1029
1 INTRODUCTION
The study of the accident-free operation of the ship's
propulsive system leads to the need to study the
operation of each unit and mechanism of this system
(see, for example, [1, 2]). Especially important in this
aspect are sliding bearings (pairs), which are present
both in the main power plant and in the auxiliary
mechanisms of the vessel [1]. Studies of the operation
of sliding pairs, which are based on the fundamental
hydrodynamic theory of sliding pairs (see, in
particular, [3]), are constantly developing [4, 5, 6, 7].
This mostly applies to the sliding bearings of the ship's
propulsion system, in particular, in work [8], the
bearings of the propeller shaft were experimentally
investigated with different lubrication methods, in
work [9], research was carried out on the operation of
the propeller shaft, taking into account its bending and
cavitation. Various aspects of the durability of the
ship's support sliding bearings were studied in works
[10, 11, 12, 13], in works [14, 15] the efficiency of ship
propellers and rudders was studied. Viscosity
characteristics of marine lubricants [16], in connection
with extreme modes of operation [17 - 19], have a non-
Newtonian character [3, 18], which is especially
noticeable when using various additives and
impurities during operation [20, 21]. Failure to take this
fact into account leads to significant errors when
calculating the durability of ship's sliding bearings.
These calculations are carried out using integral
characteristics of sliding pairs, namely dimensionless
load coefficients [1, 3, 6] and resistance to rotation and
hydrodynamic friction [3 - 6]. In [1], the impact of non-
Newtonian behavior of lubricants on the load factor
New Mathematical Models for Coefficients
of Hydrodynamic Resistance to Rotation and Friction
of Sliding Bearings of Ship Propulsion System
for non-Newtonian Lubricants
O. Kryvyi, M. Miyusov & M. Kryvyi
National University “Odessa Maritime Academy”, Odessa, Ukraine
ABSTRACT: The boundary value problem for the Reynolds differential equations for the lubricating layer in the
sliding bearings of ship power plants and auxiliary ship equipment is solved by the boundary variation method.
As a result, analytical representations for hydrodynamic pressure, shear stresses, and integral characteristics of
the lubricating layer were obtained, as well as their refinement, which considers the presence of a lubricating
layer outside the working zone of the sliding pair. This made it possible to build new, easy-to-use mathematical
models for the coefficients of rotational resistance and hydrodynamic friction of sliding bearings of the ship's
propulsion system, considering the non-Newtonian properties of lubricants, that is, in the case of the dependence
of dynamic viscosity on pressure and temperature. The obtained mathematical models consider the geometric
parameters of bearings, operational parameters: relative radial clearance and relative eccentricity, angular
velocity, as well as viscous characteristics of lubricants, in particular, dynamic viscosity and piezo coefficient of
viscosity of lubricants. A criterion for the applicability of non-Newtonian lubricants for given operating modes
of the sliding bearing has been developed, which uses the viscosity characteristics of the lubricants. The research
results are illustrated in the form of tables and graphs.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 19
Number 3
September 2025
DOI: 10.12716/1001.19.03.38
1030
was investigated, new mathematical models were
obtained that provide the load factor due to the relative
eccentricity of the sliding bearing and the viscosity
gradient. Regarding other integral characteristics, only
their tabular or graphic dependences are known [3 - 5,
22] only for Newtonian lubricants.
The purpose of this work is to study the influence
of non-Newtonian behavior of lubricants on the
coefficients of resistance to rotation and hydrodynamic
friction for marine sliding bearings and to build new
adequate mathematical models that would be
applicable for any geometric and operational
characteristics of sliding bearings and consider the
viscous characteristics of lubricants.
2 THE FORMULATION AND MATHEMATICAL
MODEL OF THE PROBLEM
Let the sliding bearing of length be L , the radius of the
sleeve be R2 and the radius of the shaft (trunnion) R1 be
in a steady state of operation, that is, the trunnion
rotates at a constant speed
0[s
-1
] (see Fig. 1). Such a
rotation will be resisted by the force of the viscous
hydrodynamic shear of the lubricating layer, which in
the coordinate system related to the normal and
tangent to the trunnion can be represented as follows:
(0; )=
TT
FF
. The value of the function
()=
TT
F F y
can
vary along the thickness of the lubricating layer (
[0; ], ( ) = y h h h
– thickness of the lubricating layer at
the angle ), and can be given due to the shear stress
φ
( , )
y
in the lubricating layer:
2
1
φ
1
( ) ( , )
2
=
T
F y y d
(1)
where
12
,
– are the beginning and end of the
working zone of the lubricating layer.
Figure 1. Slip pair model
We denote the value of FT on the trunnion as
follows: FTh=FT(h) , on the bushing as follows: FTv=FT(0).
These values, as well as the values of the moments on
the trunnion and bushing in the working zone of the
lubricating layer can be presented as follows [3]:
2
3 1 3 1
22= =
Th Th Th Th
F k LR M k LR
(2)
2
3 1 3 1
2 , 2= =
Tv Tv Tv Tv
F k LR M k LR
(3)
where
00
3
0
kg
Pa s
ms
= =
k
,
0
1
=
R
,
21
()
=−RR
–
the radial gap of the sliding pair,
0
2
Ns
m
– the
dynamic viscosity of lubricants at atmospheric
pressure. Dimensionless coefficients Th and Tv in
formulas (2), (3) are called coefficients of resistance to
the rotation of the lubricating layer on the trunnion and
bushing for the finite bearing. In the practical
calculation of the resistance to rotation of the
lubricating layer of the sliding pair and the study of the
durability of its operation, the coefficients of
hydrodynamic friction on the trunnion and bushing
are also used: fTh , fTv. These coefficients are defined as
the ratio of the hydrodynamic friction forces on the
trunnion and bushing FTh, FTv (or
Th
F
,
Tv
F
) to the value
of the radial load P [3]:
00
,
= = = =
Th Th Tv Tv
Th Tv
PP
FF
ff
PP
(5)
where P – the dimensionless load factor of the sliding
bearing in the working zone [1, 3].
Dimensionless coefficients Th and Tv related to
the corresponding coefficients of resistance to the
rotation of the lubricating layer on the trunnion and
bushing for an infinite bearing
Th
and
Tv
so:
22
.
= =
Th Tv
Th Tv
(6)
where the dimensionless coefficient
2
can be
determined in different ways [3]. Therefore, to
determine the coefficients of resistance to the rotation
of the lubricating layer on the trunnion and bushing for
an finite bearing, it is sufficient to determine the load
coefficients of an infinite bearing
Th
and
Tv
, that
is, a bearing in which there is no leakage of lubricants
at the ends, the so-called flat problem of lubrication
theory (see Fig. 1). This problem is formulated in the
form of a boundary value problem for the Reynolds
differential equation [1, 3, 11] with respect to the
distribution of the specific pressure in the lubricating
layer
1
1
( ) ( )
−
=p k p
, where
()
p
– the pressure in the
lubricating layer,
12
( , )
. For non-Newtonian
lubricants, dynamic viscosity depends on pressure and
is determined using the Barus formula [16, 19]:
( ) ( )
0 0 1
,
= = =
pp
e e k
, (7)
where
00
1
22
0
kg
Pa
m с
==
k
,
1
Pа
−
is the piezo
coefficient of lubricant viscosity. The parameters
and
0 are determined experimentally and depend on the
temperature [16 - 18].
1031
The boundary value problem of the plane
lubrication theory for non-Newtonian lubricants with
respect to the new unknown function:
()
()
−
=
p
qe
, (
1
( ) ln ( )
−
=−pq
,
( ) ( )
1
−
=hh
) has the form [1]:
( ) ( )
3
12
1 2 2
6 ,
1, ( ) 0
= −
= = =
d dq dh
h
d d d
q q q
(8)
3 DETERMINATION OF SPECIFIC PRESSURE
AND DISTRIBUTION OF VISCOUS SHEAR
STRESSES IN THE LUBRICATING LAYER.
In work [1], an exact solution of the boundary value
problem (8) was obtained, which will be written as
follows
( )
2
0
0
3
1
( ) ln ( )
ξ
cos cos
( ) 1 6 ψ
()
−
=
−
= +
pq
qd
h
(9)
where
0 the angle at which the relative hydrodynamic
pressure reaches a maximum.
Shear stresses in the lubricating layer in the plane
theory of lubrication can be presented as follows [3]
01
1
1
(2 )
2
= + −
R
dp
yh
h R d
(10)
or in dimensionless quantities so
3
11
(2 ),
2
()
= = + − =
dp y
y h y
kd
hq
(11)
Considering the formula
1 ( )
()
()
ξ
−
=
q
p
q
,
representation (11) will be rewritten in the form
0
3
()
11
3 (2 )
()
()
−
= + −
hh
yh
q
h
h
(12)
Shear stresses for non-Newtonian lubricants,
according to formula (12), vary along the thickness of
the lubricating layer. On the trunnion (at
=yh
) and on
the bushing (at
0=y
), we obtain the following
expressions:
0
2
0
φφ
2
0
4 ( ) 3
,
( ) ( )
3 2 ( )
( ) ( )
=
=
−
==
−
==
h
yh
v
y
hh
qh
hh
qh
(13)
which, considering the formula:
( )
0
1 ε cos = + h
, we
will rewrite as follows
0
φ
2
0
0
3
14
( ) 1 ε cos
(1 ε cos )
=−
+
+
h
h
q
(14)
0
2
0
0
3
12
( ) 1 ε cos
(1 ε cos )
=−
+
+
v
h
q
(15)
4 DETERMINATION OF COEFFICIENTS OF
RESISTANCE TO ROTATION AND
HYDRODYNAMIC FRICTION OF THE
LUBRICATING LAYER ON THE TRUNNION
AND BUSHING
In formula (1), let's go to dimensionless quantities,
then, considering representations (2), (3) and (11), we
will obtain expressions for the coefficients of resistance
to the rotation of the lubricating layer on the trunnion
and bushing in the form
22
11
22
11
φ
φ
1 1 1
2 2 2
11
24
()
= = =
=
h
Th
v
Tv
h dp
dd
d
hq
d dp
hd
d
qh
(16)
After integration by parts of the second integral in
(16), we obtain
22
11
0
1 1 1
ε ( )sin
24
()
=
Th
Tv
d p d
qh
(17)
Using the results of works [1,3], it is not difficult to
obtain the following formula
2
1
P εδ
1
sin ( )sin
2
=
pd
(18)
where
P εδ
,
, respectively, the load factor and the
angle of deviation of the center line of the sliding
bearing, the expressions for which are obtained in the
paper [1]. This makes it possible to obtain the following
expressions for the coefficients of resistance to the
rotation of the lubricating layer on the trunnion and
bushing
0
0
1
22
()
11
22
( ) ( )
− + − + −
−−
+
=
+
Th
c c c s s
sc
Tv
A A A A A
j
AA
(19)
Notations are introduced here
( )
( )
22
11
2
1
0 1 0 0 2 0 3
0 1 0 0 2 0 3
0
1
0
0
0
2 (3 1.5 ) 3 ;
2 (3 1.5 ) 3 .
cos
.,
( )(1 ε cos )
( ) 1 ε cos
sin
( ) 1 ε cos
= + −
= + −
==
+
+
=
+
c c c
c
s s s
s
c
k
k
s
k
k
A j h j h j
A j h j h j
dd
jj
q
q
d
j
q
It should be noted that the obtained formulas for
calculating
Th
and
Tv
somewhat underestimate
the value of the resistance to rotation, since they do not
take into account the forces of hydrodynamic friction
outside the working zone of the sliding pair. There is
1032
also a lubricating layer in which there is no radial
pressure, but there are viscous shear stresses. The
second term in representations (17) and (19)
corresponds to the hydrodynamic pressure in the
working zone of the sliding pair, and the first term,
assuming that the lubricant is supplied to the bearing
under pressure, will be extended to the entire circle of
the bearing, as a result, we will obtain the following
formulas that take into account the magnitude of the
hydrodynamic friction force outside the working area
of the sliding pair
π0
0
1
22
2
π0
1
0
0
()
11
22
( ) ( )
( )(1 ε cos )
− + − + −
−−
+
=
+
=
+
Th
c c c s s
sc
Tv
A A A A A
j
AA
d
j
q
(20)
The coefficients
Th
and
Tv
are called refined
coefficients of resistance to rotation for non-Newtonian
lubricants, with the help of the last values of
hydrodynamic friction forces and moments in the
entire bearing on the trunnion and bushing can be
written in the form
2
3 1 3 1
2 , 2
= =
Th Th Th Th
F k LR M k LR
(21)
2
3 1 3 1
2 , 2
= =
Tv Tv Tv Tv
F k LR M k LR
(22)
Refined coefficients of hydrodynamic friction on
the trunnion and bushing:
Th
f
,
Tv
f
, write as follows
00
,
= = = =
Th Th Tv Tv
Th Tv
PP
FF
ff
PP
(23)
Formulas (5) and (23), as well as (19) and (20), make
it possible, using the results of work [1], to obtain the
following expressions for the coefficients of
hydrodynamic friction
22
sc
0
0 0 c
1
22
c c s s
sc
( ) ( )
ε
2
( ) ( )
−−
−
+ − + −
−−
+
=
+
+
Th
Tv
f
AA
A
j
A A A A
f
AA
(24)
22
sc
0
0 0 c
1
22
c c s s
sc
( ) ( )
ε
2
( ) ( )
−−
−
+ − + −
−−
+
=
+
+
Th
Tv
f
AA
A
j
A A A A
f
AA
(25)
So, to calculate and study the operation of the
sliding bearings of the ship propulsion system,
together with the load factor, eight dimensionless
integral parameters (19), (20) and (24), (25) were
obtained, which we will combine into two vectors,
namely the vector of resistance to rotation and the
vector of hydrodynamic friction:
4
0
1
, , , ( , )
=
= =
T Th Tv Th Tv k
k
(26)
4
0
1
, , , ( , )
=
= =
T Th Tv Th Tv k
k
f f f f f f
(27)
For Newtonian lubricants
0
=
, the values of the
coefficients of resistance to rotation [3, 6], which are
obtained in the form of tables, are known precisely for
such a case. A further task is to obtain, in a form
convenient for use, mathematical models of integral
characteristics (26), (27) for sliding bearings for non-
Newtonian lubricants.
5 REFINEMENT OF THE CRITERION FOR THE
EXISTENCE OF A LUBRICATING LAYER
CONSIDERING THE RESISTANCE TO
ROTATION OUTSIDE THE WORKING ZONE OF
THE SLIDING PAIR FOR NON-NEWTONIAN
LUBRICANTS.
Vectors
0
( , )
T
and
0
( , )
T
f
depends both on the
relative eccentricity
0
and on the dimensionless
parameter
μ0 0
2
0
=
G
, where
μ0 0
=G
– the viscosity
gradient of lubricants at atmospheric pressure. In [1], a
criterion for the presence of a lubricating layer in the
working zone of the sliding pair is proposed:
μ
K
,
where
1
μ 0 0
()
−
=Kq
. The values of the number
K
for
some values of the relative eccentricity
0
are given
there. At the same time, if the processes of
hydrodynamic friction outside the working zone are
not taken into account, the area of parameter change
0
,
during liquid friction can be defined as follows:
0 μ
(0;1), [0; )
K
. Considering hydrodynamic friction
outside the working zone requires clarification of the
areas of change of parameter
. This is due to the
expansion of the area of change of the argument
of
the function
()
q
, to the interval
[0;2 ]
in the
integral
π0
1
j
in formulas (20), (25). Such an expansion
will be correct and the sliding pair will continue to be
in the conditions of liquid friction, as the condition
( ) 0
q
is fulfilled at
[0;2 ]
. Numerical analysis of
the specified condition led to the need to fulfill the
following criterion:
sv
K
(28)
The values
sv
K
for different values of relative
eccentricity
0
are given in Table 1.
Table 1. Values of criteria
sv
K
and
μ
K
for some values of relative eccentricity
0
0
0.01
0.1
0.3
0.4
0.5
0.6
0.7
0.75
0.80
0.90
0.95
0.99
sv
K
11.2
1.15
0.37
0.277
0.21
0.16
0.11
0.09
0.06
2.6·10
-2
9.6·10
-3
9.2·10
-4
K
14.18
1.48
0.57
0.396
0.28
0.20
0.13
0.10
0.07
2.7·10
-2
9.9·10
-3
9.2·10
-4
1033
The given results show that for all values of the
relative eccentricity
0
, the condition
μ
sv
KK
is
fulfilled. Therefore, criterion (28) clarifies the criterion
for the existence of a lubricating layer [1], obtained
using the load factor, while the viscosity gradient of
lubricants
μ0
G
must satisfy the condition
2
0
μ0
0
0
sv
K
G
(29)
Condition (29) allows specifying the maximum
possible values of the viscosity gradient of lubricants
[1], in which there are no zones of semi-dry and dry
friction in the sliding bearing.
6 NUMERICAL MODELING OF THE OPERATION
OF A SLIDING PAIR.
The results obtained above make it possible to carry
out a numerical simulation of the operation of the
sliding pair, and to determine the coefficients of
resistance to rotation and coefficients of hydrodynamic
friction on the trunnion and bushing for non-
Newtonian lubricants. The difficulty of numerical
modeling of hydrodynamic processes in a sliding pair
is that the boundaries of the working zone of the
lubricating layer
1,
2 are not known in advance. To
overcome this problem, we will apply the method of
successive approximation or the method of boundary
variation [1].
We present some results of the implementation of
the proposed numerical modeling approach. In table 2,
for different values of the relative eccentricity
0
, the
values of the usual and refined coefficients of
resistance to the rotation of the lubricating layer and
the coefficients of hydrodynamic friction on the
trunnion and bushing at
0
=
are given, respectively,
for Newtonian lubricants.
The data given in Table 2 have an independent
value and are used to calculate and predict the service
life of sliding bearings [3 - 6] of the ship propulsion
system. In addition, they can be used to build
mathematical models of integral characteristics of
sliding bearings.
Table 2. Integral characteristics of the lubricating layer for Newtonian lubricants
0
Th
Tv
Th
Tv
0
Th
f
0
Tv
f
0
Th
f
0
Tv
f
0
2.13667
2.13667
3.14159
3.14159
40.9613
37.5010
66.6619
66.653
0.025
1.92051
1.91808
3.14385
3.14131
16.4831
15.1701
26.9826
26.961
0.050
1.90679
1.89711
3.15040
3.14066
8.41423
7.79968
13.9020
13.859
0.075
1.91264
1.89137
3.16124
3.13969
5.67962
5.24869
9.3874
9.3234
0.100
1.92205
1.88440
3.17628
3.13856
4.31829
3.97110
7.1362
7.0514
0.125
1.93475
1.87762
3.19550
3.13736
3.50144
3.19881
5.7831
5.6779
0.150
1.95087
1.86979
3.21887
3.13622
2.95958
2.68144
4.8832
4.7578
0.175
1.97056
1.86083
3.24641
3.13525
2.57479
2.30950
4.2419
4.0966
0.200
1.99401
1.85063
3.27817
3.13458
2.28817
2.02828
3.76179
3.5970
0.225
2.02002
1.84197
3.31418
3.13436
2.06651
1.80820
3.39045
3.2065
0.250
2.04976
1.83243
3.35456
3.13469
1.89030
1.63002
3.09360
2.8908
0.275
2.08345
1.82193
3.39947
3.13569
1.74713
1.48220
2.85070
2.6295
0.300
2.12136
1.81038
3.44909
3.13748
1.62868
1.35702
2.64805
2.4088
0.325
2.16139
1.80079
3.50345
3.14040
1.52872
1.25014
2.47794
2.2212
0.350
2.20587
1.79049
3.56299
3.14444
1.44341
1.15658
2.33144
2.0576
0.375
2.25513
1.77942
3.62804
3.14976
1.36973
1.07356
2.20361
1.9131
0.400
2.30959
1.76752
3.69899
3.15653
1.30542
0.99904
2.09074
1.7841
0.425
2.36647
1.75844
3.77593
3.16533
1.24823
0.93231
1.99168
1.6696
0.450
2.42913
1.74895
3.85976
3.17606
1.19713
0.87108
1.90218
1.5652
0.475
2.49817
1.73905
3.95117
3.18893
1.15104
0.81439
1.82052
1.4693
0.500
2.57427
1.72873
4.05099
3.204211
1.10907
0.76150
1.74528
1.3805
0.525
2.65469
1.72124
4.15943
3.22298
1.07020
0.71255
1.67682
1.2993
0.550
2.74358
1.71385
4.27835
3.24495
1.03397
0.66612
1.61237
1.2229
0.575
2.84206
1.70667
4.40916
3.27055
0.99982
0.62180
1.55113
1.1506
0.600
2.95143
1.69987
4.55366
3.30033
0.96728
0.57927
1.49238
1.0816
0.625
3.07031
1.69565
4.71297
3.33595
0.93589
0.53878
1.43660
1.0169
0.650
3.20348
1.69268
4.89075
3.37732
0.90517
0.49943
1.38192
0.9543
0.675
3.35215
1.69113
5.08951
3.42640
0.87502
0.46138
1.32852
0.8944
0.700
3.52071
1.69202
5.31457
3.48365
0.84467
0.42406
1.27505
0.8358
0.725
3.71164
1.69544
5.57036
3.55225
0.81405
0.38766
1.22171
0.7791
0.750
3.93193
1.70374
5.86605
3.63323
0.78229
0.35166
1.16710
0.7229
0.775
4.18986
1.71285
6.21023
3.73213
0.74981
0.31631
1.11137
0.6679
0.800
4.49552
1.73131
6.62049
3.85149
0.71494
0.28120
1.05288
0.6126
0.825
4.86860
1.75049
7.11629
4.00178
0.67833
0.24626
0.99149
0.5576
0.850
5.33007
1.78991
7.73683
4.19064
0.63738
0.21136
0.92519
0.5011
0.875
5.93622
1.82512
8.35688
4.38658
0.59329
0.17665
0.85307
0.4440
0.900
6.74762
1.91303
9.62836
4.78626
0.54095
0.14170
0.77190
0.3837
0.925
7.95618
2.02841
11.23517
5.30096
0.48023
0.10669
0.67815
0.3200
0.950
9.99951
2.23619
13.94714
6.17516
0.40420
0.07145
0.56377
0.2496
0.975
14.65363
2.73676
20.11258
8.16396
0.29769
0.03591
0.40859
0.1659
0.990
24.00274
3.73706
32.42554
12.11478
0.02157
0.00169
0.26437
0.0328
1034
7 CONSTRUCTION OF MATHEMATICAL
MODELS OF COEFFICIENTS OF RESISTANCE
TO ROTATION AND HYDRODYNAMIC
FRICTION FOR NEWTONIAN LUBRICANTS
On the basis of the obtained data (Table 2), with the
help of regression analysis methods [23, 24],
mathematical models of the coefficients of resistance to
rotation and hydrodynamic friction for Newtonian
lubricants were built, that is, for the components of the
vectors
0
( ,0)
T
and
0
( ,0)
T
f
:
2
1 0 2 3 0 4 0 5
ctg( )
0
( ,0) , 1;4
+ − −
= =
s s s s s
k k k k k
b b b b b
k
ek
(30)
22
0 0 0
1 2 3 4 5 6
ctg( )
0
0
( ,0)
, 1;4
+ + − −
==
f f f f f f
k k k k k k
b b b q b b
k
f
ek
(31)
The values of coefficients
,
s
km
q
and
,
f
km
q
are given
in Tables 3, 4.
Table 3. Values of coefficients
,
s
km
q
of the mathematical
model (30)
s
km
b
m=1
m=2
m=3
m=4
m=5
k=1
0.7918
0.68977
-0.44966
2.5367
-2.74492
k=2
-0.27895
0.68275
-0.11333
1.94426
1.11316
k=3
0.68551
1.14525
-0.40149
2.36389
-2.55556
k=4
0.043376
1.15179
-0.22285
1.9894
-2.11964
Table 4. Values of coefficients
,
f
km
q
of the mathematical
model (31)
f
km
b
m=1
m=2
m=3
m=4
m=5
m=6
k=1
2.45311
-3.67829
1.29393
0.1775
3.05865
0.040243
k=2
0.19879
-2.27976
0.81077
0.2971
3.01256
0.073772
k=3
0
-0.061802
058154
0.56386
2.70177
0.12956
k=4
0
-1.81972
1.29078
0.26961
2.98797
0.05962
To verify the obtained mathematical models, their
numerical analysis and comparison with the results of
spline approximation of the data given in Table 2 were
carried out. Figures 2, 3 and 4, 5 show the dependence
on the relative eccentricity
0
, respectively, the
components of the vectors
0
( ,0)
T
and
0
( ,0)
T
f
,
which are obtained using the formulas (30), (31) (solid
lines in all figures) and spline approximations of
tabular data (dotted lines in all figures).
Figure 2 Verification of coefficients of hydrodynamic
resistance
Th
and
Tv
Figure 3. Verification of refined coefficients of hydrodynamic
resistance
Th
and
Tv
Figure 4. Verification of coefficients of hydrodynamic friction
Th
f
and
Tv
f
Figure 5. Verification of refined coefficients of hydrodynamic
friction
Th
f
and
Tv
f
The given results demonstrate the excellent
adequacy of the obtained mathematical models (30) -
(31) for dimensionless coefficients of resistance to
rotation and hydrodynamic friction in sliding bearings.
In addition, it is shown that the values of the
coefficients of resistance to rotation on the trunnion
significantly exceed their values on the bushing, and
the values of the normal coefficients on the trunnion
and bushing are significantly less than the values of the
refined coefficients, which confirms the need to use the
1035
latter when calculating the service life of the sliding
bearings of the ship propulsion system.
8 CONSTRUCTION OF MATHEMATICAL
MODELS OF COEFFICIENTS OF RESISTANCE
TO ROTATION AND HYDRODYNAMIC
FRICTION FOR NON-NEWTONIAN
LUBRICANTS
Unlike Newtonian lubricants, the integral
hydrodynamic characteristics of sliding bearings for
non-Newtonian lubricants, in addition to the relative
eccentricity
0
, also depend on the dimensionless
parameter
, which takes into account the viscosity
gradient of the lubricant
μ0
G
. The paper [1] shows how
the dimensionless load factor
P
depends on the
parameter
, figures 6-9 show the dependence on this
parameter for hydrodynamic characteristics (26) and
(27), which are obtained using formulas (19), (20) and
(24), (25), for some values of
0
.
In particular, Figures 6 and 7 are obtained for the
value of
0
0.2
=
, Figures 8 and 9 for the value of
0
0.2
=
, Figures 10 and 11 for the values of
0
0.8
=
. In
Figures 6, 8, and 10, the solid lines correspond to the
refined coefficient of hydrodynamic resistance on the
trunnion
Th
; dotted lines of the refined coefficient of
hydrodynamic resistance on the bushing
Tv
; dashed
line of the coefficient of hydrodynamic resistance on
the trunnion
Th
; dash-dotted line of the coefficient of
hydrodynamic resistance on the trunnion
Tv
. In
Figures 7, 9 and 11, the solid lines correspond to the
refined coefficient of hydrodynamic friction on the
trunnion
Th
f
; dotted lines of the refined coefficient of
hydrodynamic friction on the bushing
Tv
f
; dashed line
of the coefficient of hydrodynamic friction on the
trunnion
Th
f
; dash-dotted line of the coefficient of
hydrodynamic friction on the bushing
Tv
f
.
The graphs of the dependences of the components
of vectors (28), (29) on the parameter
ξ
shown in
Figures 6-11 show the significant influence of the
viscous properties of lubricants on the coefficients of
resistance to rotation and hydrodynamic friction for
non-Newtonian lubricants. The increase in the values
of which can reach 200% in comparison with
Newtonian lubricants, while the more significant
growth is observed in the specified coefficients of
hydrodynamic resistance and friction.
Figure 6. Coefficients of resistance to rotation, at
0
0.2
=
Figure 7. Coefficients of hydrodynamic friction, at
0
0.2
=
Figure 8. Coefficients of resistance to rotation, at
0
0.8
=
Figure 9. Coefficients of hydrodynamic friction, at
0
0.5
=
Figure 10. Coefficients of resistance to rotation, at
0
0.8
=
1036
Figure 11. Coefficients of hydrodynamic friction, at
0
0.8
=
The calculation of the values of the specified
coefficients by formulas (19), (20) and (24), (25) relates
to the need to sequentially calculate several integrals,
which is not always convenient and complicates their
application. To avoid this problem, we will build easy-
to-use mathematical models for the load factor. In
order to build mathematical models for the
components of vectors (26), (27), which would take into
account both parameters
0
ε
and
ξ
, and would be
easy to apply, following the work [1], we present the
specified components as
0 0 0
( , ) ( ,0) ( , ), 1;4 = =
s
k k k
Qk
(32)
0 0 0
( , ) ( ,0) ( , ), 1;4
= =
f
kk
k
f f Q k
(33)
For the functions
0
( ,0)
k
,
0
( ,0)
k
f
we use
representation (30), (31), for the functions
0
( , )
s
k
Q
,
0
( , )
f
k
Q
, taking into account the conditions
0
( ,0) 1=
s
k
Q
,
0
( ,0) 1=
f
k
Q
, using regression analysis
methods [24, 25], the following dependencies are
obtained
1 0 0 0
( ) ( )
0
( , ) 1 e , 1;4
+
= + =
ss
kk
qq
s
k
Qk
(34)
00
10
( ) ( )
0
( , ) 1 e , 1;4
+
= + =
ff
kk
qq
f
k
Qk
(35)
Tables 4 and 5 show values for coefficients in
representations (34) and (35) for different values of
relative eccentricity
0
ε
.
Table 4. Coefficients of the mathematical model (34)
0
ε
11
s
q
10
s
q
21
s
q
20
s
q
31
s
q
30
s
q
41
s
q
40
s
q
0.001
0.0208
-4.30313
0.02080
-4.30313
0.02284
-4.55608
0.02284
-4.5561
0.01
0.1915
-3.10068
0.19152
-3.10065
0.21617
-3.35634
0.19912
-3.1948
0.05
0.9382
-2.30208
0.93902
-2.30136
0.98667
-2.41243
1.08433
-2.5993
0.075
1.3890
-2.10250
1.39171
-2.10080
1.46413
-2.21243
1.46904
-2.2154
0.1
1.8129
-1.95832
1.81888
-1.95509
1.96775
-2.11243
1.96525
-2.1045
0.2
3.1342
-1.61430
3.18863
-1.60986
3.79965
-1.84514
3.80211
-1.8251
0.3
4.5402
-1.45396
4.61816
-1.41688
5.78581
-1.70114
5.86241
-1.6812
0.4
6.4571
-1.28261
6.66183
-1.21173
7.91429
-1.50114
8.26191
-1.5042
0.5
9.0112
-1.16015
9.47004
-1.05216
10.9536
-1.37114
11.27194
-1.3045
0.6
12.9753
-1.03838
13.96435
-0.87831
15.0108
-1.17114
15.88568
-1.1042
0.7
20.3922
-0.92060
22.61749
-0.69489
22.1662
-0.94114
24.33716
-0.8804
0.75
27.2191
-0.86450
30.74584
-0.60087
28.9554
-0.84114
31.12387
-0.6924
0.8
35.3726
-0.61821
40.39774
-0.26245
40.4798
-0.74411
43.71794
-0.5524
0.825
43.1450
-0.54821
49.82282
-0.16045
49.1789
-0.66414
53.03485
-0.4424
0.85
54.8036
-0.48421
64.09475
-0.06545
59.80970
-0.50414
66.06997
-0.3024
0.9
101.8759
-0.28421
133.56529
-0.03005
113.1045
-0.34414
127.90368
-0.1388
0.925
157.6871
-0.12421
221.46783
-0.00305
172.5020
-0.14414
205.37955
-0.0422
0.95
313.3037
-0.01421
445.62755
-0.00015
339.9728
-0.07014
514.78473
-0.0012
0.975
1014.3771
0.00021
1468.22600
0.00005
1092.743
-0.00114
1335.92775
0.0001
Table 5 Coefficients of the mathematical model (35)
0
ε
11
f
q
10
f
q
21
f
q
20
f
q
31
f
q
30
f
q
41
f
q
40
f
q
0.001
0.02783
-5.6982
0.02783
-5.6982
0.03364
-6.3551
0.03153
-6.145
0.01
0.21084
-4.6668
0.21084
-4.6666
0.29256
-5.5155
0.32754
-5.859
0.05
1.02375
-3.8338
1.0239
-3.8282
1.38360
-4.5841
1.47391
-4.759
0.075
1.51268
-3.6224
1.51301
-3.6097
2.11712
-4.4522
2.05296
-4.359
0.1
1.96945
-3.4642
1.97007
-3.4419
2.75506
-4.2522
2.77657
-4.259
0.2
3.38393
-3.0611
3.38518
-2.9818
5.56796
-3.9021
5.49161
-3.816
0.3
4.84747
-2.8637
4.8445
-2.7020
8.37827
-3.6354
8.11516
-3.4553
0.4
6.87147
-2.6682
6.84393
-2.3760
10.92305
-3.3353
10.86028
-3.1553
0.5
9.78396
-2.6009
9.6701
-2.1463
14.51684
-3.1354
13.79772
-2.7553
0.6
14.33120
-2.5183
13.9698
-1.8508
19.0495
-2.8354
18.85370
-2.4553
0.7
23.12562
-2.4670
21.9994
-1.5192
28.5020
-2.6594
27.23446
-2.0553
0.75
30.82675
-2.4185
29.3730
-1.3356
35.73759
-2.4594
33.61743
-1.7553
0.8
41.30289
-2.2250
38.3845
-0.9425
48.62480
-2.2994
45.39468
-1.4953
0.825
50.10994
-2.1450
46.6547
-0.7825
58.71969
-2.2092
56.21780
-1.4253
0.85
63.87173
-2.0950
59.8622
-0.6525
73.52462
-2.1186
66.43514
-1.1353
0.9
120.21446
-1.9350
110.7919
-0.2325
133.82293
-1.8736
123.41520
-0.8253
0.925
186.53945
-1.7850
178.8030
-0.0525
197.5991
-1.5736
179.4469
-0.4253
0.95
371.70094
-1.7250
363.9758
-0.0085
380.77676
-1.4936
341.54509
-0.2012
0.975
1095.96689
-1.45501
1261.95659
0.00008
1038.62345
-0.8513
1103.5679
-0.0001
1037
With the help of formulas (19), (20) and (24), (25),
the verification of the obtained mathematical models
(34) and (35) was carried out. In particular, Figures 12,
14, and 16 show the values of the functions
10
( , )
s
Q
and
0
1
( , )
f
Q
obtained, respectively, at
0
0.2
=
,
0
0.5
=
and
0
0.8
=
using mathematical models (34)
and (35) – solid lines, and spline approximations of the
data obtained using formulas (19) and (20) – dotted
lines. Figures 13, 15, and 17 show the values of
functions
30
( , )
s
Q
and
0
3
( , )
f
Q
, obtained,
respectively, at
0
0.2
=
,
0
0.5
=
and
0
0.8
=
, using
models (34) and (35) – solid lines, and spline
approximations of data obtained using formulas (19)
and (20) – dotted lines. In all figures, the graphs
practically coincide, which indicates the adequacy of
the obtained mathematical models (34), (35) for any
values of relative eccentricity
0
ε
.
Figure 12. Verification of the values of
1
s
Q
and
1
f
Q
, at
0
0.2
=
Figure 13. Verification of the values of
3
s
Q
and
3
f
Q
, at
0
0.2
=
Figure 14. Verification of the values of
1
s
Q
and
1
f
Q
, at
0
0.5
=
.
Figure 15. Verification of the values of
3
s
Q
and
3
f
Q
, at
0
0.5
=
Figure 16. Verification of the values of
1
s
Q
and
1
f
Q
, at
0
0.8
=
Table 6. Values of the coefficients of the mathematical model (36)
0
s
k
0
s
k
0
s
k
r
0
s
k
g
1
s
k
1
s
k
1
s
k
r
1
s
k
g
k=1
-2.65736
-1.86760
2.21040
-2.79280
2.56295
-0.56496
3.00351
-3.09635
k=2
-2.31467
-2.11704
2.05117
-2.73513
2.61995
-0.60110
2.99670
-3.08712
k=3
-3.01201
-2.03416
2.21880
-2.78207
2.82221
-0.51346
3.02088
-3.10678
k=4
-2.93559
2.21020
2.14239
-2.74060
2.85742
-0.53993
3.01546
-3.10678
1038
Table 7. Values of the coefficients of the mathematical model (37)
0
f
k
0
f
k
0
f
k
r
0
f
k
g
1
f
k
1
f
k
1
f
k
r
1
f
k
g
k=1
-5.04825
-3.44381
1.86950
-2.64420
2.72860
-0.56034
2.99328
-3.08649
k=2
-5.58041
-4.48504
1.87060
-2.51295
2.69903
-0.53122
3.01275
-3.09497
k=3
-7.19969
-5.14479
1.80720
-2.50202
3.40809
-0.37550
3.06669
-3.13289
k=4
-7.84340
-6.41556
1.70890
-2.38336
3.40199
-0.33730
3.01275
-3.09497
Figure 17. Verification of the values of
3
s
Q
and
3
f
Q
, at
0
0.8
=
Using Tables 4 and 5 and methods of regression
analysis [23, 24], we will obtain the dependences of the
coefficients of models (34), (35) on the relative
eccentricity
0
ε
in an analytical form (
1;4=k
):
, , , , ,
00
0 0 0 0 0
ctg( )
= + −
s f s f s f s f s f
k k k k k
q r g
(36)
, , , , ,
1 1 0 1 1 0 1
exp( ctg( ))
= + −
s f s f s f s f s f
k k k k k
q r g
(37)
The values of the coefficients included in formulas
(36), (37) are given in Tables 6-7.
9 CONCLUSIONS
Formulas (32) - (35), as well as formulas (30), (31) and
(36), (37) determine new mathematical models of
coefficients of resistance to rotation and hydrodynamic
friction in the sliding bearings of the ship's propulsion
system, which take into account viscous characteristics
of lubricating oil, in contrast to the existing
representations of the specified integral characteristics
of sliding bearings, which are mainly tabular or
graphical in nature [3, 4, 5, 22] and obtained only for
Newtonian lubricants.
The new mathematical models obtained for the
coefficients of resistance to rotation and hydrodynamic
friction in sliding bearings have an easy-to-use
analytical form and can be applied to the entire range
of changes in the geometric and operational
characteristics of the sliding pairs of the ship
propulsion system, such as the relative radial gap
0
,
the relative eccentricity
0
and the speed of rotation
of the trunnion
0
,
and also take into account the
viscosity characteristics of lubricants using the
viscosity gradient of lubricants
μ0 0
.
=G
In contrast
to the existing table values, the obtained new
mathematical models of coefficients of resistance to
rotation and hydrodynamic friction in sliding bearings
take into account hydrodynamic processes outside the
working zone of the sliding pair. This makes it possible
to significantly increase the accuracy of calculations of
the service life of sliding bearings of the ship's
propulsive system.
Criterion (28), (29) was developed, which
generalizes the criterion of article [1], and allows
determining at which values of the viscosity gradient
μ0
G
, the values of the coefficients of resistance to
rotation and hydrodynamic friction on the bushing and
trunnion will be finite, and the sliding bearing will be
in the conditions liquid friction. This makes it possible
to constantly monitor the trouble-free operation of the
sliding bearings of the ship propulsion system, which
are constantly in extreme operating modes. Indeed, the
viscosity gradient
μ0
G
of marine lubricants during
operation is constantly changing [1, 17, 20, 21], either
due to contamination or due to the use of various
impurities. Therefore, during long-term operation, it is
necessary to carry out the following monitoring:
− Determine, after a certain period of operation
Т
,
the viscosity gradient
μ0
G
of marine lubricants and
check the fulfillment of the criterion (29).
− Before using additives, determine the viscosity
gradient
μ0
G
of the mixture of marine lubricants
and additives and check the fulfillment of criterion
(29).
As was stated in the paper [1], this monitoring
should be performed within the shipping company in
specialized laboratories, using, for example, the work
methodology [17, 18, 19]. The time interval
Т
can be
determined experimentally for each type of engine, for
different operating conditions.
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models for the load factor of slip pairs in the ship
propulsion system for non-Newtonian lubricants.
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https://doi.org/10.31217/p.38.1.9
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of Lubricants and Lubrication. Springer, Berlin,
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22647-2_14
[3] Korovchinsky, М. V. Theoretical basis for plain bearings
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