HIBRID JOURNAL BEARINGS
WITH POINT INJECTION PORTS
V. Nosov, J. Gómez-Mancilla, I. Ramírez-Vargas
Instituto Politécnico Nacional IPN, Vibrations & Rotodynamics Lab, Zacatenco,
México, D.F. MEXICO
Abstract. This work deals with modeling and determination of the stationary
eccentricity and attitude angle for a short journal bearings with single external
pressurization injection port located at an arbitrary point. In case of hybrid journal
bearings, both hydrodynamic and injection force are simultaneously considered in
Reynolds Equation. Modeling of the point injection port is performed using a Dirac
spatial delta function. The resulting pressure field is given by an analytical expression
allows determining (also in analytical form) the pressure force components,
equilibrium eccentricity and attitude angle. Special cases which correspond to an
injection port located at the upper or lower part of the journal bearing are analyzed.
1. INTRODUCTION
One of the main problems in rotating machinery is related with instability and
excessive vibrations of different parts which conduct to premature wearing and severe
damage of the equipments. One way to attenuate and to control vibrations existing in
rotating machinery is the use of the external pressurization by means of some
injection ports (1,2). The first step in the study of this phenomenon is a correct
modeling of the dynamic behavior for the externally pressurized journal bearings and
also a correct determination of its main static and dynamic characteristics. When an
hydrodynamic journal bearing is externally pressurized, it is possible to modify the
dynamic properties of the oil film, decreasing the instability velocity and vibration
level (1). Nowadays there are companies dedicated to the manufacturing and
experimental researches of externally pressurized rotating equipment, one of these is
BENTLY-NEVADA CORP (2), which proposes to use 4 symmetrically situated
injection ports for external pressurizing of journal bearing.
In this paper we develop a principally new model of the point injection port based on
spatial Dirac delta function  ( x) (also called generalized Dirac function or Dirac
distribution) which has special properties helping to study this model (7, 4). This
concept has been proposed in our previous papers (4). The Dirac delta function is
currently used in different physical problems to model point masses, point electrical
charges, point sources and so on (4, 7). As we know, in rotodynamic problems Dirac
delta function has never been used. Special properties of Delta function simplify the
analytical solution of difficult problems related with externally pressurized journal
bearings and permit to obtain some qualitative and quantitative characteristics of such
bearings. These results can be useful to understand the process of pressurization and
also they may be used in preliminary stages of pressurized bearings design. Also it is
evident that our models of journal bearings based on delta function have theirs proper
restrictions and cannot be consider as an universal model for all type of pressurization
problems.
7-62
2. CLASSICAL OCVIRK SOLUTION FOR SHORT JOURNAL BEARINGS
In this section some known result necessary for us from Ocvirk short bearings theory
are given in condensed form. The well known Reynolds equation describing the
pressure field in hydrodynamic bearings can be written in a following form (3, 5, 6,
and 8):
  3 p 
  p 
 R2 


h
  R 2  h 3   12
 Cr  Cos  C r  (  ) Sin 
(1)






   
z  z 
Cr3 
2


L
L
z ,
2
2
L
p( )  0 ,
2
0  θ  2π ,
h(  )  1  εCosθ
(2)
L
p (  )  0 , p(  2 )  p( ) ,
2
(3)
In (1)-(3) and below we always use the notations from (6). In this work, the main
interest is connected with the short journal bearings approximation, which is valid for
sufficiently little value of the ratio L/D. The Reynolds equation for short journal
bearings in stationary state can be written as (3, 5, and 6):
  3 p 
 R2  

2
h
  12
   Sin 
R
(4)




z  z 
Cr2 
2

Passing to dimensionless variables
z
L
z,p
2
2
 
, pdim   N  R     

pdim
 
p
 R 


 2  C r 
 Cr 
2
(5)
transform Eq. (4) to the form:
  3 p 
L
h
  12    Sin


 
z  z 
D
2
(6)
The solution to (6) is (3, 5 and 6):
2
 Ocv Sin
L
z 2  1
pOcv  6  
3
 D  (1   Ocv Cos )
(7)
Solution (7) is known as Ocvirk solution, and for this solution the relation between
attitude angle Ocv and eccentricity  in stationary position is given by (6):
Ocv
  1   2 Ocv
4 Ocv

 Ocv  arcTan 

.


(8)
The pressure force components in the immovable coordinated system XOY (Fig. 1)
are (6):
2
2

 2  pres
 L   4 pres
FX ,Ocv   Fdim  
Cos  pres 
Sin  pres 

(1   2 pres ) 3 / 2
 D   (1   2 pres ) 2

FY ,Ocv
L
  Fdim  
D
Fdim  DLP
2


4 2 pres
 2  pres

Sin  pres 
Cos  pres 
 (1   2 pres ) 2

(1   2 pres ) 3 / 2


dim
7-63
(9)
3. NEW MODEL OF POINT PRESSURIZATION PORT
Consider two systems of coordinates, XOY is a fixed and X´JY´ is a moving system.
The point O in Fig. 1 is the center of the bearings and the point J is the center of the
journal. The angle  refers to the angle in the circumferential coordinate system


X´JY´ as it is shown in Fig.1. Vectors U R and U T are unit vectors of the mobile
rectangular system X´JY´, the axe Z is perpendicular to the plane XOY. The angle
between the axes of the mobile system X´JY´ and the fixed system XOY is notated by
 pres and is called attitude angle. The angular position of the injection port is given
by the angle  in the fixed system XOY and by the angle         pres in the mobile
system X´JY´.
Fig 1. Coordinate systems: XOY-fixed, XJY¨ – mobile,  pres -attitude angle,  –
circumferential coordinate of the injection port in XOY system.
Now let us consider the external pressurization point port situated in arbitrary
position.
Suppose the pressurization is performed by the injection in the point port with the
axial dimensionless coordinate z  a and the circumferential coordinate “  ”.
Suppose also the surface of the port s is sufficiently small and the pressure surplus
is p . The total pressurization force is equal: Fpres  ps . Suppose now
that s  0 , p   in such a way that pressurization force guards its constant value:
Fpres  ps  q  const . . In this case we can model a punctual pressurization port
using dimensionless pressure distribution given by spatial Dirac delta function (4)
(p ) prt  q prt ( z  a) (  (     pres )), q prt 
Fpres
DLpdim

Fpres
Fdim
(10)
Here q prt is dimensionless pressurization intensity, “ a ” is dimensionless axial
position and “  ” is circumferential position of the injection port in the fixed
coordinate system XOY .The above considerations can be also generalized to the case
of injection in “n” arbitrary situated ports. Therefore, our mathematical model based
on Dirac delta function can be used to represent the pressurization performed by
injection through an arbitrary number of ports with sufficiently small surface.
7-64
Suppose now the pressure field for our model of pressurization through a point port is
given by following Reynolds dimensionless equation with boundary conditions:
  3 p   L 
 h     q prt  ( z  a)   (     pres ),
z  z   D 
 1  z  1, 0    2
2
p ( z  1)  0,
(11)
p (  2 )  p ( )
(12)
Integrating Eq. (11)-(12), and using the properties of Dirac delta function we obtain
the pressure field due to pressurization in the arbitrary positions in the form:
 L  q prt   (     pres ) 
1  az  abs( z  a )
p pres ( , z )   
3
2(1   presCos )
D
2
(13)
Solution (13) is defined in terms of Dirac delta function, thus it is not possible to draw
in a simple way the pressure field. Nevertheless using an appropriate approximation
of delta function we can give an approximation of pressure field. For example,
suppose delta function is approximated by following function (7):
   (     pres )  
n

e

 n 2  (    pres )
2
(14)
In this case we have that total pressure field is equal to the sum of Ocvirk pressure (7)
and the pressure produced by pressurization (13). The following approximate 3D
presentation of total pressure field for n  10 in (14) is shown in the Fig. 2a and 2b.
a)
b)
Fig. 2 Total approximate pressure field a) upper injection, b) lower injection
for delta function replaced by (14) and
a  0,
n  10 ,   180 ; S  4 , L D  1 4 ,   0.3287 ,   66.08 ,
q prt  5 / 2 , f prt  10 .
Note that some integral characteristics of the pressure field in externally
pressurized bearings (such as average pressure, force components and so on) can
be easily calculated and drawing without any approximation of delta function. For
example, the average pressure (over θ) for the pressure field given by (13) is equal:
2
q prt
1 2
1 L
1  az  abs( z  a)
p prom   p pres ( , z )d   
2 0
4  D  (1   presCos(     pres ))3
7-65
(15)
4. PRESSURE FORCES COMPONENTS
Now determine the pressurized force components. Vertical force component FX is
defined in dimensionless form as
FX 
Fdim
4
 p( , z ) Cos(  
pres
)d dz
(16)
Integrating over z and θ we obtain following expressions for both vertical and
horizontal pressurized force components:
FX , pres
(1  a 2 )Cos(   )
L


 Fpres  
 D  8(1   presCos(     pres )) 3
(17)
FY , pres
(1  a 2 ) Sin (   )
L
 Fpres  
 D  8(1   presCos(     pres )) 3
(18)
2
2
The total vertical pressure force component is equal to the sum of Ocvirk pressure
force components (9) and pressurized force component (18):
L
FX , result  FX , Ocv  FX , pres   Fdim  
D
2
 4 2 pres

 2 pres
Cos pres 
Sin pres  

2
2
2
3/ 2
(1   pres )
 (1   pres )

(1  a 2 ) Cos (   )
L
Fpres  
3
 D  8(1   presCos (     pres ))
2
(19)
In the same way for horizontal component we obtain
2
2

 2 pres
 L   4 pres
FY , result  FY , Ocv  Fpres   Fdim   
Sin


Cos


pres
pres
2
2
(1   2 pres ) 3 / 2
 D   (1   pres )

(1  a 2 ) Sin (   )
L
Fpres  
3
 D  8(1   presCos (     pres ))
2
(20)
5. EQUILIBRIUM ATTITUDE ANGLE AND ECCENTRICITY FOR
PRESSURIZED BEARING
The equilibrium attitude angle and eccentricity in pressurized short bearing is
determined from forces balance equations:
W  FX ,result  0,
(21)
FY ,result  0,
For one point port Eq. (21) takes form:
2
2

 2 pres
 L   4 pres
W  Fdim   
Cos


Sin  pres  
pres
2
2
2
3/ 2
(1   pres )
 D   (1   pres )

(1  a )Cos (   )
L
Fpres  
0
3
 D  8(1   presCos (     pres ))
2
2
7-66
(22)
L
Fdim  
D
2


4 2 pres
 2  pres
Sin pres 
Cos pres  

 (1   2 pres ) 2

(1   2 pres ) 3 / 2
(23)
(1  a 2 ) Sin (   )
L
F pres  
0
 D  8(1   presCos (     pres )) 3
2
In general case Eq. (22) is a system of 2 nonlinear equations with respect to unknown
equilibrium attitude angle  pres and equilibrium eccentricity  pres . In general case
the solution of Eq. (22), (23) can be obtained only by numeric calculation
6. UPPER VERTICAL PRESSURIZATION PORT
In this case system (22) can be solved analytically. For upper vertical port we have the
angle    and for lower port   0 . From (18) follows that in these both cases
horizontal component of pressurization force is equal to zero: FYpres
 0 . Thus the
relation between equilibrium attitude angle  pres and eccentricity
same as in Ocvirk theory and is given by same relation (8):
 pres is just the
Tan  pres 
 1   2 pres
(24)
4 pres
Now using the first equation of the system (22) and replacing Cos pres and
Sin  pres by theirs expressions (3, 6) we can determine equilibrium eccentricity of the
short pressurized journal bearings. Really:
W
L
Fdim  
D
2
 4 2 pres
4 pres
 2 pres


2
2
2
 (1   pres ) 16 2 pres   2 (1   2 pres ) (1   pres )
L
Fpres  
D
2
(1  a )
2


4 2 pres

8 1 
2
2
2


16



(
1


)
pres
pres


3


  2 (1   2 pres ) 

16
2
pres
(25)
0
2
Fdim  NDL  R 

Introduce now Sommerfeld number S 
  . Note Sommerfeld
W
W  Cr 
number S depends only on parameters of bearings and is independent to external
pressurization. After some algebraic transformations Eq, (25) passes into:
2
2 2 
3

(1  ( upper
(1  a 2 )( upper
D
pres ) )
pres )




S
  f prt
upper upper 
upper
upper 2 3 
  pres  pres  L 
8 ( pres  4( pres ) ) 
upper 2
2
upper 2
 upper
pres  16 ( pres )   (1  ( pres ) ) , f prt 
7-67
(26)
Fpres
W
,
(27)
For upper position of pressurization port the behavior of attitude angle and
eccentricity given by (26) are shown in Fig. 3a). The eccentricity increases when the
Sommerfeld number S decreases for all pressurizations and that for each value of
Sommerfeld number corresponds one and only one eccentricity  upper
pres . Eq.(26)
permits to obtain Sommerfeld number S as a function of eccentricity, pressurization
and (L/D) value (in the region of Ovrick approximation validity). Remark that the
attitude angle  upper
is just the same for a fixed eccentricity  upper
and different
pres
pres
pressurizations. As to any Sommerfeld number S corresponds one and only one
eccentricity it is possible to obtain eccentricity and attitude angle as a function of
Sommerfeld number and pressurization. Performing spline interpolation of previous
data, we obtain Table 1 where equilibrium eccentricities  upper
and attitude angles
pres
are given as a function of Sommerfeld number and pressurization. Note that the
 upper
pres
eccentricities  upper
and attitude angles  pres have monotonic variation when
pres
pressurization increases. The importance of Table 1 is fundamental, it allows in a
practical and easy way to predict the possible values of the real equilibrium
eccentricity and attitude angle. It is clear that tables similar to Table 1 can be obtained
also for any other value of (L/D), because Eq. (26) allows this modification.
upper
7. LOWER VERTICAL PRESSURIZATION PORT
If pressurization port is located in lower part of vertical axis when 
0.
In this case
equilibrium eccentricity  lower
and attitude angle 
have a different behavior
pres
from the case analyzed above. In fact, if the pressurization vertical component force is
bigger than the weight, FX pres  W , when equilibrium position of the journal can
lower
pres
displace in another coordinate quadrant. Thus, there appear two cases.
Case 1. Pressurization Vertical Component Force is Less than Journal Weight,
FX pres  W .
It is clear that there exist only one possible quadrant for the journal to be in
equilibrium; it is the quadrant III. Taking into consideration the above figure and
performing a force balance such as (22) it can be obtained the following relation:
2
lower
1 2 2 
(1  ( lower
(1  a 2 )( pres 1 ) 3 
D
pres ) )




S
  f prt
lower1
lower 1 lower 1 
lower 1 2 3 
 pres  pres  L 
8 ( pres  4( pres ) ) 
Here FX
p res
(28)
 W , and so we have
2
lower 1 2 3
1
 D  8 ( pres  4( pres ) )
 
lower
(1  a 2 )( pres 1 )3
 L
lower
f prt
(29)
Determine now the admissible values of pressurization in this case. Fig. 3a) shows
right part of (29) as a function of eccentricity for ( L / D)  1 / 4 . It is clear that:
D
0  f prt  128  8  
L
2
. Fig. 3b) shows the dependence of Sommerfeld number on
eccentricity given by (28) for several values of pressurization force. Fig.3b) shows a
7-68
physically clear fact that the eccentricity decrease when the pressurization force
increase but is less that rotor system weight.
Case 2: Pressurization Vertical Component is Bigger than the Weight FX
pres
W .
In this case the pressurization force is higher than the weight of rotor system. In third
quadrant III we have Cos  0, Sin  0 and we obtain:
2
lower
lower
(1  ( pres 2 ) 2 ) 2 
(1  a 2 )( pres 2 ) 3
D 
 f prt
S
   
lower2 lower 2
lower2
lower2 2 3

 pres  pres 
8 ( pres  4( pres ) )  L  
(30)
In this case f prt  128 . Eqs. (26), (28), and (30) are similar but mot identical. To
indicate it we have used different notations for equilibrium eccentricity in these 3
cases. Fig. 3b) shows the dependence of Sommerfeld number (30) on pressurization
force.
a)
b)
c)
Fig. 3. Sommerfeld number as a function of eccentricity for inferior pressurization: a)
upper pressurization, b) lower pressurization, case 1, c) lower pressurization, case 2.
Fig.3c) shows that in case 2 the eccentricity increase when pressurization increases. In
this case the bearing has the behavior similar to hydrostatic bearing.
8. CONCLUSIONS
Modeling which uses spatial Dirac function in the Reynolds equation gives a simple
analytic way to determine the pressure field in short bearing which can be pressurized
7-69
by any number of fluid injection ports with arbitrary axial and circumferential
positions.
The dependences given in Fig. 3 between pressurization and eccentricity are
calculated for special cases of upper and lower vertical position of point
pressurization port
Table 1 allows finding eccentricity values and attitude angle for given values of
Sommerfeld number and for a different pressurization force or vice versa. This table
can be used for prediction of equilibrium position in real situations.
TABLE 1. UPPER INJECTION PORT, (L/D)=1/4.
Eccentricity and attitude angle of a pressurized bearing with upper location of
pressurization port for a given Sommerfeld number.
f prt  0
f prt  10
f prt  20
f prt  50
f prt  100
 upper
pres
upper
 upper
pres  pres
 upper
 upper
pres
pres
upper
 upper
pres  pres
 upper
 upper
pres
pres
 upper
pres
0.02
0.9397
15.82
0.9400
15.78
0.9403
15.74
0.9412
15.62
0.9426
15.43
0.05
0.9031
20.44
0.9037
20.37
0.9043
20.29
0.9062
20.07
0.9090
19.73
0.10
0.8624
24.85
0.8633
24.77
0.8641
24.69
0.8665
24.47
0.8701
24.11
0.20
0.8129
29.05
0.8146
28.88
0.8162
28.72
0.8205
28.30
0.8264
27.76
0.30
0.7706
32.91
0.7725
32.76
0.7743
32.61
0.7793
32.20
0.7865
31.60
0.50
0.7072
38.13
0.7099
37.92
0.7125
37.71
0.7197
37.14
0.7302
36.30
0.80
0.6375
43.49
0.6415
43.19
0.6452
42.91
0.6554
42.14
0.6696
41.05
1.00
0.6000
46.31
0.6047
45.96
0.6091
45.63
0.6209
44.74
0.6375
43.50
1.30
0.5522
49.85
0.5579
49.43
0.5633
49.04
0.5775
47.98
0.5971
46.53
1.50
0.5245
51.88
0.5309
51.41
0.5369
50.98
0.5526
49.82
0.5738
48.26
2.00
0.4660
56.14
0.4739
55.57
0.4811
55.05
0.5001
53.67
0.5249
51.85
2.50
0.4187
59.57
0.4277
58.92
0.4360
58.32
0.4574
56.76
0.4854
54.74
3.00
0.3793
62.42
0.3893
61.70
0.3984
61.04
0.4218
59.35
0.4523
57.14
3.50
0.3461
64.82
0.3567
64.06
0.3664
63.36
0.3914
61.54
0.4237
59.21
4.00
0.3176
66.88
0.3287
66.08
0.3389
65.35
0.3650
63.45
0.3989
61.00
5.00
0.2720
70.19
0.2835
69.36
0.2940
68.60
0.3217
66.59
0.3573
64.02
7.00
0.2092
74.75
0.2204
73.94
0.2308
73.19
0.2586
71.17
0.2956
68.48
8.00
0.1870
76.3
0.1974
75.61
0.2078
74.86
0.2350
72.88
0.2722
70.18
10.00
0.1538
78.78
0.1639
78.04
0.1731
77.37
0.1979
75.58
0.2340
72.95
15.00
0.1073
82.16
0.1155
81.56
0.1225
81.05
0.1412
79.69
0.1724
77.42
S
ACKNOWLEDGEMENT
This work was partially sponsored by the S.N.I. and the EDI scholarships granted by
the CONACyT, and the Instituto Politécnico Nacional de México, IPN, respectively.
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