit der Druckkrdfte (5.3)+1 Arbeit der FliehkrZgfte

Beschleunigungsarbeit in radialerRichtuný (uZuwachs an kinetischerEnergie)

(Reibungsarbeit......): friction effect of the shearing stresses in the wall inradial direction.

(Arbeit der Druck.'..): effect of the pressure forces(Arbeit der Flieh...): effect of the centrifugal forces(BeschleuingungsaLbeit...): effect of acceleration in radial direction

( = increase in kinetic energy).

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In this connection, centrifugal force energy means that work isperformed by a torsional moment Md and that rotation of the gasis induced thereby. The centrifugal forces acting upon the gasparticles thus work in radial direction. The equation No. 5.3can, therefore, be written in the following way:

w dw -db ± dap -- 0•8 (5.4)

It is shown by equation No. 5.1 that the energy required foracceleration in circumferential direction is assumed to be sosmall that it can be disregarded. Equation No. 5.4 representsthus the energy equation of the flow in general form. It can,therefore, be used for the calculation on bearings with laminaras well as with turbulent flow if the respective members areintroduced.

6. Radial temperature profile.

In order to obtain a general idea about the change of stateof the gas during the flow through the clearance, an estimationwith the aid of the laminar flow with heat exchange on the surfacesof the discs is attempted. According to the first principal theoremof thermodynamics, the following applies:

C dT= dq9zu + dgtu" + Ada + AdA (6.1)

where c is the specific heat of the gases in kcal/kg 'C; A isthe calgric energy equivalent in Kcal/m kp: T means the averagetemperature across the width of the clearance in degrees K;dqzul is the heat introduced or removed by heat conductionrelative to the unit of the weight of the throughput, in kcal/kg;dqzu 2 is the heat of friction produced by the tangential shearingstresses, in kcal/kg; A.daR is the heat generated by radial shearingstresses on the walls, in kcal/kg p means the static pressue ofthe gas in the clearance, in kp/m; is the specific gravity ofthe gas, in kg/m3 .

For the further studies in this connection, a simplification canbe made by disregarding the first two members of the equation No. 5.4,since they are negligible in comparison to the two other members.

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W dw db << «÷+ dao • (6.la)

An estimation showing in which cases this approximation is validwill be given in Chapter 8.1. Thus, if

_t daR - 0(6.2)

is introduced into equation No. 6.1, the first principal theoremis reduced to:

c dT - rigzu,, + dqzuz. (63CrcT (6.3)

The amount of heat transmitted by a bearing disc per annularsurface element 2 r E .dr to the gas or, conversely, from thegas to the disc by heat conduction, relative to the unit ofthe amount of gas flowing in radial direction is

dQ101 z rr dro o( -T)

G (6.4)If it is assumed that both discs which determine the clearancehave the same wall temperature Tw, the following results:

qju 1 -z rxr drO(T - T) (6.5)G

According to reference (III), for laminar flow in a plane, parallelclearance with coqstant wall temperature, the heat transferencenumber OL(Kcal/m< °C sec) is

S .5'75 (6.6)

where h is the height of the clearance (in m) and X the heatconductivity of the gas (in kcal/m 'C sec). If this value ofo( is used for the further calculations, it must be mentioned

that this is only an approximation, since in our case a flowwith change of the radial cross section is involved. Besides,a v component is superimpose,' to the radial flow; the influenceof Lhis component on the value of d. cannot be estimated here.Consequently, the following applies:

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dqu 4 . 3*7 I'5r N (Mw -T) r dr-d r G (6.7)

The sign "+" means: flow towards outside; the sign "-" means:flow towards inside.

In the equation No. 6.7, Tw is the wall temperature of the discs(in 'K' and G is the throughput per second (kg/sec).

The second heat introduced, which is generated by the rotationof a disc, is found directly from the torsional moment. By theassumption of a linear ve distribution over the height of theclearance, it was alread' anticipated that the whole energy ofthe torsional moment is transformed intoheat and that the energy which is requiredfor the torsional flow vq is small andcan be disregarded. The torsional momentfor an annular element 2 r JT .dr is

.0

dcM -zrl dr r "Cyz(u .o) I ZTrdr y - a-? j

(6.7a)

By introduction of the equation No. 5.1 into this formula, thefollowing is obtained:

dM 2.irr~drý -h (6.7b)

The heat supplied relative to the throughput per second is:

d Qi•: AdMn. 2wrrdrQ2rfrVA =+27rAqr.r.2&dr"G" hG hG (6.8)

If the members No. 6.7 and No. 6.8 are introduced into the firstprincipal theorem No. 6.3, the result is:

Cp dT •15 ) (Tw-T)r + 2a dr . (6.9)hG

where "+" applies for the flow outwards (with increasing dr, Tincreases on input of heat); and "-1" applies for the flow inwards(with decreasing dr, T increases on input of heat).

- --P ;mp:'--

-20-

If we assume that there is a very good heat conduction in the bearingdiscs and that the wall temperature adjusts itself as constant over r,and if we use the following abbreviations:

M - + 15.__r-hcrGZ " A P (6.9a)N :t ý- 81 c G

+: for flow outwards. -: for flow inwards,

the equation No. 6.9 is transformed into:

dIT M frlT N r3 + hTwr (6.10)dr

The general solution can be deduced from the homogeneous equation

dT +MrT -0T .- M(6.11)

T - CCr) eby variation of the constants C(r):

dT _M -Mr

_ P- 2- M.r. C(r) e 2 (6.12)

If No. 6.11 and No. 6.12 are introduced into the equation No. 6.10,the result is:

C(r) = M.Tw, f r eem 2.dr + N r.3e • d6.12a) ~~ 6.12a)

i'1Twi'ed N r.e dr +I-C.,After integration, the following is obtained:

N MV_Cr) - .e Y Mr2-- ] 2 + T e + c , (6.12b)

Thus, the desired solution by introduction of C(r) into equation

No. 6.11 is:

N (.

By introduction of the boundary conditions, r rla, T : Tla,and r = rli, T Tli, the course of the temperaiure for theoutward flow is:

T- T - ( -Tw - -- _ [(M I rLz)_(r rr-2-_)e-) (M 4- .)J (6.14)

S.... .. .. ... ... .=-'= '- . . . ... T h-T >

-21-

and for the inward flow:

Ta Tw-(Tw-T;L) e2(r' M Y,2-2)-(M r-2) e'(~r~ * 6.15)

The derivation of the analogous equations for the turbulentclearance flow will not be given here, because this would notinvolve an essential contribution to the principal studies madein this paper.

The course of temperature in a bearing was calculated accordingto equations No. 6.14 and 6.15 for the adopted conditions. Itis illustrated in Diagram 1. The case given here could, forexample, occur in a machine in which the gas is heated by theshaft and the casing, and cooled by the bearing gas. It canbe seen that the heat exchange between the bearing discs andthe gas is very intensive. After the gas has flowed the veryshort distance A r, the gas has reached the temperature ofthe wall and would remain constant up to the end of the clear-ance for 11= 0. The rotation causes heating of the gas whichremains, however, very low on account of the reflux of the heatinto the discs.

The basic assumption that the wall temperature remains constantmay not be quite correct in all cases, since the discs are cooledor heated in the immediate vicinity of the gas inlet, dependingon the temperature of the gas, are subject to heat supply byrevolution along the entire bearing surface and thus radialtemperature gradients occur within the bearing discs. But itcan be seen from this estimation that the temperatures of thegas and of the wall are always almost the same for the clearanceflow on account of the good heat exchange in the narrow clearance.To what value this wall temperature, remaining almost constantover r, adjusts itself, does not only depend on the temperatureof the gas at the inlet into the bearing and the number of rotationsper minute, but also on exterior conditions (such as introduction andremoval of heat by the components of the bearing).

Diagram 1. (left) assumed (right) calculated

(last line, 2nd col) tw = constant over r.

22 iurjm----I

assumed P .alculated

-rid A,3 M Gfo'0 k/ Zt50.05 POO 19 340"3 #txi if 1005 - 0

0,05 :f tu 0ee0C A 2(. ~l .300 C: U..4" ti ,

M 4W0P~ x constant over r t ~ twZ2

N I

-200--

2 - -+

-23-

On account of the intensive heat exchange between gas and discs,any hypothesis based on an adiabatic change of state of the gasduring the clearance flow in bearing materials found in practice,i.e., of good heat conduction, can be eliminated. The furtherstudies of the clearance flow rely, therefore, on an isothermalchange of state of the gas.

7. Laminar, isothermal clearance flow.

It can be concluded from the explanations given in the precedingChapter that the temperature of the gas in the bearing clearancedepends, to a great extent, on the temperature of the bearingdiscs. Assuming an isothermal clearance flow, the temperatureof the gas at the inlet into the bearing must not be used asa base for the calculation, but it must be governed by the walltemperature of the discs which is the resultant from the thermalequilibrium (heating by rotation, heat supply or removal by theshaft, outside cooling of the bearing discs, heat transport ofthe bearing gas, etc.) and remains constant.

In the following, equations for the calculation of the courseof pressure and the throughput volume are derived for radialflow through narrow clearances. Since the derivations are basedon different assumptions in each case, the range of validity ofthe equations obtained is limited.

7.1. Simple approximation - Slow flow.

Starting from the Navier - Stokes equation (No. 5.2), it isfirst assumed that the energies of the acceleration and centrifugalforces are small in comparison with the energies of the radialshearing stresses and pressure forces. An estimation on thepermissibility of this approximation will be given in Chapter 9.1.

Thus equation No. 5.2 is reduced to the following:

-a P t Vr YVr1

S.t a rar = • (7.1)

If the z-components are disregarded, the continuity equation forthe case of incompressibility is

S(7.1a)

-24-

or

or+ 1 - _ 0 . (7.lb)

We obtain by differentiation:

-Vr a Vr Vr-0r-- + rzr r2 . 0 (7.2)

If equation No. 7.2 is introduced into the second member of equationNo. 7.1, only the following remains:

dr T " (7)3)

The solution of this differential equation gives, with considerationof the boundary conditions

z = O, vr 0 and z =h, v 0 :2 dr ~hz__ci(7.A)

which is a parabolic distribution of the radial velocity in theclearance. The average value can be determined from:

W = - -j vdi• 12 - d-• r (7.5)0

Another way of writing this equation

dp ±12_ wdr 0 (h% (7.6)

or, with consideration of equation No. 6.2

daR~ i? q' ± W drdOR =' ± W."r (7.7)

represents the decrease in pressure or the work on account of theradial viscous flow, respectively, where "+" applies to the outwardflow and "-" to the inward flow. It is worth mentioning that theapproximation equation No. 7.6 is analogous in form to the equationfor the pressure drop in a channel of invariable width. The varia-bility of w with r takes the widening and constriction, respectively,of the clearance in radial direction into consideration.

I

-25.- dr

wr)(7.7a)

By using the continuity equation

6G = 2rrhrw (7.8)

wherein G represents the throughput volume per second through the outeror inner bearing clearance, respectively, and the gas equation

p r"RT (7.9)

the equation No. 7.6 becomes:

p.dp. 6 RTG (7d.1r

"P - T -h3 r 0 (7.10)

The integration gives the course of pressure over r (The indices "a"and "i"l mean connection with the outward and inward flow, respectively).The boundary conditions are:

r r r2a P = P2a r., xr r~ar 0r rla P = Pla "•

r r 2 i P = P2i

Thus the following applies to the outward flow:

PI h1 (7.11)

or

P Ra. 12 GRTLnLTh [ (7.12)

If the throughput volume is eliminated, we obtain:

P 'cL L. + Ln r n PL (7.13)

or

ILLn r__(.)= ~ - P42OL (7.14)r4 OL

77"-

-26-

The following applies to the inward flow

P•' N.+ "-•,GR LvLn~- r W rzl (7.15)

or

- LIL fl (7.16)

and for eliminated amount of flow

P"u" •+ -Ln -" Ln * (7.17)

or r4i

(7.18)

The amount of flow can be given as:

. RT.n r, (7.19)

and

12 IZrRT In r,,•T n- (7.20)

As will be shown later, the accuracy of these approximation solutionsis, in general, sufficient for the design and calculation of aerostaticthrust bearings. Thus, for instance, these equations are used in thepapers (IV) and (V).

7.2 Extended approximation

In order to get an idea of the influence of the acceleration andinertia members, respectively, in the equations No. 5.2 and 5.4,respectively, an iteration is made in such a way that the termfor the energy of the radial shearing stresses daR according toequation No. 7.7, obtained from the simple approximation (Chapter 7.1)and by disregarding these members, is introduced into equation No. 5.4.

The member of the centrifugal force energy can be obtained, if thefollowing assumption is made (see equation No. 5.1)

db*- " ' () (r (7.20a)... f VP.

-Jr-a r h- 0r V

-27-

as: r nb r dr3 9 (7.21)

Using the equations No. 7.7 and 7.21, equation No. 5.4 is transformedinto:

t iz qwdr ÷ j_ t wdw Z rdr = (7.W X, U 9 399 (7.22)

In order to determine the pressure as a function of r, the variablesw and 7• must be expressed by 6 and p. Thus the continuity equation(cf. N6. 7.8) reads:

W = Zr (7.22a)

and the gas equation (cf. No. 7.9)

PsR T (7.22b)

If equation No. 7.9 is introduced into equation No. 7.9. we obtain:

SRTz rh rp (7.22c)

and, differentiated:

iw2 h r, p 5" d (7.22d)

According to the studies made in Chapter 6, T remains almost constantover r. If the terms given above are introduced into equation No. 7.22,the following general differential equation results:

S_)dp (p(_•6r•LRT G&RT .. A' - 'pL)dr.

- -' h'g rLp• F - 4 .p r 9 Rgt r

(7.23)or, using the following abbreviations

(7.23a)

03G__ - +: for outward flow

-: for inward flow

C - 33 RT(7.24)

CIr F- rp-

drE

-28-

The integration of this equation in complete form is not feasible; if, how-ever, two assumptions are made, a relatively simple solution can be found.

Assumption "a": In the clearance flow at large radii, it is permissible -

since there is p >> b/r 2 p - to simplify the second term of the denominatorin the equation No. 7.24 in such a way that r is replaced by a constant radius,e.g. the average value of the initial and final radius,

or that the whole member is omitted. In the flow at small radii, i.e. forthe course of pressure in the immediate vicinity of a nozzle which opensdirectly into the bearing clearance, disregarding this member involves acertain error, which may be smaller or larger depending on the conditionsprevailing in the bearing. But since the second member of the numeratorin equation No. 7.24 is inversely proportional to r 3 and the second memberof the denominator is inversely proportional to r2 only, the former has agreater influence by far than the latter when the radii are very small. Thisfact will be illustrated in Chapter 8.1 by actual examples.

Assumption "b": In contrast to the equations given in (VI), appreciablysimpler terms for the course of pressure are obtained, considering at thesame time also the influence of inertia, if an interation is made in such away that, for the pressure in the third member of the numerator of equationNo. 7.24, the solution from the simple approximation (Chapter 7.1) is intro-duced.

If the equations No. 7.14 and 7.18 are used, respectively, and the assumption"a" is applied, equation No. 724 is transformed into:

TT) dp + (T 7~I II F, (7.25)

where the factors f and k for the outward flow are:

f ,CflL 0 (7.25a)

ka= CLr, a

and f6r the inward flow:

Cp --t - k R rL(7.25b)

k C A _P

If the equation No. 7.25 is integrated, the course of pressure becomes:

.f b JL k C(nr t ,(.6

- Lnp + aLnr + 2)-i" ) ,(7.26)

(.

-29-

The boundary conditions being given, for the flow outwards, as:

r = r la, P = Pla or r = r 2 a, p = '2a' the following equationresults:

r'4o. •" P"O.• (:7.27)

2- r, r%. 2 PIG.

or

Pat, (7.28)

+ i- rlý r

For the flow inwards: r = rli, p = Pli or r- rr2 i, p p2i' and thus:b

zu•ZP':-• Lrn r- l 2•÷ -(- ru r,-, z PI(7.29)

or•-(p p,•) --bL I o.n + - : r I. -:P'C -r -

L I "- r(2-(7.30)

k. L n L, L+ r L

Flow volumes 2 r

If the second boundary conditions are introduced into equations

No. 7.27 and 7.29, quadratic equations for G are obtained. For the outward

flow, the following applies: AGa 2 - BGa + K = 0, where A, B, and Kmean the following:

A=AT'r, rr,,

Thus the result is:

+-A (7.31)

In an analogous way, for the inward flow applies:

+ EG. - L 0 0, where D, E, and L have the following meaning:

In /31 -

E,- LrvjE

-30-

and the flow volume is:

E + E` + 4fl25 D (7.32)

For the square roots only those signs are applicable which give positive

throughputs.

8. Discussion of the flow equations.

8.1 Influence of the kinetic energy.

In order to estimate the individual members of the Navier - Stokesequation (No. 5.2) relative to each other, the quotient of the radial inertiaforce relative to the unit of volume 7 by the largest viscosity force • isestablished. U 3FROM dwd r w h

W W 'd r Rei (8.0a)

WE

RT und(8b)

the comparative Reynolds number is determined as:

9* -RT F 2 Tr r19 h" I • (8.1)

It can be seen that this influence of acceleration becomes the greater, thelower the radius and the higher the pressure and the clearance are.

Case 1. First, the course of pressure and velocity in a bearing which ise7juipped with an annular channel and where the flow is in both directions,along with the volume of throughput are calculated by means of the variousequations derived in Chapter 7 for the data listed below. Since equationNo. 7.24 cannot be solved, the comparative course of pressure was determinedfrom the differential equations by step-by-step approximation by means offinite differences P p/L r.

r *- 0,055 m h a 20.iO'rg R = 2c,3 m/°C*4i a C,087 p Ua u G 208.10 kp/lr T = 293 "I( (8

r 4&" C',089 - 6,365.I0• = i,85.iO" kp s/ZLz'&- C,133 104- pa. 1.I0' O 0

___ __ _- -M~7

-31-

Outward flow.radial velocity

radius r (in m) pressure p (atmospheres) w (in m/see)

Calculated differential extended simple approxi- continuity

according to equation approxi- mation equ. No. equationNo. 7.24 mation, eq. 7.13 No. 7.8

No. 7.28

09089 -6,208 6,208 6,208 15,2

0,090 6,125 6,135 6,120 15,30,095 5,707 5,705 5,694 15.60,100 5s278 5,296 5,256 16,0

04105 4,834 4,825 4.796 16)7

0110 4,368 4,345 ,)390 18,70,114 5,972 3,945 3,920 18,8011l8 5,545 3,515 3 S•'91 20 L,

0,122 5,071 3,050 5,007 25,0 (8. lb)09126 2,520 2,391 2,559 2o,4

0,128 2,205 2s163 2,139 30,80,130 1,836 1,786 1754 3649,

0,131 1,617 1%565 1,504 61s50,152 1,357 1,517 1,104 49,0

0,133 1,000 1,000 1000 63,4

Throughput volume G (gr/sec)a

Calculated (assumed) (No. 7.31) (No. 7.19)

according to 1.200 1.230 1.231

S .. ... • - ml .

-30-

L r n n_

and the flow volume is:

+ DLfl (7.32)

For the square roots only those signs are applicable which give positive

throughputs.

8. Discussion of the flow equations.

8.1 Influence of the kinetic energy.

In order to estimate the individual members of the Navier - Stokesequation (No. 5.2) relative to each other, the quotient of the radial inertiaforce relative to the unit of volume 7 by the largest viscosity force ' isestablished. 3FROM dw

bZ r% Rei (8.0a)

" PT und W zr (8.b)

the comparative Reynolds number is determined as:

9IR~ Tr 2wr~g~

It can be seen that this influence of acceleration becomes the greater, thelower the radius and the higher the pressure and the clearance are.

Case 1. First, the course of pressure and velocity in a bearing which isequipped with an annular channel and where the flow is in both directions,along with the volume of throughput are calculated by means of the variousequations derived in Chapter 7 for the data listed below. Since equationNo. 7.24 cannot be solved, the comparative course of pressure was determinedfrom the differential equations by step-by-step approximation by means offinite differences • p/L r.

r, - 0.055 m h . 20. 1 m MR = 29.3 m/°Cr44 C,087 p, - 6v208.Q10kp/re T 295 OxI6 (8.1r 4a- O,089 P4L " 6,365.104 2 1,85.10C kp s/mLrdL- 0,133 Pic - pjm= 1.10 XfL- 0

-31-

r, L

Outward flow.radial velocity

radius r (in m) pressure p (atmospheres) w (in m/see)

Calculated differential extended simple approxi- continuity

according to equation approxi- mation equ. No. equation

No. 7.24 mation, eq. 7.13 No. 7.8No. 7.28

0,089 6,208 6,208 6,208 15,20,090 6,125 6,133 6,120 15,3

0,095 5,707 5,705 5 96,c 15,60,100 5,278 5,296 5,256 16,00,105 4,834 4,825 " '79G 16?70,110 4,368 4,345 4,550 17,70,114 3,972 3,945 -,920 18,80,118 3,545 3,515 3 5L,-91 20 140,122 3,071 3,030 3,007 2350 (8.lb)0,126 2,520 2,391 2, 359 2o,40,128 2,205 2,163 2,139 .0.18

0,130 1,836 1,786 1 ,754 36,90,131 1,617 1,565 1,5 -.6 41s,50,132 1,357 1,317 1,304 49,00,133 1,000 1,000 1,000 63,4

Throughput volume 6 (gr/sec)a

Calculated (assumed) (No. 7.31) (No. 7.19)

according to 1.200 1.230 1.231

-32-

Inward flow

radius r (in m) pressure p (in atmospheres) radial velocityw (in m/sec)

Calculated differential extended simple approxi- continuityaccording to equation approxi- mation equ. No. equation

No. 7.24 mation, eq. 7.17 No. 7.8No. 7.30

0,087 6,365 6,365 6,365 1L,00,086 6,288 6,287 6,287 14940,084 6,135 69130 6,122 15,10,082 5,970 . 5,962 5,955 15,90,080 5,798 5s785 5,772 16,)0,076 5,422 5,)403 5,378 19,009072 4,992 4,986 4,923 21,9 (8.lc)09068 4,491 4,457 4,589 26900,064 3,889 3,813 3,749 32,40,060 3,120 21980 2,918 '", 5O,:758 2,612 2,432 2,360 56,80,:•57 2,298 2,177 2,012 67,9-57 210192 6719

0,056 1,905 1,324 1592 87120,055 1,000 1,000 1,000 141,4

Throughput volume Gi (gr/sec)

Calculated according to (Assumed) (No. 7.32) (No. 7.20)1.053 1.138 1.138

It can be seen that in bearings where the dimensions of the radii are large dur-ing clearance flow, the variation of tI' kinetic energy of the radial velocityhas only a small and unimportant influence. The comparative Reynolds number isin this case Re* << 1. All the equations derived in Chapter 7 yield practicallythe same result.

Case 2.

If an inlet nozzle opens directly into the clearance, the Reynolds num-ber of thf clearance flow in the immediate vicinity of the orifice is: Re* 1.

-33-

At very small radii the following applies to equation No. 7.24:

crp2 and a br r

i.e. the increments of centrifugal force and friction become small and negligi-ble compared to the kinetic energy. The equation No. 7.24 is reduced to:

dr b

P - r 2p (a

or, re-transformed into the original form p,

RT dp + wdw = O.p

This equation describes the friction-free flow in a radial diffuser. Its solu-tion is:

RT Ltnp + -, + C -0. (8.2)

The clearance flow at small radii is only interesting, in this connection, forthe flow outwards. If one does not consider the cases where the gas in the bear-ing clearance reaches the velocity of sound which changes subsequently into super-sonic speed (this case has been described in detail by reference (VII))and if onedisregards the fact that the change of state of the gas for the flow in immediatevicinity of a nozzle is not, as assumed, isothermic (this error is, however, quitesmall), the pressure after entry into the clearance had to increase correspondingto diffuser flow, according to equation No. 8.2. The theoretical pressure pro-file, determined by means of stepwise approximation through finite differences

Sp/ t r for a bearing with the values

r"-= 1 mM h =20.10-6 Mri.= 68,5 mm R = 29,5 m/°C (8.2a)S0 1 10'kp/mz T .= 293 4K

- 0, 177 Gr/s 1 1,9.10 5kp s/m24

is plotted in Diagram 2 as a function of the pressure profile which wereobtained from the two approximations No. 7.13 and 7.28, respectively. The

interaction of unfavorable factors such as (1) poor conditions at the inlet(on account of the difficulty to obtain clean nozzle edges), (2) flow inhydraulic and thermal take-off run, (3) occurrence of the highest possibleRenumbers in each clearance flow, justifies the assumption that the flowdoes not undergo this high increase in pressure which had to be overcomewithin a few tenths of a millimeter.

Another problem, however, which is important for the design of thebearings comes up here: Which pressure p1 at the inlet of the clearance isrequired in order to obtain that pressure profile which is described bythe three equations (Diagram 2) identically in the area of large radii? Forapplying the pressure p1 calculated theoretically according to equation No.7.24 or equation No. 7.28, would not guarantee reaching the desired courseof pressure. The simple approximation No. 7.13, on the other hand, givestoo high values for pl.

The radius rk where the pressure reaches a maximum (beyond this there is nolonger an increase of pressure in the flow) can be calculated from equationNo. 7.24 for 0 = . If it is assumed that dp = 0, there is, as aconsequence: T-

a - b = 0 , and, therefore:r -•(8.3)

The inherent maximum pressure Icannot be found from equation No. 7.24 in asimple way. The extended approClmation No. 7.28 yields, as can be seen inDiagram 2, a value which is somewhat too low but still applicable. It can beassumed that the pressure profile between P1 and Pk remains almost constant:thus one can establish the required pressure Pi.

A method of circumventing the questionable range between p, and Pk isthe following: Next to the nozzle, a depression isprovided of the magnitude rk, on the side of theclearance. In this case, the radius rI is identi-cal with rk, and the pressure Pl * Pk. For theexample given on page 33, the result is:rk = 1.58 mm", Pk = 6.775 x 104 kp/m 2 (cf. No. 7.28).

Diagram 3 (page 36) shows the course of the Reynolds numbers over rfor the example mentioned above. In the area of small radii, Renumbers ofover 1000 are reached, while Re 1 * increases beyond 10.

• I-

P Ot3 Diagromm 2

S'(7913) Simple approximation

S.. , (7,28) Extended approximation

•/(7,2) Step-by-step approximation*6 by finite differences

5

r4

0 5 0 '$ 20 i

-. .36..

S........ .............-..

'Re 9 2

.......... ... l-.-- - . ...-. . .---...--.---..- -'------------.-- ....... ...--...... .......

. ..

S..... .. -................... ...... .

.. ... .. .

20 is30 ".

MDe I

S~-37-

8.2 Influence of rotation.

If, according to equation No. 5.2, the quotient of the centrifugalforce 1 2 relative to the unit of volume by the highest viscosity forceis written, the following applies according to equation No. 7.21:

---- CV r n -- e

64W 3 qW Pe (8.3a)

or by 9 .und 9W (8.3b)

the second comparative Reynolds number is given:

r .xi " ? •z • r'-a' P- (8.4)

The influence of rotation depends in a high degree from the conditions ofoperation of the bearing; it increases quadratically with the factor p.r.,.

For the example on page 31, the pressure profile and the amountof flow was calculated with = 1570 1/sec.

Outward flow Inward flow

Amount of flow 7(gr/sec)

calculated according to No. 7.31 7.32

1.280 1.110

Outward flow Inward flow

Radius r Cm) Pressure p Radius r (i) Pressure p(atmospheres) (atmospheres

calculated according to No. 7.28 No. 7.30

O O•9 ~6s203 •• •C, 10 0, 0 C1 GJ 6c090 6,131 o~02o 3,303

095 5$722 kaY 00,100 5,26' .G0 0O 105 4.'33 0•C0 5U7700 110 4.577 O 07"3 5,,350,114 397L 0,,("72 L.3,C'. "3C1118 5 5L;. CG ,,3809122 C0,2 C.•064- 5 97250 126 2-,511 0L3 0% 2slo0 912 2C- 0ý058 225 45C 5 1j?798 02C,57 2,0030151i 1,579 0,056 .)5-50,1i2 1,352 0,055 1,CCO0,33 .1,000

-38-

In the case considered here, the influence of rotation is very small, asis shown by a comparison of the pressure profiles for 1 = O. (pp. 30 and31). The comparative Reynolds numbers assume values which are: Re* 2 ;l.

In reference (VI), the influence of rotation on the load capacity wasstudied in an aerostatic thrust bearing with outward flow. It is pointedout that a reduction or increase of ressure can occur in the bearing clear-ance, compared with a bearing with fL. = 0, by the rotation of a bearingdisc, depending on the conditions or the parameters, respectively r a/ra'Pla/ p2a and 3 - 2r 2la/10 RT. For the evaluation of each condition, onlythe boundary conditions for which the one case or the other may come trueare to be established here. It can be concluded that in isothermal, laminarflow in outward direction for

_.a ('- r 0- ) (8.41b)

the pressure in the bearing clearance, according to equation No. 7.28, issmaller in all places r which meet this condition when there is a rotationthan the pressure which would occur without rotation. AnalogQusly, in theflow inwards there is, according to equation No. 7.30, for

(r. r L ) > - r- (8.40

an increase in pressure compared to the bearing at rest. If the unequalitysign is reversed in the conditions given above, the pressure in the clear-ance becomes greater (in the first case) and smaller (in the second case)than when no rotation takes place. At high pressure ratios pl/p , rotationcauses, as a rule, an increase of the load capacity in the part af the bearingwhere the flow is outwards and a reduction of load capacity in the part wherethere is an inward flow. (cf. the results given above). But since theinfluence of rotation in static gas bearings is, in general, very small, rota-tion is disregarded in calculation of bearings.

In an analogous way, an increase or reduction of the volume of throughputcorresponds to a pressure reduction or pressure increase under the conditionsgiven above with r = rla and r-• rli, respectively.

8.3 Gasdynamic influence.

For the laminar, radial clearance flow, the course of the radial velocity waveraged over the clearance is derived as a function of r first. Thiscalculation is only an approximation, since thevelocity distribution vr over h had to be /taken into consideration in an accurate investi- VV. Le)gation. In contrast with the studies made in VrV" XChapter 7, it is permissible to assume an adia- -batic change of state for the clearance flow ,

J1--

-39-

in the transsonic region. From the equation No. 7.22 i* may be concludedthat, for f1 = 0, the fe.'lowLng applies if the continuity equation No. 7.6is considered:

k .0hG.t ~ - (8.5)

Transformation of the second member:

The following equation applies to adiabatic change of state:

~ (8.6)

The differentiated continuity equation No. 7.8 gives:

(8) dw

If equation No. 8.7 is introduced into equation No. 8.6, the result is:

S dW) -) .(8.8)

The temperature, expressed by the adiabatic equation, becomes:

.A. L. /S.9)T = To - A 9 (.9

where w * 0, T = To. If equation No. 8.9 is introduced into equationNo. 8.8, the following results:

R T z•c r,• W (8.10)

The introduction of equation No. 8.10 into equation No. 8.5 gives, withconsideration of

ARCr Z (8.10a)

the differential equation for w = w (r):

dir X , .Lw " a - W.(8.11w

S.24 rrw. _ RT, L+ -2 w

_40-

where the sign "+" applies to the outward flow and the sign "-" to the in-ward flow. If it is assumed that dr/dw = 0. it appears that the r-w function,as represented, runs parallel to the w-axis at a certain velocity, which isthe velocity of sound

W S R0 (8.11a)

and is obtained when the numerator is zero, since the denominator of thefraction on the right side of equation No. 8.11 cannot reach an infinitevalue. T".is the numerator of the fraction is positive for subsonic speeds,and negative for supersonic speeds.

Flow towards inside.

Since the thicd member of the denominator in equatLon No. 8.11 is, as arule, small compared to the second, the entire denominator of the .ractionremains negative. For w < w the whole fraction becomes negative, fromwhich it can be concluded that the velocity increases when the radius de-creases. Velocity of sound can only be generated at the outlet of theclearance; supersonic velocities cannot occur in the clearance.

Flow towards outside

In a subsonic flow starting at r = r a, the course of the flow depends onthe relative magnitude of the individual members of the denominator. For

24"rig r w• W-__. L_•> Ro

hr 29 r (8.11b)

which is the regular case, the flow is accelerated on account of the greatpressure reduction of the gas in spite of the fact that the clearance is tobe considered geometrically as diffuser. If the pressure of the bearinggas is sufficiently high, velocity of sound can appear at the end of theclearance (r - r2a) if the above condition is fulfilled.

Looking for the magnitude of the radius ru of the bearing where velocity ofsound is generated in the clearance - which changes over to supersonicvelocity with increasing radius - one will find the answer in equation No. 3.11.Since dr/dw must be positive in order to fulfill the above condition andthe numerator of the fraction is negative for w > ws, it can be concludedthat also the denominator of the fraction must assume a negative value.This is, however, only true if the following applies:

RT.- > zs 7 (!.nc)

lbr r = ru and w a ws the following applies:

- ~RT - ~ ~ -I-=---- - -ý (8.11d

-41-

from which the following can be derived by means of given values:

-RTo - 1.a24 IT t (8.12)

If the continuity equation No. 7.8 is considered for the respectiveplace

(8.12a)

another equation results:

x R TW" • (8.13)ra 612 .~2 S

An example using the following values

To = 293 OK V = 1,85.10"f kp S/M (".,aR - 29,3 m/C h 50.J0"'rn (8.13a)

S1,4 G& - 2,5.10 `3 kg/Sa 314 rn/s

gives: ry = 3.0 x 10- 2 m.

Thus supersonic velocities can occur in the clearance only at radii whichare smaller than, or equal to, rU. This fact is very important for thedesign of aerostatic bearings with radial flowsince, in the region of supersonicflow, the pressure in the bearing isgreatly reduced and can even assumenegative values relative to the sur-rounding pressure. It need not bespecially emphasized that, in this case,the load capacity would be reducedcorrespondingly. In order to havea bearing with proper dimensions,rla must be larger than ria.

_42-

The compressible radialsource flow from the center of twodiscs in close vicinity to each otheris described in references (VII) and(VIII). With sufficiently greatbearing gas pressures, sound velo-city is reached at the beginningof the bearing clearance, i.e. atthe opening of the inlet orifice.The radial diffuser formed by the -A_.two discs produces a further e

acceleration to supersonic flowand similar conditions as in aLaval nozzle can be observed here.The great pressure reduction inthe supersonic region can go sofar that the pressure in theclearance can become lower thanthe surrounding pressure. In -t "--such cases, there occurs a pres-sure impact which is connected rwith a sudden change from super-sonic to subsonic speed. __ .

The Figures on the right side show: BOTTOM p. 41: section of athrust bearing with bilateral flow. UPPER, this page: Course of the velocity;ws = velocity of sound. Supersonic velocity for the outward flow between rand rst. Pressure impact at rst. Subsonic flow between rst and r2a, and alsofor tfe inward flow. LOWER, tils page: CORRESPONDING PRESSURE PROFILE.

8.14 -Conditions of separation.

With small heights of the clearance and large radii, the frictionforces are so strong that, inspite of the enlarged cross section in the out-ward flow, there is a reduction of pressure rather than an increased pres-sure. Thus no positive pressure gradient is imparted to the layers in proxi-mity to the wall, as it is the case with larger widths of the clearance, anda separation will hardly occur. The measured friction values for laminarand turbulent flow through narrow divergent clearances (IX) confirm thishypothesis.

Ak4

-43-

8.5 Conditions at the transition from laminar into turbulent flow.

In the literature, the clearance flow is designated by a Reynoldsnumber which is simila.r to Rel*, which is, however, formed with thehydraulic diameter of the radial clearance flow

d6 - 4f .2rrh r (8.13b)

and therefore reads:

Re . ZHhpw?ŽL G z r - (8.14)

*4 9 7 u9ýRT - r9 'e I

At constant viscosity 11 and volume of flow G, there is a hyper-bolic relation between Re and the radius in such a way that the Renumberincreases with decreasing r.

Although it is not necessary for the calculation, the friction co-efficient )L "-r the ciearance flow will be discussed here supplemen-

tarily. If the radial pressure decrease is defined as

dp- A, "r 9 w-hW 2- (8.14a)

where dh = 2 h means the hydraulic diameter of the clearance flow, and

Eqw2 /2 the pressure head of the average radial velocity and if the aboveequation is equated to No. 7.6, then

dr L• V? W'dd2.- 'AL. 7 y - (8.14b)

gives the following result- L = 96

WIT" 7

For the flow of air through narrow clearances with constant rectang-ular cross sections, reference (VIII) gives a synopsis of the criticalRe-numbers found by measurements. The transition value is, for example1800 - 2000 in a channel with sharp edges at the inlet, but 2400 in achannel with rounded-off edges at the inlet opening. In reference (IX),values of 2120-- 3810 were found for clearance heights between h= 50-250 x 10- 6 m.

The results, reported in references (VIII) and (IX), of experimentson air flows through parallel clearances with expanded lateral borderssuggest a connection between the critical Renumber and the degree ofdivergence of the channel. In the extreme case of the divergent clearanceflow, i.e. in the flow between two parallel circular discs with eccentricgas feed across a nozzle, the transition point occurred, according toreference (VIII) already at Re = 1060. If one studies, for comparison,the calculated example on p. 31 and the Diagram 3, this low transitionvalue can be easily understood. With small radii and high Renumbers,the flow starts as turbulent. It is a fact which is generally known thata change from turbulent to laminar flow takes place only at considerablylower Re-numbers than in the inverse case, i.e. transition from laminarto turbulent.

According to equation No. 8.14, it is to be expected, consideringthe variation of the Renumber over r, that the transition will be fromlaminar to turbulent in inward flow, but from turbulent to laminar in out-ward flow.

The flow in the bearings described in the present paper does notshow, as a rule, high degrees of divergence or convergence on account ofthe geometrical shape (radial flow from a closed annular channel). Con-sidering the poorly defined border line between laminar and turbulentflow, it is advantageous to avoid the transition region in the designof bearings. On the other hand, one can assume that, in the bearingsmentioned above, there will be laminar flow when the highest Renumbersoccurring there are below 2000, and turbulent flow when they are above3000.

The experiments described above referred to fixed boundary wallsof the clearance. In the practical operation of a bearing, the criticalRenumbers are influenced by the rotation of a disc by the action of thevq components which are superimposed to the radial flow in such a waythat there occurs an earlier transition from laminar into turbulent or,respectively, a delay at the transition from the turbulent to the laminarflow.

-45-

8.6 Influence of labyrinths in the bearing discs.

In a pair of discs of the testing setup (Figs. I and 2), experimentswith recessed throttle labyrinths at the clearance outlets were performed inthe region of laminar flow. The picture which follows shows the bearingdiscs used in the experiment and illustrates the size of the throttle sites.It was believed that there would be a greater pressure drop at the labyrinthsites than could be obtained in a smooth bearing clearance.

It was found, however, that the grooves produced a decrease in theload capacity of the bearing at an equal throughput volume, or an increasedthroughput volume at constant load capacity, compared to the bearing withoutlabyrinth grooves. The fact that the smooth clearance induces the greatestresistance in laminar flow, has been verified again in this case. Only inthe region of turbulent flow, baffle plates and labyrinths are effective.

Fig. 1. The pair of bearing discs with recessed throttle grooves(labyrinths) did not prove to be useful.

Resonator

______________Annular space

t = 1, b = 2.

ý\l .19Labyrinths

251, Feeding nozzle""W d= 0.7 diameter.

2.6"601*

-46-

9. Turbulent, isothermal clearance flow.

Whereas the friction equation for the laminar clearance flow could bedefined in a simple way by the introduction of the viscosity, the frictionmember daR of the flow equation (No. 5.4) is governed by a more complex lawin turbulent flow. It is useful to introduce a friction coefficientwhich depends on the Re-number and can be experimentally determined.T~efriction energy of the radial shearing stresses of the wall, therefore,becomes:

daR - 'A r T- (9.1)

"÷" for outward flow: "-" for the flow inwards.

If the centrifugal force term of (No. 7.21) is not taken into eansidera-tion, the above equation in connection with equation No. 5.4 gives:

Wtd +- -+0 . (9.2)

In reference(IX), measurements of air currents in narrow clearances with rectan-gular cross sections are reported. Three cases were studied which differ fromeach other by the shape of the channel.

a) parallel plates, b) parallel plates, c) widening plates,parallel edge widening edge parallel edgeboundaries, boundaries boundaries

NNC

Xhi

Re = constant over x Re # constant over x Re • constant over x.

The ) T values were given, for the cases a) and c), as a function of theRe-numbers. These values are, however, not generally applicable, because onlythe first two members of equation No. 9.2 were considered in the back calculationOf IT, whereas the member w.dw/g was disregarded. ýhe influence of this memberwas most comspicuous in case c). It was found that A.T decreased in the experi-memts with increasing divergence of the plates. Actually, the flow is sloweddown, which causes a recovery of pressure which was considered to occur at theexpense of a reduction of friction.

In the case b) where Re varies over the length of the clearance, neitherlocal nor average IT values were determined in the paper quoted above.

-47-

The measuremerts for the case a) are illustrated in Diagram 4.Whereas the conformity with the theory is very good in the laminar region,the measured points in the turbulent area scatter somewhat more. Inthe paper mentioned above (IX) the influence of the roughness of the wallwas not emphasized. The roughness measurements which I made in a finelyground thrust bearing disc are compiled on p. 49. I suppose that theplates used in the experiments described in reference (IX) had approxi-mately the same average roughness (Rm = 1.5 x 10-6 m). Whereas theexperimental points in (IX) are in rather good agreement in case a) forh a 147 x 10-1 m (h/Rm = 98) with the Law of Blasius for smooth boundarywalls

0316

S(9.2a)

i.e., at this height of clearance roughness does not have any influenceyet, practically speaking, higher I T values were obtained at h = 127x 10-6m (h/Rm = 85) on account of the greater relative roughness. Itcan be, furthermore, seen in Diagram 4 that the X T values for constantrelative roughness tend towards constant values with increasing Re-numbers- which is a known fact, - these values depending only on the magnitudeh/Rm.

In order to make an approximate calculation of a thrust bearing asfar as course of pressure, load capacity and throughput volume are concerned,in the turbulent region, the term w.dw/g in equation No. 9.2 is disregardedand, besides, a constant value -T deduced from Diagram 4 based on anaverage Re-number and a known relative roughness is introduced for IXT.It has already been estimated in Chapter 8.1 for the laminar flow, inwhich ranges it is allowable to disregard w.dw/g. In the later calculations,in most cases bearings are considered where the clearance flow occurs at largeradial dimensions. Likewise, the ratios of the radii of the bearing betweenthe inception and the termination of the flow

(9.2b)

as well as the ratios of the corresponding local Renumbers do not becomevery large. The -6T value which is assumed disregarding an influence ofroughness r-J Re-h does no longer change very much with r.

-448.

For example, for r2a * 1.50 Ž - 1.11rl Rei

The average Reynolds numbers Re for -

which the corresponding I T values are to be ,r.C-determined are found for the outward flowfrom: F -a

Res (9.2c) Pa

, "

and for inward flow from:

t t(9.2d)

The equation No. 9.2 is transformed by thesimplifications described above to:

9 t (9.3)

-49 -

Roughness measurement in

disc of a thrust bearing

(finely ground) , >

n. average roughness

umaximum depth of the roughness

1: in transverse direction to thedirection of grinding

.. .

"": "* :IT : , ' ;

2: in the direction of grindng m"

-77

3: 4*'0 to the direction of I s

grinding, . .. .. . ... ...

, . . .•

* ': *. .

Magnification, horizontal , vertical

11 r . . . .. - -. . . ;. . .

-50-

From the continuity equation No. 7.8 and the gas equation No. 7.9, the follow-Lng can beIconcluded: G L 0',- '161r&9h' ÷ dp -0 (9,7) h

after integration with consideration of the bounda;-y conditions

r a r,, P = P10 oder r= r• p = po

r a rL P - pL OTr r- ra P = PA (9.4a)

the following is obtained for the flow outwards:& IT RTd2 •

pL pb .'T-LRTGa. (. (,5)

or

L+'AT RT G.9 6

If the volume of flow is eliminated, one obtains:

Plo"p ' I- - 1-) iO (98)a.I _i r 97

or

p~ L L

and for the inward flow:

a. __- R_1- __ (_P AL (9,9)

or

"p B , 2 %,- (9,10)

and, when the volume of flow is eliminated:

(p AL -1 - (T,, TO (91

rL rT -or.

I M x_ (9912)

The amounts of flow are

and

_______ Fill)

E xpleritm Intal-:e~sults firom. reforence (IX r. , (I',,~

...Ver~uch5s.erýe~ b sie..d QU$Lý

.... ......-

j .. ... .. .

..z I ........ ..~ .. ... ... ...

.... . .. . . . .

OJ 4ý zo.. '

-52-

10. Load capacity.

The effective axial thrust Kges which the bearing can take up canbe determined from the difference between the force acting in the carryingbearing clearance and the force acting upon the backside of the movablebearing disc.

Ksges= K- Kx ~ (lol)

where KI is the axial force of the carryingside of the disc, and KII is the counterthrust of the non-carrying side. If thebearing is built in such a way that onlyone side can carry, the counter thrust is Kzto be formed by the surrounding pressure . .

P2 and the area of the bearing (cf. theupper drawing.) P2 !

K - Pa... (10,2)

where P 2 is the pressure surrounding thebearing and FL = (r- a - r~ i) is thebearing area of one side o0 the disc. In ' l

a bearing which can be loaded bilaterallyand is built symmetrically the counterthrust is a magnitude which depends on theclearance h11 , or on the pressure in theclearance on the non-carrying side, respec-tively (cf. the lower drawing). Since,however, hI is appreciably larger than hin a fully loaded bearing, the equationNo. 10.2 can be applied here also in mostcases.

The axial force of the carrying bearing _ _ _ _ _

clearance can be obtained by integrationof the absolute static pressure in theclearance over the whole bearing surface.

So. (10.3)

An integration of equation No. 10.3 can be performed.analytically onlyfor the solution of the simple approximation (cf. Chapters 7.1 and 9.)

-53-

10.1 Radial isothermal laminar flow

If the approximation equations No. 7.14 and No. 7.18, respectively,are introduced into equation No. 10.3, the result for the outward flow is:

_rV - _- In (10.4)

The results for the inward flow is:

Kn -2-Tp1i~.Jr Vk -i- - ( L~ * .r (10.5)K3• -2 p••rV V L-' In g .dr .(os

n r

Outward flow.

The solution of the integral for the outward flow was already given

by references (IV) and (X), but will be repeated here for the sake of com-pleteness. With the abbreviation

I -- ) (10.5a)CL M -

and the substitution

4- Le

d r 4, e -- (lO.5b)

U 1--aL n d r 2.- (1du~

rQ

the equation No.10.4 becomes:

2 . A U-.L (10.5c)

The above form can be simplified by partial integration, and the result is:

P-"o (10.Sd)

K, U. - - •.r.e Tue} u

• '1 L,'

13

-5 -

If the remaining integral is transformed by further substitution, namely

-.i (lO.5e)

the following is obtained '_I(, 0,,r' P4(l0.5f)

The integral still remaining is the error integral

e-eOt * (10.5g)

which is shown in reference (XI) for given limits in table form.

Thus the following transformation is obtained:

or

Kjc. - r r ,• CO., (10,6)

where ~1 ~~[(~ l.ar,,. ,1 (,. 1- N If) -<,o1 1.6a),

is a dimension-less load factor, which depends only on the pressure ratio

P2a/Pla and the ratio of the radii r2a/rla. In the following Table, the Ckavalues are listed which were calculated as functions of the two parameters.

P10. /P46.

0,00 0,20 0,40 0,60 0,80 1,00

0 0,000 0,200 0,400 0,600 C0800 1,0001

i009000 - 0,216 0,42 0,626 ,8 099910,000 0,403 0,453 0,559 0,689 0,825 0o 90 (10.

5,000 0,451 0,491 0,581 0,694- 0,815 0 960 63,333 0,465 0,499 0,575 0,675 0.,775 0,910 b)

PO 2,500 0,459 0,486 0,552 0,633 0,728 0, 34-0"2,000 0,431 0,453 0,508 0,579 0,661 0,750

r,. 1,666 0,382 0,402 0,445 0,502 0,568 0,6401,428 0,317 0,315 0,562 OPZ05 0,455 0,5101,250. 0,230 0,240 0,260 0,289 0,321 0,5591,111 0,124 0,126 0,140 0,154 0,172 0,1891

-55-

Diagram 5 (p. 60) shows the graphical plotting of the load capacity fac-tors for the flow in outward direction.

Inward flow

By means of the abbreviation P 2.

- (10.6c)

and the substitution

2. -e

drv -- e b .v

V -•- eng v dv

the axial force deduced from equation No. 10.5 is

-I z(v4L-i)

or[i . il _V_

T " Wp4 r.Y eA'L "v.v.e -dv (l0.6f)

which gives, after partial integration,

K - e-16- dv (10.6g)By fvrther substitution, namely

V J 1 (l0.6h)

the integral is simplified and, with the limits introduced, it reads:

2~.r ALj r.e e - y (10.6i)

-56-

The remaining integral represents, according to reference (XII), theprogression:

je SLds .+ + .. (s) (l0.6j)

For the boundary conditions between 0 and 2, the values carL be obtainedfrom a Table given in reference (XI). For limiting values larger than 2,

the series was discontinued, for the present calculations, after the 10thmember. Thus the force Ki becomes

~ } (10.6k)

or

K? -- PW rkI CK. (10,7)where

° 2-

represents again a dimension-less factor which depends only on the ratio of

the pressures and radii. The Cki values were calculated as functions of thetwo parameters; they are compiled in the Table which follows and plottedgraphically in Diagram 6 (p. 62)

0,00 0,20 0,40 0,60 0,80 1,00

1,000 1,000 1,000 1,000 1,000 1,000 (10.10,000 0,881 0,895 0,911 0,956 0o962 0,,'90 7b)5,000 0,806 0,816 0,855 0,864 0,908 0,960

3,33 0,753 0,744 0,769 0,808 0,856 0,910'-",L 2,500 0,653 0,664 '0,693 0,736 O, 7841 0, 8'O

2,000 0,565 0,576 01605 0,645 0,695 0,750r' 1,666 0,468 0,478 0,506 0,545 0,589 0,6 0

1,428 0,366 0,373 0,396 0,428 0,466 0,5101,250 0,253 0,258 0,275 0,299 0,327 0,3601,111 0,129 0,134 0,143 0,156 0,172 0,190

10.2 Isothermal turbulent radial flow.

If the equations No. 9.7 and 9.11, respectively, for the course of pres-

sure, derived for simplified conditions (w.dw = 0, SL= 0), are introduiced intoequation No. 10.3, again an analytical solution of the integral is possible.

-57-

Outward flow

The equation

Kj- 21 1) (10.8)

using the following abbreviations

F-- and rl= -- rn (10.8a

becomes

Kro, = a. Vrr r rnm dr • (10.9)

According to reference (XII), the solutions depend on the magnitude ofcoefficients.

a) for n < 0, i.e. if (.I. y (10.9a)

the result is

01, G, 2-, W&(10.10"•

"I-rn. arcsivi +-a"cs'+

or (10.11)

b) for n > 0, i.e. if '_6)_ > '0 (l0.11a)

the following applies

V-" Y 10..H6

-- 0 -Cos

or1 or K, T rnl P.,. C'• keoL

(101_

-58-

Depending on the sign of n, the first or the second term must be used for thedetermination of the load capacity. The factors Cka are dimension-less valueswhich depend on the ratios of the pressure and the radii. They are compiledin the following Table, and in Diagram 5a.

Pa. /Pla.

0,00 0,20 0,40 0,60 0,80 1,00

100,000 0,075 0,220 0,410 0,6L5 0,801 0,999 (10.10,000 0,248 0,329 0,473 0,636 0,811 0.990 I3a)

5,000 0,337 0,396 0,515 0,651 0,802 0,9603,333 0,385 0,048 0,525 0,645 .0,771 0,9102,5C0 0,4u4 0,446 0,515 0,620 0,722 0,840

r2k 2,000 0,392 0,425 0,482 0,564 0,651 0,750"- 1,666 0,359 0,380 0,428 C,491 0,576 0,640

r 1,428 0,305 0,321 0,354 0,405 0,459 0,5101,250 0,223 0,230 0,254 0,289 0,322 0,3591,111 O,120 0,13U 0,146 0,157 0,172 0,.189

Inward flow.

The integral

_____ dr-j -2- 104Z r d)r (10.14)

using the following abbreviations

k -( ) and t = -i'"k (lO.14a)

becomes "Kz IT Po41 j er- rL k r dv' (10.15)

Since the coefficient is always greater than zero, there is only onesolution:

k~ (10.161LIO4.{"•12.e)�(r~4, )•V/.-- 4 - (10.16)

ký [rcCos(. .1--i-Ardos(7.e r. _1k-a R r

-59-

SThe following is obtained if . load capacity factor:, C ki, is introduced:

r r4 iL C(10.17)

The following. Table and uiagram 6a shows the relationship of these C kivalues:

P2.4 /P4

0,00 0,20 0,40 0960 0,ý8 1CO

10,000 0,941.1 0 v911.2 0,949 C 0,960 0 973 0• 905,00 0,:38 0,872 u,885 0,905 02930 02960 (10.

3,333 0,785 0,791 09809 0,396 Uo871 0,9i0 17ay 2,500 0,693 0,702 0,723 0;755 0.A794- O,4u

2,000 0,594 0,603 0,629 0,661 0,'702 0,7501,666 0,488 0,498 0,520 0, 55 5 0 2 594'- 060401,428 0,374 0,93S 0 406 0,ý-4 09469 025101,250 0,256 0,258 0,282 0,502 0,529 0,ý55

1,111 0,130 0,131 0,149 0,158 0,168 0,189

11. Optimization and economy

11.1 Optimum geometry for bearings without compensating space.

S If the bearing gas is fed into the clearance of the bearing at the siterla= r r0 - this can be done by a circular clearance as feeding organ, orby individual nozzles. which open into a compensating channel of close to zerowidth - and if the flow is always in radial direction, ro can assume any valuebetween r2i and r2a. Therefore, criteria must be found for which ro has themost economical or favorable value, respectively.

A) Bearing for the smallest flow volume

If a bearing shall have a gas consumption which is small as possible,i.e. the output of the bearing gas compressor shall be at a minimum at givenpressures, namely:

PB pressure of the bearing gas in front of the throttling devices,

Pla = Pli = p, highest pressure in the clearance

P2a = P2i S P2 pressure surrounding the bearing

and if the radii r2i and r2a are given, the following must apply:

6z - "6a G- -6lifi (ll.Oa)

orr

-- v-{

-60- Diagraimm .5

00'4

03CL

12. 1. 11 .2. 2f___

.... Vigr195q

Os's

01

06.

01s

4..*.16 2.0 V2

.. ~~~62-D~rfl

...........

04 0

01.

41L IV

-63- Dorm g

006

Olt

42414 16 4*4o~ ~

-64-

Laminar flow.

If the equations No. 7.19 and 7.20 are introduced into equationNo. 11.1, the following will occur:

*'a ( h3(i~Ž ) 0 (11.1a)

If the constant factors are omitted and the equation is further transformed,the result is

+Ln ) 0 (11.1b)

and

S=" 0 . (1r.lc)

The differentiation with respect to ro thus gives:

-"-Ln J (11.1d)

for which the following results:

r. . L2 or (11.2)

In a bearing with laminar flow and lowest gas consumption, the radius of theinlet site rO must equal the geometrical average of largest and smallest bear-ing diameter. This formula (No. 11.2) is illustrated,by way of comparison, inDiagram 7. It can be easily seen,besides, that the amount of throughput is aminimum when both partial flows are of equal size.

Ga = Gi.

On p. 68, the course of pressure in the bearing clearance, the volumesof throughput and the load capacity for constant initial and terminal pressuresand given main dimensions of the bearing tre shown diagrammatically for varia-tions of ro between r2i and r 2 a whereas GI is infinitely large at the sitesr 2 i and r2a, and has the smallest value at ro = v a - r2i, the load capacityK1 increases continuously from r21 to r 2 a.

Turbulent flow.

If the equations No. 9.13 and No. 9.14 are introduced into the equationNo. 11.1, the consequence is:

I .F•11tll .. L. Aell 1,/I 1 0 (11.2a)

,7,

-65-

If the constant magnitudes are omitted and it is, furthermore, assumed that

'Ta-' iTi, the following remains:

_________ __________(11.2b)

Partial differentiation with respect to r and further simplification gives:

I - = - �(ll.2c)

and thus for the radius in question:

(11.2d)

or the following equations:

up and (11.3)

The amount of throughput of the bearing with turbulent flow has, likewise, aminimum at Ga = Gi. By equating No. 9.13 and No. 9.14 directly, the same re-sult is obtained. In Diagram 7, equation No. 11.3 is illustrated in the formof a curve. It can be seen, as a striking feature, that radius rO is appreci-ably cbser to r2i in turbulent flow than it is in laminar flow.

B) Bearing for load capacity/flow volume= maximum.

In the following discussions. again PB, Pla = Pli = Pl' P2a = P2 i = P2'

as well as the radii r 2 i and r are constant and given. The diagrams on p. 6 8

show G, and KI as a function oiathe gas inlet radius ro. The inclusion of theabsolute load capacity KI in the discussions on optimization makes only senseif the equivalent comparative value of KI versus G, has been clarified. In addi-tion to the frequent cases where GI is to be kept as small as possible, there areapplications of aerostatic bearings where the amount of throughput is rather un-important. In these cases, however, the bearing must have the highest possibleload capacity at increased gas consumption. Thus, if one arrives at a compari-son between optimum load capacity and minimum throughput and if the criterion

"" -- (11.4)

- 66.

is selected for the determination of ro, an optimization can be performed ina relatively simple way. If the equations No. 10.6, 10.7, 7.19 and 7.20 ofthe isothermal laminar flow are introduced into formula No. 11.4, the follow-ing result is obtained:

K 1 + _ _ Max I112 RTv(PL ) [-7- L Ln A

12. PT L~ro r,.L

If the magnitudes which are kept constant are eliminated, the followingremains:

(y~Z~+ .U] -Lr-~ M ax I (11.4b)

%here Cka and Cki are functions of the ratios r 2a/ro, ro/r2i and, respec-tively, r2a . ro/r 2 i . r2a. The maximum of this function could be obtainedby equating the partial differentiation with respect to ro to zero. Since,however, this cannot be performed analytically, the function for variousratios r2a/r 2. over ro/r 2 a was plotted graphically and the optimum valuesobtained for the individual curves

La (11.4c)

were recorded in Diagram 7. For the determination of the Ck values, a pres-sure ratio P2/Pl = 0.20 which remains constant was assumed. Since the Ckcurves (cf. pp. 60 and 62) show similar courses for various pressure ratios,it can be assumed that the present result depends on pressure to a limitedextent only.

11.2 Optimum geometry for a bearing with compensation space.

The annular compensation space which is the fundamental principle ofthese studies serves for the uniform distribution of the gas prior to theclearance flow and for increasing the load capacity. The feeding nozzles openinto this space. If the depth of the compensationspace s >> h is designedmuch larger than the bearing clearance, p1 is constant over the area of therecess. Besides, the effective throttling length of the bearing clearance isshortened by the compensating space in such a way that the throughput is in-creased; on the other hand, it reduces also the friction in the bearing whenthe shaft is rotating.

As far as the interrelationship of the limiting radii of the compensa-tion space, r1i and rla is concerned, a law similar to equation No. 11.2 hasbeen established:

(11.5)

-67-

This selection was made insuch a way that, with isothermallaminar flow which is presumed tooccur in the discussions below,,the two partial amounts 6a and Gi r_. _

are equal.A

The following questionarises now: How large shouldthe compensating space be in abearing the limiting dimensionsof which are given, and whichpressure of the bearing gas is - -a -

required in order to reduce thetotal power input which is com- 'e

posed of the effect of the bear-ing gas compressor N1 and of thefriction due to the rotation N2

of the shaft, to a minimum, ifthe load capacity KI has beenestablished?

Diagramm 7

CL

15' A -[n

JGi

t

;L

-68-

i i~

PP

Paa

Gp 1

K1

-;as consumption and load capacity as function of the gas inlet radius r%.-. ...

-69-

P Let us suppose that in a bearing which can be loaded unilaterallythe following is given:

(11.5b)

and that the following should be determined:

p4 , pU N, + 0 .( r /ra• ) (II.5c)

and that the energy of the compressor of the bearing gas N1 be givenby:

where 'fko is the adiabatic efficiency of the bearing gas compressor,Had is the adiabatic delivery head of the compressor and k is the through-put volum. of the bearing per second. Then the delivery head can be cal-culated from

- RT2.- [(.( )''.- 1 3 (11.7)It is presumed that the compressor aspirates from the surroundings ofthe bearing and then furnishes a final pressure of PB" The amount ofthroughput d, is determined with the aid of equations No. 7.19 and No. 7.20for the laminar isothermal clearance flow.

Tr ____ + -GL~@~'L iZ 2 R TL fl~ Lr I

In the following it is assumed that the aspiration temperature T2 of thecompressor is equal to the gas temperature of the clearance flow. if theequations No. 11.7 and No. 11.8 are introduced into equation No. 11.6, theresult is:

Sor using the condition No. 11.5 and the resolution

(11.8b)

-70-

the following is obtained:

The ratio pl/p 2 must be determined, in compensating spaces of differentsize, from the stipulation of a constant load capacity KI. Consideringthe equations No. 10.6;and 10.7, the absolute load capacity of the wholebearing area is:

Kr w1 [ ( r,.- L4) 1' ~~± ~CL (11.10)

and, according to equation No. 11.5, the following applies:

the pressure ratio

-L Kx

for a certain size of.the compensation space can be obtained by itera-tion: select pl/P 2 and determine Cka and Cki from the diagrams 5 and 6(P. 60 and 62)calculate Pl/P 2 according to equation No. 11.12.

If the assumed and the calculated presslxre ratio pl/P 2 are in goodagreement, N1 can be determined from equation No. 11.9.

The friction effect N2 of the bearing can be derived, if equation No.5applies, from the equation No. 7.21. It is for s h:

.N.m M~.-= - -r. r r~ddr I (11.12a)

or, after integration

). 1n-- 1Ly L d K (11.13)

Considering equation No. 11.5, the following can be concluded:

irMr 0N I - ( (11.)

-71-

Since a general optimization according to the stipulation

U N + 0 (11.l4a)

cannot be performed analytically, it is useful to form the sum of NI+N2for the different sizes of the compensating space and to determine theoptimum graphically. For an actual case with the following values:

S0,133 1/r0,055 m Z 1i i 9.10-6 'p s/M'

K - 1579 Ip h = 30.10"6 MP1/PA 0,80 "o = 0,80p1. 1.10 I04 kp/l"

the effects N1 and N2 were calculated as function of r2a/rla, the r.p.m.were determined and both represeuted in Diagram 8.

Result: (11. 1L4c)

-zia /r~a p4, /Pa Pb /P2. N,~ (r~kp/,s) ii (w'M k S)11 1C00O -.(= 1750 aO= 200C

1,556 5,00 6,25 77? 65 3S1; , L2 1c51,51,4b0 4,83 6,05 79713 2c, ;0L. 78,21,330 4,39 5,49 d7?85 23,,G- .4,ý 13Jo1,210 4,01 5,02 1 0 US762 C-4c2 105c 9i,110 3,72 4,65 139 50 i!,15 5I,0 67,11,000 3,43 4,30 cpo 0,00 C0, 0.,0

Although a general conclusion referring to the most economical bearing as faras the compensation space is concerned cannot be reached, it is, nevertheless,evident from the example that, if the number of rotations is zero or small, abearing without compensating space requires, at constant load capacity, thesmallest power input. This can be explained the following way: If a compensa-ting space is present, a lower pressure may suffice, but the volume of through-put becomes appreciably larger on account of the reduced throttling in theshortened bearing clearance. From this it can be concluded that the load capa-city in bearings with a low number of rotations or large heights of the clear-ance should not be obtained by large compensating spaces but rather by highpressure of the bearing gas. If the machine works at very high r.p.m., theeffect of friction plays a larger role and, in this case, a larger compensatingspace is justified.

11.3 Optimum height of the bearing clearance.

Another possibility of determining the smallept power input of a bearingwith given dimensions (including compensating space) and pressures can be de-rived from the formula: N -N)

A otm vae= 0 tc (b.15)

An optimum value for the bearing cie•t'9Lwte can be calculated this way.

-72-

If the equations No. 11.9 and 11.13 are written in abbreviated form, oneSobtains eA""N, u (ll.15a)

i a ,

where "A" and "BeW have the following rueaning:

A Tr r 90 P 1..., F("�"A 6 L (11.15b)

no~ - Y40

and, from the following equation:B(N~l--Ni,• F ,,, N- ."O( 1 ) - 0 (11.15c)

the optimum height of the clearance is obtained:

o T -I T =. (11.16)

For an example with the same data as compiled on p.71, the optimum height ofthe clearance, hopt, was calculated as function of the compensating space.The following values resulted:

rL', /r 4 P 4 /PL h.pt

1,556 5,00 2o,63.10' m I1,480 4,83 24,21.10" 6 (11.16a)1,330 4,39 23,02.10-'19210 4,01 C20,I0. 107'1,110 3,72 17,04.10"'1,000 3,43 0 III

- A ~ A

ll.4 Power input for different gases.

In the following, the influence of different gases on the power inputrequired for the operation of an aerostatic thrust bearing is illustrated bya fundamental comparison. In this comparison, one and the same bearing isexamined in different gases, but with constant axial thrust. The nozzle

-73-

pressure ratio p./p = 0.80 is, likewise, constant with all gases. The totaltheoretical power iLput is composed of N1 and N2 , where the compressor energyN1 for adiabatic conditions is determined from equation No. 11.9 and the bear-ing friction effect from equation No. 11.13. The examples with the values:

ra - 0,055 m h - 20.0C-' m-'i 0,087 m p4 = pS = 4,80.10" kp/mL

S= 0,089 m = P 1j f i,00"104 (11.16b)•%Q 0,133 m pS = 6,00.10"S = 1570 '/s T = 293 OK

gives the following table with various gases:Gas ae *m6"t '6"ý P1A -- N-"•,-

Gasi0 ~ ~ ~:) i N4 Nk N1,NkSwtV m k~p/s mfls mkr/s

i Lu;t 1,40 1,85 1,890 0,670 27o5 106 133,5N2 1,40 1,78 1,968 0,670 2g,6 102 1•O..6 (ll.l6,ie 1,66 1,0 1,327 1,040 V 0 V.). /C02. 1,30 1,50 2,o90 0,510 32,0 86 113,0

Sattdampf 1,33 1,30 3,100 0,560 37,6 71,5 112,1Freon 11 1,12 1,09 8,322 0,219 39,5 62,5 102,0

NHa 1,31 0,950 4,450 0,529 5190 54,5 105,5H1 1,41 0,896 3,840 0,685 57,1 51,3 105,1

I Air- nitrogen - helium - carbon dioxide - saturated steam - Freon 11 -ammonia - hydrogen.

It can be concluded from this table that the pow• input for the com-pressor of the bearing gas N1 (or the total effect at . 0.=0) depends,practically, onl, on the viscosity of the gas. The smaller viscosity I is,the larger an amount excapes through the bearing clearance and the larger isN1 . The influence of 2C is only minor. The order of the gases examinedcorresponds to the power input for .l= 0.

If, however, the bearing friction which is generated by rotation is* also taken into consideration, the order is changed, since N2 increases with* increasing viscosity. What importance the friction effect has compared to

the compressor effect does not only depend on I1 but also on the size of zhecompensating space and the angular velocity . . In the example consideredhere, the order of the gases with respect to minimum total power input isalmost reversed.

11.5 Optimum nozzle pressure ratio in double-acting bearings.

The studies of optimum conditions made so far referred to bearingswhich are supplied only unilaterally with bearing gas and, therefore, cantake up the required axial thrust in only one direction.

-74-

It happens, however, frequentlyin practice that the thrust of machineswhich are subject to non-stationaryconditions (starting and arrestingperiod, fluctuating level of pressure,pumps, compressors, etc.) works tem-porarily in both directions. Thefollowing discussions are based on abilateral bearing of entirely symmetri-cal design which has, at the sameheight of the clearance, the sameload capacity in both directions. Abearing designed according to thisprinciple has the disadvantage ofa high gas consumption. On the - -

side I of the bearing which isloaded by Kges, a small clearancehI is formed, while, on the un-loaded side, a much larger clear-ance, hll in correspondence withthe total axial clearance, is pre-sent. Since the unloaded side must be continuously prepared for a possibletake-up of thrust, an unused quantity of gas Gil escapes here steadily.

ODtimation consist here in the following: To find the nozzle pressureratio PB/P1I or, respectively, the required pressure PB at which the compres-sor output N, for a given bearing pressure ratio pIl/P 2 becomes a minimum forthe preparation of the total bearing gas.

In this connection, the following abbreviations are used: G. is theamount of throughput per second through the carrying clearance hI, (Kg/sec);b11 is the amount of throughput per second through the non-carrying clearanceh1i (kg/sec); P¶I is the highest pressure in the carrying clearance hI(kp/m 2 ), which is determined by the magnitude of the axial thrust; Pil'Z? P2,

since hiT >> h., is the counter-pressure of the nozzles on the non-carryingside of -he bearing, approximately equal to the pressure surrounding the bear-ing, P2 (kp/m 2 ); PB is the pressure of the bearing gas in front of the inletnozzles V final pressure of the bearing gas compressor (kp/m 2 ); m is thenumber of stages of the compressor; Had is the adiabatic delivery head of acompressor stage (i).

For the derivation, the following values are considered as given: Kges,KI KII' P2 P4 I, 61I

Assumptions: The nozzles are designed as cylindrical throttles orapertures. The outflow from t.! nozzles is considered as adiabatic change ofstate. On the non-carrying side of the bearing, the outflow from the nozzlesis critical, provided that

> (11.16d).

-75-

In a similar way as in equation No. 11.6, the compressor output, free oflosses, for m stages becomes:

Nm = (6 t ýz.) m H - . (11.17)

The amount of throughput, in adiabatic change of state, through cylindricalnozzles or aperatures for the carrying bearing clearance is:

T c. pa & /Rro z P~) (11.18)-~ ~ ~1 RT.;(.a C-1) hk W) -- \•wI I. 8

and, considering the assumption mentioned above, for the non-carrying clear-ance:

RTZ g Lx 2- \X-1 2- a. - 1 (11.19)

The partial delivery head of the m-stage compressor is, with adiabatic com-pression:

HaPRT 4, [(t / - (11.20)

where (pB)F is the ratio of the stage pressures.p2

If equations No. 11.18, No. 11.19 and No. 11.20 are introduced into equationNo. 11.17, the follow•ing result is obtained:

1E'j (11.21)

Nim ['R) -- ] •L

If the following condition is met:

pis x _ (11.21a)

the power input Nlm is now only a function of the desired pressure ratio

P4I/PB and the optimization N 0

.. '0 (11.22)

can be performed. For m = 1 and t = 1.40 the minimum effects at variouspressure ratios P'I/P 2 by variation of P1I/PB were determined by graphicalplotting. The result is represented in Diagram 9.

I f "-,, ... .~ .-. ...- .. ••i • ,,, . .• ? -"" - ' ,. .

*i;-76-Digau 1 .

*~~ .. .. ....1 57 ,

IPRO

raa

80-.5 9 k :.. .

2-

F ~ fi) riN \ \ . .. .. . .. .. ..

-.- - - - - -- - - - - -. ~ - ~ .1 N ' - - - - - - .ls-

-77-

_____

1 I00.1I~7t 7-

''d :It: 7. F I

3 4 5 7 8 _ _110 4 0 bI 6 0_( .~

-78-

Determination of the optimum pB1P4I and PB:

From the given load capacity K1 the pressure ratio P5 /P2 is determined(For example, PlI/P 2 = 6.00). From Diagram 9 it can be seen that, for m = 1,

PB/PlI equals 1.316. This allows the design of the required nozzle area. Thecorresponding pressure PB is in this case:

__- (11.22a)

For example, PB = 7.90 P 2 -

12. Properties and design of the throttle elements.

In order to impart to the bearing the property of increasing its loadcapacity with decreasing height of the clearance, the bearing gas must be ad-mitted to the clearance by means of throttle elements. (cf. the explanationson pp. 10-14 ). These throttle elements may consist of individual nozzleswhich open directly into the bearing clearance or into a compensating space,

or gap-shaped throttles (e. g. annular gap) or inserts of porous material.

If, in a bearing of geometrically fixed magnitudes, also hI and KI aregiven, then also pI1 is determined. Since the amount of throughput throughthe bearing clearance must be equal to the quantity of feeding gas, Gclearance =Gthrottle, the operating point P1 of the bearing is the intersection of thecharacteristic of the throttle elements aAd the characteristic of the bearingclearance.

G_ Charakteristik

des Lagerspa-ates

IKennlinie -i

d. DrosselI F-

/ P /.

paa

(top) characteristic of the bearing clearance -

(bottom) characteristic of the throttle

-----------------------------------

-79-

In bilateral thrust bearings, the curve hI indicates the carrying,the characteristic h11 the non-carrying bearing clearance.

The characteristics of the throttle elements, which will be investi-gated more in detail later for various cases, depend, above all, on the kindof throttles and on their effective cross section area. The characteristicof the bearing clearance is mainly determined by the height of the clearance,geometry of the bearing, and the viscosity of the gas (in laminar flow), and,respectively, by the friction conditions in the clearance (in turbulent flow).

The characteristic of the bearing clearance is determined for theradial isothermal laminar flow. The amount of throughput through the bearingclearance is given by the equations No. 7.19 and 7.20:

L2 nL120a

Thus the characteristic is:

G- (121

The constants are here:

-12. ~R T (12.1a)

and

( (12.1b)b CLa~

The equation No. 12.1 is here a parabola of second order which isshifted downwards from the origin in the direction of the ordinate by themagnitude b.

12.1 Short cylindrical nozzles and apertures (I = d).

The figures at right show ____ ,,________.__satisfactory designs of nozzle(left) and aperture, 149d (right). /The aperture, though prepared /1 0 0with somewhat greater efforts, C , •'•° d

hahowever, no fundamentaladvantages over the short 'cylindrical nozzle. L / \-7F"

Dilse L d Blonde L(<d

-80-

The area FD of the required throttle cross section of the nozzlescan be determined for a bearing with given values PlI, P2 and 6, from thenozzle characteristic which, assuming an adiabatic change of state, of theflow, gives the following result:

) A.? - ( -) (12.2)

The effective throttle area FD depends not only on the diameter of the bore-

holes of the nozzles, but also on the arrangement of the nozzles in the bear-ing. The coefficient P- is a flow factor depending on the shape of thenozzles (sharp or rounded-off edges) and on the Renumber of the nozzle flow.

Another way of writing equation No. 12.2 is

G.VTS R (l2.2a)

It is represented in Diagram 10 for some frequently used gases, assuming idealgas properties. At a given effective throttle area F? and constant pressurePB, the amount of throughput 4 increases with decreasing pressure P, until thevalue

(12.2b)

is attained where, since velocity of sound has been reached in the throttlecross section, the amount of discharge reaches a constant value which doesno longer change with further decrease of pI.

A) Opening of nozzles into a compensating space.

If a uniform radial flow shall occur in the bearing, the individualnozzles which are evenly distributed along the circumference of the diameter2r 0 must open into a compensating channel which is recessed in a bearingdisc. This compensating space is preferably designed as continuous annularchannel.

In order to guarantee asgreat a stability as possible inthe bearing relative to self- In a.

induced vibrations (cf. Chapter .~

16), the volume of the compensa-ting space should be small. Thelower limit for the cross section17of the compensating space isestablished by the requirementthat the highest velocity of theI gas in the channel be not greaterthan the velocity at the inletinto the bearing clearance.

-80a-

Diagramm 10

'04

C12

* I _

I l6 06 0;Ot 0*9

S~-81-

In the following, b is the width --

of the compensating channel; s is thedepth of the compensating channel; hIis the height of the clearanceat normal operating conditions; andn the numbers of aozzles.

If it is assumed that theentry velocities into the outer andinner bearing clearance are approxi-mately equal and if the width ofthe channel,

= rla rli

is small, the requirement stated aboveis met by equating the cross section of thechannel and the cross sections at the entry into the bearing clearance.Thus the following is obtained:

2-r, r- hriL (12.3)

It can be concluded from equation No. 12.3 that an increase in the number onozzles has the advantage of a reduction of the channel cross section andthus of an improvement of the stability of the bearing.

The effective throttling area FD of nozzles or apertures which openinto an compensating space can be figured to te the whole amount of thecross section.

F d 7r (12.4)

B. Opening of the nozzles directly into the bearing clearance.

(Nozzles with variable throttle cross section, depending on the clearan

If the nozzles are arranged in sucha way that their cylindrical borehole oper.s "directly into the clearance, the effectivethrottle area is not equal to the crosssection of the nozzle, d 2 /4, but is dformed by the edge of the nozzle and theheight of the clearance. When the follow-ing applies _

() < ',.. (12..4a)

. . . ....

-82-

the effective throttle area is hb~ V\i dW-i (12.5)

For reasons of design, dmin= lO h. For nozzles which open directly intothe bearing clearance, the characteristic

depends also on the height of the clearance. The consequence is that, uponreduction of the clearance by an additional load on the bearing, the loadcapacity can only be raised insignificantly on account of the strongerthrottling effect. In order to impart good load capacity to such a bearingin the region of low clearance widths, the nozzle must have an appropriatelylarger borehole diameter d on account of its small effective throttling area.In such a case, however, the amount of throughput would increase in an undesir-able degree if the clearance were enlarged by reducing the load of the bearing.

C. Nozzles with boreholes on the side of the clearance.

In order to avoid the danger of self-induced vibrations (cf. Chapter16) which appear when a compensating spaceis present, and the disadvantages mentioned iO d.sub B), individual nozzles with a bore- 'Shole at the side of the clearance weresuccessfully installed in various experi- 12omental devices and machines. The largestdiameter D of the borehole is, in thiscase, selected in such a way that, forthe smallest height of the clearance

hmin occuring during operation, the \requirement

cLZ Tr 1_0= Dii hmiv (12.6) 1

is fulfilled. The effective throttling

area FD is, for h > hmin, identical withthe cross section area o• the nozzle. Thus equation No. 12.4 applies also here.

The effect of the borehole on the side of. the clearance in cylindricalnozzles, compared to nozzles which open directly into the bearing clearance,on the load capacity and amount of throughput was ascertained clearly by anexperiment. Diagram :27shows measurements of the load capacity Kges and theamount of throughput 61 in a bearing which can be loaded unilaterally, with-out compensating space. In the first experiment, the four cylindrical nozzlesopened directly into the bearing clearance, whereas, in the second test, thenozzles had a borehole on the side of the clearance. It was found that, withconstant nozzle cross sections, the load capacity characteristic Kges = f(h)of the bearing with counterbored nozzles was shifted into a region of greaterclearance heights while the course of the curve was similar. The load capa-city at equal heights of the clearance is markedly greater than in a bearingwith nozzles without counterbore. The amounts of throughput show a similarbehaviour. The bearing with counterbored nozzles, has, the clearance being

-83-

equal, (at least in the region of small h) an increased gas comsumptionA corresponding to the larger effective throttling area,

Rclcot D _ - = 285 ) . (12.5b)

(anges) with borehole(nicht anges) without borehole.

The values obtained in the experiments verify this ratio satisfac-torily, at least for small heights of the clearance.

In a synopsis of the three cases A), B) and C), let me add a compari-son of the dependence of the effective throttle area FD on the bearingclearance h.

The design according to C) is quite similar to the case A), whereasthe direct opening of cylindrical nozzles into the clearance according to B)is unfavorable.

SK).1.CL Q "

12.2 Capillary nozzles 1 >) d.

The friction of the gas entering /through a capillary and theresulting great reduction of the pres-sure p1 cause an appreciably flatterpattern of the throttle characteristic,i.e. below the nozzle characteristic,compared to the short nozzle, at anequal effective throttle area FD. Theconsequence is that, at equal throttl-ing area and equal height, of the clear-ance, the load capacity of the bearing Short nozzlewith short cylindrical nozzles is Xmarkedly higher than that of the bear-ing with capillary nozzles.

On account of the flat throttlecharacteristic, the use of capillarynozzles is also disadvantageous as far

~ as stability at self-induced vibrationsis ioncerned (cf. Chapter'16).

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12.3 Venturi nozzles.

In most aerostatic bearings, the inlet nozzles were designed as cylindrical boreholes. A disadvantage of these nozzles is, however, that,at agiven feeding pressure PB' the amount of flow decreases comparativelyfast, if the counter-pressure p1 increases above the critical value, Petiwhere the velocity in the nozzle is still that of sound. In order to obtaina high load capacity, it is desirable to select the inlet pressure p1 of thebearing clearance which corresponds to the counterpressure of the, nozzle asclose as possible to the available feeding pressure. At a given bearingclearance, the amount of flow is also given and, on account of thedecrease of the amount at increasing counterpressure Pl, it is necessary toselect comparatively large feeding nozzle cross sections. This involves notonly the disadvantage that a large amount of gas escapes in bilaterally load.able and symmetrically built thrust bearings in the wide clearance on the noicarrying side, but it interferes also with the rigidity and the stability ofthe bearing. In order to avoid these disadvantages, venturi nozzles for theadmission of gas were installed in one of the experimental devices. (Figs. 1and 2). In contrast to the cylindrical nozzle with the narrowest cross sec-tion area FDa, the counterpressure p1 in the venturi nozzle with an equalnarrow~est cross section can be increased considerably without an appreciablereduction of the quantity on account of the subsequent diffuser. The use ofventuri nozzles is, of course, only justified if they open into a compensa-ting channel or if the diffuser 4s counterbored at the side of the clearancein such a way that the effective throttling area is formed by the narrowestnozzle cross section and the effect of the diffuser can be fully exploited.

The following diagram illustrated schematically a comparison of the throttlecharacteristics (volumes of throughput as a function of the nozzle pressureratio pl/PB) for regular nozzles and venturi nozzles.

The meaning of the curves is the following: Curve a) is the characteristic of venturi nozzles in a flow involving a loss ( 7•_ = 0.65; narrowestcross section area FDa). Curve b) is the characteristic for cylindricalnozzles (with a larger narrowest cross section area F1,). Curve c) is thecharacteristic of cylindrical nozzles without diffuser (narrowest cross sectiarea FDa). Curve d) is the characteristic of the carrying bearing clearance(cf. p. 78).

-85-

It can be concluded from this comparison that the operating point Pcannot be reached by cylindrical nozzles (charactertistic "c) of the samecross section FDa as the venturi nozzles. Using cylindrical nozzles of largercross section area FDb (characteristic "b") the operating point could bereached, but the volume of throughput increases considerably with an enlarge-ment of the clearance by reducing the load of the bearing. In symmetricallydesigned bearings with bilateral load capacity an amount which is larger by

a Gab than in the case of venturi nozzles with the cross section FDa wouldcontinuously escape on the non-carrying side in the presence of nozzles withthe cross section area FDb.

10i

pCL IN

Determination of the characteristic for venturi nozzles.

The velocity occurring in the smallest cross section of the venturinozzle is retarded again in theadjoining diffuser. It is assumedin the calculation that the changeof state in the narrowed part ofthe nozzle and thus also the course -of velocity is adiabatic and free -6

of losses. Otherwise, the calcula- " - -tion would be more complicated. Ina nozzle which is not loss-free andwidens subsequently, the maximumvelocity is not in the narrowest C Velocitycross section. At least in Lavalnozzles with supersonic flow, soundvelocity appears •nly after thenarrowest cross section.

)

-- - -d • - • • . . . .• ' - -- • " .. .-- - - ' ... ..... . .. -7mw . ;.....' , J,+

-86-

,

The throttling characteris-tic of the venturi nozzle shouldhave an almost rectangular coursewith a flow free of losses and avery small cross section ratio

(A i (12.6a)

(ef. the curve marked "with diffuser ) 1" in the sketch which follows).

The two curves without and with diffuser show the pressure recovery inthe loss-less diffuser pl' - P1i" At an efficiency factor of the diffuser of

o1 d < 1, the pressure recovery would be only'j D(Pl' - p ). Thus one cantain, without difficulties, the whole course of pressuremfor )D < 1 from

the characteristic of nozzles without diffuser, where the compressibility hasalready been taken into consideration.

Nozzle with Z=1diffuser

Nozzle with

(T~-~i)diffuser

Nozzle without

diffuser

p, P.

IIDS

In an experimental example (cf. diagram 11) with 4 venturi nozzleswith an aperture angle of 100, an area ratio 01 = 0..45 (32 -0021),and an diffuser length of 7.5 mm, an efficiency factor of = 0.65 wasmeasured. This value appears to be correct, considering tIe large wideningangle, the poor rounding-off of the narrowest part of the nozzle and thehigh Mach numbers, compared to the results measured in reference (XIII).The outflow coefficient was determined from the quotient of maximumvolume of throughput, calculated theoretically from the nozzle area present,by the largest measured throughput volume.

'A G max orchoei W. TO (12.6b) (gemessen) measured -

(gerechnet) calculated.

-87-

The efficiency factor of the diffuser is defined by:

. P-1- PM (12.6c)

where p1 is the pressure actually measured in the back of the nozzles: pm isthe pressure in the narrowest cross section; Pl' is the pressure in the backof the venturi nozzles in a loss-free flow.

For 9, p1' -'pB, and thus the pressure ratio is:

- q (12.7)

The throttle characteristic of the venturi nozzle for a very large area ratiocan be obtained from the characteristic of nozzles without widening, equationNo. 12.2, by introduction of the formula No. 12.7 into the nozzle pressureratio of equation No. 12.2. The result is:

RT'F-)[ Z9 1 (12.8)

The general conclusion is: Bearings equipped with venturi nozzles affordeither, at a given amount of throughput, a higher load capacity or, at agiven load capacity, a saving of consumed bearing gas. In diagram 12,measurements of the pressure p1 as a function of the height of the bearingclearance h in the same bearing are compared for venturi nozzles and cylin-drical nozzles of identical narrowest cross section. The following can beseen: In those regions where the pressure ratio pl/PB is higher than thecritical, the pressure p1 is higher in the venturi nozzles, at equal heightof the clearance, i.e. the load capacity is improved; with large bearingclearances, however, where the flow through the venturi nozzles is over-critical, the load capacity is lower than in a bearing with short cylindri-cal nozzles without expansion on account of higher losses by supersonicflow, and formations of shocks and eddies. Another advantage of theventuri nozzles can, likewise, be seen in Diagram 12. In the region whichis most important for bearings, i.e. at heights of the clearance which areneither too small nor too high, the curve p1 = f(h) runs steeper than thatin bearings with regular cylindrical nozzles.

This results in a greater rigidity of the bearing, i.e. the height of theclearance does not change as much with increase or decrease of the load as inthe bearing with regular nozzles. The higher rigidity has an advantageouseffect on the stability connected with self-induced vibrations (cf. Chapter 16).

12.4 Throttle elements with variable pressure-dependent cross section.

Still better characteristics than those of regular cylindrical andventuri nozzles can be obtained by throttle elements where the effective

"OPO

-88-

throttling area FD increases with increasingnozzle pressure ratio pl/p . The design andconstruction of such throttles involves, however, Ecertain difficulties.

WIn reference (XIV), the use of individual

nozzles of elastic raw material (plastics, sili- p,.f •€0Ss F•rKIeicone rubber) is described where the cross sectionarea changes as a function of the differencebetween the feeding pressure pB and the pressurein the back of the nozzle p1. In the region oflarge nozzle pressure ratios pl/pB the elasticmaterial is compressed in such a way that a con-

striction of the cross section is produced. If, \on the other hand, pl/PB is small, no deforma- F6I-k|ef F.= •fO55tion of the nozzle occurs and the whole cross •',' -

section initially provided is available for theflow through it.

Two more designs of nozzles with variable, pressure-dependent crosssection are sketched in the drawings on the right side. In the first pic-ture, a needle body N is supported by a spiral spring Fe. The greater thedifference of pressures PB - Pl is, the smaller is the effective throttlingarea FD between needle body and seat i.e. at a small nozzle pressure ratioless bearing gas escapes through the nozzle. In the second figure, a leafspring fulfills this task. At an increasing pressure difference PB - P1,the spring is bent to a greater extent, reducing, at the same time, theeffective throttle cross section.

An exact analytical determination ofthe characteristic for such throttling elementsis not feasible in most cases. It is, there- IF.

fore, advantageous to obtain the throttlecharacteristic by experimental methods. Thedesign of a bearing or the determination ofthe operating point, and the entire courseof the load capacity over the height of theclearance h, respectively, can than be per-formed readily. The advantages of suchthrottling elements appear immediately bystudying the diagram drawn below. Thevolume of throughput of the nozzles de-scribed above either remain constant over

Pl/pB or even increase somewhat (curve e),while the throttle characteristic for acylindrical nozzle (curve f) with constantcross section drops sharply in the region oflarge pl/PB. The characteristic of the bear-ing clearance (curve g) intersects the throttlecharacteristic in the operating point P of the

* bearing. !

*

- _ *_ _ w

-89-

....- . biagramm Ii.W! :34

-I A

: "~) o0 • 7

NuN4 U

o .0

~0

- /

"')' / s-:-I

4 .. .

0

41J

0 ,

WOW O

- N*S •....

U) Nt - - I- -

-90-

S S .E~i ~....... ........... †† †††† ††.

r=E E E EE 12~OD CV

r.J -

.44

0

4. 0

Cc u

o~ 00

-91-

A bearing with regular nozzles k9es(FD = const) consumes appreciably moregas at increasing clearance, on account 0F,- (.)of reduction of the load, than a bearingwith the throttle elements describedabove. As a consequence, the rigidity,i.e. the increase of the load capacityKges at a certain decrease of the bear-ing clearance h, is considerably greater \ F0 .ko°itin the latter case. In this comparison, <the bearing geometry, the operating pointP and the feeding pressure PB are assumedto be constant.

Uh

12.5 Annular inlet clearance.

The compensating channel for equalizing the gas (cf. page 67)can be eliminated when the bearing gas is fed through a ring-shaped inletclearance which extends over the whole circumference of the diameter 2 ro.The sketch at right suggests a design forsuch a feeding through an annular clear- r.ance. Such bearings without compensatingspace are stable as far as self-inducedvibrations are concerned. (cf. Chapter16). The boundary walls of the throttleclearance may consist either of parallelplanes (corresponding to the cylindricalnozzle) or of widening planes (correspon-ding to the venturi nozzle). Since the

-92-

required throttle cross section area FDis very small compared to the area of the £ r.bearing - in most cases FL/FD '! 5000 &

7000 - and the circumference 2rT Jt iscomparatively long, the width b of theannular clearance is of the same order ofmagnitude as the carrying bearing clear-ance. The precise adjustment of such afine clearance which remains uniform overthe entire circumference involves difficul-ties in most cases. Besides, special con-structions such as composite bearing discsare required which can provoke a distor-tion of the bearing surface.

The throttle characteristic of theannular inlet clearance with parallelwalls is identical with the nozzle characteristic (equation No. 12.2), whereasthe characteristic for the venturi annular clearance can be calculated fromequation No. 12.8,

12.6 Gas feed through porous inserts.

An intrinsicaliy different way of feeding gas into the bearing clearanceconsists in introducing the gas through individual cylindrical or annular insertswhich are arranged in the fixed bearing disc and are made of porous material.The properties and possibilities of use of a porous metal (bronze, stainlesssteel) sintered together from minute balls is described in reference (XV). The

distribution of the gas can be done in sucha way that a radial flow occurs over theentire bearing, whereby the compensatingspace which can provoke self-induced vibra-tions is eliminated. In certain cases, anincrease of load capacity or a reduction of --

the volume of throughput can be achieved inbearings with feeding elements of poroussurfaces, compared to a bearing with conven-tional throttle elements. The throttlecharacteristic of a porous material is simi-lar to that of a capillary nozzle (cf. p. 83)and depends above all on the porosity and thethickness of the material.

13. Non-radial clearance flow.

In order to avoid the danger of the appearance of self-induced vibra-tions, (cf. Chapter 16) it is advisable to allow the inlet nozzles to opendirectly into the bearing clearance without increase of the clearance volume(annular or compensating spaces). Thus each inlet nozzle must feed a certain

4i

-93-

sector of the bearing area; the clearance flow is, therefore, no longeruniform and radial to the axis of the bearing. Since only local pressurepeaks form around the nozzles, relative to their number, the load capacityis reduced compared to a bearing with an annular channel.

13.1 Potential flow.

In the following, the mathematical interrelationships will be studiedand compared which apply to the incompressible and the compressible isothermalflow with any distribution whatsoever in narrow clearances. in this connec-tion, it is assumed that the influence of inertia of the flowing medium isnegligible in comparison with that of friction. This prerequisite is notcompletely applicable, however, to the immediate vicinity of the inlet nozzle,since the influence of acceleration outweighs that of friction there. Theh.ollowing studies are performed in cylindrical coordinates. The symbols havethe following meanings:

r = radius; = angle of the coordinates; Vr, q = theradial velocity or tangential velocity, respectively, averaged over theheight of the clearance; • density of the flowing medium; p = abso-lute pressure; * = potential function; q = flow function; C1 - C4 =

constants calculated from the boundary conditions.

Incompressible clearance flow Compressible clearance flow

Lm',0,ulation from equation No. 7.6

Vr 9 (13.1)

P (13.2)

Continuity

n i fu(9nVi __ o

~r Pa ar r rZY (12a

Density function

~mkonst _ __(13.2b)

-94-

By introduction of the equations No. 13.1 and No. 13.2 into the con-tinuity equation, the following is obtained if the density function is takeninto consideration:

Incompressible clearance flow Compressible clearance flow

•r16 r• a -r + r-ra 0 (13.2c

Correlation

p '-CIO+.C.' . Jp2 C3 + CV (13.3)

Potential equation

S+ 1A+r rTr 0 (13.4)

Solution of the potential equation

Velocity components Flow density components

-P -vr ý =L ay Zr~ rzy (134b

Field of flow

= const potentials lines or isobars, respectively

S= const flow lines

This leads to the following conclusion: Every laminar flow in narrowclearances can be considered as potential flow if the influence of accelera-tion is disregarded, and the potential lines are assumed to be isobars, bothin incompressible and in compressible media. Th, flow lines and the isobarsform an orthogonal network. The difference b1tween incompressi e and com-pressible flow is only the correlation bet!,7den p, !, 4 and . Whereas,in incompressible flow, the pressure iv a linear function of the potentialvalue, this relationship is expressed by a quadratic function in compressibleflow. An analogous connection between velocity and flow density is obtainedby differentiation of the potential function.

-95-

13.2 Load capacity.

If the boundary values p, and p are given and the magnitudes ofand 0 are determined from the bearing geometry, the constants C3 and C0can be calculated from equation No. 13.3; the result is:

C3 -P. (13.5)0. g.1- ('" 13.6)

From this the pressure distribution in the clearance is found to be:

P L., z.__-__4) (13.7

If equation No.13.7 is introduced into the integral for the load capacity,equation No. 10.3, the following result is obtained:

1<~ ~r~dr iT * (13.8)A 0- 0 4€) r dr. dy • <3a

13.3 Amount of flow.

The volume flowing between two flow lines very c1c-:\ to each other is:

dV - - V dr) (13.8a)

The corresponding weight/second is, if the gas equation No.7.9 is used:

dd = ý (V prd --V p dr) ( (13.8b)

If the velocity components are eliminated by means of the equation No. 13.1and No. 13.2 the consequence is as follows:

d6. RT rp -r. dy - p d r) (13.9)

The partial differentiations of equation No. 13.3 with respect to r and

respectively, have the following results

20E.C." and WC(13.9a)

Po 8r s

-96-

Introduction into equation No. 13.9 yields:

k 3 CG3 Rdf dr)d~~ d? ~ r (13.9b)

Integration affords the amount of gas escaping along a potential line:

G- rd - f dr1 (13.10)

f~l rj - rF

In a bearing which is fed by individualnozzles which open directly into theclearance, the volume ofbe easily determined in another way if thecounter-pressure p, in back of the nozzlesis known. On account of the small heightsof the clearance and the relatively muchlarger nozzle diameters, the area formed Vby the boundary of the nozzle and theheight of clearance should be consideredas site of throttling, rather than the nozzle cross section. (cf. Chapter12.1B). For small nozzle cross sections, in comparison with the bearing ar.he potential lines in the immediate vicinity of the nozzle are circles, anthe nozzle orifice itself is potential line corresponding to pl. Therefore, the pressure gradient next to the nozzle orifice is uniform, and thesame amount of gas per unit of boundary flows into the clearance over thewhole circumference.. The amount of gas flowing through a nozzle can, therefore, be calculated in a similar way as equation No. 12.2, from:

4 I'AoL dh I 2gae A at_ -~ 1 (13.11)G P iRT. 106I

where PB is the absolute pressure in front of the nozzle; P1 is the absolhpressure in back of the nozzle; h is the height of the clearance- )Athe outflow coefficient; d is the diameter of the nozzle; GD isweight of throughput per second through a nozzle.

In a bearing where the drawing at theright represents a sector, a summation of

& 6D had to be done according to separatecounter-pressures p1. In the following eequation,nis the number of the nozzle groups.

,AG.,L)i .'U%)* (13.11a)

-97-

13.4 Application of the potential theory.

A) Analytical method.

A complete integration of the load capacity of the bearing and theamount of throughput can only be performed forsimple flow patterns. For example, in the caseof radial source flow from one single nozzle inthe center of a circular thrust bearing, wherethe following applies:

the potential function and the flow function are given by

4- Inr (13.llc)

With the boundary conditions* -

0 0 01' (13.11d)P P4 P P..~ a r 'IV

the already known equations No. 10.4 and No. 7.19 which can be solved, areobtained for the load capacity and the amount of throughput, respectively.

B) Numerical investigation of bearings with individual nozzles uni-formly distributed on the circumference 2 ro.

The symbols used have the following meaning: ra is the outside radius ofthe bearing; ri is the inside radius of the bearing; ro is the radius ofthe bearing, on the circumference of which the nozzles are arranged: n isthe number of nozzles on the circumference 2roK..; d is the diameter of thenozzle, or the largest diameter of the counterbore of the nozzle (cf. Chap-ter 12.1.C); k is the modulus of the elliptic functions; K is the halfperiod in direction of the circumference; K' is the half period in radialdirection; *l is the potential value at the nozzle boundary; 02 are thepotential values at the boundaries of the bearing; V is the adjustment factor.

The mathematical formulation of the problem can'be done by a ray-likearrangement of alternate sources and sinks at a specified distance, originatingfrom a center. The number of the rays must correspond to the number of nozzlesdistributed over the circumference. Thus two adjoining boundary potential lineswhich contain n sources and form circles, surround a bearing with annularsupport surface.

-98-

• - Trough

Boundary

flow lines

As can be seen in the drawing above, a boundary flow line runs fromthe source 1 to the trough 4 across two points 2 and 3 where the flow linehas to change its angle suddenly by 900 in the same direction in each case.This fact suggests .that a flow field with two periods is involved, wherethe period in radial direction is given by the alternate arrangement ofsources and troughs; whereas the period in the direction of circumferenceis given by the continuously recurrent image which is formed by reflectionby the walls which are impermeable to the flow (connecting lines 2 -3, 2' - 3').

For greater clarity, the most essential features of fields with twoperiods will be pointed out briefly. If 2, j, u, w, § are differentplanes between which conformal representations are performed, it can be con-cluded according to reference (XXIV) that the upper half-plane of the t -

plane is transferred into the interior of a rectangle of the i - plane by thetransformation

_" ____- ____- ____ ' (13.11e)

where the term on the right side represents an elliptical integral of firstkind.

-99-

D> E C

SD A 0 C A ii I I , I - -

-1/k-1 1 4/k

By the simplifcation, si = s 5 u, the integral is transformed into:

dii_ _ _ _ _ _ _ _- -- ( 1 3 . l 1 f )

sea

The inversion function of this elliptical integral, i.e. the correspondingelliptical function of first kind reads

S- am ( Z, k ) (amplitudinis i ) • (13.11g)

This function which was introduced and calcuilated by Jacobi represents atransformation which transfers the interior of a rectangle of the i - planeonto a half-plane of the j - plane. It can be concluded, furthermore, fromreference (XXIV) that the function "sinus amplitudinis 2"

il a ; m (Ilk) - r. (ZI k)S[~ ( k . . (13.12)

represents a simple source flow of the w - plane into the two-period (sourceand sink) flow of the z - plane, which is illustrated on p. 100 (left).

A source with the complex potential ln w in the w - plane istransformed, by the function No. 13.12 into:

W (•) = Lvi sL (i, k). (13.13)

Thus this function represents the two-period source and sink field which isillustrated on p.100 (left). In order to arrive at the potential flow whichis looked for, another transformation is required:

= n4 (13.14+)

by which all straight lines of the 2 -plane which are parallel to the x -axisare conformally mapped in concentric circles of the ý -plane, and allstraight lines parallel to the ý -axis in radial rays of the g - plane whichpass through the center. If equation No. 134 is introduced into equationNo. 13.13, the following result is obtained:

-100-

This complex function represents in its real. portion the potential linesiand in its imaginary portion the flow lines 0 . The last transformatior(No. 13.14) attains not only a transition of parallel strips into closedcircular bands, but also a logarithmic distortion of the periods which areequal in the f - direction in such a way that, with increasing radius, theperiods increase steadily.

e L +l e )

- LLvr- • (13.

I~' 1 -7 o

A circle of the radius • = e° =1 is coordinated to the line y = 0 of theS- plane by the projection 2 -i ln in the - plane. The straight

line in infinite distance parallel to the R - axis of the 2 - plane corre-sponds to the zero point of the - plane through 2 = i Wn - i :-%The entire lower half-plane of the 2 - plane would be projected to theinterior of the circle r = I of the Y - plane, while the upper half planeof the z - plane is projected on the exterior of the circle F = 1. Sincethe circle S - S (figure at the lower right) corresponds to the straight lile (in the left figure below), this circle is the flow line for the bearing

wC.9-C.

_T11

f ' " -A

d -Ie ,j

C4~Ca4

ITI-A A B C D

-101-

The constant k is designated as modulus of the elliptical functionand depends on the elementary periods K and K' or their ratio K/K'.

In order to have the boundary conditions, selected for the bearing,fulfilled by the transformation function No. 13.15, the following coordina-tions between the figures above must be observed.

Y period r - period

a... 0 a.. . r.( i/r,,)(•

b... KY2 b... 40e)rL )ra)

e . . 3/2 elogarithmic. * * r

e K2 coordination e__*.___4(r

f . . 5KV2 f ...*o(•-57-L (fr-e/re. r,,

S.e. 6KY2 .*. r.o( I )/ra•(Ir )'-

arithmetical progression geometrical progression

If, on the outset, the site of the gas feed r0 is chosen in such a waythat the following applies: G. - ra .r(

(cf. equation No. 11•2), it is feasible to insert the r -periods of the -

plane into a geometrical progression with the quotient /ra/r.V

- period - period

A... 0 A... 0B.. K B... W/nC. *. 2 . C... 2•r/nD. 2K linear coordination D , , , T i/nD 3 4K D... 3r/n (13

•... 2nK . .2r

Thus an angle of 3600 corresponds to the distance 2nK. If this value isexceeded in the R - direction, the cycle is resumed in the ' - plane (2ndleaf according to Riemann).

By comparison of the corresponding periods, the constants for theJacobi function

-102-

K L)A r L Zin y (13.16b)

and k = are obtained. For the ratio* n

K WK' f '(k) (13.16c)

the modulus k2 for the transformation function aan be obtained from one ofthe refetences (XXIV) or (XXV). The comodulus k' which will play a role inlater studies is determined by the equation:

k2 + k' 2 = 1.

Since the coordination between K and k and, respectively, between K' and k'is unequivocally defined by integrals (XXIV) and the magnitudes of the K andK' values calculated from the given bearing conditions are not in agreement,in their absolute values, with the values of the basic progressions given inthe function tables (XXV), another transformation must be made in such a waythat the arguments and periods in the direction of the coordinates are en-larged or reduced. This conversion factor will be designated as adjustmentfactor V. Thus the following applies:

V us (13.16d)

where T is an index indicating the relationship to the Table. The argumentsfor this coordinate directions are found, in analogy to the periods, as:

r - direction . . . V ln r/rO WK'T = V ln r a/ri)

¶- direction . . V (KT = V .l/n)

By introduction of this last transformation into equation No. 13.15

S(13.16e)

The final adjusted function of representation results:

E~ ) b[sn Li~ j In (13.17)

The solution, or separation of equation No. 13.17 into real and imaginary por-tions, is done the following way: If the inner logarithmic function is resolved,one obtains:

S+ -(.

-103-

Then the "sinus amplitudinis" is solved according to the addition theoremdescribed in reference (XI), and the result is:

(13.18)

" IRsn'•kl•VL•kl÷c(-•Wdn('V•' )lsn (Vin F',k!)cn (Vtn•''• e.(v Ln ., I ) + t- Sy-(tv•,V 4) W.(v L r., W')

where cn and dn represent the cosinus (cosine) and delta amplitudinis func-tions, respectively. Since, however, the following applies:

S -Vy, k) SV d (VI, k)C Y1-Vy, k) C cV1(V~1 k) (13.18a)

and equation No. 13.18 is resolved according to the following principle

l. C~ i~)" I'vI0.'-t,:-i- . av..t3-• (13.18b)

the results are as follows:

VSnl, V, dvy, k) (V In r- ).i.'+ c '(Vy,, k) d ."CV .,k)5 ,Vf .e ,') Of"Vtf . ,k') (13.19)

cA"(VLn , i •)') + k', S'(v, k) SVI'(VL Lk')

.and

aCrc to cfCV•,k)dn(V?,k)SV(VLn-,Ik')Cvl(VLl Wk')- sn (V k) , (V Un.r .,k') (13.20)

Boundary conditions: The boundary values of the potential + as a func-tion of the bearing geometry can be easily found.

1) If the nozzle diameter or the largest diameter of the counterbore ofthe nozzle (cf. Chapter 12.1 C) d << ro, one obtains for small argumentsv?

sn V V? (13.20a)

cn V. dn Vy Z 1

)#V

-104-

and for r = r the result is:

sn (0) %"0, en (0) = dn (0) ''. 1.

From this one obtains, by introduction intoequation No. 13.19:

-r V-d (13.20b)

2) The following can be given as potential value of all circular potentia]lines (limiting potential lines)

__ _ -- (13.20c)whereas the potential for the stagnation point S situated between two adja-cent flow fields is ( r = ro, C = iL/n)

The coordination of pressure and potential was given already before by equa

tion No. 13.3.

Example of calculation.

In a bearing of the following given dimensions and operating values

flow medium: airI

n = number of nozzles which open directly into the clearance.

r a a 60,0 mm Strbmungredium Luftr. - 38,75 mm P, 9 8.104 kp/m'rL 25,0 mm pi 1.104 kp/m&d 0,7 mm q 1,85.1C'6 kp.s/m(FL a 93,4 cm h - 20.10`6 m (13n . . Anzahl der direkt in T = 293 OK

den Spalt miindenden R = 29,3 m/*KWiisen

the load capacity and the flow volume were determined at a varying num-ber of nozzles, n. Since the integrals No. 13.8 and No. 13.10 cannot besolved mathematically, only a numerical procedure could be applied, using ttpotential function found, No. 13.19. By subdivision of the half-sector ofthe carrying surface in the r- and I -direction into many small parts •rand , the load capacity is obtained as:

-105-

rn,. ifilnKT %=. I p.r &rAy (13.21)

and the amount of flow as:

2 V (I;L r) (13.22)a " ?/ RT (oP3.-,, 1P"2 L• oi AQ ra 0fA, "•

where t,•1, and 4AT; represent potentials dependent on at a dis-tance of Ara and Ari from the outer and inner boundary, respectively.

Results

n K K' 0c k'2 V KT KT0. 02.

4 0,785 0,875 0,390 0,610 2,25 1,770 1,960 -3,897 0,236

6 0,524 0,875 0,080 0,920 3,06 1,604 2,684 -3,550 0,63110 0,314 0,875 0,004 0,996 5,01 1,574 4,385 -3,010 1,58220 0,157 0,875 0,000 1,000 10,00 1,570 8,750 -2,405 5,697

•g/s .. g/S . gr/6n l C G0.G G 6a /GYk. a)& ki/An = 004 233 0,312 0,394 0,363 0,757 0,2075 o,1911

6 284 O, 380 0,625 0,576 1,202 0,3295 0,3035

10 348 0,465 0,942 0;870 1,812 0,4970 0,4590

20 422 0,564 1,482 1,365 2,847 0,7805 0,7190

00 486 0,650 1,899 1,899 3,798 1 1

(13.22a)On page 106, the isobars calculated for the data given are illustrated forn = 4, 6, and 20 nozzles at a magnification of 2 : 1.

Load capacity.

The results of the calculated example are plotted in the Diagrams 13and 15 as functions of the number of nozzles. The load capacity of the bear-ing was represented there with the aid of a dimension-less load factor

"KIC -- F .. - 0, (13.23)

where KI is the absolutI carrying capacity of the bearing; FL is the areaof the bearing .[.(r - r. ); P1 is the absolute pressure directly

a Lin back of the nozzles. The factor C is, of course, not applicable every-where but is again a function of the ratios p /p and r /r.. In order to1 2 a L"allow a comparison to a bearing with radial flow only (annular clearancefeed n = • ) with respect to load capacity, the load factor C On is

-- n

-106-

Calculated Isobars

200

10,

/ S..w4

-107-

calculated. From the equations No. 10.6 and 10.7 it can be deduced thatthe absolute load capacity for such a bearing is:

L - C (13.23a)K1 Iir y.LNc,(. +r.c )

By the equation No. 13.23 and consideration of equation No. 13.16, this istransformed into

Cko. + XL ra * (13.24)

In the comparison of individual bearings to each other, not the absolute num-ber of the feeding nozzles is decisive, butrather a proportionality factor obtainedfrom the distribution of the nozzles andwidth of the ring; this factor givesapproximately similar potential distri-butions for the individual bearing sec-tors. In order to be able to arrive at amore generally applicable result from thecalculated examples given above, a sectorratio T is introduced which can be deter-mined from the quotient

, L 2 7r (13.24a)- ~n(r.- rj

Using the equation No. 13.16, the following is obtained:2 7"Er 2 (.1_rý (13.25)

In Diagram 15, the ratio of the load capacity factors C/Cnn= in the calcu-lated example is plotted as a function of I . In order to answer the ques-tion how many nozzles should be arranged on the circumference 2 r0 - thisquestion will come up when a bearing is being designed - the fact that theload capacity increases steadily with an increasing number of nozzles, mustbe remembered. As can be seen in Diagram 13, however, the course of C showsfirst a steep rise with an increasing number of nozzles, but increases onlyslightly beyond a point which is established to a greater or lesser degree;Thus in that region, an enlargement of the number of nozzles does not bringabout a considerable increase in load capacity. In the calculated example(ra/ri = 2.4) this critical number of nozzles would be n = 20 - 24. The lossin the calculated load capacity compared to a bearing with annular clearancefeed would be about 11% (if it is based on the measured values, only about 5%).Corresponding to this number of nozzles, the following is obtained:t = 0.25 - 0.35, and, for the design of new bearings:

,3ý

-108-

18- 2s (13.26)

is the optimum number of nozzles.

Amount of flow.

The results of the calculations of the flow weights of bearingair are represented in the Diagrams 14 and 16. The outward and inward flow-ing portions are divided by the amount of throughput Gn =ao which is require(at a strictly radial flow (cf. equationsNo. 7.19 and 7.20). The fact is worthmentioning that, in spite of conformance awith equation No. 13.16, the values forGa and Gi are different in the presenceof individual nozzles. In order to makethis fact more clearly visible, a nozzle 5with the boundary flow line (circle2 ro• ) is drawn in a magnified scale.

Since the boundary of the nozzle i4 formed bypotential line 1 and a uniform majs flow perpendicular to *l prevails, alarger portion of the circumference of the nozzle is available for the flowoutwards.

Measuring results.

In the Figures 1 and 2, in which the experimental equipment is illus-trated, the course of the flow lines and the pressure distiibution in a bear-ing with four individual nozzles which open directly into the clearance,could be measured. The actual sites of the pressure taps where the measuremewere taken, and those reduced to a quarter sector, respectively, are shown orp.112. The picture of the flow lines could be taken by means of small amountof oil and water of condensation which entered the bearing, at the beginningof the measurements, on account of the absence of a satisfactory separator,together with the compressed air and left behind yellowish colored traces onthe bearing surfaces. One half of a bearing with the detected course of theflow lines is illustrated on the top of p.113. Since the radius of the feed-ing nozzles r is only slightly different from the conditions according toequation No. 13.1 6 , the line which separates the mass flows outwards fromthose inwards is, as expected, a circle. Likewise, the stagnation point S isclearly visible. The isobars measured for two different bearing gas pressure

PB are also plotted on p. 113 (middle and bottom). A comparison by super-imposition of the flow lines and isobars shows a very satisfactory orthogo-nalitv of the two groups of curves.

In Diagram 13, measuring points for the load capacity factor C areplotted as a function of n. The points at n = 4 and n= 30 were measured bymeans of the experimental equipment which is illustrated in Figures 1 and 2.

The point at n = 8 could be ascertained on the thrust bearing of the experi-mental compressor (cf. Fig. 12). This shows clearly that the measured valuesare always higher than the theoretical results. This deviation can be traced

-109-

back, in part, to the fact that, in the theoretical studies, the retardingportion w.dw of the flowing medium which occurs at high entrance velocitiesinto the clearance, was disregarded. The total energy which is introducedinto the bearing consists not only of the static pressure P, which wasmeasured and was the base for the comparisons of the calculations, but alsoof the velocity component which is transformed into pressure energy and fric-tion energy in the course of the clearance flow.

The area of the nozzle cross sections has no influence on the abso-lute load capacity of the bearing, unless the character of the flow is signi-ficantly changed, but it has an influence on the correlation between heightof the clearance and load capacity. If, for example, cylindrical feedingnozzles which open directly into the clearance are counterbored at the sideof the clearance (cf. Chapter 12.1 C), a shift of the load capacity charac-teristic into the region of larger clearance widths is obt.ined. The advan-tage of achieving a larger clearance and thus increased safety of operationat equal bearing load is counterbalanced by an increased volume of through-put. Measurements in a bearing with and without counterbored nozzle on theside of the clearance are shown in Chapter 15. Summarizing, it can be statedthat the number of individual nozzles has an influence on the absolute loadcapacity; and that, on the other hand, the effective area of the feedingnozzles has an influence on the correlation between KI and h.

14. Bearing characteristics.

In the present chapter, the characteristic magnitudes are representeddimension-less for various arrangements of the bearings and flow patterns bymeans of the continuity requirement between the amount of gas flowing intothe bearing through the nozzles and that escaping from the clearance.

The functions designated as bearing characteristics in the followingwere derived under these conditions: I) The change of kinetic energy ofthe flowing gas was disregarded; 2) The rotation of the disc was disregarded;3) Short cylindrical nozzles, throttling clearances of apertures as feedingelements were present.

The characteristics

f AC14.0a)R. Ps

represent there pressure dependent functions the values of which are deter-mined by a dimensionless number which characterizes the kind of bearingand nozzles. The bearing characteristics are very useful for the determina-tion of the total course of Kges or 5 over h. By this method proposed forrepresentation of the bearing, furthermore, not only the condition of thebearing at any time can be shown, but also the influence of change of thesize of the bearing, of the area of the feeding nozzles, of the kind andtemperature of the gas and of the height of the clearance.appear immediately.

4 ~" .. Diag'rum m 13

.1d

p l-0

.1 ~ ,,tea

G06 Cale',

I ~20 2

.7 .. ........ r

......... ... .....

raeaELred

0.,.1te

. ............ ..t .

-112-

Actual distribution of boreholes in which thepressure was measured

I

Boreholes reduced to one sector

ro040 mm

Sr i •25 mmD = nozzle....m.Number = Radius of the

borehole

-113-

Measurements

Flow Lines

i sobars

P1 -..10 ato

K~w 373 kp

-114-

14.1 Radial flow.

A) Laminar.

In order to have equilibrium in the bearing, the amount of gas whichenters through the feeding nozzles must be equal. to the amount which flowsout through the clearance. Thus, if the equation No. 12.2 is equated to thesum of the two equations No. 7.19 and 7.20, the following is obtained aftercontraction of the individual members to dimension-less terms:

(14.1)where the following equation was applied

S(14.1a)

For constant pressure ratios pBB/p2 and adiabatic exponents d variation of

Pl/P2 yields different values for the term on the right side of equatioi No.14.1; they must always be equal to the bearing characteristic A Thus thefollowing applies to the characteristic for the laminar radial clearance flow4 nwards and outwards:

A 1" hIP. (14.2)

It should be considered in this connection that the term of the square rootin equation Ni.. 14.1 for

has the following limiting value on account of the critical outflow from thenozzles (velocity of sound):

Diagrams 17 and 18 show the A values as a function of the pressure ratiosfor At = 1.4 (air) and at = 1.66 (helium). It can be seen that the influ-ence of 3;, is very slight.

The same characteristics apply, of course, also to bearings where theflow is only in one direction. Theý L values, modified accordingly, are,for the outward flow:

AL -rWP ~(14.3)12 jFavjYLgRT W

"-115-

and for inward flow, exclusively, in a bearing

"it P.? V R I (14.4)AL =

B) Turbulent.

In a similar way as in the laminar flow, the amount of gas which entersthrough the throttle elements (cf. equation No. 12.2) is equated to the amountflowing through the bearing (cf. equations No. 9.13 and 9.14). Thus the di-mension-less characteristics are obtained from:

- e. (14.5)14t

For the term

the same limitation applies as in the case of laminar flow. The function ofthe different pressure ratios on the right side of equation No. 14.5 variessomewhat from the conditions at laminar flow. For ? = 1.4 and 1.66, thecharactertics for several pressure ratios of the bearing gas, PB/P 2 , wererepresented in Diagrams 19 and 20. The following bearing characteristicsresult for the bearing with flow in both directions

=~ lr= '1 _ _ _ 1 (14.6)

for the bearing with outward tlow only

Ar u _____(14.7_

and for the bearing with inward flow only

_ -1Ar P~r (Y2 ',014.8)

)

_ _ _ _ _ _ _ _ _ _ _

-116-

14.2 Parallel flow.

A) Laminar

The amount of throughput through• a parallel clearance of constantheight and the length 1 is, when the change of the state of gas is isothermal:

Zq •RT L

SIf, therefore the equation No. 14.9 is equated to the equation No. 12.2, thedimensionless characteristics are after a transformation:

i• In•b IN- k,••-Fl (- o, 14.10)

£Zqt Fs, RT L r

The right side corresponds again to the pressure function which has been de-rived in radial flow; The Diagrams 17 and 18 apply here, too. Thus thebearing characteristic for parallel flow in one direction is

h AL k) PL M i'RTL (14.11)

and, if it occurs in both directions

L* bj L

(14.12) -- ,

B. Turbulent

The amount of throughput for the compressible, isothermal flow throughla parallel clearance is

(14.13)"•Y Ar, RT L.

i1

I

-117-

Thus, by equating to equation No. 12.2, the characteristics represented inDiagrams 19 and 20 can be applied in an analogous way. The bearing charac-teristic for the parallel flow in one direction only is, therefore:

Ar ý2/hl (14.14)6 V RTAyL

For the flow in two directions from a central channel, the characteristicis as follows:

AT b.__ (14.15)

UtAFb V R T L Fxrj i VTA:L L

14.3 Potential flow.

The bearing characteristics calculated before for the laminar clear-ance flow apply also to a bearing equipped with nozzles which open directlyinto the clearance where the flow pattern is given by a potential function(cf. Chapter 13).

A) Annular shape

If the amounts of throughput through n nozzles (cf. equation No.12.2) and through the bearing clearance (cf. equation No. 13.10) are equated,the consequence is:

0 (14916)-m':f fV(=-) -rLt d ; + '-J7)Jd(&

If equation No. 13.5 is also taken into considera-tion, the result for the characteristics is:

24 M To VO29RT (d L)

A Vt/. 4 , - (14.17)

*L .P

T poI,~~ 0) '

'Her I- I

00

I* SeL '

r- I

II I-~ -'1+

s0fl ~ 9lojq N')ua 14 ostso~

-118-

The bearing characteristic can also be written in abbreviated form:

Jkf3~ (14.18)At. an24 v FoV2.R T+

where q represents a value which depends on n, pl/p 2 and ra/ri. In anexperime,,t with n = 4, r /r. = 2.4, d = 0.7 and j = 0.78, for example,the following results were ottained:

P1/PL a 3, 8= 14,85 9,96 (14.18a)

B) Rectangular shape.

The laminar isothermal flowthrough a clearance with an arrange-ment according to the drawing at theright, requires an amount of through-put of

i•L~-P~ J• 4•dx (14.19)

12. RT ( 4 .)_

By equating this formula to equationNo. 12.2, the bearing characteristicresults as: bPi

0/

(14.20)

Its dependence on the pressure ratios is again shown inDiagrams 17 and 18.For the effective throttle area FD the following applies bcth to annular (A)and rectangular (B) bearing surfaces but depending on the design of the nozzle:

j Cir 4h (14.20a)

IN a

-J-1~

.. .. .. .. . .... .. - -- -

ia

In

.1-

410'

I I

...... . ----------

... .. .. .. .. ...

a)E-4-

.... ...... .

i -

1%I

p'a

-. . . . ... .- . . . . . . . . .! .. ... .. [ •. ...... .. . .0 .. • ..

oz ul i

t -

'I , ,

4 . .I

. .........

.. .. .. ...

.. .. .. .. . .I

........

0 I0

-----------Ie

!M* Ooiq~I.'~ ~ *

-124-

15. Results of the experiments.

All the experiments were performed in air at regular ambient tempe •ture. The required compressed air was tapped partly from the factory pip(system (7 atmospheres) and partly from a special valve testing station (4Lwhich was fed by a two-stage "Rotasko" compressor system. The air was pufied by a filter with layers of cotton, silica gel and activated carbon.the outlet of the filter, a sintered porous steel plate was inserted in oto prevent the penetration of solid materials into the bearing.

15.1 Heating.

In the following, it will be shown in an experimental bearing howaverage temperature of the fixed axial disc I changes from the initial vaas a function of time and what the stationary final temperature of the diwill be if the removal of heat is taken into consideration. The experimewere performed with the gas bearing testing apparatus which is shown, assection, on p. 137(cf. also Figs. 3 - 11). The double-acting thrust bear '6

loaded by means of a pressurized air cushion on the front side of the rigend of the shaft. On the left, carrying side of the thrust bearing, a clance h, which was large enough for operation was adjusted by the admissiothe bearing air. On the right, load-free side II of the bearing no air wfed. In the initial stage ( n = 0), the temperature of all parts of theing was equal to the temperature of the bearing gas at the inlet; it waskept constant during the experiment. At the start of the measurements, trotation of the shaft was very quickly adjusted to the desired r.p.m. valwhich remained constant during the test. On account of the heat of frictin the carrying bearing clearance, the parts of the bearing underwent a sheating.

Calculation model and assumptions!

Since the heat transfer in the

clearance (cf. Chapter 6) is very good, Bit is assumed, at any time, the tempera-tures of the gas at the outlet from thebearing and those of the parts I and IIIare equal. The heat which is generatedin the clearance h, causes heating ofthe discs I and III and, besides, com-pensates the different losses of heatto the. outside. Between the fixed discI and the spacing ring K, several cali- ..brated shim-discs B (of 3 mm thickness)were inserted i-i order to increase the htotal free axial space and to reduceheat conduction between disc I and thecasing. Since the ratio hii/h=I'!'3000/20 150, the amount of heatgenerated in the clearance h1i byrotation is smaller by the same factorand can thus be disregarded.

-125-7

The meaning of the symbols used in the following is:a + 6i (kg/sec) is the total amount of gas flowing through

the bearing;S(kcal/kg 0C) is the specific heat of the bearing gas;

(kg) is the weight of the discs I and III;"dp (kcal/kg °C) is the specific heat of the bearing discs;t (see) is the time;T, (0 C) is the temperature of the bearing gas at

the inlet;T2 = f(t) (0 C) is the temperature of the bearing gas at

the outlet = temperature of the discs I andIII;

TR = TI, (o C) is the temperature in the surroundings ofthe test equipment;

(o C) is the stationary final temperature of the2 discs I and III;

47 (kp sec/mr) is the dynamic viscosity of the gas;R (m/O C) is the gas constant;hI m) is the height of the clearance on the

loaded side of the bearing;(1/sec) is the angular velocity;

A (kcal/m kp) is the energy equivalent;r () is the radius of the bearing;

(kcal/sec °C) is the factor of the portion of heat removedby shaft and casing.

Heat balance

1) The heat generated in the clearance h, by rotation can be deter-mined from equation No. 6.8

dQaL d + 2- i r'dr ( 15.1)

(a, + : flow outwards i, -: flow inwards)

Integration of this equation using the limits rla - r2a and r 2 i - rli gives:(NB. The bearing has a continuous ring channel; cf. Fig. 8)

* - + A ± 2 ' [it I +rL rL.] (15.2)

2) The heat removed by heat conduction across shaft and casing, andby convection (ventilation, air currents) is almost proportional to T2 - T1;

Po

-126-

on the other hand, the gotion of heat removed by natural convection is pr'portional to (T - T ) 5/. Since the exponent of the second item does ncdeviate very mugh fr& 1 and, besides, its value is low, it can be assumec

. that the total heat removed v can be defined by the following term:

S," (T36- Ti) (15.2a)

The factor represents a constant whose value depends on the cooling c

ditions in the bearing.

3) The heat removal per second by the flow of the bearing as is:

1ý " •Cr (% - T4) •(15.2b)

4) A portion of the heat produced is accumulated in the bearing cI and III. dT, 1 .c

Thus the heat balance can be derived from:

- dtL

Written in another way, it is

A. dt (15.4)i b - Ga-where a and b have the following meaning

a and JD" , a'r + (15.4a)

Integration of equation No. 15.4 yields the dependence of the bearing temptureT 2 on time (aT - b) t +C (15.4b)

The integration constant C becomes, if .the limiting conditions are t = 0,T2 = T : I • n (a T., - b) I l.c-~ U OLrIA4 J (15.4c)

This gives, for the course of the temperature:

T - T, + -e, (1-.r_.5)

After a sufficiently long period of time C t--) 00 ), the following applie

-at (15.5.)OI- :

k

$i -127-

I and the final temperature is: GtUZ (15.6)T. 4C1 +

The linear rise of temperature at the beginning of the measurement can befound from equation No. 15.4 assuming t = 0, 4 = 0, T2 = T1.

(dTr\ _•(15.7)

The measurements reported in the following are io show the order of magnitudeof the heating and of the starting period in aerostatic thrust bearings. Inthe different series of experiments, only the r.p.m. were changed, while theother operating conditions, namely load capacity and amount of throughput re-mained constant. The dimensions and operating conditions of the experimentalbearing were as follows:

rl = 0,055 m G - 69975 kgr4L 0 0,087 m cr - 0911 kcal/kg 0C

r 4a = 0,089 m G = 1126.10-3 kg/secr=e 0,133 m c, - 0,24 kcal/kg 0Cs =0,7 mm R - 29,3 m/°C(Ringkanaltiefe s) Tj= To - 17,0 @Ch, 20.10-6 m W 1,90.10"6kp sec/m'.h, 5.10-3 m

Ringkanaltiefe = depth of the annular channel. (15.7a)

SDiagram 21 shows the measured temperature of dics I. during the pre-heating period as a functijn of the r.p.m. The results of the partly measuredand calculated values are: (tl= length of starting period)

u/min = r.p.m. - gerechnet = calculated - gemessen = measured.

n Qau, (dT 2/dtX.o T" J t" Anlaufzeitu/min gerechnet gerechnet gemessen gerechnet gemessen10 000 0,1149 0,898 41,2 0,04720 13012 300 0,1731 1,355 53,0 0,04745 140 (15.7b)15 000 0,2560 2,000 67,4 0,05050 135

kcal/sec "C/min 00 kcal/sC mrinThe measured temperature gradients at the beginning of the experimental seriesare in complete agreement with the calculated values. The factor u wascalculated from equation No. 15.6 by means of the measured final temperaturesTX. The measurements show that the starting period tx is almost the same at

different numbers of rotation. The equation No. 15.6 can be written inanother way:

JOT (Tay-T) G- T+'- T,) 1 (15.7c)

A B

-128-

in order to illustrate the portions of heat which are removed, on the onehand, by the bearing discs and, on the other hand, by the bearing gas itself

n U/min A % B % (15.7d)

10 000 o 1P". 93,64 6,3712 300 93,67 6,3315 000 93,06 5,95

Summary: The experiments show that heating of aerostatic bearings must betaken into consideration because the heat removed by the bearing gas, in cortrast to fast running .:ydrostatic oil bearings, is only a small portion ofthe amount of heat generated. The components of the bearing and of the adjacent casing must, therefore, be designed in such a way that satisfactory re-moval of heat is possible.(if necessary fins or a cooling system should beprovided). On account of the relatively small heat of friction, preheatingthe thermally stationary state takes a very long time.

The problem of heating of the bearing becomes, however, less importEat a higher pressure level of the gas surrounding the bearing (e.g. in encasmachines or those operating under higher pressure). The amount of heat intrduced by bearing friction (cf. equation No. 15.2) depends only on the viscoEof the gas when the dimensions of the bearing, the height of the clearance Ethe r.p.m. remain constant. But since 4 is independent of the pressure 1(the amount of heat introduced is the same at low and high ambient pressure.But the effect of the amount of bearing gas through 6 is quite different.According to equations No. 7.19 and 7.20, 6 increased, at constant height ofthe clearance, significantly with pressure p1 and p2 ' respectively, i.e. thEheat removed by the bearing gas

QG = 6 Cp AT (15.7e)

increases with the pressure level while Q 2 remains constant. From this i"zu 2can be concluded that the excess temperature of the bearing

T "- (15.7f)Gc,is substantially lower at a highes pressure level than at normal surroundinspressure.

15.2 Pressure patterns.

The pattern of the radial pressure was measured, with the shaft infixed position, in the thrust bearing of the testing setup (cf. Figs, 3 - 1)and the longitudinal section of the testing apparatus on p. 137). The fixe(bearing disc (cf. Fig. 8) was equipped with a circular annular groove intowhich, in order to render the gas feed more uniform, 20 cylindrical inletnozzles of 0.7 -m hole diameter opened. On account of the high absolut(pressures, 7 mercury high-pressure differential manometers, which are illus.trated in Fig. 5, were connected in series. Fig. 6 shows the connections ofthe instrument leads from the fixed bearing disc. The measuring pressure ti4 mm diameter which open into the carrying bearing clearance can be clearly

-129-

seen in Fig. 10. In order to avoid mutual interference of the pressure taps.their arrangement in the direction of the circumference was somewhat stag-gered. (cf. Fig. 8). In Diagram 22, the measuring results for differentinitial pressures p at constant surrounding pressure p, are plotted for aheight of the clearince h = 20 x lO- 6m. The fully draws curve representsthe theoretical course of pressure as it can be deduced from the approxima-tion equations No. 7.14 and No. 7.18.

15.3 Bearing characteristics.

By means of the large gas bearing testing apparatus (Figs. 3- 11),the bearing characteristics AL were measured for laminar air flow in theclearance as a function of pl/p 2 and plotted in Diagram 17. A preliminaryexperiment was performed in order to determine of the nozzle outflow coeffi-cient • . From the amount of flow in critical flow, through thenozzles, LAF could be obtained.' In order to adhere to the desired dimen-

Dsions of the clearance, several spacing foils of equal thickness were insertedinto the carrying bearing clearance at equal distances; besides, the wholeshaft or the movable bearing disc, respectively (cf. Fig. 7) was subjected toa strong pressure during the measurement by means of a heavy thrust which wasproduced by the application of a pressurized air cushion on the front side ofthe shaft (cf. the longitudianal section on p.137 , on the far right). Thepressure p1 could be determined very accurately - in a similar way as in themeasurement of the distribution of pressure - by means of the differentialmonometers in series. As can be seen in Diagram 17, the concordance betweencalculation and experiment is rather good. No experiments were performed inQ the turbulent region of the flow.

15.4 Amount of throughput.

A) Bearing with annular compensating space

In an experimental bearing with the following dimensions

rL = 0,055 m P= - 7,70"10-6 m1 (15.7g)r,,L = 0,087 m n - 20r,, = 0,089 m S - 0,70 mmrl - 0,133 m

where r is the radius of the bearing, FD is the area of the nozzle, n is thenumber of nozzles and s is the depth of the ann,,lar compensating channel, theamount of throughput was measured for laminar air flow at a temperature ofT = 2910 K, a dynamic viscosity 1J = 1.85 x 10-6 kp sec/m2 and an ambientpressure of P 2 = 1 x 104 kp/m 2 and comparedwith the values calculated according to equa- r-!\tions No. 7.19 and 7.20. Diagram 23 showsthe results of these experiments, i.e. the 6--amount of throughput as a function of theheight of the clearance. From the prelimi-nary experiment, the average value of thenozzle outflow coefficient &.- was found tobe 0.78. The twenty cylindrical nozzles dis-tributed on the circumference of the compensa-

10 tint channel have a total area of F = 7.70 x0-o m2. For the comparative calculation, the

-130-

£ connection between h and p1 was obtained by the bearing characteristics iDiagram 17 (p. 105). The result for the given values is:

A L = 2.685 x 1012 x h3.

Thus the calculation of G over h is most advantageously done the followinway: h is chosen, AL is calculated, the corresponding pressure ratio

PI/P 2 is determined from Diagram 17 and, finally, the amount of throughpuis calculated from equations No. 7.19 and 7.20.

The measurement of the volume of throughput was done by means ofDIN (=German Industry Standards) orifice of 6 or 9 mm, respectively, diamwhich was inserted in a measuring section. The pressure difference of thorifice was indicated by the high-pressure differential manometer which iillustrated in Fig. 6 (left front).

B) Bearing without compensating space.

In a later design, the annular channel of the bearing describedsub A) was filled with a hardening plastic, then the surface of the beari.was leveled again and the nozzles were bored again through the plasticlayer. Thus the nozzles opened directly intothe bearing clearance through the cylindricalborehole. As expected, the measurements 88

Diagram 24) has now a considerably flatter

t slope with increasing bearing clearance. . doThis fact can be traced back to the throttl-ing effect between boundary of the nozzleand bearing clearance. A comparison of thetwo measurements A) (Diagram 23) and B)Diagram 24) shows that the maximum amount ofthroughput is larger in the second case. Thereason for this is that, after filling the annular channel with plastic, inozzles were drilled for a second time and thus got a somewhat larger dian

15.5 Load capacity.

A) Bearing with annular compensating space.

In Diagram 25, the effective load capacity of the bearing describeChapter 15.4A is shown for the same gas states as a function of the heightthe clearance. The calculated load capacity, which is represented in saicgram as fully drawn line, is, as can be deduced from equations No. 10.6 ar10.7:

S~(15Kis. - F, 7r [r,,, -t- r,,c, Ck,+rt (cL- )-I r (r...~) l

where the area of the bearing is:

FL - jrA, .. il 461 cm" (15

B} , ,•

'" " i . .. .. .. .. . . ' - .. .. -1 3 1 -

O* I ; , , :,* + i .. . " I .... 4! . .. . ....... m-n" 2 1

* * ,. I; i.

* , OI, , I* : . + . ,, . .

* . S : '

*.. . . . . . . . . . " . . .. . .. . . . i

.9. " o ... ' .. . • -,

S. . .. .i . .. ... ', • ... . .. . . .. . . : +" ... . ,. 'SI0uj;.I

S! 41 + "S. . . .. . : .. .. ... ' ' " ... . .

• ":• +, •S... .. i . .. .. • +i.• i ... . .. • : : + :'I •

.. .* .. .• . . * . . . : . .! " :C :I

• . .: ..

* • . . . .•. . . .+ ... . ..n . l ,

• . ' .. : - - . . . . . i• • . .. . " . . ' " "

*6 4 1

• . : .•. : .:.:.. .... • .... . ... . .. . . .,

* +"+. . • I

4 4.

+ , t• , *. 6 . .+ . 4 .

I' -132-

I... ....................... .. .. ..

-7 7. . . . . .. .... .. ... .. .

i ~ -.calculated according

'4to equations 7.13 and

4- f ... .... ..71

I j... .......

*1 .... .......... .. .

... ... .. .. .. .

.. ... .

L ....

.. .. ...

71'""- j.

J.N2

.. ~~133- . .z

..... ... ....... . .(... .....

.. .. .. . .. .

b.. ..... .. ... .. ... .

94 ......

_____Calculated according:.to equation .0 \.No. 7.19an7.0

So 40 *0.0 O 6

.. *i 1 !vLd~I .......4...

4 1L .. ....

...- .~ .. . . . .-I .. .

... .. ..

.. . .. . .

.. ... ... . .. . .. . .

.1h

lo 12

-135-

"I Via . ara m

I, ! i [ t! i 2 , 2

ii. .. .... ... . .. ... i ... ....... ....:.. ........ •... ........ . -.. . ' • •. . r• • .. • .. . .

- - ... .m. .. 0

: 55 in yn t 0 mM

-. ----• Calculated according' .to equations

* No. 10.6 and 10..7.

* ' - - , --- ". -.--- ".. --I -4-- • • '-- .. '-''"" --

* - -- - -- *- .- -. *-- *- -. -..h " ..--....-.-.?.-. .. i .. i. ...

* -- - -. . -. -- - ----- .- . .- -.- -

_ . .. ... .... .

- • , - : ---4.--'-7

S. ..................

30 30 to O o9I

X rn

W.i

tr w

0.h (D -

0 CD •

m 0

-E

-" L5 -L -

-136- _________

~~~~- m__ _ _ _ _

r~~'r 1 13

.......... _ ... __.... . . . . . .

. .... .. .

hQ!M

10 r0 aO 6,0 i0 to 90

-138-

For the mathematical determination of K over h, one chooses advantageouslyagain the height of the clearance h, whireby the corresponding pressure ratioPl/P 2 for the given bearing gas pressure PB is obtained by means of the bear-ing characteristic from Diagram 17. The two load capacity factors Cka andCki can subsequently be determined by interpolation from the Diagrams 5 and 6(p. 60 and 62). It can be seen in Diagram 25 that the measured points ob-tained are in every case lower than the calculated values. This deviationcan probably be traced back to insufficient precision of the manometer usedfor measuring the pressure acting upon the free end of the shaft.

B) Bearing without compensating space.

The measuring results for the load capacity of the changed bearing(cf. Chapter 15.4B) are plotted in Diagram 26 as a function of the height ofthe clearance. It can be seen, in this connection that K s is substantiallysmaller than in the bearing with annular channel. The lola was again appliedby means of a pressurized air cushion on the front side of the free end of theshaft.

15.6 Nozzles with a counterbore at the side of the clearance.

By means of the testing device which is illustrated in Figs. 1 and 2,a comparative measurement was made in one and the same bearing with differentnozzle designs. In the first experiment, the four cylindrical nozzles openeddirectly into the bearing clearance, whereas, in the second experiment, theywere slightly chamfered on the side of the clearance. Diagram 27 shows theload capacity Kges and the amount of throughput G for both designs as a func-tion of the height of the clearance h. By the counterbore on the side of theclearance and the enlargement of the effective throttling area connected there-with ( d 3t h -+ d Yt /4) the load capacity characteristic was shifted intoa region of larger widths of the clearance, the flow characteristic, onthe other hand, into a region a smaller clearance widths. This design has theadvantage that a bearing can be operated, at the same load, with a largerclearance and thus guarantees a safer operation. This improved load effect is,however, counterbalanced by an increased flow of bearing gas.

It could be shown in this experimental device, and also in other ones,that the counterbore on the side of the ýlearance (enlargement o4 the clear-ance volume) of the magnitude D d does not cause an unstable

operation as far as self-induced vibrations are concerned.

15.7 Measurement of the axial shift in a machine.

The determination of the effective axial shift in a machine duringoperation can be performed in aerostatic axial bearings according to a verysimple method. If, firstly, it is assumed that the feeding pressure p forthe bearing gas remains constant and that the bearing discs do not perpormany rotatory motion, there develops, in the presence of an axial shift inthe bearing clearance, a pressure peak whose total value corresponds to theshift prevailing at that time over entire area of the bearing. Thus, t .e

-139-

greater the axial shift is, the greaterpressures must be present in the bearing -<1clearance. If a bore for pressuremeasurement is made in the fixed bear-ing disc at any place whatsoever (pre-ferably in a region with high pres-sure), exterior application of aknown pressure (either mechanically _ _ _

e.g. by a stretching device andspring balance, or pneumatically bya pressurized air cushion) allowsto measure the corresponding refer- Ience pressure PR and to plot it inthe form of a calibration curve as a f (Kqofunction of this axial shift KIt is immaterial, in this connection,whether the bearing disc has a con-tinuous annular channel for gas con-pensation on the side of the clearanceor only individual nozzles which open r) I¶e$o)directly into the clearance. Thistaethod can also be applied in double-acting bearings (cf.the testing deviceon p. 137) where the carrying side is •LLKs()influenced by the non-carrying side. 1%

The only precaution to be taken is

that the calibration cuiAve be drawnunder the same conditions (equal bear-ing gas pressure also on the non- rcarrying side of the bearing) as theywill occur later during operation.If the calibration curve is drawn fora bearing, the axial shift produced by the machine can now be determined, cot,-versely, by measurement of the reference pressure PR.

In practical cases, however,the presumptions described above arenot satisfied. The bearing gas pres-sure PB fluctuates always in a higheror lesser degree, while the componentsof the bearing perform a motion relativeto one another. As can be seen, however, PYin Chapters 8.1 and 8.2, the influence C - -

of the kinetic energy of the gas flowingin the clearance - this energy being sub- pitject to changes on account of pressurefluctuations - and the influence of therotation of the disc are very slight and can be disregarded, especially forsmall pressure levels (P 2 = I atmosphere). At increasing pressure PB andconstant axial shift only the volume of flow and the bearing clearance

j are enlarged, whereas the pressure peak and the reference pressure must remain

-140-

the same, corresponding to the shift. The influence of rotation becomes onlynoticeable at a high pressure level P2; it can, however, be expressed arith-metically and corrected in the calibration curve. By means of this refer-ence pressure measurement, an inclined position which might occur in therotating bearing disc can be detected (fluctuation of PRat slow rotation ofthe shaft) and can be corrected.

..... ? .. ... f t -

ID- - 4 i.I ~ - - - -

H, ~ ~ --- ~--p.1?.P)

4J4414 TEEi

S . , .1j

. IV!

C,.,

f Lý ULL

CA. ) I5 1 i . 0 1 00 5

-141-

Diagram 28 shows the calibration curve for axial shift measurementin the four-stage radial compressor "Zurich 52" which is illustrated inFig. 12. The borehole for reference pressure measurement is located betweentwo of the eight nozzles which open directly into the clearance. The pres-sure pB of the bearing air introduced fluctuated during the measurefdetibetween 7.2 and 7.8 at. The load was applied to the resting shaft by meansof a cable line and was measured with a spring balance. The pressure PR wasdetermined with a tube spring manometer. In Diagram 29, the axial shiftwhich had been determined with the aid of the calibration curve (Diagram 28)and the measured pressure PR' is plotted as it occurs in actual operation atdifferent r.p.m.

16. Self-excited vibrations.

A self-excited vibration is defined as a free vibration with negativedamping. Every vibration with negative damping is dynamically unstable, i.e.the amplitude become larger with time. The changing force which maintains thevibration is generated by the mutual movement of thebearing components themselves and is controlled bythem.

6Ae

In aerostatic bearings, self-induced vibra-tions occur especially in those cases where addi-tional spaces are provided in one of the bearingdiscs on the side of the clearance, in back of thethrottling elements. The installation of compensa- - -

ting channels or pockets serves mainly for theimprovement of the gas distribution in the clear-ance and affords thus a considerable increase ofthe load capacity compared to a bearing with punctiform gas feed. Thisinstability causes an undersirable very noisy vibration of the movable bear-ing disc and of the adjoining free masses (rotor).

The reason for the appearance of such a vibration can be traced backto the discontinuity between the amount of gas which flows into the compensa-ting space through the feeding elements and that which escapes from the bear-ing through the clearance, and also to the fact that the compensating volumecan be accumulated on account of the compressibility of the gas. From thisfluctuation of quantity, a corresponding pressure fluctuation results whichis decisive for the exciting force.

In the bibliography listed in p. 8, a linear differential equation ofthird degree is shown which relates to the height h of the clearance as a func-tion of time. This ibration equation was derived from the change of theamount of gas in th- bearing with time.

16.1 Equation of vibration.

In the following, the equation of vibration is established first in asimilar way as in paper (XVII), but for the case of a thrust bearing which con-sists of annular supporting s-irfaces and with consideration of the actual con-ditions relative to pressure distribution and amount of flow. In thisconnection, it is assumed that the vibration of the bearing interferes withthe characteristic parameters to a small extent only. Besides, the simplifiedequations for the clearance flow are used which were determined with the assump-tions: w.dw = 0, 0, T = const. In order to simplify the calculation, the

-142-

temperature TB of the entering gas is equated to the temperature of the carrying gas flowing in the bearing clearance. The symbols used have the followinmeaning:

PB bearing gas pressure in front of the feeding nozzles (absolute)

p static pressure in the bearing clearance (absolute)

Pl pressure in the compensating space (absolute)

P2 ambient pressure (absolute)

h bearing clearance

m vibrating mass

Gaus weight of gas escaping from the bearing through the clearance persecond

6ein weight of gas entering the bearing through the feeding elementsper second

GL total weight of gas in the bearing

TB temperature of the entering gas a temperature of the gas duringclearance flow

r radius of the bearing

FD effective throttling area of all feeding nozzles

D depth of the annular compensating space

s width of the annular compensating space

VA volume of the compensating space

sign for a small deviation from the equilibrium position

terms containing this crossbar denote the state in equilibriumposition

The following drawing shows the arrangement of the bearing and thecourse of pressure in the radial clearance as basis for the determination ofthe self-induced vibrationk The inclusion of the resonator chambers describbelow, which are connected to the compensation channel, in the vibration equation would be necessary for the exact study, but is too complicated for ananalytical treatment. The calculations are, therefore, limited initially tothe determination of the properties of a bearing (influences of the differentparameters on the stability, frequency of the bearing vibration) without anyprecautions for removing the vibrations.

-143-

In formulating the basic equations, it is assumed that the pressure atC the start of the clearance flow (i.e. at rla and r 1 i) is equal to the pressure

PL Pa.P

.vn

in the compensating space. Furthermore, the differentials are replaced byfinite differences on account of the small changes of the magnitudes p1' Gemn,daus and h during the vibration. Thus we obtain the following:

ALf M R, L~ Geij, t AGeir,(W

in. (16. 1)

h~L) = + -Gos•

The first basic equation is a consequence of the fact that the difference be-tween the amount of gas flowing in and out per unit of time must be equal tothe change of the weight of the gas in the whole bearing with time.

dtL0G' ÷ - •8uC] "(61

-144-

In the case of short cylindrical feeding nozzles or orifices, the amountentering, at constant pressure PB' depends only on pl(t) and is, accordingto equation No. 12.2:

SGe. 0)I fop a. +' )+.,), XF (16• , .2)

From this the following can be derived:

46 6eiv,+ V Z 7r4L4 (16.3)

and the time-independent constant is:

The amount of gas excaping from the bearing through the clearance depends,however, on the pressure p (t) and the height of the clearance h(t). Forisothermal laminar radial flow, Gaus can be determined by means of theequations No. 7.19 and 7.20:

7r-.G) , (16.5)"G"O G& + L- RTS .7,

Thus a small difference Gaus becomes:

(~G ~ Mb Ga(us

or

where the constants have the following meaning:

1T, Y" (16.9)

The amount of gas between Lhe twu bearing discs results from:

GL 27Trh [Spr.dr + p.r.dr + 2. 0 + ._)• j (

R Ts

-145-I If the equations No. 7.13 and 7.17 which apply to laminar isothermal clear-ance flow are introduced for the course of pressure, the integrals give thefollowing solutions according to Chapter 10.1:

rr,f rrd p, W~ 14. Pt.) t4 f) (16.10a)

Sp-r-dr - .- rýL p,-(i) p

Besides, the following applies: b§3• - r,1t (16. 10b)

Since the factors Cka and Cki which depend on the ratios of pressure and radiido not vary too much for small pressure fluctuations (cf. Diagrams 5 and 6) thevalues I and ý2 which can be obtained from the equilibrium data, wereassumed to Be independent of the fluctuations with time. The compilation ofthe constants gives:

2~ ~ L s Tr /ý L (16.10c)

where C is a load capacity factor referred to the whole bearing. Thus equa-tion No. 16.10 can be written in a simpler way:

G1. ~*. 1 +)+~~1 - ( i) + +~s ~ 16. 11)

where } has the following meaning:

' 1Irs (r VA (16.11a)R1 T -k-iRT%, RTB

According to equation No. 16.11, the amount of gas in the bearing dependsboth on the pressure and on the height of the clearance. The differentiationwith respect to the time t gives: (NB. In the following all terms markedwith a period sign on top mean differentiation with respect to time).

- ~ (G\(16.*12)

where the coefficients are:

e . .. - ' and - (16.12a)

If the individual equations derived, No. 16.3, No. 16.7 and No. 16.12, areintroduced into equation No. 16.1, it follows that:

( 4F••m -,,,÷- '•%+ + • t • (16.13).

0

-146-

For the calculation of the height of the clearance h as function of thetime t, another equation is required whereby the pressure in equation No.16.13 can be eliminated.

The equilibrium of forces between acceleration of the mass (free bear-ing components and shaft) and the spring action of the bearing can be ex-pressed, without outside damping or friction, by the following equation:

VY1h& - 21Jpw )ridr - 2 Jfarw r-dr

Considering the data given above, the following can be concluded:OO

M Av (.') - 2 (§ 1 + +•§ 3 ) (16.14a)

which can also be written in another way:

_ _ _ _ (16.15)

The desired vibration equation is obtained by iLtroduction of this equationinto equation No. 16.13

000 0 0

Ah -t a.ah + b.A-& +- c.ah - 0 . (16.16)

where a, b, and c have the following meaning:

e (16.16a)

o ...2Tr(fi1-ftgs)A C FL A

16.2 Influences on the stability.

The stability criterion for the vibration equation of third degree,given by Routh (cf. reference XXVI) is based on the condition of positivedamping. The bearing drawn diagrammatically on p. 143 is stable, as far asself-excited vibrations are concerned, if the following applies:

a~b > C (16.17)

A) Feeding nozzles.

In order to get a high value for "a", OC. must be as small as possi-,ble. But since 0. can only assume negative values between 0 and - 00the requirement stated above is satisfied if the tangent to the characteris-tic of the throttling elements has as steep a slope as possible in the

-147-

operating point. In order to amplify the comparison of 5hort cylindricalnozzles (characteristic D) and capillaries (characteristics K), which isgiven in reference (XVII), the venturi nozzle (characteristic V) has beenincluded. For the comparison, the effective throttling area FD and thepressure p1 in the point of interpretation towards the feeding nozzles ismaintained constant. (Equal Gmax and equal load capacity in the point ofinterpretation).

Characteristic of. 4\1

the clearance D

As can be concluded from the aboIe comparison, the capillary nozzlehas the tangent with the lowest slope in P. The venturi nozzle, which hasa strongly bulging characteristic on account of the diffuser adjacent to thelowest cross section, at high pressure ratios pl/pB, (cf. Diagram 11) has,in certain regions, a better stabilizing effect on the bearing taan the shortcylindrical nozzle.

B) Volume of the compensating space.

If all the individual terms are introduced into the Routh stabiliza-tion equation No. 16.17, the following applies:

(r--CK)• F __ 2t 1 (16.17a)

This is transformed, by introduction of the abbreviations and considerationof the volume between the bearing discs, VL

VL'= Phi(16;17b)

into

'ýVA -I (16.17c)

C W /3

07

.1

-148-

Thus the following result is obtained

CV. Vc,)-i3(Pi'LgP,, -( -1-/ 1)

(16.17d)The division of the fraction gives:

VAs..-1 +p.1. ____ (16. 16.)

It can be concluded from this equation that the stability increases with

decreasing ratio between compensating volume and bearing volume. In themost unfavorable case of bearing operation1 O• = 0, i.e. the mass flow throuIthe feeding nozzles does not change if the pressure p1 is varied. Thusequation No. 16.18, written in another way, is transformed into:

V^ T 2 (16.19)

+ W, Pr -- l ki )iý'_(( w,

Another conclusion to be reached is that also bearings without compensating

space can be unstable. When VA is chosen as zero, this can occur at apressure p1 ) V'T3-P 2 .

The introduction of pressures and bearing clearances as they occurpractically in static bearings into the stability equation No. 16.18 sug-gests that the compensating volume should be kept very small. In most case,

the depth s of the annular channel must not be larger than the clearance hitself. By such a limitation, the compensating space lose s foriginal

purpose, i.e. improvement of the load capacity by uniform distribution ofthe carrying gas entering through the nozzles.

With the testing equipment shown in Figs. 1 and 2, experiments re-lating to the influence of the shape of the compensating spaces were per-

formed, using a bearing with the following principal dimensions: D2a =120 am, D = 80 b, D2 i = 50 W n n, and which was equipped with four shortcylindrical nozzles at an angle sf 900 which were arranged in the fixedbearing disc. The pressure of the carrying gas in front of the bearing wasvaried between 2 and 20 atmospheres (absolute), whereas the limits of thetheight of the clearance h were 0 and 100 gm. Some of the cases

itself• ". B • such a limitation, the compensating....e l . o ". .ial

-149-

"investigated are compiled on p. 151 . The experiments showed that the in-stability of the bearing cannot be eliminated by the shape of the compensa-ting channel. At equal operating conditions, (Kges, p h), the only con-clusion that could be reached was that the volume of t~e compensating spacehas a great influence on the frequency of the vibration. The larger thevolume VA was designed, the lower was the inherent frequency of the bearingvibration.

C) Other parameters.

The influence of the pressure p1 upon stability cannot be deriveddirectly from equation No. 16.18. But if the simplified limiting case accord-ing to equation No. 16.19 is considered, it appears that, on account of theincrease of the load capacity factor C with decreasing pressure ratio

pI/P2, the influence of the pressure acts in two ways, both working in thesame direction. A greater stability is thus achieved by reducing the pres-sure ratio pl/p2. It was found as a result of these experiments that thevibration frequency increases with increasing pressure (at equal width ofthe clearance).

Summarizing, it can be stated that the stability is improved or in-creased, respectively, with increasing nozzle FD, viscosity I of the gas,temperature-TB and gas constant R, and with decreasing height of the clearance.

16.3 Frequency of vibration.

0 For reasons which will be described later, the calculation of thevibration frequency of the bearing and the components connected with it isrequired. From the vibration equation No. 16.16 and the formulation ofsolution

ah =e` (15.19a)

the frequency equation can be obtained:

X -- A + b A C c 0 • (16.19b)

The solutions ofthese cubic equations are:

UI= • V - -Y I (16.19c)

where "u" and "v" are

U T- I+161d0

/1

-150-

where, again, "q" and "p" mean:

• . ab C (16.19e)1- 6 2.

b Q

Thus the 6eneral solution of the vibration equation is:

Ah , +2e C,-)+Wt 3 eE(MU~v+)-L ~((4-v)]L (16.24

The cyclic frequency of the vibration of the bearing is desired from theimaginary part of the exponents of e (NB: "Hz" = Hertz or cycles)

2- (16.21)or

As could already be concluded from the experiments described before, thisgeneral statement can be made: The larger the pressure p the smallerthe compensation volume, the bearing clearance and the vibrating mass arethe greater becomes the frequency.

Example:

For the purpose of comparison, the frequency of the thrust beariniof the large testing device (ef. section on p. 137 and Figs. 3 - 11) arerecomputed.

Dimensions of the bearing: (NB. Disenzahl = number of nozzles)ra - 0,133 m L 461.10- mr.r 4• - .0,089 m Pb = 6,$2.10"'mi (16r4 0,087 m n = 20 (DMsenzahl)r M O,055 mh - 44.10" mb a 0,002 m U 41,3 k9 s8/m5 0,7.10"S m

Flow medium: airTa o 291 °KR = 29,3 m/K (16.21b)

S- 1,85.i0"6 kP s/m'

Calculated values:

- 0 = 19 676. 10"0 q - 3,042.102 (l* 17,t40 - 0,1006 p - 19483: 10O (1

2,242.10-" a 134 U 3 ,245. IO•3 3,8"1 10"6 b a570.108,775.;10.I - 8,175.10l- = ,172.10~ v

_ _ _ _

-151-

Section through one of thed. = 097 mm four feeding nozzles, In the

C d experiment without compensatingspace, stability of the bearingfor all operating conditions.

_ DO Chamfering of the nozzles ind 0,7 / order to obtain a better distri-d = 20 'f bution of the gas. The bearing

mm vibrates only at pressures pe d 954k• • above 20 atmospheres (abso-

CLIOA= 0 lute).

Non-chamfered nozzles withd = 0,7 MM flat central recess. Instabi-

S D = 59C mm lity of the bearing within a

L S =091 MM wide operating range.'r VA= 1,96 mme

d = 097 111M Hz = Hertz = cycles._,__ __b = 5v0 mm in these cases, the bearing has

___ .s = 0,03 mm an annular channel which con-VA 37,7 mm' nects all nozzles with each

- = 1. 0 mm other. Strong, self-inducedV// b 1,0 mm vibrations over almost the

S = 03 MM hole range of operationrlltYl V. 67 ammXN 1000 Hz

b = 1,0 mmS = 1,5 mmVA= 1,12.10" mmO-ist 600 Hz

b = 6,0 mms = 12 mmVA = o,1.10 mm39 80 Hz

r -Five annular channels of equalOd = 0,7 mm size, separated from each

a = 05 MM other. Strong vibrations in

VS=0 2 mm the whole range of operation.

I0 1200 Hz

Five annular channels, seperatedro S, CL a = 0,5 MM from each other; the one in theS4 = 0,2 rilm center has an enlarged volume.SL= 095 MM Strong vibrations in the wholeVA- 59,6 -,I range of experiments.

-152-

From the values given above, a calculated vibration frequency of )P = 108cycles (Hertz) is obtained, whereas the frequency neasured with the audio-frequency generator and cathode-ray oscillograph was 123 cycles (Hz).

16.4 Elimination of self-excited vibrations.

As a rule, the self-induced vibrations occuring in bearings are

excited spontaneously without outside influence. The amplitude increasesthere on account of the negative damping until the energy which is requiredfor a further growth of the amplitude is consumed by interior damping forces.(cf. the Fig. on p. 141). An elimination of the vibration can be achievedif the damping effect can be amplified in the bearing itself or if the vibra-ting mass can be influenced suitably by external influences.

A) Damping by resonators.

The bearing illustrated on p. 143 (without damping elements) is asystem capable of vibrating which, on account of self-excitation at a cer-tain frequency, performs undamped vibrations or even vibrations with nega-tive damping and increasing amplitude. It is irrelevant for the study ofthe attenuation of such a vibration, which follows, whether self-excitedor foreign-excited vibrations are involved. Aforeign excited mechanical vibration of a system(principal system M - C in the drawing at right) P P.sin tcan be attentuated, as it is known, also in caseof resonance ( () = W 0 I by the attachmentof a second spring c and a'second mass m, if theinherent frequency A0 of the system m - c is Iequal to the inherent frequency of the principal -Csystem M - C (cf., e.g. reference XXVI). In ananalogous way to the previously described dynamic mdamping of mechanical vibration , an acousticalelement i.e., the Helmholtz resonator, is appliedfor the elimination of vibrations in aerostatic bearings. Transferred tothe case of.the bearing, the mass of air in the neck H of the resonatorcorresponds to the mass m and the volume V of the resonator to the springconstant C. In order to damp the principal vibration, the inherent frequencyof the resonator must be equal, referringto the mechanical analogy, to the frequencyof the self-excited vibration in this case.It is advantageous to branch the resonatorsoff the compensating channel A in order toachieve as good a damping effect as possi- Able in the bearing. For the auxiliary sys- - P "tem, the following equation can be formu-lated if the interior damping forces are

disregarded:

Y + C.Y - i (16.22) / fwhere y is the deflection of the air mass miin the neck of the resonator, cR is thespring constant of the resonator, f is thecross section area of the neck of the

4.i -153-

resonator, and p1 Ct) the pressure in the compensating space which fluc-tuates with time. Since the foreign excitation pl(t) for the auxiliarysystem vibrates at the frequency Q •, the inherent frequency of theauxiliary system must be, in order to achieve optimum vibration damping:C.) (iL. From equation No.16.22 it can be deduced that the inherentR

frequency of the resonator is:

C (16.23)

The spring constanL of the resonator becomes: ,p

C dp (16.214) )Pdyrd

At a displacement 4y of the mass m in theneck, the amount GR in the resonator is yincreased by the magnitude

dG = f.r. dy .- V dr (16.25)

If an adiabatic change of state of the gas is assumed in the resonator, wherethe following applies: . - (16.26)

Pa combination of equation No. 16.25 and No. 16.26 gives:

G f0 P (16.27)

If the mass in the neck of the resonator is

m =f P (16.27a)9 R T_

the inherent frequency of the resonator will be:

0 If.Q (16.28)

where as = V--.gRT Iis the sound velocity in the gas, and

1* = 1 + •/4L = the effective length of the neck of the resonator,which is always higher than the actual length, since te gas must also beaccelerated in the regions of th mouth; d(f = d2 -40.' is the diameter ofthe neck duct.

Results of the experiments

1) The original design of the bearing discs of the testing apparatusillustrated in Figs. 1 and 2 (D2 a = 120 mm, D2. = 50 mm, with an annularcompensating channel b = 3 mm, s 1 mm) did not allow any measurementsof load capacity and amount of throughput on account of strong self-excitedvibrations with a frequency of 200 to 600 cycles. By the installation of0

-154-

of 4 equal, symmetrically arranged resonators in one of the bearing discs

(stationary disc in F. 2) with the following dimensions: d 1 mm,1 = 2 mm, VR = 13.2 cm , tuned to an inherent frequency of R = 227cycles (Hertz), all vibrations in all desired ranges of operation couldbe completely eliminated. The sketch on theright shows diagrammatically the bearing discsused with feeding nozzles and installedresonator chambers. D-240 •

2) According to the design of the largegas bearing testing apparatus (cf. Figs. 311 and the section on p.. 137) the thrust r 10 _3bearing of the dimensions D2a = 266 mm,D = 176 mm, D2 i = 110 mm, was equipped with r r -

an annular compensation channel of the dimen-sions Dla = 178 mm, Dli = .174 mm,. b = 2 mm, 'Ksos = 0.7 mm, and several differently tunedresonators. In this arrangement, two resona-tors in pairs with equal dimensior and inher-ent frequencies were in diametrically opposite positions. On p.158 , theworkshop drawing of the fixed axial disc is illustrated without cover. Th,installed resonators have the following dimensions and calculated inherentfrequencies: (NB. Anzahl = number; HZ = cycles (Hertz))

Anzahl D L VR- DL d f e e.Im mm ema mm mm• mm mm Hz

6 56 1,58 .1 0,785 3,3 4,08 54714 56 8,62 1,5 1,77 3,3 4,5 336

2 14 56 8,62 0,8 0,504 3,0 3,93 1922 14 51 7,85 0,8 0,504 8,3 8,93 1331 26 2x5 5491 00 0:,504 8,3 8,93. 5114 (60) (105) 351 1,5 1,77 8,3 9,5 36,2

(16.

x) with resonator attached outside (cf. Fig. 10) and enlarged neck borehol,compared to the drawing on p. 158. The velocity of sound was uniformly as.tO be as = 312 m/sec.

The shaft which was, likewise, carried by aerostatic radial bearin.could follow, without friction, the vibrations induced by the thrust beari;since the clutch elements for rotating operation were omitted. The thrustwas produced by means of a pressurized air cushion acting upon the front sof the free end of the shaft. The experiments were performed with air atinlet temperature T of about 2930K. In consideration of the annular compition channel, as described above, and the presence of all the resonators,thrust bearing proved to be rather unstable. (The channel can ae clearly

*= in Fig. 8). Diagram 30 shows the individual stable and unstable regions a;

.155-

* fjnction of the pressure ratio p1 /p_ and the height of the bearing clearanceh. The axial vibration amplitudes af the 410 kp shaft were recorded in a fewcases only and amounted to 4 to 10 )1 m at the high frequencies and up to 50&m at the low frequencies. As can be seen in Diagram 30, the bearing has

two stable and one unstable main regions. It is interesting to note that theunstable region of operation includes two special bands. The first bandappears at frequencies between 27 and 47 cycles. The bearing is stable inthis region. The second place is, strictly speaking, only a separating lineand is characterized by the fact that, upon passing this line, the vibrationof the bearing of 52 cycles fades out almost completely and then suddenlyincreases to 63 cycles. It was not feasible to adjust to a vibration of afrequency between these two values. If the inherent frequencies of the in-stalled resonators are considered (cf. p. 154), an explanation can be givenfor the behavior just described. The two last-named resonators with-V,R = 51and 36.2 cycles attentuate the vibrations of the bearing at the respectiveplaces more or less satisfactorily. Though, damping is limited here to a verynarrow frequency band.

For the next experiment, the neck ducts of all resonators were pluggedup in such a way that only the ring channel was responsible for the vibrationbehavior of the bearing. Some measuring results with this arrangement areincluded in Diagram 31, using the same way of representation. The inherentfrequency of the system was considerably higher in this case and was in therange between '- = 108 and 210 cycles. A comparison of the two series ofmeasurements shows (cf. Diagrams 30 and 31) that the vibration frequency isconsiderably lowered by the installation of a great number of differentlytuned resonators and the lower boundary of stability was further reduced,i.e. shifted into a region of smaller pressure ratios and clearance heights.In this respect, operation without resonators proved to be more advantageous.An arrangement with several resonators of different inherent frequency rendersthe conditions poorer, since the resonators which are not in resonance haveno damping action and contribute only to the enlargement of the compensationvolum VA and thus to the reduction of the vibration frequencies. It mayalso be assumed that the exciting forces are enhanced by the presence of notaccurately tuned resonators and that this overcompensates, in a way, thefavorable effect of the resonators.

When, after these experiments and results, those two resonators inthe bearing in which the inherent frequency was closest to that of the shaft(the resonators with V' R = 135 cycles, cf. p. 154) were reopened and thusbecame operating, no vibrations occurred any longer in the practical workingrange of the bearing.

Conclusion.

The preceeding experiment shows that it is not useful to install, atthe outset, a great number of resonators of different inherent frequency inthe bearing discs. Instead, it is advisable to determine, either by experi-ment or calculation, the exact inherent frequency of the system and to designand install the resonators in accordance with these results, There is, ofcourse, also the possibility to select such a resonator construction where at

Sleast one of the important magnitudes (e.g. VR) can be tuned to the operatingconditions. It could be seen in all experiments performed that high frequency

.1M

-156-

vibrations could be damped much more easily than very low frequency vibra-tions. It is, therefore, advantageous to design, first of all, the volumeof the compensatLng space as small as feasible (cf. also p. 81). Anotherreason for tending towards high vibration frequencies is the fact thatlarge resonator chambers are required for low inherent frequencies, whichinvolves difficulties in the construction of the components of the bearing.It is also shown in Experiment 1) that the elimination of vibrations athigher frequencies is not limited to the frequency for which the resonatorwas designe,& bui. extends also to adjacent frequencies. It should also bepointed out that the apertures of the resonators (cross section of the neckduct) must be sufficiently large. If the opening is too small, the influencof the viscosity of the gas is felt; this increases the acoustical resistancof the resonator, decreases the inherent frequency and lowers the attenuatineffect and the transmission of force, respectively (cf. also reference XXVII

B) Damping by interference.

The idea of generating several vibrations of different frequenciesin the same bearing in such a way that the vibrations are superimposed andcanceled, was followed up by an experiment with the components of the smalltesting apparatus (Figs. 1 and 2). A disc with four feeding nozzles wasused for this purpose, each nozzle opening into a separate compensation spac(cf. the drawing below).

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-161-

The compensation spaces were first designed in pairs of differentAV size, then all four were constructed at different depths s and arrangej.in

various combinations. The experiments showed that no superimposition ofthe vibrations, as desired, did occur and that the bearing remained unstablein all cases. The frequency of the vibrations was always that of the largestcompensation space; thus the bearing vibrated at the lowest frequency. Thepredominance of the lowest frequency does not allow any influence by higherfrequencies where the energy of vibration is lower. The generation of vibra-tions '• equal frequency, but with a phase shift of 1800, would be desirablebut cannot be realized in one and the same bearing.

C) Damping by friction.

The self-excited vibrations generated by the annular compensationchannel of the axial bearing in the large testing apparatus (Figs. 3 - 11,or section, p. 137), with simultaneous admission. of compressed air intoboth radial bearings - allowing completely free motion of the shaft - dis-appeared immediately after the gas feed was cut off to one of the two radialbearings. These experiments were performed while, of course, the shaft didnot rotate. Thus the vibration was completely attenuated by friction in theradial bearing as soon as the shaft got into contact with the bearing seat.

When the shaft was entirely free (i.e. pressurized gas was suppliedalso to the radial bearings), the self-excited vibration of the thrust bear-ing could be eliminated also by application of a medium heavy lead hammer tothe front surface of the shaft. Though, the separation between the frictionO damping effect proper and the dynamic damping action is not feasible in anaccurate way.

In another case, connection of the swinging shaft to the fixed driv-ing aggregate was already sufficient to create a satisfactory attenuationand thus stability of the system. It must be mentioned, however, that theconnection was not rigid but allowed, for reasons of measuring technique,shifts in axial directions; thus only the friction in the connecting sleevewas decisive for the attenuation.

16.5 Vibrations of double-acting thrust bearings.

In some machines, the thrust bearing must take up equal or differentforces in both directions. The most simple method consists in using in suchcases a bilateral symmetrical bearing (cf. left drawing on the next page)where the total axial free space of the shaft hI + hli is determined by theoperating conditions of the machine. As soon as, however, the bearing gasis admitted on both sides of the rotating axial disc, it depends on theabsolute axial free space to what extent the two sides of the bearing influ-ence each other in the particular positions as far as load capacity isconcerned.

01 ___

-162-

ci b

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. . . . ... -K ' -K

The Drawings A and B above illustrate diagrammatically the bearing charac-teristics Kges = f(h) at very small (=a) and very large (=b) free axialspace. With the small axial free space, there is an appreciable reductionof Kges on account of the counter-thrust. With a large total free space,however, the counteracting force affects the bearing side I only with theportion KII = FL.P 2 . In order to maintain the load capacity of the bilatebearing high, it is advantageous to select the total free space in such athat the points 01 and 02 (cf. Drawing b) just coincide. Without axialthrust of the machine, the widths of the clearances hI and hI are equal cdifferent in their final adjustment depending on the parameters PBI' pB...FDT and FDII. In the case of the characteristic a, 0 is the point of equibrium, whereas in Drawing b the position of the shaft of the machine betwe01 and 02 is not defined.

Diagram 32 shows the measuring results of the axial thrust ratiosthe bearing in a four-stage radial compressor which is illustrated on p.in section and in Fig. 12 in top view. The section of the thrust bearingillustrated on p. 167 shows the relative sizes of the components as wellas the direct opening of the 8 nozzles into the bearing clearance. Therotor of the machine was, likewise, carried by aerostatic radial bearings.In urder to avoid damping from outside, the connection to the gear wasremoved for the vibration experiment. With bilateral gas feed and without

-163-

* axial thrust, strong axial vibrations of the whole rotor around to O-posi-5 tion occurred inspite of the absence of a compensation space in back of

the nozzles. Vibrations were even found when the free space was 'nlargedj to such an extent that, as shown in Drawing b, there was no longer any

mutual influence of the two sides of the bearing. The vibrations did aotdisappear when one or the other feeding pressure pB was varied, i.e. at ahorizontal shift of the point 0 in the Drawing a. Diagram 32 illustratesclearly how the rigidity of the bearing increases at the site hl = baat PBI = PBII with increasing pressure ratio pB/P2. This fact became alsoevident by an increase of the vibration frequency, the vibration amplitudebecoming smaller with increasing rigidity. It may be conclude,' from theconsiderations which follow that it is not a regular vibrating spring - masssystem with the "spring constant" (rigidity of the bearing) c. = LKPes/i hand the mass of the r'otor m1_'` 16 which is the reason for this instability,-at that a self-excited vibration is involved which is caused by the feedingsystem and the reversal of the thrust:

1) Although the bearing was stable at unilateral gas feed and awide range of loads and widths of the clearance, the vibration is excitedspontaneously without any external influence at bilateral gas admissionfor Kges = 0.

2) The vibration does not fade out any more, i.e. the interiorattenuations which are due to the air cushion between the bearing discs arecompensated by the self-excitation.

3) A rough calculation of the vibrating frequency without con-siderat 4 - of damping and self-excitation, from the rigidity values inDiagram - lives, eg. for PBI = p .1 6 x 104 kp/m 2 a spring value of

CL = 2.46 x 106 kp/m, from which tRelfollowing formula can be derived:(NB. Hz = Hertz = cycles)

S(16.28b)

In contrast to this, a frequency of 36 cycles was measured. This discre-pancy proves that we are not facing here a simple vibrating spring - masssystem.

Elimination of vibrations in double-acting bearings.

A) In order to remove the instability which appears at Kges = 0, it provedto be effective to apply a relatively low axial thrust into one directionin such a way that 0'. (cf. Drawings a and b on p. 162) becomes the new equili-brium point. This could be done pneumatically, in a very simple way, in thecase of the bearing of the four-stage radial compressor which is shown as ase-tion on p. 167 , by throttling the exhaust air of the left thrust bearingdisc and the adjoining radial bearing slightly by means of a stopcock. Inthis way, an increased pressure was generated in the collecting chamber, andthis pressure exerted a thrust on the front side of the radial bearingjournal pin.

0

-164-

B) An increase of the total free space h, 4 h,, caused initially a reduc-tion of the vibration frequency. The reason for this pnenomenon is thatthe stroke between 01 and 02 (cf. Drawing b on p. 162) which must bepassed by the shaft as an inactive distance, was increased. But if a verylarge total free bearing space is selected, stability can be achieved.

C) As has already been pointed out in Chapter 16.4 C, only a small exter-nal damping of friction is necessary in order to eliminate the instabilityof the bearing. In the case of the radial compressor illustrated on p. 164all that was required was a connection of the shaft to the gear by means oa Jaw clutch coupling.

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-168-

Literature Cited.

I SCHLICHTING, H. Boundary layer theory.Published by Braun, Karlsruhe, 1951.

II SO0, S.L. Laminar Flow over an EnclosedRotating Disk.Transactions of the ASME. Febr.1958

III GRIGULL, U. Fundamental laws of heat transfer.Published by Springer, 1961.

IV LICHT, L. and FULLER, D.D. A Preliminary Inves-tigation of an Air-LubricatedHydrostatic Thrust Bearing.ASME Paper No 54-Lub-18

V PIGOTT, J.D. and MACKS, E.F. Air Bearing Studiesat Normal and Elevated Tempera-tures.Lewis Flight Propulsion Labora-tory, National Advisory Committeefor Aeronautics, Cleveland, Ohio

VI OSTiRT.I% J.P. and HUGHES, W.F.The Affect of Lubricant Inertiain Hydrostatic Th,-ust-BearingLubrication.Wear, Vol.1 No 6, June 1958

VII D ,UKER9 A. et WOJTECH, H.;Radial flow of a viscous liquid

between two discs which are veryclose to each other. Theory ofthe air bearing.Revue g~n6rale de l'HydrauliqueNo 65 Sept.- Oct. 1951

VIII COMOLET, R. Flow of a liquid between twoparallel planes. Contributionto the study of air abutments.Scientific and technical publi-cations of the French AirDepartment, No. 334, 1957.

IX SHIRSS, G.L. The Viscid Flow of Air in a NarrowSlot.Aeronautical Research Council Cur-rent Paper No 13 (12329) 1950

X LAUB, J.H. Hydrostatic Gas Bearings.

Transactions of the ASME June 1960* Journal of Basic Engineering

XI JAHNKE und EMDE Function tables. Third edition.

XI MEYM ZUr CAPELLEN Integral tables.:Published by Springer, 1950

-169-

XiiT NAUN.NN, Degree of efficiency of diffusers..- Research report No. 1705, 1942.

Aerodynamic Institute of the TechnicalUniversity, Aachen,(West Germany)

XIV LAUB, J.H. Elastic Orifices for Gas Bearings.Transactions of the ASME Dec. 1960Journal of Ba-sic Engineering

XV POTET, P. Filters of sintering metal.Technische Rundschau Kr. 12, 1960

XVI HUGHESW.F. and OSTERLEJ.F.Heat Transfer Effects in HydrostaticThrust Bearing Lubrication.Transactions of the AS1-• No 6 Aug.1957

XVII LICHTL., FULLER,D.D. and STERNLICHT,,B.Self-Exited Vibrations of an Air-Lubricated Thrust Bearing.Transactions of the ASME. Febr. 1958

•XVIII ROUDEBUSH, W.H. An Analysis of the Effect of SeveralParameters on the Stability of anAir-Lubricated Hydrostatic ThrustBearing.NACA Technical NJote 4U95

XIX RICHARDSONki.h. Static and Dynamic Characteristicsof Compensated Gas bearings.Transactions of the ASME. Oct. 1958

XX LICHTL. Axial, Relative Motion of a CircularStep Bearing.Transactions of the ASPE. June 1959Journal of Basic Engineering

XXI LIGHTL. Air-Hammer Instability in Pressurized-Journal Gas Bearings.ASME Paper No. 60 - WA-lO

XXII LICHTL. and ELRODH.A Study of the Stability of ExternallyPressurized Gas Bearings.Transactions of the ASRIL. June 1960Journal of Applied Mechanics

XXIII GOTTWALDF. und VIEWEGR.Calculations and model experimetnts

in water- and air bearings.Zeitschrift fUr Angewandte Physik2.Bd. Heft 11, 1950

XXIV BETZ, A. tonforma! mapping.Springer Verlag Berlin, 1948

XXV IMLNL-TH014SONL.M. The elliptical functions of Jacobi.

Published by Springer, Berlin, 1931.

?-l 70-

XXVI DTN HARTOGI,JP, Mlechanioal Vibrations,Mc. Graw-Hill] Publishins Company ,Inc.Now York III. TBdition 194.7

-171-

Lp

0 Fig. 1. Total view of the small test Lng apparatus for aerostatic thrust bear-ings. The experiments were performnd in non-rotating discs. Loading deviceand measurement of the clearance are in the center of the picture. On theright: Bearing gas compressor; on the left: Instruments for measuring ofthe pressure distribution in the bearing clearance.

Cl

Fig. 2. Different bearing discs and accessories. Left front: Disc with

:pressure compensation space (annular channel). Center: Discs with 4 and 20,

respectively, feeding nozzles. Left back: Stationary disc in back-view witt

the lid removed and 4 resonator chambers.

i1 -172-

Fig. 3. Large testing apparatus for aerostatic and aerodynamicbearings.

Fig. 4. Individual components of the testing apparatus.Front: Shaft with thrust bearing disc (weight approx. 400 kp).Center: Casing of the testing apparatus.Left: Movable and fixed radial bearing.Right: Fixed radial bearing and all axial discs.

-173-

Fig. 5. View of the side of the* thrust bearing and the driving

elements. Left back: High pres-sure Hg manometer for measuringthe pressure distribution in thebearing clearance.

"SamI Fig. 6. View of the siof the thrust bearing.Left back: Valves foradmission of compressedair. Left front: Highpressure differencemanometer for measuringthe amount of throughpuFront center: connecticof the 12 measuring

pressure taps for detez

nation of the pressuredistribution.

Fig. 7. Thrust bearing discwrth collet and nut on theshaft. In the back, fixed

bearing disc with compressed

air duct and feeding elements.

0

-174-

Fig. 9. Stationary bearing disc (bac'side) with the lid removed. The bore-

Fig. 8. Stationary bearing disc with holes of the nozzles, the measuring ccannular channel. Twenty feedifg nozzles nections and 10 resonators, equal inand 12 measuring boreholes on different pairs, are clearly visible.radii.

Fig. 10. Stationary bearing dis"with mounted lid. Left: Connections of the pressure distributimeasurement. Right: Resonator,arranged outside the disc for increasing the volum...

Fig. 11. Stationary and rotatingbearing disc after mutual contact at13,000 r. p. m. On account of thermaldeformation, the damage occurred onlyon the interior portion of the bearingsurface.

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