Earlier work has shown that oil-film forces can be modelled by linear coefficients. Identification techniques were used to generate numerical values for these coefficients. This paper has shown the invalidity of applying the perturbation techniques normally used in bearing studies to derive expressions for linearized coefficients to represent a cavitated oil-film. An alternative approach was developed based upon energy techniques to obtain estimates for linearized coefficients. Some current work being undertaken suggests that an alternative analytical approach is possible. These results will be reported in due course

Burrow, C. R.

Kucuk, N. C.

Sahinkaya, M. N.

Taylor, D.

English

NASA Technical Reports Server

N87-22218
DYNAMIC PERFORMANCE OF SQUEEZE-FILM BEARINGS
C.R. Burrows, M.N. Sahlnkaya, and N.C. Kucuk
University of Strathclyde
Glasgow, Scotland
O. Taylor
Cornell University
Ithaca, New York 14853
I. INTRODUCTION
Squeeze-film elements are widely used for vibration control and force
isolation. The dynamic characteristics of these components are of great importance
in the design and analysis of rotor-bearlng systems.
It is generally assumed that Reynolds equation can be used to provide an
adequate model for a bearing oil-film. Various simplifying assumptions are used,
e.g. the short bearing approximation, to enable the equation to be solved and thus
yield an expression for the pressure distribution in the oil-film (I). The oil-
film force is obtained by integrating the pressure distribution circumferentially
and along the axis of the bearing, and this gives both positive and negative
pressure regions. It is frequently assumed for simplicity that negative pressures
cannot be sustained in a cavitated oil-film, hence the oil-film force is obtained by
performing the integration only in the positive pressure region. It follows that
the limits of integration are of great importance (2,3).
It is often assumed that squeeze-film bearing coefficients can be deduced from
those obtained from a journal bearing simply by suppressing the angular rotation.
This leads to the conclusion that the stiffness coefficients are zero whereas in
practice a squeeze-film bearing can support a dynamic load without the use of
centering springs. The limitation of this assumption was noted by Holmes (3) in
relation to the velocity coefficients. He suggested that the only case in which
the damping coefficients for a squeeze-film bearing and a journal bearing would be
equal is a full film of oil because the limits of integration for both bearing films
are then identical. This limitation has been frequently overlooked. Thus in
general the linearized coefficients used to model a squeeze-film bearing cannot be
deduced from journal bearing coefficients. The problem determining squeeze-film
coefficients has been tackled by various authors using several different approaches
e.g. (4,5).
In this paper the squeeze-film force equations, with the correct integration
limits, are used to show that the classical linearization process cannot be adopted
to derive oil-film coefficients for a squeeze-film bearing. This leads to a
discussion of the physical meaning and usefulness of linearized models to represent
squeeze-film bearings.
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https://ntrs.nasa.gov/search.jsp?R=19870012785 2020-03-20T11:50:45+00:00Z
I .2 Nomenclature
c ... etc.
ce¢
d
e
F ,F
e ¢ ,Fej ,Fcj
gl 'glj .... etc.
h
1
oil-film damping coefficients
radial clearance
mass unbalance eccentricity of the shaft
shaft displacement from bearing centre line
oil_film forces in n , n directions for squeeze-film
bearing and journal _ear_ng respectively
particular integral solutions for squeeze-film bearing and
journal bearing respectively
oil-film thickness
bearing length
ne ,n_
P
R
V
Z
Z
g
g
_o
_o
el,e 2
q_
rl
al
II
(')
( )T transpose
direction vectors defined in Fig. I
oil-film pressure in the clearance
bearing radius
velocity vector in Fig. I
axis along the bearing length
state vector
eccentricity ratio e/c
static value of
attitude angle
static value of ¢
oil-film limits
angle defined in Fig. I
small change in E
small change in
angular velocity of journal
fluid viscosity
differential with respect to time
2. OIL FILM FORCES
The pressure distribution p in a short bearing of length 1 is given by
Reynolds' equation as
(h3 3_P) = 12 _
3z _z c3
(e cos e + e ($ - _/2) sin e) (I)
where z is the distance along the longitudinal bearing axis, c is the clearance and
is the oil viscosity. Variables e, ¢ and e are defined in Fig. I, and _ is the
angular velocity of the journal (zero for a squeeze-film bearing). Integration of
equation (I) twice with respect to z and insertion of the boundary conditions p = 0,
z = ±1/2 gives the pressure distribution
p (e) = ---
c3h 3
(e cos B + e ($ - m/2) sin e) (2)
Thus the oil-film forces along the orthogonal axes, defined in Fig. I are
366
e2
F e - R1 I p cos e de
el (3)
2
F¢ - R1 p sin e de
eI
For an uncavltated film the limits are 0 to 27. For a ruptured film the
force is computed by using only the positive region and assuming zero pressure
elsewhere. A positive pressure occurs in the arc eI to e I + _ defined from
equation (I) by
cos e + e (¢ - m/2) sin 8 < 0
That is tan el--- (4)
e(q_ - _/2)
Equation (4) is central in explaining the different characteristics of journal and
squeeze-film bearings.
2.1 Journal Bearing
For small changes in the attitude angle ¢, as occurs in a journal bearing ¢ <<
_/2.,
tan e. _ 2_/e_ (5)
1
Since _ is positive e. is always positive, thus the positive pressure arc
I =
oscillates with a small amplitude around e. 0. Hence the limits of integration
can be taken as 0 and _ and the oil-film forces are
wR13
Fej= c2 (_ glj ($ - m/2) + _ g2j)
wR13
FCj= 2
c
(E g3j ($ - m/2) + _ glj)
(6)
where glj
g2j
g3j
=- 2 e (I - _) -2
_ (I ÷ 2 e2)(I _ c2) -5/2
2
_E2) -3/2
= _ (1
If _ is set to zero in equation (5) then eI = 7/2 whereas in practice it is known
that in a squeeze film the positive pressure region rotates around the bearing.
2.2. Squeeze-film Bearing
For a ruptured squeeze_film, equation (4) becomes
tan eI _ -e/e$ (7)
367
Angle ¢ can be positive or negative depending upon the position of the journal
in the clearance circle (see Fig. I), thus 81 is finite, and can take positive or
negative values. It was at this stage in hfs analysis that White (2) incorrectly
set 81 ; 0 for small values of e. This is equivalent to suppressing angular
motion o the journal, that is to derive the squeeze-film bearing coefficients from
those for a journal bearing by setting _ to zero.
If equation (7) is used, to define the limits of the positive pressure arc,
then as the journal describes an orbit in the clearance circle the cavitation region
rotates. This region is determined by the squeeze velocity and makes a complete
rotation for each rotation of the journal. The maximum pressure occurs in the
direction of the velocity vector V shown in Fig. I. Thus the oil-film behaves in a
totally different manner to that in a journal bearing where there are only small
oscillations of the cavitation region.
With the correct variable limits inserted the oil_film forces for a _ squeeze-
film bearing become (6)
uR13
F -
e 2
C
($ g1+ g2)
uRI 3
C
(8)
= 2 2 _2
where gl - 2_ cos3B I (I - E cos e I
g2 E sin eI [3 + (2-- 5 2) cos2e c2)-2 2 2 )-2= i] (I - (I _ _ cos eI
+ m (1 + 2 2) (1 -" 2 ?5/2
tan B
I
= _
g3 = _ sin 81[I - 2 co 2 81+ e2cos 2
+ _ (I -2) -3/2
+ tan-lie sin e (I - 2)-_]
I
eI -I(I - 2)-I 2 28 _2(1 - _ cos 1
since 8_ { O, gij { gl etc. and the squeeze-film forces Fe,
from Fej] and FCj.
F cannot be derived
It can be seen from equations (6) and (8) that the oil_film forces are non_
linear functions of the states (_, &, ¢, ¢)
3. LINEARITY
There are many analytical benefits to be gained if linear models can be derived
which adequately represent these non-linear forces. If we define a state vector as
_, one possibilSty is to seek to linearize the system about the equilibrium position
Z defined by Z__ = (s o, O, ¢o' 0). We then assume small perturbations q,_ about Z o,
--0
368
that is
(co + , , ÷ ' (9)
The procedure defined in equation (9) is used in most of the literature which
is concerned with deriving llnearized oil-film coefficients. When this approach is
applied to a _-film Journal bearing it yields expressions for four stiffness and
four damping coefficients (7). As noted earlier some workers have incorrectly
suggested that by setting m- 0 in these expressions the resulting stiffness and
damping coefficients are obtained for a squeeze,film bearing. To llnearlze about
Z for a squeeze'film we must use equation (8) Consider one coefficient
-o
aF
e
e¢ a¢
Z
-o
aF
e
Z
-o
c a¢ a¢
Z
-o
ag I
z
-o
[6 E cos 2 O I sin e I (I - c2cos 2 oI)
i
2 28 2 2 a eI I+ 8 e cos 3 eI (I - E cos I)(E cos eIsin eI )]
a¢ I,z
- o
An expression can be derived for ag2/a$, but this tedious operation need
-I
not be performed. Now eI --tan -_T-' but at the equilibrium point
thus c cannot be evaluated. Hence the classical approach toeI is undefined, e
obtaining linearized oil-_im coefficients cannot be applied to a cavitated squeeze-
film bearing. The question arises: do linear oil-film coefficients have any
meaning for a cavitated squeeze-film bearing? To answer the question it is
necessary to appreciate the physics of the situation.
The linearization described above is performed about a point which is usually
defined as the centre of the orbit. In practice state Z is not achieved, that is
-o
there is no point on the orbit that both velocities are slmultaneously zero. Thus
we must reject the concept of linearized coefficients for a ruptured squeeze-film
bearing, or adopt an alternative approach to obtaining equivalent linearized
coefficients (4,5) or seek a new analytical approach to the problem.
4. FURTHER CONSIDERATIONS
Consider a perturbation _ in E with _ - O.
369
If _ is positive the shaft moves against the thin part of the oil-film and a
large negative radial force is produced. If _ is negative the shaft squeezes a
thicker film and the magnitude of the radial force is lower, as shown in Fig. 2.
The force is linear in velocity and the slope depends upon E . However the force
also depends upon the sign of _ (or _) and o
F (_) _ _ F (-_)
e e
This essentially non'linear behaviour is not reproduced by setting
the journal bearing expressions as shown in Fig. 2.
_ = 0 in
Now consider perturbations in with c = O. If is positive a radial force
is produced which seeks to centralise the journal. The magnitude of the force
depends upon _ and e . As shown in Fig. 3 if _ is negative the same centralising
force is produced, th°t is
e e
Once again this highly non-linear behaviour disappears when we compute the
bearing forces from the journal bearing expressions with _ = 0 (Fig. 3).
Figures 2 and 3 demonstrate why the classical approach to linearization breaks
down, namely that the principle of superposition is violated.
CONCLUSIONS
Earlier work by the authors (5) has shown that oil-film forces can be modelled
by linear coefficients. In that work they used identification techniques to
generate numerical values for these coefficients. This paper has shown the
invalidity of applying the perturbation techniques normally used in bearing studies
to derive expressions for linearized coefficients to represent a cavitated oil-film.
Hahn (14) has developed an alternative approach based upon energy techniques to
obtain estimates for linearized coefficients. Some current work being undertaken
by the authors suggests that an alternative analytical approach is possible.
These results will be reported in due course.
REFERENCES
I) Mohan, S., Hahn, E.J. "Design of Squeeze Film Damper Supports for Rigid
Rotors", ASME, Journal of Eng. for Industry, Vol. 96, 1974, p976.
2) White, D.C. "The Dynamics of a Rigid Rotor Supported on Squeeze Film
Bearings", Conference on Vibrations in Rotating Systems, London, Feb. 14-15,
1972, p213.
3) Holmes, R. "Vibration and its Control in Rotating Systems". IUTAM Symposium
on Dynamics of Rotors, Lyngby, Denmark, August 12"16, 1974, p.156.
4) Hahn, E.J. "Equivalent Stiffness and Damping Coefficients for Squeeze Film
Dampers", I. Mech.E. Conf., Vibrations in Rotating Machinery, York 1984, Paper
C325/84, p507.
370
_)
T)
Burrows, C.R, Sahlnkaya, M.N. and Kucuk, N.C. "Modelling of Oil-Film Forces
in Squeeze-Film Bearings". ASME/ASLE 1985 Trlbology Conference, October 8-I0,
1985, Paper 85-Trlb. 19. To be published in Journal of Lubrication
Technology.
Caprlz, Q. "Sulle Vibrazlonl Delle Aste Roberti", Ann. Scuola Normale
Superlore, Vol. 17, 1963, p31.
Holmes, R. "The Vibration of a Rigid Shaft on Short Sleeve Bearings", Journal
of Mech. Eng. Science, Vol. 2, 1960, P337.
Y
q x
Line
C
@
Cb- Centre of the bearing
C1. - Centre of the journal
Fig. I Bearing coordinate axes
371
\
\
\
\
\
F_
\
\
\
\
"6oo -
600
400
200
--- Joumal Bearing
-- Squeeze-Firm Bearing
3
"_ 3o_o 6oo6
_,oo
- 400
- 6OO
Fig. 2 Oil-film force in n direction plotted against
e
!
-0.2
!
-0.I
2000
--- Jourr_l Bearing
-- Squeeze-FiLm Beoring
_. o._' o._ 0.3'
-2000
Fig. 3 Oil-film force in n
e
372
direction plotted against
C-_


Dynamic performance of squeeze-film bearings

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