hydrodynamic instability 1 ?lain journal bearings · 2014. 4. 24. · hydrodynamic instability 1...

HYDRODYNAMIC INSTABILITY 1 ?LAIN

JOURNAL BEARINGS

by

NICHOLAS Jon HUGGINS

A thesis submitted for the degree of Doctor of Philosophy

in the University of London

and for the Diploma of Imperial College

NOVEMBER 1962

4-

Department of Mechanical Engineering,

Imperial College, London, S.W.7.

2

ABSTRACT

An entirely new theoretical concept of hydrodynamic instability, oil

whirl, is presented. Starting from Reynolds' Equation and employing the

short bearing approximation due to Ocvirk the equations of motion of a

journal on an oil film are deduced. Only plain journal bearings and rigid

shafts are considered but an out-of-balance force is included. Instead of

linearization of the system, as in every previous investigation, all the

non-linear terms are retained. The complex equations are solved on an

analogue computer and produce some notable results. Non-linear modes of

vibration occur which have the characteristic frequency of,whirl, namely

half that of rotation. This is an important discovery and shows the

inadequacy of linear stability theory. Also the frequency changing

properties of an oil film are demonstrated and some of the apparent

contradictions of previous experimental work explained.

An attempt is made to establish the non-linear theory in practice and

to investigate the effects of imperfections in the geometry of the shaft or

bearing. it is evident from earlier work that misalignment, ovality, etc.

have an appreciable influence on the occurrence of whirl. A large clearance

bearing based on Dr. Cameron's idea is used for it is pai.ticularly suitable

to these studies. .However the thick oil film of the model approach is

found to produce undesirable distortions of the oil film forces. The main

cause is the high oil inertia, an effect which is normally negligible.

Even so it is found that its influence can be beneficial in preventing whirl.

A CYNO1LEDGEMITTS

The author would like to thank Professor Hugh Ford, D.Sc(Eng.),

Ph.D., r.I.Mech.., M.Inst.C.L:., Professor of AiOied Mechanics in

the Department of Mechanical engineering, Imperial College, for his

useful and critical supervision of the research, and the many people

who have contributed in numerous ways.

Thanks are also due to the Central Electricity Generating Board

for their generous support.

3

CONTENTS

Chapter 1.

Abstract

Acknowledgements

Contents

Introduction

Page No.

2

3

4

1.1 Definitions 8

1.2 The Problem B

1.3 Basic Bearing Theory 10

1.4 History of the Theory 16

1.5 Previous Exprimental Work 23

Chapter 2. The Non-linear Concept

2.1 Some general remarks 28

2.2 Non-linear Indications 30

2.3 Basis for the Non-linear Theory 31

2.4 Derivation of the Non-linear !quations of 33

Motion

2.5 'excursions of the Variables 45

2.6 Analysis of the Equations 46

Chapter 5. The Analogue Simulation of the Non-linear

Equations of Motion

3.1 Introductory Remarks 48

3.2 Programming the Comiuter 49

3.3 Checking the Analooue and its accuracy 54

4

CalITTS (contd.)

:o.

3.4 Computing rethods

3.4.1 Initial Conflitions 55

3.4.? rethods for the free and forced 56

solutions

3.4.3 M2thod for Variable Speed of Excitation

57

Chapter 4. Analogue Solutions of the '.questions of notion

4.1 The form of the results 59

4.2 Analogue Results

4.2.1 For Equilibrium 'ccentricity katio of 0.1 60

4.2.2 If ft II 0.2 61

4.2.3 It It 0.3 61

4.2.4 II ft It

0.4 74

4.2.5 I, fl II If 0.5 74

4.2.6 ,, fl ft 0.6 75

4.2.7 II It It If 0.7 75

4.2.8 fl If II 0.8 76

4.2.9 Amplitude Contour and Fr2quency raps 76

4.2.10 Solutions where the Journal and Excitation 79

speeds are different

4.3 Discussion of the Analogue Results 82

4.4 Application of the Theory 88

5

CONTENTS (contd.)

Page No.

Chapter 5. Experimental Apparatus

5.1 Introduction 90

5.2 Design Criteria 91

5.3 Practical Requirements 93 5.4 The Final Design 95 5.5 Vibration Tests on the Bearing Supports 100 5.6 Experimental Procedures

5.6.1 General Remarks 100

5.6.2 'S1 Parameter versus Eccentricity Ratio 102

5.6.3 Friction versus the 'S' Parameter 103

5.6.4 The Onset of Whirl, Frequency ratio, 105

and Cessation of Whirl

Chapter 6. Experimental Results

6.1 Data 107

6.2 Notation 107

6.3 The Onset of Turbulence 108

6.4 Description of Results

6.4.1 Journal Centre Locus 113

6.4.2 'S' Parameter versus Eccentricity Ratio 114

6.4.3 Friction Characteristics 114

6.4.4 The Onset of !Thirl

115

6.4.5 The Cessation of Whirl

116

6.4.6. Frequency Ratio 116

COTITETiTS ( contd

PaEe No.

6.5 Gil inertia and other Considerations 130

6.6

Discussion of Experimental Results 143

6.7

Application of the Results 157

Chapter 7.

7.1 General Conclusions 155

7.2 Suik.octions for 1-Urther 156

References 157

7

CHAPTER 1 - INTRODUCTION

1.1 Definitions

It is generally recognised that there are three types of vibration

of a plain journal bearing. Firstly there is half speed whirl, or oil

whirl, which is characterized by its frequency being approximately half

that of rotation: it is a property of the journal bearing alone. There

is the response of the journal bearing to an out-of-balance force; the

frequency is usually assumed to be exactly equal to that of the force.

Thirdly a vibration caused by the interaction of the shaft and the oil

film and generally occurs at rotational speeds greater than twice the first

shaft critical. This phenomenon is called resonant or oil whip and has a

frequency approximately equal to that of the first shaft critical.

1.2 The Problem

None of the vibrations of a plain journal bearing are fully understood

and it is still impossible to predict their occurrence with certainty.

Both whirl and whip can be of large amplitude and constitute a considerable

danger to the machinery. The first problem is oil whirl and the consequent

possibility of hydrodynamic instability since it avoids the added

complexities of a flexible shaft. Although resonant whip will not be

specifically discussed it will be necessary to make reference to it as

there are some points of interest common to both subjects.

Oil whirl was first observed by Newkirk and Taylor in 1925 (29) while

studying the critical speeds of shafts, but since then very little

experimental work has been performed. What has been done often appears

contradictory for the researchers rarely discuss their results in terms of

the conventional non-dimensional parameters. Thus Pinkus (35) discussing

the 'desirable conditions for the prevention of oil whip' finds that a high

viscosity oil is required whereas Newkirk and Lewis (30) conclude that a

low viscosity oil is better. Another cause of difference may be that little

or no attention has been paid to such effects as misalignment of the bearings,

though it is apparent from several papers that they play an important role.

Newkirk and Taylor in their initial studies found that deliberate

misalignment of the bearings prevented the vibrations occurring. Other

influences may be the position and pressure of the ,oil supply, ovality

of the bearing surfades etc. but in no case has a thorough investigation

been made although it is clear the effects are not negligible.

A complete experimental investigation was therefore proposed with

special reference to the imperfections which may well occur in practice.

In the first instance it was necessary to design an apparatus where these

influences were negligibae in order to form the basis of comparison. The

design was to be based on Dr. A. Cameron's large clearance idea where

radial clearance of up to 0.25 ins. on a shaft radius of about 3.5 ins.

are used. Originally this model approach was employed to reduce the speed

at which whirl commenced, thereby reducing the power input and heating

effects and was found to be satisfactory. However though advantageous

in these ways, it was considered even more beneficial in the present case

as it would in effect reduce the comparative sizes of the geometrical

imperfections in relation to the thickness of the oil film.

The theories for predicting the onset of whirl are inflexible and

cannot readily be adapted to account for any but the ideal journal bearing.

The primary cause is the difficulty of integrating the basic Reynolds

Equation with suitable boundary conditions. The theories are all linear in

character and deduce that under certain conditions of operation instability

will result. There is some evidence from the previous experimental work

that whirl is a non-linear phenomenon and it was decided to investigate

the possibility with the object of ascertaining whether or not self-excited

or subharmonic solutions were theoretically admissible. Out-of-balance

forces were to he included but peculiarities in geometry still could not be

taken into account. It was realized from the start that such an analysis

would be mainly qualitative because of the assumptions necessary, yet it

was considered that it would give useful information on the nature of oil

whirl.

1.3 Basic Bearing Theory

Before discussing the historical background of the subject it will be

useful to first consider the basic theory.. The foundation of hydrodynamic

lubrication is Reynolds Equation which was originally deduced by

Irofessor Osborne Reynolds and published through the Royal Society in

1884 (38). The equation rests on the following assumptions:-

(i) The flow in the fluid film is laminar;

(ii) The thickness of the fluid film is small compared to its

length and bn:adth;

(iii) The fluid inertia forces are small compared to the viscous

forces;

(iv) Compared to the velocity 7radients and-r), all others are

negligible - see fig. 1. for the notation;

(v) There. is no pressure variation through the thickness of the

oil film, that is in the y-direction;

I0

-Nom, Li

J 0 OR.Ni A.1-.

SCA.A.1N1

ti, v-

yr- Jr.. , Li.

ft

V

PAGE II.

co .0 Rap I NAA.-ria SYSTEM F-OR. 'THE. 01 1-- F-1 I-.1\11 .

k .. Fii-M THICKNESS U, V- J OUR-NIA-I- .5‘.3R-F Ac- E.

1/al-00 rr I as

PIG. I .

12,

(vi) There is no slip between the fluid and the surface of the

bearing or journal;

(vii) The lubricant is Newtonian, the shear stress being

proportional to the rate of shear;

(viii) No external forces act on the film, such as gravitation.

That the flow is laminar is a necessary starting point and implies that

the Reynolds Number must be low. Assumption (ii) allows the curvature of

the film to be ignored and thus the rotational velocities to be replaced

by translational ones. It can be shown that the fluid inertia is negligible

if the first two assumptions are satisfied but it is necessary to include

(iii) in the first instance.

In a normal journal bearing the flow is laminar. A 12 ins. diameter

turbine bearing running at 3000 r.p.m. has a Reynolds' Number of about

150 well below the value for transition to turbulence of about 900. The

clearance ratio, that is the radial clearance divided by the radius, will

be about 0.0015. Assumptions (i) - (iii) are therefore reasonable and

indeed valid.

Assumption (iv) follows from (ii) and that the dominant velocities

are in the x- and, to a lesser. extent, z-directions. Assumption (v) is

also consequent on (11) and is usual in connection with thin films as is

(viii). No slip between the oil and the bearing surface is necessary in

order to define some of the boundary conditions of the film. A Newtonian

fluid is a reasonable assumption since oils are mostly used and in journal

bearings they are not very highly stressed. In addition it is usual to

assume the fluid has constant density and viscosity. Density is unlikely

to vary much but viscosity is greatly dependent on temperature and the

assumption cannot readily he acccted,although it is necessary in order to

simplify the development of the theory.

Reynolds' Equation is, with notation as indicated in fig.1A-

( 41.3 ) T I \7 6x I+ Tzk = GL-) TiL 6 2)x_ rlv

where p is the pressure at the point (x,z) and

r2 is the viscosity of the lubricant.

The equation (1) is exceedingly difficult to integrate to obtain

the pressure distribution and load carrying capacity of a bearing.

Simplifying approximations can be made by considering an infinitely long

bearing (sometimes called the Sommerfeld case) or a very short bearing

(Ocvirk's solution) in which either`;19,cz is zero orIbecomes negligible

respectively. Numerical methods have also been used to provide solutions

to the complete equation but whichever approach is employed it is necessary

to insert suitable boundary conditions for the oil film and those in the

circumferential (x) direction present a special problem.

In much of the early work on bearings it was assumed that the oil

could withstand large negative pressures, for example Sommerfeld (42), but

this is unrealistic. They arise if the boundary conditions of the film

are assumed to be zero pressure (gauge) at the maximum film thickness and

that the pressure profile is periodic with respect to the angular

co-ordinate of the bearing surface, i.e.

(e) (e 21) ; 13 ° e=o

In practice where there is a possibility of negative pressures developing

cavitation occurs and the film is broken. Floberg (10) found by experiment

that if air were dissolved under pressure in oil a slight drop in pressure

(17

14

was sufficient to release some of it. He concluded that cavitation which

is probably initiated by air coming out of solution would take place when

the film pressure dropped just below the ambient pressure - generally

atmospheric. Smith and Fuller (41) found experimentally that there were

slight sub-ambient pressures existing in the film immediately before it

ruptured - see fig.2. Such conditions are difficult to handle theoretically

so normally it is assumed that cavitation begins when the pressure falls to

zero. This alone is not sufficient to establish the boundary of the film,

for continuity must be satisfied demanding that the pressure derivative

with respect to the distance along the bearing surface should be zero as

well.

Various pressure profiles are illustrated in fig.2 and have been drawn

so that the maximum pressure is the same in each case in order that the

shapes may be more easily compared. Sommerfeld's actual solution (42)

is shown in the bottom diagram and is compared to the theory into which

the more precise boundary conditions have been inserted. The difference is

marked. For a bearing of length/diameter ratio (L/D) of unity the

discrepency is reduced though here the numerical solution of Walther and

Sassenfeld (49) is contrasted to the half-Sominerfeld solution, half because

the negative pressures are ignored. Comparison to Smith and Fuller's

results (41) shows that the theory is still lacking accuracy. However

the short bearing approximation due to Ocvirk (31),illustrated in the top

diagram of fig.2 where again negative pressures are neglected, demonstrates

that the simplified boundary conditions have a much smaller effect with

this geometry.

Po-GE IS com PA.R. I CPP4 OF PRESS t-Igt_ 1=4:Z.OF-11-5

Cce..a.r-s-rAtic.try Li/-T10 ---.-- 0- ES

Lif) 0

Co0 t1

FIG. 24

The equilibrium of the forces of the oil film is shown in fig.3.

The resultant couple caused by the forces on the bearing and journal

surfaces being out of line is balanced by the difference in the friction

moments.

\Nr.e. slts4 -t- Rb = .Fi

W s the load on the journal

e = the absolute eccentricity

0 m the attitude angle

R - the radius of the journal or bearing (since the clearance

between them is very small the radii may be considered- eqaal)

Fb = friction moment on bearing

Fj mg friction moment on journal

There are also friction forces arising from the assymmetry of the pressure

profile but these are very small when compared to the load, being of the

order of the clearance ratio.

1.4 History of the Theory

Considerable difficulties were experienced in integrating Reynolds'

Equation even for stable running of the journal, as has been explained

in the previous section, and the development of oil whirl theory is closely

related. The early investigators employing the unrealistic full Sommerfeld

conditions found that the journal position was always unstable. Various

methods of deducing the restoring forces on the disturbed journal have

evolved since then usually culminating in criteria for stability but it is

Ian

Po...G.= I T

FIG. 3.

rg-I Ca. it

only in recent times that the complete form of these forces has been

realized:

Fc„ = -t- b b .

All the terms are significant and influence stability. As some authors

have neglected certain terms their theories lack generality.

Harrison (16), Robertson (39),and Swift (45), were the first people

to notice the instability of the journal position. They all treated the

infinitely long bearing and included the negative pressures. Harrison and

Swift's work was in connection with dynamically loaded bearings and both

found that under a constant load the shaft position was unstable. Harrison

made an algebraic slip in his calculations for the frequency of the

movement. Swift corrected it and found that the frequency ratio (vibration/

shaft frequency) varied between 0.4 and 0.5 depending on the eccentricity.

Robertson improved on their methods and included the journal inertia and

viscous drag terms though he found the latter of second order. He again

found the journal position unstable. However inclusion of an elastic shaft

modifies his argument and he concluded, from a qualitative discussion of

the force diagram, that large amplitude vibrations could be substained only

when the shaft speed was twice the first shaft critical.

In 1926 Hummel (19a) published a theory of whirl which although it

appeared seven years before Robertson's was in many ways superior. The

displacement coefficients for the restoring forces were calculated from data

obtained under stable running conditions but the velocity terms were over-

looked. Even so some of the difficulties of integrating Reyholds' Equation

8

had been overcome and the unrealistic Sommerfeld conditions abandoned.

Hummel found that stability could exist in the range of eccentricity

ratios greater than 0.7 (Eccentricity ratio is the ratio of the distance

between the bearing and shaft centres to the radial clearance. Its

maximum value is unity and in the present notation is equal toe/(:).

Hagg (12) pointed out that in a bearing with a complete film whirl

must occur at half the shaft speed by considerations of the continuity of

flow. His argument is as follows. Assume that there is no side leakage

of the oil from the bearing and that the path of the journal centre is a

circular locus around the bearing centre, as shown in fig.4. The radial

clearance is '0, that is the difference between the radii of the bearing

and journal. If a linear velocity gradient through the film is also

assumed then, considering a unit length of the bearing, the oil flow

through the gap AB (fig.4) - ,c2.4 a)/2

Likewise the flow through CD is

P- - e-)/2 The whole shaft has a translational velcity of (L e) producing a void under

it at the rate of 2Runt.For continuity of flow

— S2-R(c.- e)/2, - 2, P. = 0 and hence 1.A.:7- =

The analysis, though simple, indicates how fundamental the half shaft speed

vibration is to a journal bearing.

Hagg, in the same paper, produced another theory using an approach

similar to Hummel's but inserted the direct velocity terms. These were

based on formulae relating to the tilting-pad journal bearing but, although

realizing that some lack of accuracy would result, he declared that the

19

2.0

order of magnitude would be sufficient. A stability criterion was deduced.

He carried on to show by a qualitative argument that instability would

come about if the shaft speed exceeded twice the first shaft critcal.

Hagg concluded that both of his criteria must be satisfied for smooth

running. Together with Warner (14) using similar equations (the origin of

the velocity terms is not given) Hagg employed an electric analogue device

to establish a stability criterion, this time including the effects of a

flexible shaft. Although their analysis was linear they suggested that

whirl might be a subharmonic vibration. However it is thought unlikely

because, it is remarked, unbalance generally inhibits whirl.

In the same year Poritsky (37) using Harrison's equations deduced

complete instability for a vertical shaft where the eccentricity is small.

Realizing the falseness of the theory he introduced an unknown radial force

to effect some compensation. Poritsky reached the same conclusion as Hagg

for stability with regard to oil whip. He continued with an examination of

the roots of the characteristic equation for when the shaft speed was three

times the first shaft critical and found an unstable root corresponding to

the shaft critcal - essentially in agreement with experiment. Furthermore

he remarked that stable operation in these regions was unlikely even if the

non-linearities of the oil forces were to be considered. Boeker and

SternlictiM2), although disapproving of the imaginary radial force,

demonstrated that such a force could in fact exist in bearings which,

although not plain, still possessed a high degree of symmetry.

Pestel (34) in 1954 deduced all the coefficients for the equations

of motion by progressive analogies commencing with two flat plates. He

assumed that the shaft rotation would not influence the velocity terms

at

(contrary to the findings of later analysed) and found that the cross-

velocity terms were negligible. The coefficients were given in graphical

form calculated from Needs (28) data for a 1200 partial bearing under stable

operating conditions. Cameron (3) neglected the velocity terms altogether

in his treatment of the rigid shaft system. Essentially it was an

extension of Hummel's work and was also based on results obtained under

stable running conditions. In the eccentricity ratio range below 0.7,

which Hummel had rejected as unstable, Cameron found that there was a

resonance accompained by an amplification factor. This factor, it was

argued, would be removed when the damping was introduced and so stability

could exist. Instability, however, would occur when the resonant frequency

was equal to half the shaft rotational frequency.

In 1956 Korovchinskii (23) published a very long treatise on journal

bearing instability. By a purely analytic approach of an extremely complex

nature he deduced the equations of motion for a journal disturbed from its

equilibrium position in a finite bearing. The equations were linearized

and stability criteria found for various length/diameter ratios. The

boundary conditions for the oil film were (apparently) taken as zero

pressure and pressure derivative in the circumferential direction.

. It is difficult to assess the accuracy of Korovchinskii's method

because of the mathematical assumptions introduced. There is one point

on which his theory differs from others and from experimental work. It ie

that he found shorter bearings are less stable than longer ones and,although

the variation is small,it is in the opposite direction to that anticipated.

However this paper is certainly a landmark in the history of the subject

being the first complete analysis based solely on Reynolds' Equation. It

is unfortunate that it has been overlooked in the Western literature.

22

Prederiksen (11) starting with Harrison's equations made an attempt

to account for the non-linearitics of the oil forces. He used an averaging

technicre though it is clear he lacked understanding of the oil film.

Halton (15) returned to the fully flooded bearing with a 360° unruptured

film. He showed that if the whirl locus was assumed to be elliptic the

frequency of vibration was equal to the natural frequency of the 'free'

cylinder's suspension - which could be interpreted as the elastic shaft.

The analysis is in line with observations though based on doubtful

assumptions. Orbeck (32) produced a relationship between amplitude and

frequency df whirl for a vertical shaft. It is similar to Robertson's

analysis.

Three years after Korovchinskii's work was published,another complete

theory appeared. Hori (19) treated the infinitely long journal bearing

but assumed the oil film ruptures at the minimum clearance. As explained

earlier this is not a good approximation. Starting from Reynolds'

Equation he found the linearized restoring forces and deduced a stability

criterion: both rigid and flexible shafts are considered. As is common

to this subject the shaft is simplified assuming a central massive disc on

a massless, perfectly elastic shaft. Hori argued that instability would

occur when both the shaft speed was above twice the shaft critical and also

the stability criterion was violated - subaely different to Hagg's opinion

(12). Holmes (18) produced a very similar theory based on the short

bearing approximation but without the inclusion of shaft flexibility.

Sternlicht, Poritsky, and Arwas (43) published a complete analysis which

was later reprinted in the book of Pinkus and Sternlicht (36). It is the

best one produced so far. Reynolds' Equation was integrated numerically

2,3

for finite bearings for various journal centre velocities. The boundary

of the oil film was defined by zero pressure and pressure derivative.

Coefficients for the linear equations of motion were deduced assuming

that the journal centre had no velocity and was located in the equilibrium

position. They produced a stability criterion which inextricably involves

the shaft flexibility. However one result which is independent of shaft

stiffness, is the frequency ratio at the onset of vibrations. (Frequency

ratio equals the vibration divided by the rotation speed). This ratio is

far removed from the usual half varying between 0.37 and 0.22 in the

eccentricity ratio range of 0.2 to 0.8. It is possible that this analysis

indicates that the assumption of linearity is unsatisfactory when better

approximations to the oil forces are obtained.

Kestens (22), reverting to the long and short bearing approximations,

introduced a refinement by allowing viscosity to vary. He assumed it was

proportional to the film thickness. Drawing heavily from the work of

Tipei(48) he concludes that for stable running the eccentricity must be

greater than 0.4. Capriz (6) has reached some interesting conclusions as

regards oil whip though his equations for the oil forces are valid for low

eccentircity ratios only. An analysis by Morrison (27) for the modifi-

cation of the natural frequency of a shaft supported on an oil film provides

a useful discussion of the oil forces.

In all the theories it has been assumed that the journal and bearing

are of impeccable geometry. Also it is assumed that whirl is purely a

linear phenomenon. Both require further investigation.

1.5 Previous Experimental Work

Since very little work has been done on oil whirl resonant whip will

also be discussed in this section as it may help to elucidate the problem.

a+

The first investigation of bearing vibration was performed by Newkirk

and Taylor in 1925 (29). Initially using a horizontal shaft in two bearings

they found that the shaft whipped violently at all speeds above 2,300 rp.m.,

the speed of the vibration remaining between 1205 and 1280 r.p.m. which was

approximately equal to the shaft natrual speed of 1210 r.p.m. Although

the bearings were suspected of being the cause other possibilities were

eliminated. The first positive evidence was produced when it was found

that the vibration stopped when the oil supply was reduced and returned when

the supply was restored. Different shafts and bearings behaved similarly

but deliberate misalignment of the bearings prevented whip. Nel,kirk and

Taylor built another apparatus this time having a vertical shaft, a ball

bearing at the top and the test journal at the bottom. Whip did not start

until approximately a speed of three times the shaft critical was reached

though a large resonance peak occurred at twice the shaft critical. Only

with a large clearance bearing (0.008 ins. on la in.dia.) and at speeds above

250 r.p.m. a half speed vibration of the journal in the bearing was

observed, the shaft not flexing. Increase of the frictional restraint on

the shaft was found to decrease the frequency of vibration. An explanation •

of the phenomena was formulated as follows. The resonance at the speed of

twice the shaft critcal is due to the natural half speed vibration of the

journal. Failure to pull through and become stable again may be because the

increased frictional resistance and side leakage of oil reduces the

journal's natural frequency thus still acting as excitement to the shaft.

They presumed that in the case of the vertical shaft the damping of the

upper ball race was a major influence. When a little extra restraint was

applied to the horizontal shaft system by means of a carbon ring similar

behaviour was observed.

2.5

Hagg and Warner (14) used a single disc turbine to study whip using

different shafts and bearings. They found that at the onset of whip the

frequency ratio was about 0.43. Their results compare reasonably with

their predictions in form but not in magnitude. They discuss the

possibility that the onset of whip is related to some resonance of the

system, often that of the shaft but sometimes that of the bearings or

something else. They conclude that every part of the machine has some

influence and that friction could be decisive. Hagg and Sankey (13)

attempted to produce in a generalized form experimental results for the

spring and velocity terms of the equations of motion. Unfortunately they

overlooked the possibility of cross velocity terms so that the results are

only applicable to bearings of similar geometry. Boeker and Sternlicht (2)

using a rigid vertical shaft found that with a plain bearing whirl

persisted from the lowest speed attainable.

Newkirk published some more results on resonant whip together with

Lewis in 1956 (30). The apparatus was similar to that used before having

a very flexible shaft. They reported that slight misalignment of the

bearings gave completely different results but gave no details of degree.

Another investigation of resonant whip was performed by Pinkus in

1956 (35). He found that whip commenced when the shaft speed was

approximately twice the shaft natural speed and had a frequency ratio of

about one half (i.e. vibration/rotation frequency). As the shaft speed was

increased the vibration continued,remaining at the same frequency except

under certain peculiar circumstances. At these times the speed of vibration

jumped to be synchronous with that of rotation. It is likely that

resonances of the test stand were influencing the observations. Pinkus

also tested other types of journal bearing, such as lemon shaped and

three lobed, and found that they had better characteristics with regard

to vibration. He also found that oil starvation reduced the tendency to

whip.

At the 1957 Conference on Lubrication and Wear, two papers were read

on journal bearing instability. Cole (s) obtained results which he himself

regarded as exploratory, and tentatively concluded that whirl occurs at

any speed where the eccentricity ratio is less than 0.2. He associated

whirl with the unruptured film condition which existed in this range rather

than any other parameters. Cameron and Solomon (5) described some

preliminary work done on the very large clearance bearing with clearances

of the order of I ins. on 3;1 in. radius. They found that the frequency

ratio was always about 0.49 and that it was pos'ible to drive through the

whirl region into stable running conditions beyond. It is concluded that

whirl is a resonance. However it is probable that turbulence was changing

the conditions in the oil film, thereby producing forces sufficient to

restore smooth running.

In all the experiments it is assumed that the bearings and shaft are

perfectly cylindrical, though, from the work of Finkus it is evident that

departure from the circular does cuppres vibration. Finkus, and Newkirk

and Taylor both found that oil starvation was beneficial to the maintenance

of smooth running; Newkirk and Lewis indicated the importance of alignment;

but none of these effects have been investigated. The bearing supports

are assumed to be rigid and excitation from adjacent machinery arriving

through the foundations ignored. There is no thorough investigation of oil

whirl or whip where every possible effect has been examined.

26

Comparison of the results which do exist is difficult because of a

lack of information and also very few investigators have urged the non-

dimensional parameters appropriate to bearing work. Me meaning of

curves of amplitude versus speed are lost since the eccentricity is

continuously varying. Also it is unlikely that the resonant whip phenomenon

will be understood until oil whirl is conquered because of the added

complexities.

17

CHAPTER 2 - TRE NON-LINEAR CONCEPT

2.1 Some General Remarks

Up to the present all theories of hydrodynamic instability have been

based on the assumption that it is purely a linear phenomenon. Little

thought has been given to this point though occasionally it is remarked that

only small oscillations are of interest from the practical standpoint, there-

fore linear theory is adequate. The argument is seeminfAy confirmed for the

theories indicate that at the boundary of instability the frequency ratio will

be a little less than a half as found experimentally.

Linearization however denies the possibility of certain types of

vibration. A velocity term which is negative for small displacement but

positive for large ones produces self sustained vibrations. The most famous

case is the Van der Pol equation (see, for instance, Stoker (d0). Any

initial disturbance will produce a vibration which tends towards one periodic

solution called the limit cycle. With complex forms of coefficient for the

velocity term several limit cycles can exist and then the initial conditions

will determine the final solution. Hon-linearity in the displacement forces

can lead to periodic solutions at frequencies which are submultiples of the

exating force - subharmonics. Stoker (44) puts forward a simple explanation.

Since a non-linear free oscillation contains higher harmonics of its main

frequency it is to be expected that a force at one of these harmonics is

capable of causing a large amplitude vibration at the basic frequency.

Ludeke (24) has built some simple models which,demonstrate sub-harmonic

response.

Another non-linear characteristic is the 'jump' phenomena.' In fig.5A

a typical response curve is shown for a hard spring with a little damping.

As the frequency of excitation is increased the response follows the line

OAB, until at B a sudden jump in amplitude occurs down to point D. On

1/3 Z.S e1I-4 15.4a4"00.46C 6 i12.

/

/ /

i

/

PAGE 29 JUMP PI-IE.NOM E.NIA.

Li 0 3 I. :i a. 1 d

FIG. 5e.

Li 0 3 I.

30

decreasing the frequency the response follows the curve DCAO, a jump

occurring between C and A. The result is a hysteresis effect. Jumps can

also occur between different solutions of the same equation as illustrated

in fig. 5B - after Ludeke (25).

Linear systems can also produce subharmonics if the spring force is

an explicit function of time and the damping is small. The vibrations occur

when the applied force has a frequency approximately twice that of the

system. It is Npry interesting that Reynolds Equation for a gas bearing

contains a term in which on integration might lead to the above

conditions. With out of balance as an exciting force and natural frequency

of approximately half shaft speed a gas bearing will then be particularly

susceptible to whirl.

2.2 Non-Linear Indications

It is possible that half speed whirl could be a subharmonic vibration

with the residual out-of-balance acting as the exciting force. Hagg and

Warner (14) have already rejected this argument because they say unbalance

usually inhibits whirl, but this is not necessarily logical in a

complicated non-linear system. Shawki (40) obtained values of the frequency

ratio of about 0.499 and the explanation of a subharmonic is indeed

reasonable. However in practice the ratio can be as low as 0.43 or

further in which case, even allowing for experimental error, it is an

unlikely solution. Nevertheless the possibility of subharmonics occurring

in certain circumstances cannot be denied.

,On the point of the frequency ratio at the boundary of instability

the linear theories are also deficient. For instance, Holmes (18) using

the short bearing approximation predicts that this ratio will be between

0.4 and 0.5 depending on the eccentricity. He has made assumptions as to

the extent of the oil film which Sternlicht, Poritsky, and Arwas (43) have

managed to avoid by their numerical method. For this reason the latter

is.superior. But for the frequency ratio at the onset of instability they

obtain values which are always leas than 0.4, contrary to normal experience.

It is therefore apparent that when better approximations to the oil forces

are employed the linear theories show definite signs of weakness.

Very occasionally a third speed whirl has been observed. It is quite

feasible that in these cases an exceptionally strong resonance is occurring

in one part of the system. However a subharmonic vibration of order one

third could also be an explanation.

Several investigators have noticed that the shaft speed for cessation

of the oscillations is lower than that for the onset - for instance Pinkus

(35) and Cole (8). The effect is a sort of hysteresis and is common in

non-linear systems. It provides another indication as to the importance of

the non-linearities of the oil forces but it is again easy to refute.

Once whirl has been initiated the oil flow pattern, oil flow rate,

temperature gradients, etc. change considerably. Therefore it is not

surprising that whirl• ceases at a different speed.

None of these arguments are at all conclusive. Yet it was considered

there was sufficient evidence to warrant further investigation.

2.3 Basis for Non-Linear Theory

To set up the non-linear equations of motion expressions for the oil

forces on the journal are required. Two approaches are possible. One is to

use one of the approximate theories for the infinitely long or short

bearing or alternatively to use the computer results published by

Sternlicht (43). The latter would require the use of empirical laws

based on the calculations which would, of course, produce some error.

Also there are not adequate results to cover the eccentricity range. The

infinitely long bearing theory was immediately rejected because of its

known inaccuracy. The short bearing approximation however was considered

for as DuBois and Ocvirk (9) have shown it predicts the performance of such

a bearing with reasonable precision if the eccentricity ratio is less than

0.5.

A solution to the problem of finding suitable expressions for the oil

forces would have been to re-compute from Reynold's Equation, on the lines

of Sternlicht, but to use much smaller intervals. In this way quite

reasonable equations of variation could have been produced. It was

considered that the amount of work involved was prohibitive especially

as the investigation was of a preliminary nature, the object being mainly

to establish whether or not whirl was a non-linear phenomenon. Further-

more an examination of Reynold's Equation revealed that the character of the

oil forces was dependent to a large extent on the terms in film thickness

cubed on the left hand side. As long as these terms were retained it was

considered that the nature of the forces would be preserved, and an analysis

based on such an approximation would be qualitatively correct. The short

bearing theory was therefore chosen to form the basis of the investigation.

32

2.4 The Derivation of the Non-Linear Equations of Motion

Reynolds' Equation, which is given below, must first be integrated

to find the oil forces on the journal.

( 23-U

1 3 4)- Gr2D-3 The notation has already been introduced, but it will be more convenient

to replace x by R6 where R is the radius of the journal and eio defined below. Also,use will be made of the following:-

c.n . the absolute eccentricity - the distance between the journal

and bearing centres;

• . the radial clearance between the journal and the bearing;

n ▪ the eccentricity ratio;

the angular speed of journal (shaft) rotation; .

0 the angle between the direction of the resultant oil force

and the line of centres (journal and bearing);

e the angular co-ordinate around the oil film measured from

the point of .maximum film thickness;

h the film thickness at any point. As the axes of the bearing

and journal are assumed to be parallel h is a function of a

(or x) alone.

represents differentiation with respect to time.

The bearing centre, which is fixed in space, is the origin of the

co-ordinate system. post of these quantities are demonstrated in fig.6.

The right hand side of Reynolds' Equation (1) involves the surface

velocities of the journal which may be rewritten in terms of the journal

centre velocities (see fig.6),

See over -

33

L. I N C Of" C.EfrailTRICS

PAGE 3.4-

/V 0 TATI 0 FOR. THE EQUACTI oNss OF MOT I ON(

F .

ar...."-Au NAG c cpair ca.E. ( rox..c.o

Di la-CC:1'10N COP" 1..001.0 •

I—Y — EA,4E3

V - crt . cos e . SIN e

Also = C (I -t-- r\S-05e).

to a good approximation as long as the clearance is very small in

comparison to the radius. (!:ormally c,-/R o 002).

As the short bearing approximation is to be used, allowing )13/2))c_

to be neglected when compared to '2A57, substitution of (2) and (3) in

(1) gives

?1,z. (4)

Terms of the order of the clearance ratio have been neglected when

compared to unity, and x has been replaced by RED

From (3} 31N-s e

and since h is not a function of z, (4) becomes

,21D 62

)z z ca (t e)31—

= ek

(LLT- 23Z) -4- Z r'vc.-osej (,5)

The right hand side of equation (5) is independent of z so that the

integration for the pressure is not difficult. The boundary conditions

are that at 2 = t L/2, p = 0; where L is the length of the bearing.

35

(a)

(6)

36

As has already been explained the short bearing approximation assumes

that the oil film only exists in the range

The forces on the journal will then be given by

Q, 9.R &O. . LiL/2,

r

'-‘ R_L r Q, - — - .-.-T1-7s -f n t-G-ic_z,s8cLe • 5 Q -- la 2 4511--lede. (7) j 0 o

Integration of the equations (7) is complicated by the factor

-+-rtc...05e occurring in the denominator of the function 6'4. A

substitution method must be used which is called after its originator,

Sommerfeld. It will not be reproduced here as it can be found in the

standard textbooks - for instance Pinkus and Sternlicht (36).

Equations (7) become

11 ,...,2. where Q.is along the line of centres and Q2 perpendicular to it - see

fig.7A. As the pressure p is a product of two parts, one dependent one

independent of:z, the whole integration can be effected in two parts.

e. cle

L•12‘2

f- ADC uat ix-Nteafel ( A Tff, "72_

2. r‘a ( — r \-1)4

a) h CI an!) ( — r\9.) 5

7a

(e)

Q2.- cz a 0 (-to-- 20) -+

At this point the present analysis departs from the theory of

stability for a short bearing as deduced by Holmes (18). He next

linearized the oil forces whereas here all the non-linear terms are

retained.

'37

Having produced expressions for the oil forces the problem may now

be treated as one of particle dynamics. There is virtually no choice of

co-ordinate system, the forces being described in terms of the angle

and displacement (eccentricity) cn. It is unfortunate since a problem is

more difficult to visualize in the polar form. However the complications

which arose when conversion was made to a rectangular system were quite

aecisive.

Figures T demonstrate the various forces acting on the journal. The

first one shows the oil forces, Q t and Q2, for the journal position defined

by eccentricity, en, and angle 0. 0 is as yet arbitrary though later it

will be aefined from the line of action of the static load on the journal,

usually the vertical. Under equilibrium conditions, that is zero journal

centre velocity and displacement, the oil forces •Q,0 and Qao will be •

exactly equal and opposite to the components of the static load, 131 and P2.

The vectorial sum of P1 and P2 will give the magnitude and direction of

the static load. The equilibrium attitude angle 00 and eocentricity_cno

can then be precisely defined.

2 2r 2 c_a — r‘;')z.

az° ".-L2 'T-t ca a (i

01 a C.A. M.111%.1 42. C.C.Pb4 -r

de.= c:rt,

.J 0 Voa."‘A.t.-. C.C.P.Alrore.E.

BreQs

OA. • C.-Y1.0

P. Qs.

Pr i Qs,

Cc) F-0 P.C. CS I -rs-4c. (D') -rg-ta Fa..crrfrzr I NG mos-I-us:Loco Pos-n".

B C.F.'

ti.25)

\ (ust-i-n—cc

r-Coca..0

P-OR.Cr.. T:31 os...GP-A.- SAS PAGE. 38

(A.) THE OIL FORCES. 13 • at.Ec.-7" • c.r.i op-

S-r-Ak...1- I C

(B) EctuiLi 136:kW NA CrC:).4 Corr I ONE ,

The suffix zero refers to the equilibrium position. Q,, is negative

because it is, in fact, an upthrust suppong the :journal.

The resultant oil force iff Q0

EQ 2 to -4- Q20

P.1.2r2. 7-7 YA.c> p; .... 2. .4_ Lo aCa 2(1- r‘-g- Y4

to

The total oil force, Q0, is equal in magnitude to the static load on the

journal. It will be assumed that there are no others forces beyond its

own weight. Taking M as the mass of the journal

Q0 mi Mk

Substituting for Qo from (10) gives

2L?R 9 I"(

where K = "moo ~l G ‘rk a ( 1—

40

An out-of-balance force will also be considered and it is illustrated in

fig.7(D). It is shown separately for clarity though in reality it is

superimposed on the system of fig.7(C). The rotating force has an angular

velocity about the journal centre exactly equal to that of journal rotation.

At time t = 0 the force is at an angle to the equilibrium line of centres

and its magnitude is given as followa:-

The distance between the centre of gravity of the journal and its

axes of rotation is 'q'. Such a quantity is not only mathematically

convenient but is also used to describe the fineness of the balance of a

shaft. Thus the magnitude of the force will be

crr,32 = NA 5

It has been written in this form in preparation for the non-dimensional-

ization.

Hence the complete force system may now be written down and equated

to the journal's inertia.

tV1 c. ( n. -4-

Q2 — 2 cosO( -- Rsit-a0k. r M. t+25) _c

Ac ._ — r\.6.(2)

Q24)

Q,— P, cos a— Pa -I- 1\43 g —

where Q4, and Q2 are both functione of n, n, 0, 0 and Pi and P2 are

constant for given equilibrium conditions (being components of the

load, Mg). Inserting the expressions for the forces into the equations

(12) gives:-

41

Mc (n.F.c-÷ a rt,_6() = •

vt-LT-a56)\

ati-n-n-6/2- _ 2r\.! To- 04.1

4 ,\_r.. (I- r`'-)2 )Z/2.C-C) 5 C3C

HI- N/15 g a si t ZS) - a.] ('3)

) 2"2' (I+ a r-?--) t!.‘

(1 _ (1 r\:i)s/-2.

Y\ c>.—t-j-3. 511,4 043

-+- M9 C-OS E(sLo _ (sL]

R r_ L -t-

a r-L2.-Ez- c....os CY- ( -

'These equations can be considerably simplified. Firstly, non-dimensional-

izing time thus

P -r 0.

if The notation h, n, etc. will be used to represent differentiation with

respect to 27. Putting

- 12, (9/c)112

424

and substituting the factor K into equations (13) gives

r\C,, -t- 2kOk r 4"-'-

Since \r'aness are both exceedingly small powers above the first may be

neglected. Equations (14) can then be written in the form (omitting the

exciting force),

b,, br.5

(Ls) — b„. 1655 — sass =0

where the a's and b's are all constants. Expanding the coefficients of

(14) by Taylor's theorem in terms ofrianesi but neglecting powers above

the first

--r11- .4.. 4,_ .1-1K (1 -1- 2 r -) _ S4n., K.

'CI' ti- r\tffk S/(1— (A3,)1

1- -1.._ 4 I-

AN D

01 / 4.,s

b rs

:Sr

bps 1:D s

4 IN.1012

= 2 cx.s

_ SL

_ 20 S2,

1

By comparing equations (15) and (16) the coefficients, brr etc. can be

found. Substituting the value of K from (11) and, for the sake of

brevity, writing

THEN aTT

No = T12(1-- rI2')

E3(1 r\;:n

(1— r\) r•,-1o2

.11 (1— rQ..) 2' rA., NJ g2

A-4

These coefficients are identical to those deduced by Holmes (18). The

non-linear equations (14) can therefore be presumed to be correct.

2.5 Excursions of the Variables

The quantities involved in the equations (14) are:-

The eccentricity ratio, n, is physically restricted by the bearing

surface so that its maximum value is unity, i.e. the journal and bearing

are in contact. The angular co-ordinate has no limits whatsoever.

K is a constant for a given value of the equilibrium eccentricity ratio,

and the phase angle of the force, has values between 0 and 3600

The speed of rotationta is related to the actual speed of the

journal N r.p.m. through the relation

=GO

21-1 c — = 0- 00 5 2 i\-1 C-j4 -

where c is the radial clearance in inches.

TABLE I

Shaft Radial Diameter ins.

Clearance ins.

Speed r.p.m.

SI,

2 0.0022 20,000 4.99 5 0.005 10,000 376 8 0.010 12,000 6.39 14 0.012 3,000 , 1.75

Table I demonstrates possible values for 524. It will be seen that the

excursion of the variable of up to 6 or 7 should amply cover the practical

range of speed.

The residual out-of-balance quoted by the distance between the centre

of rotation and the centre of r7ravity is usually of the order of

0.0001 ins, which in he present notation is 0.001 ins. as

4-6

the minimum value of the radial clearance then the factor (42 qjc) will

have a maximum value of 0.1 units.

2.6 Analysis of Equations

There is very litLle published work on the solutions of non-linear

systems having two degrees of freedom. That which has been done is mainly

concerned with very simple mass spring systems without damping. The type

of non-linearity in the displacement terms is similar to that normally

associated with the Duffing equation, i.e. force proportional to

x + bx3 (see for instance Stoker (44 where the coefficient lb/ can be positive or negative, but always small.

A formal analysis of equations (14) was attempted. The variable

coefficients were expanded in terms of the displacements but convergence

could not be obtained without severely restricting the amplitude. The

non-linearity not being small made the present equations incomparable with

previous investigations and furthermore the occurrence of terms in

displacement squared introduces extra complications.

Inspection of the cross velocity terms shows that the coefficients

are always negative (n, Ronal' positive) whereas the direct terms will

always be positive and act as damping. Hence the likelihood of self-

sustained vibrations is not obvious. But as Morrison has pointed out when

commenting on his linear analysis (27) the displacement terms also govern

the stability of the system. It is therefore apparent that it is

important to consider not only the individual contributions of the

displacement and velocity forces but their interaction as well. Realizing

the intricacies of the problem and its uniqueness as regards to previous

work an analogue computer was programmed for the solution. It was

. considered that in this way the salient properties of the equations

could be easily found.

4-7

fi

CHAPTER 3 THE ANALOGUE SIMULATION OF THE NON-LINEAR EQUATIONS OF MOTION

3.1 Introductory Remarks

An analogue computer is an analogue in terms of voltage. It has as

its basic unit a high gain d.c. amplifier which, used in combination with

resistors or capacitors, can function as a summer or an integrator.

Multiplication of a variable by a constant is performed by a fixed

potentiometer. The output from this operation will obviously'be less than

the input and will necessitate correction when the multiplier is greater

than unity. It is done by 'scaling' which entails choosing the ratio of

the input resistors on the succeeding amplifier to maintain a true analogue.

Scaling is also used to keep the maximum voltages as close as possible to

the limit of t 100v. in order to preserve accuracy. Overloading an

amplifier is automatically detected and displayed on the control panel.

Multiplication of one variable by another is again performed by a

potentiometer but in this case it must be set by a servo-mechanism. The

whole unit is called a servo-multiplier. The introduction of mechanical

linkages and the resulting inertia requires that the frequencies involved

must not be high. The time parameter must be scaled to suit the computer.

On the other hand it is necessary to reduce the time taken for a particular

solution to eliminate, as far as possible, the drift of the integrators.

Sine and cosine functions for a constant angular speed can be easily

produced. A closed loop circuit of two integrators, with multiplication

for speed between, is self controlling and the functions can be tapped off

as required. Sines and cosines of angles are simulated by special

generators which approximate to the actual functions by a series of straight

lines over the range t 900. To cover 360° it is necessary to have two

units for each function coupled through a switching device. It is a

'noisy' operation - that is, produces unwanted voltages - and is better

avoided. It was therefore proposed to restrict the excursion of the

angular co-ordinate to ! 900. This was also considered to be practically

sufficient.

The output of an amplifier is always reversed in sign:to correct it

an inverter may have to be inserted into the circuit.

3.2 Programming the Computer

The equations were simulated on an Elliott G...PAC, Mark II, analogue

computer. The following equipment was used:-

4 Integrators;

9 High gain amplifiers;

22 Inverters and Summers;

8 Servo-multipliers;

2 Function generators (sine and cosine)

For recording purposes:

2 Inverters;

1 x-y plotter;

1 Double-channel strip recorder

The computer itself imposed some restrictions. One has already been

mentioned which was the limitation of the excursion of the angular

co-ordinate to ! 900. Secondly, the eccentricity ratio, n, was only

permitted to be positive because of the complications which would have

49

50

arisen if it passed through zero. When the displacements exceeded these

limits the computer became unstable, brought about by the servo-mechanisims.

Solutions in which this happened are quoted as being 'beyond the range of

the computer' or merely 'unstable'.

It was initially specified that the equilibrium eccentricity ratio

should be variable between 0.05 and 0.9. A value of zero was impossible

because of the restrictions above, and unity from physical considerations.

Variations of the parameters within this range could not be handled by the

computer unless the programme was extensively resealed. The case of

n.Q.0.9 was therefore omitted but it was still found necessary to perform

a limited amount of resealing for the cases of u.7 and 0.8. A few

solutions at 0.9 were obtained but their reliability is not certain.

The restrictions unfortunately Lleant that the cessation of whirl could

not be investigated. Once instability had started it was necessary to stop

computing and return to the initial conditions.

The magnitude of the force was limited to O.08,also because of

scaling difficulties.

See over -

( 1-2 ) /̀' LL = r1..2 •D('

(11_, r-1- i""° ) - r\7-)

je, ( nai cos csct

Zoo

(1— r\-V- b =

To obtain the most economical layout of the equipment the

equations were simulated in the form:-

r C a. -t, — K — L.124

- 2. r"\,4 -t- b COS0(—Yrt:Sir\I -'i

1-\.(34: = - a rv:. COS CX b C(\,.1

51

r J Z ) gS N 111-0.2"-r- 0(1

'n, and fal are the variables and

andare constant for a particular solution.

Y, bo,q, are functionsof no.

Figure 8 shows the symbols for fig.9, the analogue simulation

schematic. it is programmed by F.ssuming f to be known and then

integrating twice to obtain n and n. In combination with(andcX, also

assumed to be know, the richt ;and side of the first of the equations 117)

can be evaluated and ri is then found. This is the first circuit loop

and is called the radial equ,tion in fig.9. The second loop is formed

PAGE Sa

SYMBOL—S USE.0 IN .ANA6-1-0G.I.AE SC-NEI...4AT i C

r 1G 8

•-ucie-o a Amtial ApoiCIPLo or's= Ar. .

--Ar>----- $up ..sr.4 • ilea ....thegivi....•-soCin .

I NO -ir- CoGOILArrat=1. A. 00 ,i1:3 6,. i ni c.,04.,

il. 5 > SCsa.se0 - Mc. ii-rr. 6 Ppd.-IC.12 . 4 .11.✓ni:Pe.... • r- i r..4=t Ai...Sp 6.40-1-04;1..

SERVO-mui..."-• P.1.4=5;t w01..i-OW- VP Par="1"T" I ONIC-1-E-gt. , Co Crr-r GO 1-siNtE OCA‘Crr CIS ovtC..Cs-eibbik.i I c.^.1- 1........4fre.A.c.C.. ,

64....,..•c•-sr...-r Pcrrir.,ffiarr • eivtiCTC.a..

rlauti.C.S aspo ghbovIPL.111,40.3 Or."10'71r. 43^.1045.

in a similar way, the value of &being initially assumed and eventually

calculated. Secondary circuits are used to find various terms. The

imposed force is generated in another loop whose operation has already

been described.

The diagram is largely self-explanatory and does not warrant further

discussion. The labelling is in accordance with equations (17)•

3.3 Checking the Analogue and its Accuracy

The checking of the analogue presents many difficulties and in the

present case, where no approximate analytic solutions were possible, the

situation is somewhat worse. The normal checks, static and dynamic, were

performed but they cannot he reAily employed to estimate accuracy.

For the static check the output of each integrator is disconnected

from the circuits it feeds and replaced by a known voltage. The

integrators are converted into summers but the ,,7ains remain the same. ti

Under these conditions the computer is static and simple calculations can

be made to determine the correct voltages at each point in the simulation.

Discrepancies lead to the discovery of faulty amplifiers or errors in the

circuit. The dynamic check is similar except that the integrators remain

as such. The computer is allowed to run for a specified time and then put

into the 'hold' condition, freezing the solution. Since all the inputs

are known the integrators can be checked.

The recording equipment was calibrated against the computer. For

amplitude, a known voltage was applied and the internal gains adjusted to

give the required displacement. For frequency, the output of the forcing

54

55

function was connected to the strip recorder, the period of oscillation

measured in terms of machine time and compared to the nominal set values.

In this way, errors in the chart speed were eliminated.

The overall accuracy of the analogue cannot be precisely specified.

From experience and using the checks as a guide the operator estimated

that the margin of error was 4- 4;. Accuracy was limited because of the

resolution of the servo-multipliers. Also integrator drift introduced ,

some error especially when solutions took a long time to reach a steady

state. Regular checking of particular solutions showed good repeatability

within the estimated margin.

3.4 Computing Methods

3.4.1. Initial Conditions

The choice of the initial conditions is an important part of

a non-linear analysis as the final solution may well depend on it.

For instance, with a system capable of self-sustained vibrations

several limit cycles may exist, but which one is to be the steady

state solution is determined solely by the initial disturbance.

In a two dimensional problem the velocities and displacements in both

directions must be chosen, and there is obviously a large number of

combinations for even a few values of each. To avoid the

complications the initial conditions of zero velocity and zero

displacement from the (presumed) equilibrium position were imposed;

random noise and any slight unbalanced voltages in the computer being

used as the disturbance. This is not a rigorous approach but it was

found, in most cases, to give repeatable results and was thus considered

satisfactory. The exceptions arose :her. the amplitudes of vibration

56

were so small that the voltage swinge became comparable to integrator

drift. however there was still a fair degree of consistency.

In one or two instances experiments were made to ascertain

whether or not large disturbances would e fact alter the steady

state solution. Up to 20 volts were injected at various joints in

the circuit while it was computing but the solutions remained.

unchanged.

3.4.2. :cthods for the free and forced solutions

Once the difficulties of scaling the analogue had begin overcome

obtaining solutions wae'reasonably simple. Values for the various

parameters were chosen and set on the appropriate potentiometers. It

was then only necessary to release the analogue from the imposed

initial conditions to start computing.

The programme had been designed to solve the equations in polar

co-ordinatesbut, in order to make comparison to experiment easier,

the results were recorded in a rectangular system. Conversion to

x • n sinO( and y = n eesekwas automatically made by the computer.

Continuous traces of x and y against time were taken but only the

steady loci of the shaft centre were recorded.

During the preliminary studies of the forced solutions it was

found that variation in the phase angle of the force only affected

the initial transients. It was thus set at 1K0 and remained so

throughout the tcsts. Occasionally a very long transient occurred

in which case the results were always confirmed by runnins the

solution again.

Sometimes unstable transients unexpectedly appeared. It

became obvious that in these particular instances stable solutions

could exiet, and the procedure to find them was as follows.

Computing was started at a slightly different speed to avoid the

initial instability. The speed was then slowly returned to the

required value while the machine was still running. The process

took some time and therefore the integrators begain to drift

introducing some error. Where such phenomena occurred has been

indicated in the results.

3.4.5. Method for variable speed of excitation

While the free and forced solutions were being obtained it

was decided, mainly for the sake of interest, to briefly investigate

the system Aere the speed of the force was different to the speed

of rotation, but both still in the same direction. The analogue had

not been designed to take this problem yet it was found that by only

a slight modification some results could be obtained. Primarily it

meant that the servo which set the speed had to be divided in two,

one to set the speed of shaft rotation (1/R) and the other to set the

speed of the force (Sir). Also it was considered that the force

should have a constant magnitude and not vary with speed squared as

in the previous tests. The parameter was chosen so that the total •1

coefficient of the force, (;S"), would be maintained constant and

equivalent to the case where eaireneand = 0.04. Variation of the

speed of the force was restricted because of limitations inge

5-7

The testing procedure followed similar lines to that described

in the previous section. Long initial transients frequently occurred

especially at low fractional values of the rationw/E/R. It is

hoped that constant checking has eliminated any freak solutions.

58

CHAPTER At- ANALOGUE SOLUTIONS OF THE EQUATIONS OF MOTION

4.1 The Form of the Results

Solutions to the equations of motion (14) were obtained for

eccentricity ratios in the range 0.1 - 0.9, though the reliability of

the case of 0.9 is not certain because of possible overloading. For the

ratios 0.1, 0.3, 0.5, 0.7 and 0.8 the free and forced solutions were

found with 6 = 0, 0.02, 0.04, 0.06 and 0.08. At the intermediate

eccentricity ratios of 0.2, 0.4, and 0.6, only the free solution and that

for . 0.06 were obtained as it was considered thatt interpolation would

be quite satisfactory. The results are given in various forms, as will be

described, but the nomeclature is consistent.

. Angular speed of the vibration based on non-dimensional

time.

= Angular speed of the journal and of the faire where both

are the same.

Angular speed of the journal.

CLF = Angular speed of the force.

Amplitude of the rotating force.

rLo Eccentricity ratio of the presumed equilibrium position.

DC Displacement in the direction at right angles to the

presumed equilibrium position of the line of centres.

Displacement in the direction along the presumed

equilibrium position of the line of centres.

The displacements are measured as a ratio in terms of

the radial clearance and are therefore equivalent to

units of the eccentricity ratio.

59

c0

Because of integrator drift and very slight unbalance of the voltages

in the analogue the precise 'equilibrium' position could not be recorded

on the charts, and the maximum amplitude from that position could not be

measured. Amplitudes given in the results are found by halving the maximum

displacement in the given directin.

4.2 The Analogue Results

4.2.1. Equilibrium Eccentricity Ratio 0.1

The response curves are shown in•figs.l0 and 11. It must be

pointed out before describing them that the amplitudes were very small

and difficult to measure, especially at the lower end of the speed

range where they approached the thickness of the recorder's inkline,

about 0.003 units.

The free solutions contained a self-sustained vibration which

remained at a low amplitude until a speed of about 1.8 was reached.

The amplitude then rapidly increased exceeding the limits of the

computer (in this case 0.1 units). The frequency ratio (vibration/

shaft frequency) was approximately one half though it increased slightly

with speed. From the point where the ratio was exactly one half the

amplitude suddenly increased. This is seemingly a coincidence for

there is no obvious connection.

Imposition of a rotating force tended in general to increase the

amplitude and lower the stability limit. But at low speeds, of

about unity, there was some evidence of a larger force diminishing the

amplitude. The differences, which were small, may have been errors,

but since the trend could be seen in both x and y the results were

probably correct.

61

The oscillations of the forced solutions had speeds of both

the force and of the self-sustained vibration. Harmonic analyses were

not attempted because with such small amplitudes it was considered

worthless. However, by a rough sketching method the changes appeared

thus. A small force reduced the strength of the self-sustained

vibration but added a synchronous component making the total amplitude

slightly greater. A larger force also tended to suppress the natural

oscillation yet above a certain speed excited it subharmonicly. In

these solutions the two frequencies were of about the same magnitude.

Modulation of the x and y traces occurred between certain speeds

but only when the force had a magnitude of 0.02. This was -unexpected

since frequency entrainment is more likely in a non-linear system.


Self-sustained vibrations of small amplitude again existed. Their

frequency ratio was rather less than one half until the speed reached

2.0. The ratio then rapidly increased and the amplitude became unstable.

The addition of the rotating force completely suppressed the

natural oscillation up to a speed of 1.8. An exact half-speed vibration

then appeared, possibly a subharmonic, though the synchronous component

was not eliminated. At the same time the amplitude jumped, and

continued to increase with speed until instability tool: over.


Only very close to the free boundary of instability did self-

sustained vibrations occur, as illustrated by fig.13. Tests with

speeds of up to 9 units were performed but there was no sign of a

higher stable region. (Similar tests were not carried out in the two

previous cases because of the amplitude limitations). The speed of

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the unstable oscillation at the boundary was estimated to be half

that of rotation.

A rotating force introduced harmonic oscillations whose

amplitudes were roughly proportional to its magnitude. At a speed

dependent on the size of the fame a subharmonic appeared together with

a jump in the response. The behaviour was similar to the previous

case.

With the smaller forces it will be seen from fig.13 that there

was a range where the amplitude remained more or less constant. It

is in fact a resonance which is more obvious at higher eccentricity

ratios.


The behaviour of the system was very similar to the previous case.

There was a small speed range in which a self-sustained vibration

existed and, with a force, a subharmonic was generated which quickly

upset stability. The rotating force reduced the stable region but

here to a slightly greater extent - fig.14.


The stability of the free system was most clearly defined, no

self-sustained vibrations being produced - figs.l5 and 16. Above the

boundary a further stable range could not be found.

The rotating force produced a resonance at a speed of 1.5 and

its position vas confirmed by examining the initial transients. As

before the subharmonic oscillation was produced,but well below the

boundary of instability for the free solution. It was found impossible

to obtain any results in the speed range 2.4 to 3.2.

With the force at its lowest magnitude of 0.02 beats were

observed in the strip traces as indicated in fig.15. Their cause

cannot be discerned.


Instability of the free system did not commence until the speed

reached 6.9. From a speed of about 2 an exceedingly small self-

sustained vibration appeared in the y-direction only. It is shown

on fig.17. Its speed was between 2.2 and 2.4 but this dropped at the

boundary to 1.5, corresponding to the forced resonance.

With the force of magnitude 0.06 the resonance and the jump to

the subharmonic followed the pattern of the previous cases. The

occurrence of the subharmonic appeared though, in this case, to be

closely connectedito the speed of instability.

A stable region could not be found in the speed range 2.2 to 6.9

even when the magnitude of the force was substantially reduced.


The response curves for the equilibrium eccentricity of 0.7 are

shown in figures 18 and 19. Very small oscillations were produced by

the free system in the y-direction alone but they were only just

noticeable. There was no boundary of stability in the range tested,

but at a speed of 7.5 there was a sudden burst of self-sustained

vibrations, in both directions, which almost disappeared again at a

speed of 8.4. The frequencies and amplitudes are illustrated on the

graphs though no explanation can be offered.

The harmonic resonance et a speed of 1.3 was pronounced. At

higher speeds, though, the resconse was largely dependent on the

magnitude of the force, and governed whether or not the

-75

-7r

subharmonic or the second harmonic resonance would appear.

The peculiarities at this eccentricity ratio are probably due

to the extension range of the test range and the appearance of a

second resonance. However a system with two degrees of freedom does

not necessarily have two resonances, as shown by Arnold (1) but here

there is a definite possibility. Also because of the behaviour of the

free system unusual effects are almost bound to be produced.


The free equations again exhibited very small oscillations in the

y-direction only. The speeds of the vibrations are indicated on fig.20.

Addition of a rotating force produced the response curves of

figs.20 and 21 which are similar to the last case. However a second

resonance was not ohlserved, yet, because a force of 0.08 produced

instability throughout the higher speed range, it is considered that

it does exist.

The figures illustrate the significance of the regions of unstable

transients. They correspond to where one:half, or one third, of the

shaft (forte) speed is equal to either the resonant speed or that of

the self-sustained vibrations.

4.2.9. Amplitude Contour and Frequency Maps,

For one particular value of the magnitude of the force, 0.06, an

amplitude contour map has been drawn; fig.22. The results obtained

above have been used. Only the amplitude in the x-direction is given,

for it is apparent from the previous graphs that the maximum amplitude

can be expected in this direction even though it varies with

19

eccentricity ratio. The particular magnitude of the force was

chosen as it is representative of the others, and the amplitudes, in

general, are large. This would make the contours more obvious. Results

obtained from the solution at an eccentricity ratio of 0.9 have been

included but, as stated before, they cannot be regarded as completely

reliable.

The frequency map - fig.23 - gives the main speed of vibration

of the free and forced solutions. Cnly one force has been illustrated

and its magnitude is the same as for the amplitude contours. Unless

otherwise rtated vibrations occur at the speed of the force or, in the

free state, there is no oscillation.

The two maps are together self-explanatory and illustrate the

amplitude and frequency of any vibration to be expected under given

operating conditions.

4.2.10. Solutions where the Journal and !-_;xcitation Sneeds are

Different

Solutions were only obtained at the eccentricity ratio of 0.5 and

for four values of the journal rotation. The curves were remarkably

similar though complicated. For clarity, the results for one speed

alone are presented - fig.24. The upper two graphs show the maximum

amplitude in the x- andy-directions against the speed of the exciting

force. Underneath are illustrated the various speeds of vibration

that could be seen in the traces.

VEenever possible the natural frequency of the oil film was

excited by the force, whether by sub- or super-multiplication. v:ith

a high speed force there was only a small amplitude oscillation but

a slight peak did occur when the ratio of the force to the vibration

speed was one third. At a value of this ratio of one half, the

amplitude went beyond the limits of the computer. Two solutions were

obtained at SLF - 2.8 depending on which frequency occurred.

.Ao the *peed of the force was further decreased the resonance was

harmonically excited. Just below this region the solutions included

a modulation at the speeds indicated, probably produced by the proximity

of the resonance. At lower speeds of the force several minor peaks

were found. Their presence is difficult to explain for it is unlikely

experimental errors were the cause. However it may be that the

equations in this range are very sensitive which, combined with the

fact that the analogue was not designed for the purpose, gave rise

to the peculiarities. The peak at S),, . 0.65 is quite distinct caused

by the force having half the natural speed of the resonance.

With the other speeds of rotation the response curves were

similar except on one point. At the lowest testedvam. 1.84, no

unstable solutions were found there being only a very small peak when

the force was at twice the natural speed. The most likely reason for

the difference is that the force was of insufficient magnitude to

produce the effect.

ea

4.3 Discussion of the Analogue Results

Only a brief outline of the results has been given on purpose. The

equations of motion are very interesting in themselves and warrant much

discussion but the aims and assumptions of the work must not be overlooked.

There is another point to be considered. E-A Automation Systems Limited,

who programmed the computer, admitted that this was the most complicated

hydrodynamic instability 1 ?lain journal bearings · 2014. 4. 24. · hydrodynamic instability 1...

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