hydrodynamic instability 1 ?lain journal bearings · 2014. 4. 24. · hydrodynamic instability 1...
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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.
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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.
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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
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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
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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
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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
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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
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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
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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
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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 .
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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
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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
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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.
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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
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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
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Po...G.= I T
FIG. 3.
rg-I Ca. it
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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
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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
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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
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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.
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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
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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.
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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.
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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
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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.
4.2.2. Equilibrium Eccentricity Ratio 0.2
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.
4.2.3. Equilibrium Eccentricity Ratio 0.3
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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-
PAGE. 70 ca..E.5L)
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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.
4.2.4. Equilibrium Eccentricity Ratio 0.4
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.
4.2.5. Equilibrium Eccentricity Ratio 0.5
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.
4.2.6. Equilibrium Eccentricity Ratio 0.6
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.
4.2.7. Equilibrium Eccentricity Ratio 0.7
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.
4.2.8. Equilibrium Eccentricity Ratio 0.8
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