The paper is concerned with the effect of friction in drive couplings on the non-sychronous whirling of a shaft. A simplified model is used to demonstrate the effect of large coupling misalignments on the stability of the system. It is concluded that provided these misalignments are large enough, the system becomes totally stable provided the shaft is supported on bearings exhibiting a viscous damping capacity

Morton, P. G.

English

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ASP_ OFCDUI.(3V_ [I_VPING IN ROIKI_
SUP_ON}_IC BEARIb_S
P. G. Morton
G.E.C. Power Engineering Ltd.
Lichfield Road, Stafford,
England ST17 4LN
ABSTRACT
The paper is concerned with the effect of friction in drive couplings on the
non-sychronous whirling of a shaft. A simplified model is used to demonstrate the
effect of large coupling misalignments on the stability of the system. It is
concluded that provided these misalignments are large enough, the system becomes
totally stable provided the shaft is supported on bearings exhibiting a viscous
damping capacity.
SUMMARY
Rotating damping is well known to be a source of instability in shaft systems
and many physical and mathematical explanations of the phenomenon exist. Published
work seems mainly to be concentrated on a viscous or hysteretic damping model for
the rotor, with simple dissipative bearings. The present paper considers non-linear
damping of the Coulomb type in a simple rotor supported on 'realistic' linearized
bearings, characterized by 4 stiffness and 4 damping coefficients.
The paper examines the nature of the sub-synchronous limit cycles resulting
from interplay between the Coulomb damping within the rotor and viscous damping in
the bearings. The rotating damping in the present model is assumed to arise from
rubbing within gear type couplings at both ends of the rotor. The qualitative
findings are however applicable to any mechanism producing friction forces depending
on rotor flexure. It is also shown that as the couplings are misaligned in any
plane, a level is reached above which the sub-synchronous oscillations collapse.
A map of these limiting misalignments is compared with the shaft orbit under
conditions of perfect alignment.
INTRODUCTION
Rotating damping has long been recognised as a source of instability in shaft
systems. Both physical and mathematical explanations of the reasons for this
instability have been advanced. A good summary has been given by Crandall (Ref. I).
Viscous and hysteretic damping is easily incorporated in a linear mathematical
model. In many practical cases, however, the damping arises from relative motions
within the rotating structure, the force-motion relationships obeying non-linear
laws. A Coulomb damping model gives a reasonable representation of the energy
exchanges resulting from slipping mechanisms, although to accurately incorporate
such a model in a rotor/bearing system presents some problems in analysis.
45
https://ntrs.nasa.gov/search.jsp?R=19830007362 2020-03-21T23:43:47+00:00Z
Intuitively, one might suppose that instability due to any saturable type of
rotating damping would be subject to limit cycles. This is only so if the system
also contains positive damping elements with a more powerful energy-amplitude
ratio. Such a system is a shaft with rotating Coulombtype damping supported on
realistic, albeit linearized, hydrodynamic bearings. Very little published work
exists on this very commonlyoccurring system and the present paper deals with a
particular phenomenonof considerable practical significance.
The system dealt with is a rotor supported on two hydrodynamic bearings and
driven by a gear type coupling. The rotating damping arises from movementsbetween
mating teeth of the coupling.
THEMODEL
The paper deals with an idealised model (Fig. I). The degrees of freedom have
been limited to 4, this being the minimumnecessary to give a qualitatively adequate
solution. The system comprises an isotropic non-dampedshaft s_etric about its
centre of span and characterised by a single flexural mode. The shaft is
supported on two linearized hydrodynamic bearings, notionally represented as 8
stiffness and damping coefficients, each varying with running speed. Each end of
the shaft moves within a coupling which has a fixed slope but is free to move in
translation. The laws governing the force-velocity relationship within the
couplings can be either of the viscous or Coulomb type. Provision is made for
altering the coupling slope in both vertical and horizontal planes.
SYMBOLS
Axxil etc. kg
Bxx, 1 etc. Nsm -l
Exx11 etc. Nm -I
B
Bxx etc.
Blxx etc. Nsm -_
b Nsm -I
C m
Exx etc.
EIxx etc.
ex, ey tad
F N
f N
-2
g ms
L m
Elements of Inertia Matrix
Subscripts denote
Elements of Damping Matrix position in matrices.
Elements of Stiffness Matrix
Parameter concerned with viscous rotating damping (Eqn.7)
Non dimensional oil film damping
Dimensional oil film damping
Dimensional viscous damping of all coupling teeth moving
axially
Radial clearance of bearings
Non dimensional oil film stiffness
Dimensional oil film stiffness
Angular misalignment of couplings, Planes x, y
Coulomb damping force generalised (Eqn.12)
Coulomb damping force of all coupling teeth moving axially
Acceleration due to gravity
Half span of shaft
46
M kg
m kg/m- 1
qlX etc.
r m
t s
-I
Vx, Vy ms
We (Nm)
Wv (Nm)
x, y
Y
A
6x, _y
¢o m
Ca 1 0 (rad)
×
k rad •s- I
_0 rad. s-I
rad. s- I
Half mass of shaft
Mass of increment of shaft
Generalised co-ordinate. Shaft centre x plane
Radius of coupling teeth
Time
Generalised velocity of rubbing of gear teeth
Work due to Coulomb damping/cycle
Work due to viscous damping/cycle
Horizontal and vertical planes
Viscous rotating damping factor (Eqn. 7)
Modal parameter (Eqn. 4)
Misalignment parameters (Eqn. 9)
Mode shape of shaft
Modal co-ordinate at shaft centre
Modal slope at shaft end
Co-ordinate vector (Eqn.2)
Modal parameter (Eqn.6)
Natural frequency of pinned-pinned shaft
Lowest natural frequency
Running speed
EQUATIONS OF MOTION
The system co-ordinates (Fig. 2) relate to movements in the horizontal and
vertical planes. The coefficients of the equations of motion are expressed as
elements of square matrices using a subscript code where x, y determine the plane
and I, 2 relate to the shaft and bearing respectively. Thus Axy12 would appear in
a matrix as follows:-
I-- - _ fAxy 2
'- -- --I Y
(l)
In the systems with viscous rotating damping the equations of motion are:-
e# •
AX + Bx + Ex ffi 0 (2)
47
Dimensional
AXX11 = AyYll = Zm(_o) 2
Axx22 = Ayy22 = M
AXXl 2 = Ayy21 = Axx21 = Ayy21 = lm(_ )
u/0
BXX22 = BIxx
Bxy22 = B Ixy
Byx2z = B1yx
Byy22 = Blyy
br 2 ._oal zBXX1 1 = Byyl I = T ( )
Exxl I = AxxI IA2
Eyyl I = Ayyl I _2
Exx22 = EIxx
Exy22 = E Ixy
Eyx22 = Elyx
Eyy22 = Elyy
Exy11 = -Eyxl l2= ( )_
Non Dimensional Inertia
2
l
z___= _
M_o
g Bxx/_
g Bxy/_C
g Byx/_C
g Byy/_C
br 2 (_a I
B = _-_ ._0 )
X2 lX
X2 lX
g Exx/C
g Exy/C
g Eyx/C
g Eyy/C
br 2 _._1 22--if( )a
(3)
Regarding the shaft geometry, since the equations of motion are set up in terms
of the pinned-pinned shaft modes _, then it is convenient to assume a simple shaft
geometry. The choices are a disc/weightless shaft combination or a uniform shaft.
The latter model has been chosen and this results in a sinusoidal modal shape for
which:-
A = Zm@2/M@02 = 0.5 (4)
= Zm_/M@0 = Z/_ (5)
_al = w/2L (6)
@0
Roots of equation 2 give both system frequencies and dampings. If the system
damping factor is plotted against the rotating damping (Fig. 3) it is possible to
find the reserve of viscous damping possessed by the system. The viscous rotating
damping factor is conveniently expressed as:-
r2 _a I 2 b B
Y = 4-A (_o ) " _" = 2>,'--_ (7)
y being given as a proportion of critical damping of the shaft vibrating in flexure.
48
The damping B arises from the axial rubbing of the gear teeth in the coupling.
These teeth are assumedto be continuously distributed on a pitch circle of radius
r. The value b relates to the total viscous force/unit velocity obtained from
sliding the coupling axially. The values of all parameters used in plotting the
curve of Figure 3 are shownon that sheet and will be used for all further work in
the paper. The bearing coefficients relate to a specific non-circular bearing
profile of the fixed arc type.
Solutions of the linear equation of motion yield not only the systemfrequencies
and dampings but the orbit shape, the size being, of course, arbitrary.
The coupling damping operates on the relative velocities of teeth arising from
small vibrations in the system. It is possible to generalise these velocities in
terms of the shaft co-ordinates as follows:-
0
Vx = q,x + _ (qlY + _Y)
Q
Vy = qlY - _ (qlx + _x) (8)
where 6x, _y relate to the angular misalignments of the coupling ex, ey thus:-
6x = ex (_a,) _y = ey (_al) (9)
The energy reserve/cycle in viscous damping at the coupling is given by:-
Bf2_/m(Vx •Wv = qlx + Vy qly)dt (I0)
0
Coulomb Damping
Suppose now that the damping mechanism in the coupling is of the Coulomb type
and that f is the force resisting axial movement of the teeth. The Coulomb energy/
cycle is:-
Wc = Ff2_/_(Vx _ix + Vy qly)dt
0
(Vx 2 + Vy2) _
(II)
___i f r fwhere F = 2__r ( ) ..... (12)M L M
Note that strictly speaking Vx, Vy are not the same in the Coulomb case as in
the viscous case, but for reasons which will be discussed later, the error involved
in assuming both orbits to be identical ellipses is very small.
Plotting the work function (Wc - Wv) against orbit size for zero misalignment
(Fig. 4) yields the magnitude of a limit cycle at the cross-over point. The fact
that the cross-over slope is negative indicates that this is a stable cycle. The
orbit size which refers to the length of the major axis of the ellipse at the shaft
centre is proportional to f/M.
49
Misalignments
Having fixed the orbit size for the system under consideration, the values ex,
ey are now systematically varied. Figure 5 shows that depending on the values of
these parameters the character of the work function changes. Under somecircum-
stances the function shows 2 cross-over points. The smaller orbit is unstable and
the larger one is stable. Certain combinations of the misalignments give a totally
stable system, there being a threshold at which a collapse from a finite orbit to
zero takes place.
It is possible to plot a mapof constant orbit combinations as in Figure 6.
Here it will be seen that there exists a boundary in the ex/ey plane beyond which,
for the model assumed, instability cannot exist. Within the boundary, limit cycles
exist which are smaller than the limit cycle of the zero misaligned case.
DISCUSSION
It is first necessary to examine the validity of the model with regard to its
general applicability. The semi-rigid modelling of the shaft implies certain
constraints. This does not, however, affect the qualitative behaviour of the system,
particularly that relating to the stabilizing effects of large misalignments. It
does, however, meanthat the system frequency and the orbit size will be incorrect,
but to obtain a more accurate model it is only necessary to incorporate more shaft
freedoms. Either a finite element or a modal representation could be used.
The present mathematical treatment of Coulombdamping also requires justifi-
cation. It is reasonably easy, if tedious, to arrive at limit cycles by time
marching methods or other slightly more elegant algorithms. Alternatively, a
carefully constructed analogue computer circuit can solve the problem. All of these
techniques have been used by the author on the particular problem presented in the
present paper and for this case it can be shownthat the orbits are very nearly
elliptical with little harmonic content. Thus the assumptions used in deriving the
energy expression (Wc - Wv) are valid. The reasons for this are fairly obvious.
The energy supplied by the Coulombdamping is a small proportion of the inertial
and strain energy in the system so that the orbits are largely controlled by these
latter energy exchanges. This circumstance results from the fact that the shaft is
very flexible, as evidenced by the small ratio of frequency: running speed (_/_
Figure 3).
The question arises as to what happens if the shaft is stiffer. As shaft
stiffness increases, so does the stability margin. This margin is expressed in the
present context as the viscous rotating damping necessary to reduce the system
damping to zero. The author has found that reasonably accurate results can occur
when this viscous damping margin is well over 15%.
Any rotating damping, whether viscous, hysteretic or Coulomb, reduces the
stability margin of a rotating system. This means that non-sychronous whirling
will start at a lower speed than that dictated solely by consideration of the oil
film. Since most rotors contain some rotating damping mechanism, then extractions
of the oil film coefficients from rotor/bearing systems may well be subject to
inaccuracies.
50
!More important, however, is what, in the authors view, is the erroneous
deduction from many tests and case histories, that oil film non-linearities are
responsible for the small limit cycles often experienced prior to large non-
sychronous orbits. The probability is that these limit cycles arise from the
existance of friction in the drive mechanism. To be sure oil film non-linearities
operate to limit journal orbits which are large compared with the bearing clearance,
but this is a gross condition in which the oil film becomes effective around the
complete bearing circumference.
In the present mathematical example, a single running speed has been considered
so far. Obviously as this running speed rises and its associated oil film co-
efficients change, the stability margin reduces. According to our simple model the
orbits should increase with running speed and at the 'resonant oil whirl' threshold,
should grow without limit. Under these circumstances, however, it is obvious that
constraints associated with bearing clearances etc. will take over so that the
increase in orbit size may not be as dramatic as Figure 7 predicts. In any case
the present theory deals with small vibrations and above a certain level the force/
spatial relationships within the coupling may become much more complex than those
assumed here.
The nearer the running speed approaches the 'true' oil film instability
threshold, the easier it is to supply the small amount of friction damping necessary
to promote limit cycles. Figure 8 shows a shaft weighing 700 kgs supported on 2,
I00 mm dia hydrodynamic bearings. Figure 9 shows friction being introduced by 3
small steel 'fingers' supported on a freely rotating layshaft. The pressure between
the fingers and the main shaft was introduced by elastic bands. This very small
amount of damping produced from the dynamic slope of the shaft was quite sufficient
to lower the instability threshold and produce stable limit cycles. The stabilizing
effect of large misalignments was also demonstrated by varying the angle of the
layshaft.
CONCLUDING REMARKS
The author has dealt with the subject of limit cycles in a shaft/hydrodynamic
bearing system with rotating damping present in the drive coupling. The same
conclusions, i.e. that limit cycles exist when saturable damping mechanisms are
present, would apply whenever friction was caused by shaft flexure. The point
brought out in the present paper is that the destabilizing effect of Coulomb damping
can be 'biased out' by friction arising in a fixed plane and resulting from the
rotation. In many practical situations such a biasing would be impossible in that
very heavy fretting wear would be brought about by the gross misalignments necessary
to achieve stability. Nevertheless, the mechanism is of some interest and may not
only apply to couplings, but to other sources of rubbing in rotating elements.
REFERENCES
I. Crandall, S.H.: Physical Explanations of the Destabilizing Effect of Damping in
Rotating Parts. NASA Conference Publication 2133, May 12-14, 1980, Pages 369-382.
51
F 1
[] [ []
/Z,V /7;7
Figure I. - Diagrammatic arrangement of shaft and couplings.
Figure 2. - Coordinate system.
52
1.0
0"5
Q
/
System Damping /
: o.2S_ o.zs_ o.2ss p' o.2_3
-- "-- "" _237
8
+ . . o'2311_1,
Rotating Damping Z
O'I 0._ 0.3 0"_ O'S
Figure 3. - Relationship between system damping and rotating viscous damping.
IO
5
0
Net Work
Per Cycle/ F 2
w | v
IO 20 3O l_o_Orbit
].ze / F
Figure 4. - Net work ~ orbit size.
53
Net Work Per Cycle F _
/
I
I
I
I
I
I
I
I
I
i
I
\
\
\
\
\
\
\
\
Orbit Size F
! !
Curve F F
l I0 0
2 0 5
3 20 0
Figure 5. - Effect of misalignment on work function.
54
Stable
tiles
Critical Misalignment
Figure 6. Critical misallgnment boundary compared with
orbit of fully aligned system.
30'
2o
|o
o
Orbit Ratio
Ret 50 Hz
Resonant Whirl
Running Speed Hz
!
so _o _o io_
Figure 7. Growth of orbit size with running speed.
55
////
Figure 8. -Model rotor supported on lOOmm dia. hydrodynamic bearings.
56
Figure g. - Method of introducing friction damping into model rotor.
57


Aspects of Coulomb damping in rotors supported on hydrodynamic bearings

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