ISO 7902-1:1998
(Main)Hydrodynamic plain journal bearings under steady-state conditions — Circular cylindrical bearings — Part 1: Calculation procedure
Hydrodynamic plain journal bearings under steady-state conditions — Circular cylindrical bearings — Part 1: Calculation procedure
Paliers lisses hydrodynamiques radiaux fonctionnant en régime stabilisé — Paliers circulaires cylindriques — Partie 1: Méthode de calcul
Hidrodinamični radialni drsni ležaji za neprekinjeno obratovanje - Valjasti ležaji - 1. del: Postopek dimenzioniranja
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INTERNATIONAL ISO
STANDARD 7902-1
First edition
1998-04-01
Hydrodynamic plain journal bearings under
steady-state conditions — Circular
cylindrical bearings —
Part 1:
Calculation procedure
Paliers lisses hydrodynamiques radiaux fonctionnant en régime stabilisé —
Paliers circulaires cylindriques —
Partie 1: Méthode de calcul
A
Reference number
ISO 7902-1:1998(E)
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ISO 7902-1:1998(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide
federation of national standards bodies (ISO member bodies). The work of
preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which
a technical committee has been established has the right to be represented
on that committee. International organizations, governmental and non-
governmental, in liaison with ISO, also take part in the work. ISO
collaborates closely with the International Electrotechnical Commission
(IEC) on all matters of electrotechnical standardization.
Draft International Standards adopted by the technical committees are
circulated to the member bodies for voting. Publication as an International
Standard requires approval by at least 75 % of the member bodies casting
a vote.
International Standard ISO 7902-1 was prepared by Technical Committee
ISO/TC 123, Plain bearings, Subcommittee SC 4, Methods of calculation of
plain bearings.
ISO 7902 consists of the following parts, under the general title
Hydrodynamic plain journal bearings under steady-state conditions —
Circular cylindrical bearings:
— Part 1: Calculation procedure
— Part 2: Functions used in the calculation procedure
— Part 3: Permissible operational parameters
Annexes A and B of this part of ISO 7902 are for information only.
© ISO 1998
All rights reserved. Unless otherwise specified, no part of this publication may be reproduced
or utilized in any form or by any means, electronic or mechanical, including photocopying and
microfilm, without permission in writing from the publisher.
International Organization for Standardization
Case postale 56 • CH-1211 Genève 20 • Switzerland
Internet central@iso.ch
X.400 c=ch; a=400net; p=iso; o=isocs; s=central
Printed in Switzerland
ii
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INTERNATIONAL STANDARD ISO ISO 7902-1:1998(E)
Hydrodynamic plain journal bearings under steady-state
conditions — Circular cylindrical bearings —
Part 1:
Calculation procedure
1 Scope
This part of ISO 7902 specifies a calculation procedure for oil-lubricated hydrodynamic plain bearings, with
complete separation of the shaft and bearing sliding surfaces by a film of lubricant, used for designing plain
bearings that are reliable in operation.
It deals with circular cylindrical bearings having angular spans W, of 360°, 180°, 150°, 120° and 90°, the arc
segment being loaded centrally. Their clearance geometry is constant except for negligible deformations resulting
from lubricant film pressure and temperature.
The calculation procedure serves to dimension and optimize plain bearings, in turbines, generators, electric motors,
gear units, rolling mills, pumps and other machines. It is limited to steady-state operation, i.e. under continuously
driven operating conditions, with magnitude and direction of loading as well as the angular speeds of all rotating
parts constant. It can also be applied if a full plain bearing is subjected to a constant force rotating at any speed.
Dynamic loadings, i.e. those whose magnitude and direction vary with time, such as can result from vibration effects
and instabilities of rapid-running rotors, are not taken into account.
2 Normative references
The following standards contain provisions which, through reference in this text, constitute provisions of this part of
ISO 7902. At the time of publication, the editions indicated were valid. All standards are subject to revision, and
parties to agreements based on this part of ISO 7902 are encouraged to investigate the possibility of applying the
most recent editions of the standards indicated below. Members of IEC and ISO maintain registers of currently valid
International Standards.
ISO 3448:1992, Industrial liquid lubricants — ISO viscosity classification.
ISO 7902-2:1998,
Hydrodynamic plain journal bearings under steady-state conditions — Circular cylindrical
bearings — Part 2: Functions used in the calculation procedure.
ISO 7902-3:1998, Hydrodynamic plain journal bearings under steady-state conditions — Circular cylindrical
bearings — Part 3: Permissible operational parameters.
ISO 7904-2:1995, Plain bearings — Symbols — Part 2: Applications.
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ISO 7902-1:1998(E)
3 Basis of calculation, assumptions and preconditions
3.1 The basis of calculation is the numerical solution to Reynolds' differential equation for a finite bearing length,
taking into account the physically correct boundary conditions for the generation of pressure:
¶ ¶p ¶ ¶p ¶h
33
h+ h =+6huu . . . (1)
()
JB
¶x ¶xz¶ ¶z ¶x
The symbols are given in clause 5.
See [1] to [3], and [11] to [14] in annex B, for the derivation of Reynolds' differential equation and [4] to [6], [12] and
[13] for its numerical solution.
3.2 The following idealizing assumptions and preconditions are made, the permissibility of which has been
sufficiently confirmed both experimentally and in practice.
a) The lubricant corresponds to a Newtonian fluid.
b) All lubricant flows are laminar.
c) The lubricant adheres completely to the sliding surfaces.
d) The lubricant is incompressible.
e) The lubricant clearance gap in the loaded area is completely filled with lubricant. Filling up of the unloaded area
depends on the way the lubricant is supplied to the bearing.
f) Inertia effects, gravitational and magnetic forces of the lubricant are negligible.
g) The components forming the lubrication clearance gap are rigid or their deformation is negligible; their surfaces
are ideal circular cylinders.
h) The radii of curvature of the surfaces in relative motion are large in comparison with the lubricant film
thicknesses.
i) The lubricant film thickness in the axial direction (z-coordinate) is constant.
j) Fluctuations in pressure within the lubricant film normal to the bearing surfaces (y-coordinate) are negligible.
k) There is no motion normal to the bearing surfaces (y-coordinate).
l) The lubricant is isoviscous over the entire lubrication clearance gap.
m) The lubricant is fed in at the start of the bearing liner or where the lubrication clearance gap is widest; the
magnitude of the lubricant feed pressure is negligible in comparison with the lubricant film pressures.
3.3 The boundary conditions for the generation of lubricant film pressure fulfil the following continuity conditions:
— at the leading edge of the pressure profile: p(j , z) = 0
1
— at the bearing rim: p(j, z = – B/2) = 0
— at the trailing edge of the pressure profile: p[j (z), z] = 0
2
— and ¶p/¶j[j (z), z] = 0
2
For some types and sizes of bearings, the boundary conditions may be specified.
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ISO 7902-1:1998(E)
In partial bearings, if the following expression is satisfied:
p
jp−−<()b
2
2
then the trailing edge of the pressure profile lies at the outlet end of the bearing:
pz(,jj==)0
2
3.4 The numerical integration of the Reynolds' differential equation is carried out — possibly by applying
transformation of pressure as suggested in [3], [11] and [12] — by a transformation to a differential equation which
is applied to a grid system of supporting points, and which results in a system of linear equations. The number of
supporting points is significant to the accuracy of the numerical integration: the use of a non-equidistant grid as
given in [6] and [13] is advantageous. After substituting the boundary conditions at the trailing edge of the pressure
profile, integration yields the pressure distribution in the circumferential and axial directions.
The application of the similarity principle to hydrodynamic plain bearing theory results in dimensionless magnitudes
of similarity for parameters of interest such as load-carrying capacity, frictional behaviour, lubricant flow rate,
relative bearing length, etc. The application of magnitudes of similarity reduces the number of numerical solutions
required of Reynolds' differential equation (see ISO 7902-2). Other solutions may also be applied, provided they
fulfil the conditions laid down in ISO 7902-2 and are of a similar numerical accuracy.
3.5 ISO 7902-3 includes permissible operational parameters towards which the result of the calculation shall be
oriented in order to ensure correct functioning of the plain bearings.
In special cases, operational parameters deviating from ISO 7902-3 may be agreed upon for specific applications.
4 Calculation procedure
4.1 By calculation is understood determination of correct operation by computation using actual operating
parameters (see figure 1) which can be compared with operational parameters. The operating parameters
determined under varying operating conditions must therefore lie within the range of permissibility as compared with
the operational parameters. To this end, all operating conditions during continuous operation shall be investigated.
4.2 Freedom from wear is guaranteed only if complete separation of the mating bearing parts is achieved by the
lubricant. Continuous operation in the mixed friction range results in failure. Short-time operation in the mixed friction
range, for example starting up and running down machines with plain bearings, is unavoidable and does not
generally result in bearing damage. When a bearing is subjected to heavy load, an auxiliary hydrostatic
arrangement may be necessary for starting up and running down at a slow speed. Running-in and adaptive wear to
compensate for deviations of the surface geometry from the ideal are permissible as long as they are limited in area
and time and occur without overloading effects. In certain cases, a specific running-in procedure may be beneficial,
depending on the choice of materials.
4.3 The limits of mechanical loading are a function of the strength of the bearing material. Slight permanent
deformations are permissible as long as they do not impair correct functioning of the plain bearing.
4.4 The limits of thermal loading result from the thermal stability of the bearing material but also from the
viscosity-temperature relationship and by degradation of the lubricant.
4.5 A correct calculation for plain bearings presupposes that the operating conditions are known for all cases of
continuous operation. In practice, however, additional influences frequently occur, which are unknown at the design
stage and cannot always be predicted. The application of an appropriate safety margin between the actual
operating parameters and permissible operational parameters is recommended. Influences include, for example:
— spurious forces (out-of-balance, vibrations, etc.);
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ISO 7902-1:1998(E)
Figure 1 — Outline of calculation
4
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ISO 7902-1:1998(E)
— deviations from the ideal geometry (machining tolerances, deviations during assembly, etc.);
— lubricants contaminated by dirt, water, air, etc.;
— corrosion, electrical erosion, etc.;
Data on other influencing factors are given in 6.7.
4.6 Reynolds' number shall be used to verify that ISO 7902-2, for which laminar flow in the lubrication clearance
gap is a necessary condition, can be applied:
C C
RR,,eff eff
rU pDN
J J
D
22
Re== < 41,3 . . . (2)
h v C
R,eff
In the case of plain bearings with Re > 41,/3 DC (for example as a result of high peripheral speed) higher loss
R,eff
coefficients and bearing temperatures must be expected. Calculations for bearings with turbulent flow cannot be
carried out in accordance with this part of ISO 7902.
4.7 The plain bearing calculation takes into account the following factors (starting with the known bearing
dimensions and operational data):
— the relationship between load-carrying capacity and lubricant film thickness;
— the frictional power rate;
— the lubricant flow rate;
— the heat balance.
All these factors are mutually dependent.
The solution is obtained using an iterative method; the sequence is outlined in the flow chart in figure 1.
For optimization of individual parameters, parameter variation can be applied: modification of the calculation
sequence is possible.
5 Symbols and units
See figure 2 and table 1.
Minimum lubricant film thickness, h :
min
DD−
J
h = −=eD05, ye1−
()
min
2
where the relative eccentricity, e, is given by
e
e =
DD−
J
2
5
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ISO 7902-1:1998(E)
If
p
jp−−()b<
2
2
then
hD=+05,cye1osj
()
min 2
6 Definition of symbols
6.1 Load-carrying capacity
A characteristic parameter for the load-carrying capacity is the dimensionless Sommerfeld number, So:
2
Fy
B
eff
So== So e,,W . . . (3)
DBhw D
eff h
Values of So as a function of the relative eccentricity e, the relative bearing length B/D and the angular span of
bearing segment W are given in ISO 7902-2. The variables w , h and y take into account thermal effects and
h eff eff
the angular velocities of shaft, bearing and bearing force (see 6.4 and 6.7).
The relative eccentricity e describes, together with the attitude angle b (see ISO 7902-2), the magnitude and
position of the minimum thickness of lubricant film. For a full bearing (W = 360 °C), the oil should be introduced at
the greatest lubricant clearance gap or, with respect to the direction of rotation, shortly before it. For this reason it is
useful to know the attitude angle b.
Figure 2 — Illustration of symbols
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ISO 7902-1:1998(E)
Table 1 — Symbols and their designations
Symbol Designation Unit
2
A Area of heat-emitting surface (bearing housing) m
b Width of oil groove m
G
B Nominal bearing width m
Specific heat capacity of the lubricant J/(kg K)
c �
C Nominal bearing clearance m
C Effective bearing radial clearance m
R,eff
d Oil hole diameter m
L
D Nominal bearing diameter (inside diameter) m
D Nominal shaft diameter m
J
D Maximum value of D m
J,max J
D Minimum value of D m
J,min J
D Maximum value of D m
max
D Minimum value of D m
min
e Eccentricity between the axis of the shaft and the bearing axis m
E Modulus of elasticity 1
f Coefficient of friction 1
F Bearing force (nominal load) N
F Friction force in the loaded area of the lubricant film N
f
F′ Frictional force in the unloaded area of the lubricant film N
f
G Shear modulus 1
h Local lubricant film thickness m
h Minimum permissible lubricant film thickness m
lim
h Minimum lubricant film thickness m
min
h Waviness of sliding surface m
wav
h Effective waviness of sliding surface m
wav,eff
h Maximum permissible effective waviness m
wav,eff,lim
2
k Outer heat transmission coefficient W/(m �K)
A
l Length of oil groove m
G
Length of oil pocket m
l
P
L Length of bearing housing m
H
N Rotational frequency of the bearing m/s
B
N Rotational frequency of the bearing force m/s
F
N Rotational frequency of the shaft m/s
J
p Local lubricant film pressure Pa
Specific bearing load Pa
p
p Lubricant feed pressure Pa
en
p Maximum permissible lubricant film pressure Pa
lim
p Maximum permissible specific bearing load Pa
lim
P Frictional power W
f
P Heat flow rate W
th
P Heat flow rate to the ambient W
th,amb
P Heat flow rate due to frictional power W
th,f
P Heat flow rate in the lubricant W
th,L
3
Q Lubricant flow rate m /s
3
Q Lubricant flow rate at the inlet to clearance gap m /s
1
3
Q Lubricant flow rate at the outlet to clearance gap m /s
2
3
Q Lubricant flow rate due to hydrodynamic pressure m /s
3
*
Q Lubricant flow rate parameter due to hydrodynamic pressure 1
3
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ISO 7902-1:1998(E)
Symbol Designation Unit
3
Q Lubricant flow rate due to feed pressure m /s
p
Q* Lubricant flow rate parameter due to feed pressure 1
p
Rz Average peak-to-valley height of bearing sliding surface m
B
Rz Average peak-to-valley height of shaft mating surface m
J
Re Reynolds number 1
So Sommerfeld number 1
T Ambient temperature °C
amb
T Bearing temperature °C
B
T Assumed initial bearing temperature °C
B,0
T Calculated bearing temperature resulting from iteration procedure °C
B,1
T Lubricant temperature at bearing entrance °C
en
T Lubricant temperature at bearing exit °C
ex
T Assumed initial lubricant temperature at bearing exit °C
ex,0
T Calculated lubricant temperature at bearing exit °C
ex,1
T Shaft temperature °C
J
T Maximum permissible bearing temperature °C
lim
T Mean lubricant temperature °C
L
U Linear velocity (peripheral speed) of bearing m/s
B
U Linear velocity (peripheral speed) of shaft m/s
J
V
Air ventilating velocity m/s
a
x Coordinate parallel to the sliding surface in circumferential direction m
y Coordinate perpendicular to the sliding surface m
z Coordinate parallel to the sliding surface in axial direction m
-1
a Linear heat expansion coefficient of the bearing K
l,B
-1
a Linear heat expansion coefficient of the shaft K
l,J
b Attitude angle (angular position of the shaft eccentricity related to the direction of load) °
d Angle of misalignment of the shaft rad
J
e Relative eccentricity 1
h Dynamic viscosity of the lubricant Pa�s
h Effective dynamic viscosity of the lubricant Pa�s
eff
n Kinematic viscosity of the lubricant Pa�s
x Coefficient of resistance to rotation in the loaded area of the lubricant film 1
x¢ Coefficient of resistance to rotation in the unloaded area of the lubricant film 1
x Coefficient of resistance to rotation in the area of circumferential groove 1
G
x Coefficient of resistance to rotation in the area of the pocket 1
P
3
r Density of lubricant kg/m
j Angular coordinate in circumferential direction rad
Angular coordinate of pressure leading edge rad
j
1
j Angular coordinate of pressure trailing edge rad
2
y Relative bearing clearance 1
y Mean relative bearing clearance 1
y Effective relative bearing clearance 1
eff
y Maximum relative bearing clearance 1
max
y Minimum relative bearing clearance 1
min
-1
w Angular velocity of bearing s
B
-1
w Hydrodynamic angular velocity s
h
-1
w Angular velocity of shaft s
J
W Angular span of bearing segment °
W Angular span of lubrication groove °
G
W Angular span of lubrication pocket °
P
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ISO 7902-1:1998(E)
6.2 Frictional power loss
Friction in a hydrodynamic plain bearing due to viscous shear stress is given by the coefficient of friction f = F /F
f
and the derived non-dimensional characteristics of frictional power loss x and f/y .
eff
Fy
feff
x= . . . (4)
DBhw
eff h
f x
= . . . (5)
y So
eff
They are applied if the frictional power loss is encounted only in the loaded area of the lubricant film.
It is still necessary to calculate frictional power loss in both the loaded and unloaded areas then the values
f
fF,, x,
f
y
eff
are substituted by
f′
fF′′,, x′,
f
y
eff
in equations (4) and (5). This means that the whole of the clearance gap is filled with lubricant.
The values of f/y and f¢/y for various values of e, B/D and W are given in ISO 7902-2. It also gives the
eff eff
approximation equations, based on [15], which are used to determine frictional power loss values in the bearings
taking account of the influence of lubricating pockets and grooves.
The frictional power in a bearing or the amount of heat generated is given by
PP==fF . . . (6)
fth,f
Pf′ = ′F . . . (7)
f
6.3 Lubricant flow rate
The lubricant fed to the bearing forms a film of lubricant separating the sliding surfaces. The pressure build-up in
this film forces lubricant out of the ends of the bearing. This is the proportion Q of the lubricant flow rate resulting
from build-up of hydrodynamic pressure.
3
QD= y wQ * . . . (8)
3 3
eff h
where QQ**= (eW,BD, ) is given in ISO 7902-2.
331
There is also a flow of lubricant in the peripheral direction through the narrowest clearance gap into the diverging,
pressure-free gap. For increased loading and with a small lubrication gap clearance, however, this proportion of the
lubricant flow is negligible.
The lubricant feed pressure p forces additional lubricant out of the ends of the plain bearing. This is the amount Q
en p
of the lubricant flow rate resulting from feed pressure:
3
3
Dpy
en
eff
= * . . . (9)
Q Q
pp
h
eff
where QQ**= (eW,,BD ) is given in ISO 7902-2.
pp
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ISO 7902-1:1998(E)
6.3.1 Lubricant feed elements are lubrication holes, lubrication grooves and lubrication pockets. The lubricant feed
pressure should be markedly less than the specific bearing load , to avoid additional hydrostatic loads. Usually
p p
en
p lies between 0,05 MPa and 0,2 MPa. The depth of the lubrication grooves and lubrication pockets is
en
considerably greater than the bearing clearance.
6.3.2 Lubrication grooves are elements designed to distribute lubricant in the circumferential direction. The
recesses machined into the sliding surface run circumferentially and are kept narrow in the axial direction. If
lubrication grooves are located in the vicinity of pressure rise, the pressure distribution is split into two independent
pressure “hills” and the load-carrying capacity is markedly reduced (see figure 3). In this case, the calculation shall
be carried out for half the load applied to each half bearing. However, because of the build-up of hydrodynamic
pressure, Q , only half of the lubricant flow rate shall be taken into account when balancing heat losses (see 6.4),
3
since the return into the lubrication groove plays no part in dissipating heat. It is more advantageous, for a full
bearing, to arrange the lubrication groove in the unloaded part. The entire lubricant flow amount Q goes into the
p
heat balance.
6.3.3 Lubrication pockets are elements for distributing the lubricant over the length of the bearing. The recesses
machined into the sliding surface are oriented in the axial direction and should be as short as possible in the
circumferential direction. Relative pocket lengths should be such shall b /B < 0,7. Although larger values increase
p
the oil flow rate, the oil emerging over the narrow, restricting webs at the ends plays no part in dissipating heat. This
is even more true if the end webs are penetrated axially. For full bearings (W = 360°), a lubrication pocket opposite
to the direction of load as well as two lubrication pockets normal to the direction of loading are machined in. Since
the lubricant flow rate, even in the unloaded part of the bearing, provides for the dissipation of frictional heat arising
from shearing, the lubricating pockets shall be fully taken into account in the heat balance. For shell segments
(W < 360°) the lubricant flow rate due to feed pressure through lubrication pockets at the inlet or outlet of the shell
segment makes practically no contribution to heat dissipation, since the lubrication pockets are scarcely restricted at
the segment ends and the greater proportion of this lubricant flow emerges directly.
If the lubricant fills the loaded area of the bearing and there is no lubricant in the unloaded part then the heat
dissipation counts as lubricant flow rate in the loaded part only.
Key
1 Lubrication hole
2 Lubrication groove
Figure 3
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ISO 7902-1:1998(E)
The influence of the type and the arrangement of the lubricant feed elements on the lubricant flow rate are dealt with
in ISO 7902-2.
The overall lubricant flow rate is given by
QQ= . . . (10)
3
for lubricant filling only the loaded area of the bearing.
QQ=+Q . . . (11)
3 p
for lubricant filling the whole circular lubrication clearance gap including unloaded part, i.e. 2p.
6.4 Heat balance
The thermal condition of the plain bearing can be obtained from the heat balance. The heat flow, P , arising from
th,f
frictional power in the bearing, P, is dissipated via the bearing housing to the environment and the lubricant
f
emerging from the bearing. In practice, one or other of the two types of heat dissipation dominates. By neglecting
the other, an additional safety margin is obtained during the design stage. The following assumptions can be made:
a) Pressureless lubricated bearings (for example ring lubrication) dissipate heat mainly through convection to the
environment: P = P
th,f th,amb
b) Pressure-lubricated bearings dissipate heat mainly via the lubricant: P = P
th,f th,L
6.4.1 Heat dissipation by convection
Heat dissipation by convection takes place by thermal conduction in the bearing housing and radiation and
convection from the surface of the housing to the environment. The complex processes during the heat transfer can
be summed up by:
Pk=−ATT . . . (12)
()
th,amb A B amb
where
2
k=⋅15 to 20 W m K
()
A
or, by ventilating the bearing housing with air at a velocity of V > 1,2 m/s
a
k =+712 V . . . (13)
Aa
(See [3] and [14])
Should the area of the heat-emitting surface, A, of the bearing housing not be known exactly, the following can be
used as an approximation:
— for cylindrical housings
p
2
2
AD=−2 D+pDB . . . (14)
()
HH
H
4
— for pedestal bearings
H
AH=+p B . . . (15)
H
2
— for bearings in the machine structure
AD=()15 to 20B . . . (16)
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where
B is the length of the axial housing;
H
D is the length of the outside diameter of the housing;
H
H is the length of the total height of the pedestal bearing.
6.4.2 Heat dissipation via the lubricant
In the case of force-feed lubrication, heat dissipation is via the lubricant:
Pc=−rQT T . . . (17)
()
th,L ex en
For mineral lubricants, the volume-specific heat is given by
6 3
rc=×18,J10 m⋅K . . . (18)
()
From the heat balance, it follows that
P = P for pressureless lubricated bearings
th,f th,amb
P = P for pressure-lubricated bearings
th,f th,L
This gives bearing temperature T (see [15]), and lubricant outlet temperature T (see [15]). The effective film
B ex
lubricant temperature with reference to the lubricant viscosity is
a) in the case of pure convection: =
T T
eff B
b) in the case of heat dissipation via the lubricant: T = T = 0,5 (T + T )
eff L en ex
At high peripheral speed, it is possible to select, instead of these mean values, a temperature which lies nearer to
the lubricant outlet temperature.
The values calculated for T and T shall be checked for their permissibility by comparison with the permissible
B ex
operational parameters T given in ISO 7902-3.
lim
In the sequence of calculation, at first only the operational data T or T are known, but not the effective
amb en
temperature T , which is required at the start of the calculation. The solution is obtained by first starting the
eff
calculation using an estimated temperature rise, i.e.
a) T - T = 20 K
B,0 amb
b) T - T = 20 K
ex,0 en
and the corresponding operating temperatures T . From the heat balance, corrected temperatures T or T are
eff B,1 ex,1
obtained, which, by averaging with the temperatures previously assumed (T or T ), are iteratively improved
B,0 ex,0
until the difference between the values with index 0 and 1 becomes negligibly small, for example 2 K. The condition
then attained corresponds to the steady condition. During the iterative steps, the influencing factors given in 6.7
shall be taken into account. As a rule, the iteration converges rapidly. It can also be replaced by graphical
interpolation in which, for calculating P and P or P ; several temperature differences are assumed. If the
th,f th,amb th,L
heat flows P = f(T ) or P = f(T ) are plotted, then the steady condition is given by the intersection of the
th,amb B th,L ex
two curves (see figure A.1).
6.5 Minimum lubricant film thickness and specific bearing load
The clearance gap, h, in a circular cylindrical journal bearing with the shaft offset is a function given by
hD=+05,cye1osj . . . (19)
()
eff
starting with j = j , in the widest clearance gap (see figure 1).
1
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ISO 7902-1:1998(E)
The minimum lubricant film thickness
hD=−05,ye()1 . . . (20)
min eff
shall be compared with the permissible operational parameter h specified in ISO 7902-3.
lim
The specific bearing load
F
p = . . . (21)
DB
shall be compared with the permissible operational parameter p specified in ISO 7902-3.
lim
6.6 Operational conditions
Should the plain bearing be operated under several, varying sets of operating conditions over lengthy periods, then,
they shall be checked for the most unfavourable p, h , and T . First, a decision shall be reached as to whether
min B
the bearing can be lubricated without pressure and if heat dissipation by convection suffices. The most unfavourable
thermal case shall be investigated, which, as a rule, corresponds to an operating condition at high rotary frequency
together with heavy loading. If, for pure convection, excessive bearing temperatures occur, which even by
increasing the dimensions of the bearing or of the surface area of the housing to their greatest possible extent
cannot be lowered to permissible values, then force-feed lubrication and oil cooling are necessary.
If an operating condition under high thermal loading (low dynamic lubricant viscosity) is followed directly by one with
high specific bearing load and low rotary frequency, this new operating condition should be investigated while
keeping the thermal condition from the preceding operating point.
The transition to mixed friction is due to contact of the roughness peaks of the shaft an
...
SLOVENSKI STANDARD
SIST ISO 7902-1:2002
01-marec-2002
+LGURGLQDPLþQLUDGLDOQLGUVQLOHåDML]DQHSUHNLQMHQRREUDWRYDQMH9DOMDVWLOHåDML
GHO 3RVWRSHNGLPHQ]LRQLUDQMD
Hydrodynamic plain journal bearings under steady-state conditions -- Circular cylindrical
bearings -- Part 1: Calculation procedure
Paliers lisses hydrodynamiques radiaux fonctionnant en régime stabilisé -- Paliers
circulaires cylindriques -- Partie 1: Méthode de calcul
Ta slovenski standard je istoveten z: ISO 7902-1:1998
ICS:
21.100.10 Drsni ležaji Plain bearings
SIST ISO 7902-1:2002 en
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.
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SIST ISO 7902-1:2002
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SIST ISO 7902-1:2002
INTERNATIONAL ISO
STANDARD 7902-1
First edition
1998-04-01
Hydrodynamic plain journal bearings under
steady-state conditions — Circular
cylindrical bearings —
Part 1:
Calculation procedure
Paliers lisses hydrodynamiques radiaux fonctionnant en régime stabilisé —
Paliers circulaires cylindriques —
Partie 1: Méthode de calcul
A
Reference number
ISO 7902-1:1998(E)
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SIST ISO 7902-1:2002
ISO 7902-1:1998(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide
federation of national standards bodies (ISO member bodies). The work of
preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which
a technical committee has been established has the right to be represented
on that committee. International organizations, governmental and non-
governmental, in liaison with ISO, also take part in the work. ISO
collaborates closely with the International Electrotechnical Commission
(IEC) on all matters of electrotechnical standardization.
Draft International Standards adopted by the technical committees are
circulated to the member bodies for voting. Publication as an International
Standard requires approval by at least 75 % of the member bodies casting
a vote.
International Standard ISO 7902-1 was prepared by Technical Committee
ISO/TC 123, Plain bearings, Subcommittee SC 4, Methods of calculation of
plain bearings.
ISO 7902 consists of the following parts, under the general title
Hydrodynamic plain journal bearings under steady-state conditions —
Circular cylindrical bearings:
— Part 1: Calculation procedure
— Part 2: Functions used in the calculation procedure
— Part 3: Permissible operational parameters
Annexes A and B of this part of ISO 7902 are for information only.
© ISO 1998
All rights reserved. Unless otherwise specified, no part of this publication may be reproduced
or utilized in any form or by any means, electronic or mechanical, including photocopying and
microfilm, without permission in writing from the publisher.
International Organization for Standardization
Case postale 56 • CH-1211 Genève 20 • Switzerland
Internet central@iso.ch
X.400 c=ch; a=400net; p=iso; o=isocs; s=central
Printed in Switzerland
ii
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INTERNATIONAL STANDARD ISO ISO 7902-1:1998(E)
Hydrodynamic plain journal bearings under steady-state
conditions — Circular cylindrical bearings —
Part 1:
Calculation procedure
1 Scope
This part of ISO 7902 specifies a calculation procedure for oil-lubricated hydrodynamic plain bearings, with
complete separation of the shaft and bearing sliding surfaces by a film of lubricant, used for designing plain
bearings that are reliable in operation.
It deals with circular cylindrical bearings having angular spans W, of 360°, 180°, 150°, 120° and 90°, the arc
segment being loaded centrally. Their clearance geometry is constant except for negligible deformations resulting
from lubricant film pressure and temperature.
The calculation procedure serves to dimension and optimize plain bearings, in turbines, generators, electric motors,
gear units, rolling mills, pumps and other machines. It is limited to steady-state operation, i.e. under continuously
driven operating conditions, with magnitude and direction of loading as well as the angular speeds of all rotating
parts constant. It can also be applied if a full plain bearing is subjected to a constant force rotating at any speed.
Dynamic loadings, i.e. those whose magnitude and direction vary with time, such as can result from vibration effects
and instabilities of rapid-running rotors, are not taken into account.
2 Normative references
The following standards contain provisions which, through reference in this text, constitute provisions of this part of
ISO 7902. At the time of publication, the editions indicated were valid. All standards are subject to revision, and
parties to agreements based on this part of ISO 7902 are encouraged to investigate the possibility of applying the
most recent editions of the standards indicated below. Members of IEC and ISO maintain registers of currently valid
International Standards.
ISO 3448:1992, Industrial liquid lubricants — ISO viscosity classification.
ISO 7902-2:1998,
Hydrodynamic plain journal bearings under steady-state conditions — Circular cylindrical
bearings — Part 2: Functions used in the calculation procedure.
ISO 7902-3:1998, Hydrodynamic plain journal bearings under steady-state conditions — Circular cylindrical
bearings — Part 3: Permissible operational parameters.
ISO 7904-2:1995, Plain bearings — Symbols — Part 2: Applications.
1
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ISO 7902-1:1998(E)
3 Basis of calculation, assumptions and preconditions
3.1 The basis of calculation is the numerical solution to Reynolds' differential equation for a finite bearing length,
taking into account the physically correct boundary conditions for the generation of pressure:
¶ ¶p ¶ ¶p ¶h
33
h+ h =+6huu . . . (1)
()
JB
¶x ¶xz¶ ¶z ¶x
The symbols are given in clause 5.
See [1] to [3], and [11] to [14] in annex B, for the derivation of Reynolds' differential equation and [4] to [6], [12] and
[13] for its numerical solution.
3.2 The following idealizing assumptions and preconditions are made, the permissibility of which has been
sufficiently confirmed both experimentally and in practice.
a) The lubricant corresponds to a Newtonian fluid.
b) All lubricant flows are laminar.
c) The lubricant adheres completely to the sliding surfaces.
d) The lubricant is incompressible.
e) The lubricant clearance gap in the loaded area is completely filled with lubricant. Filling up of the unloaded area
depends on the way the lubricant is supplied to the bearing.
f) Inertia effects, gravitational and magnetic forces of the lubricant are negligible.
g) The components forming the lubrication clearance gap are rigid or their deformation is negligible; their surfaces
are ideal circular cylinders.
h) The radii of curvature of the surfaces in relative motion are large in comparison with the lubricant film
thicknesses.
i) The lubricant film thickness in the axial direction (z-coordinate) is constant.
j) Fluctuations in pressure within the lubricant film normal to the bearing surfaces (y-coordinate) are negligible.
k) There is no motion normal to the bearing surfaces (y-coordinate).
l) The lubricant is isoviscous over the entire lubrication clearance gap.
m) The lubricant is fed in at the start of the bearing liner or where the lubrication clearance gap is widest; the
magnitude of the lubricant feed pressure is negligible in comparison with the lubricant film pressures.
3.3 The boundary conditions for the generation of lubricant film pressure fulfil the following continuity conditions:
— at the leading edge of the pressure profile: p(j , z) = 0
1
— at the bearing rim: p(j, z = – B/2) = 0
— at the trailing edge of the pressure profile: p[j (z), z] = 0
2
— and ¶p/¶j[j (z), z] = 0
2
For some types and sizes of bearings, the boundary conditions may be specified.
2
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In partial bearings, if the following expression is satisfied:
p
jp−−<()b
2
2
then the trailing edge of the pressure profile lies at the outlet end of the bearing:
pz(,jj==)0
2
3.4 The numerical integration of the Reynolds' differential equation is carried out — possibly by applying
transformation of pressure as suggested in [3], [11] and [12] — by a transformation to a differential equation which
is applied to a grid system of supporting points, and which results in a system of linear equations. The number of
supporting points is significant to the accuracy of the numerical integration: the use of a non-equidistant grid as
given in [6] and [13] is advantageous. After substituting the boundary conditions at the trailing edge of the pressure
profile, integration yields the pressure distribution in the circumferential and axial directions.
The application of the similarity principle to hydrodynamic plain bearing theory results in dimensionless magnitudes
of similarity for parameters of interest such as load-carrying capacity, frictional behaviour, lubricant flow rate,
relative bearing length, etc. The application of magnitudes of similarity reduces the number of numerical solutions
required of Reynolds' differential equation (see ISO 7902-2). Other solutions may also be applied, provided they
fulfil the conditions laid down in ISO 7902-2 and are of a similar numerical accuracy.
3.5 ISO 7902-3 includes permissible operational parameters towards which the result of the calculation shall be
oriented in order to ensure correct functioning of the plain bearings.
In special cases, operational parameters deviating from ISO 7902-3 may be agreed upon for specific applications.
4 Calculation procedure
4.1 By calculation is understood determination of correct operation by computation using actual operating
parameters (see figure 1) which can be compared with operational parameters. The operating parameters
determined under varying operating conditions must therefore lie within the range of permissibility as compared with
the operational parameters. To this end, all operating conditions during continuous operation shall be investigated.
4.2 Freedom from wear is guaranteed only if complete separation of the mating bearing parts is achieved by the
lubricant. Continuous operation in the mixed friction range results in failure. Short-time operation in the mixed friction
range, for example starting up and running down machines with plain bearings, is unavoidable and does not
generally result in bearing damage. When a bearing is subjected to heavy load, an auxiliary hydrostatic
arrangement may be necessary for starting up and running down at a slow speed. Running-in and adaptive wear to
compensate for deviations of the surface geometry from the ideal are permissible as long as they are limited in area
and time and occur without overloading effects. In certain cases, a specific running-in procedure may be beneficial,
depending on the choice of materials.
4.3 The limits of mechanical loading are a function of the strength of the bearing material. Slight permanent
deformations are permissible as long as they do not impair correct functioning of the plain bearing.
4.4 The limits of thermal loading result from the thermal stability of the bearing material but also from the
viscosity-temperature relationship and by degradation of the lubricant.
4.5 A correct calculation for plain bearings presupposes that the operating conditions are known for all cases of
continuous operation. In practice, however, additional influences frequently occur, which are unknown at the design
stage and cannot always be predicted. The application of an appropriate safety margin between the actual
operating parameters and permissible operational parameters is recommended. Influences include, for example:
— spurious forces (out-of-balance, vibrations, etc.);
3
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ISO 7902-1:1998(E)
Figure 1 — Outline of calculation
4
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— deviations from the ideal geometry (machining tolerances, deviations during assembly, etc.);
— lubricants contaminated by dirt, water, air, etc.;
— corrosion, electrical erosion, etc.;
Data on other influencing factors are given in 6.7.
4.6 Reynolds' number shall be used to verify that ISO 7902-2, for which laminar flow in the lubrication clearance
gap is a necessary condition, can be applied:
C C
RR,,eff eff
rU pDN
J J
D
22
Re== < 41,3 . . . (2)
h v C
R,eff
In the case of plain bearings with Re > 41,/3 DC (for example as a result of high peripheral speed) higher loss
R,eff
coefficients and bearing temperatures must be expected. Calculations for bearings with turbulent flow cannot be
carried out in accordance with this part of ISO 7902.
4.7 The plain bearing calculation takes into account the following factors (starting with the known bearing
dimensions and operational data):
— the relationship between load-carrying capacity and lubricant film thickness;
— the frictional power rate;
— the lubricant flow rate;
— the heat balance.
All these factors are mutually dependent.
The solution is obtained using an iterative method; the sequence is outlined in the flow chart in figure 1.
For optimization of individual parameters, parameter variation can be applied: modification of the calculation
sequence is possible.
5 Symbols and units
See figure 2 and table 1.
Minimum lubricant film thickness, h :
min
DD−
J
h = −=eD05, ye1−
()
min
2
where the relative eccentricity, e, is given by
e
e =
DD−
J
2
5
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If
p
jp−−()b<
2
2
then
hD=+05,cye1osj
()
min 2
6 Definition of symbols
6.1 Load-carrying capacity
A characteristic parameter for the load-carrying capacity is the dimensionless Sommerfeld number, So:
2
Fy
B
eff
So== So e,,W . . . (3)
DBhw D
eff h
Values of So as a function of the relative eccentricity e, the relative bearing length B/D and the angular span of
bearing segment W are given in ISO 7902-2. The variables w , h and y take into account thermal effects and
h eff eff
the angular velocities of shaft, bearing and bearing force (see 6.4 and 6.7).
The relative eccentricity e describes, together with the attitude angle b (see ISO 7902-2), the magnitude and
position of the minimum thickness of lubricant film. For a full bearing (W = 360 °C), the oil should be introduced at
the greatest lubricant clearance gap or, with respect to the direction of rotation, shortly before it. For this reason it is
useful to know the attitude angle b.
Figure 2 — Illustration of symbols
6
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Table 1 — Symbols and their designations
Symbol Designation Unit
2
A Area of heat-emitting surface (bearing housing) m
b Width of oil groove m
G
B Nominal bearing width m
Specific heat capacity of the lubricant J/(kg K)
c �
C Nominal bearing clearance m
C Effective bearing radial clearance m
R,eff
d Oil hole diameter m
L
D Nominal bearing diameter (inside diameter) m
D Nominal shaft diameter m
J
D Maximum value of D m
J,max J
D Minimum value of D m
J,min J
D Maximum value of D m
max
D Minimum value of D m
min
e Eccentricity between the axis of the shaft and the bearing axis m
E Modulus of elasticity 1
f Coefficient of friction 1
F Bearing force (nominal load) N
F Friction force in the loaded area of the lubricant film N
f
F′ Frictional force in the unloaded area of the lubricant film N
f
G Shear modulus 1
h Local lubricant film thickness m
h Minimum permissible lubricant film thickness m
lim
h Minimum lubricant film thickness m
min
h Waviness of sliding surface m
wav
h Effective waviness of sliding surface m
wav,eff
h Maximum permissible effective waviness m
wav,eff,lim
2
k Outer heat transmission coefficient W/(m �K)
A
l Length of oil groove m
G
Length of oil pocket m
l
P
L Length of bearing housing m
H
N Rotational frequency of the bearing m/s
B
N Rotational frequency of the bearing force m/s
F
N Rotational frequency of the shaft m/s
J
p Local lubricant film pressure Pa
Specific bearing load Pa
p
p Lubricant feed pressure Pa
en
p Maximum permissible lubricant film pressure Pa
lim
p Maximum permissible specific bearing load Pa
lim
P Frictional power W
f
P Heat flow rate W
th
P Heat flow rate to the ambient W
th,amb
P Heat flow rate due to frictional power W
th,f
P Heat flow rate in the lubricant W
th,L
3
Q Lubricant flow rate m /s
3
Q Lubricant flow rate at the inlet to clearance gap m /s
1
3
Q Lubricant flow rate at the outlet to clearance gap m /s
2
3
Q Lubricant flow rate due to hydrodynamic pressure m /s
3
*
Q Lubricant flow rate parameter due to hydrodynamic pressure 1
3
7
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Symbol Designation Unit
3
Q Lubricant flow rate due to feed pressure m /s
p
Q* Lubricant flow rate parameter due to feed pressure 1
p
Rz Average peak-to-valley height of bearing sliding surface m
B
Rz Average peak-to-valley height of shaft mating surface m
J
Re Reynolds number 1
So Sommerfeld number 1
T Ambient temperature °C
amb
T Bearing temperature °C
B
T Assumed initial bearing temperature °C
B,0
T Calculated bearing temperature resulting from iteration procedure °C
B,1
T Lubricant temperature at bearing entrance °C
en
T Lubricant temperature at bearing exit °C
ex
T Assumed initial lubricant temperature at bearing exit °C
ex,0
T Calculated lubricant temperature at bearing exit °C
ex,1
T Shaft temperature °C
J
T Maximum permissible bearing temperature °C
lim
T Mean lubricant temperature °C
L
U Linear velocity (peripheral speed) of bearing m/s
B
U Linear velocity (peripheral speed) of shaft m/s
J
V
Air ventilating velocity m/s
a
x Coordinate parallel to the sliding surface in circumferential direction m
y Coordinate perpendicular to the sliding surface m
z Coordinate parallel to the sliding surface in axial direction m
-1
a Linear heat expansion coefficient of the bearing K
l,B
-1
a Linear heat expansion coefficient of the shaft K
l,J
b Attitude angle (angular position of the shaft eccentricity related to the direction of load) °
d Angle of misalignment of the shaft rad
J
e Relative eccentricity 1
h Dynamic viscosity of the lubricant Pa�s
h Effective dynamic viscosity of the lubricant Pa�s
eff
n Kinematic viscosity of the lubricant Pa�s
x Coefficient of resistance to rotation in the loaded area of the lubricant film 1
x¢ Coefficient of resistance to rotation in the unloaded area of the lubricant film 1
x Coefficient of resistance to rotation in the area of circumferential groove 1
G
x Coefficient of resistance to rotation in the area of the pocket 1
P
3
r Density of lubricant kg/m
j Angular coordinate in circumferential direction rad
Angular coordinate of pressure leading edge rad
j
1
j Angular coordinate of pressure trailing edge rad
2
y Relative bearing clearance 1
y Mean relative bearing clearance 1
y Effective relative bearing clearance 1
eff
y Maximum relative bearing clearance 1
max
y Minimum relative bearing clearance 1
min
-1
w Angular velocity of bearing s
B
-1
w Hydrodynamic angular velocity s
h
-1
w Angular velocity of shaft s
J
W Angular span of bearing segment °
W Angular span of lubrication groove °
G
W Angular span of lubrication pocket °
P
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6.2 Frictional power loss
Friction in a hydrodynamic plain bearing due to viscous shear stress is given by the coefficient of friction f = F /F
f
and the derived non-dimensional characteristics of frictional power loss x and f/y .
eff
Fy
feff
x= . . . (4)
DBhw
eff h
f x
= . . . (5)
y So
eff
They are applied if the frictional power loss is encounted only in the loaded area of the lubricant film.
It is still necessary to calculate frictional power loss in both the loaded and unloaded areas then the values
f
fF,, x,
f
y
eff
are substituted by
f′
fF′′,, x′,
f
y
eff
in equations (4) and (5). This means that the whole of the clearance gap is filled with lubricant.
The values of f/y and f¢/y for various values of e, B/D and W are given in ISO 7902-2. It also gives the
eff eff
approximation equations, based on [15], which are used to determine frictional power loss values in the bearings
taking account of the influence of lubricating pockets and grooves.
The frictional power in a bearing or the amount of heat generated is given by
PP==fF . . . (6)
fth,f
Pf′ = ′F . . . (7)
f
6.3 Lubricant flow rate
The lubricant fed to the bearing forms a film of lubricant separating the sliding surfaces. The pressure build-up in
this film forces lubricant out of the ends of the bearing. This is the proportion Q of the lubricant flow rate resulting
from build-up of hydrodynamic pressure.
3
QD= y wQ * . . . (8)
3 3
eff h
where QQ**= (eW,BD, ) is given in ISO 7902-2.
331
There is also a flow of lubricant in the peripheral direction through the narrowest clearance gap into the diverging,
pressure-free gap. For increased loading and with a small lubrication gap clearance, however, this proportion of the
lubricant flow is negligible.
The lubricant feed pressure p forces additional lubricant out of the ends of the plain bearing. This is the amount Q
en p
of the lubricant flow rate resulting from feed pressure:
3
3
Dpy
en
eff
= * . . . (9)
Q Q
pp
h
eff
where QQ**= (eW,,BD ) is given in ISO 7902-2.
pp
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6.3.1 Lubricant feed elements are lubrication holes, lubrication grooves and lubrication pockets. The lubricant feed
pressure should be markedly less than the specific bearing load , to avoid additional hydrostatic loads. Usually
p p
en
p lies between 0,05 MPa and 0,2 MPa. The depth of the lubrication grooves and lubrication pockets is
en
considerably greater than the bearing clearance.
6.3.2 Lubrication grooves are elements designed to distribute lubricant in the circumferential direction. The
recesses machined into the sliding surface run circumferentially and are kept narrow in the axial direction. If
lubrication grooves are located in the vicinity of pressure rise, the pressure distribution is split into two independent
pressure “hills” and the load-carrying capacity is markedly reduced (see figure 3). In this case, the calculation shall
be carried out for half the load applied to each half bearing. However, because of the build-up of hydrodynamic
pressure, Q , only half of the lubricant flow rate shall be taken into account when balancing heat losses (see 6.4),
3
since the return into the lubrication groove plays no part in dissipating heat. It is more advantageous, for a full
bearing, to arrange the lubrication groove in the unloaded part. The entire lubricant flow amount Q goes into the
p
heat balance.
6.3.3 Lubrication pockets are elements for distributing the lubricant over the length of the bearing. The recesses
machined into the sliding surface are oriented in the axial direction and should be as short as possible in the
circumferential direction. Relative pocket lengths should be such shall b /B < 0,7. Although larger values increase
p
the oil flow rate, the oil emerging over the narrow, restricting webs at the ends plays no part in dissipating heat. This
is even more true if the end webs are penetrated axially. For full bearings (W = 360°), a lubrication pocket opposite
to the direction of load as well as two lubrication pockets normal to the direction of loading are machined in. Since
the lubricant flow rate, even in the unloaded part of the bearing, provides for the dissipation of frictional heat arising
from shearing, the lubricating pockets shall be fully taken into account in the heat balance. For shell segments
(W < 360°) the lubricant flow rate due to feed pressure through lubrication pockets at the inlet or outlet of the shell
segment makes practically no contribution to heat dissipation, since the lubrication pockets are scarcely restricted at
the segment ends and the greater proportion of this lubricant flow emerges directly.
If the lubricant fills the loaded area of the bearing and there is no lubricant in the unloaded part then the heat
dissipation counts as lubricant flow rate in the loaded part only.
Key
1 Lubrication hole
2 Lubrication groove
Figure 3
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The influence of the type and the arrangement of the lubricant feed elements on the lubricant flow rate are dealt with
in ISO 7902-2.
The overall lubricant flow rate is given by
QQ= . . . (10)
3
for lubricant filling only the loaded area of the bearing.
QQ=+Q . . . (11)
3 p
for lubricant filling the whole circular lubrication clearance gap including unloaded part, i.e. 2p.
6.4 Heat balance
The thermal condition of the plain bearing can be obtained from the heat balance. The heat flow, P , arising from
th,f
frictional power in the bearing, P, is dissipated via the bearing housing to the environment and the lubricant
f
emerging from the bearing. In practice, one or other of the two types of heat dissipation dominates. By neglecting
the other, an additional safety margin is obtained during the design stage. The following assumptions can be made:
a) Pressureless lubricated bearings (for example ring lubrication) dissipate heat mainly through convection to the
environment: P = P
th,f th,amb
b) Pressure-lubricated bearings dissipate heat mainly via the lubricant: P = P
th,f th,L
6.4.1 Heat dissipation by convection
Heat dissipation by convection takes place by thermal conduction in the bearing housing and radiation and
convection from the surface of the housing to the environment. The complex processes during the heat transfer can
be summed up by:
Pk=−ATT . . . (12)
()
th,amb A B amb
where
2
k=⋅15 to 20 W m K
()
A
or, by ventilating the bearing housing with air at a velocity of V > 1,2 m/s
a
k =+712 V . . . (13)
Aa
(See [3] and [14])
Should the area of the heat-emitting surface, A, of the bearing housing not be known exactly, the following can be
used as an approximation:
— for cylindrical housings
p
2
2
AD=−2 D+pDB . . . (14)
()
HH
H
4
— for pedestal bearings
H
AH=+p B . . . (15)
H
2
— for bearings in the machine structure
AD=()15 to 20B . . . (16)
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where
B is the length of the axial housing;
H
D is the length of the outside diameter of the housing;
H
H is the length of the total height of the pedestal bearing.
6.4.2 Heat dissipation via the lubricant
In the case of force-feed lubrication, heat dissipation is via the lubricant:
Pc=−rQT T . . . (17)
()
th,L ex en
For mineral lubricants, the volume-specific heat is given by
6 3
rc=×18,J10 m⋅K . . . (18)
()
From the heat balance, it follows that
P = P for pressureless lubricated bearings
th,f th,amb
P = P for pressure-lubricated bearings
th,f th,L
This gives bearing temperature T (see [15]), and lubricant outlet temperature T (see [15]). The effective film
B ex
lubricant temperature with reference to the lubricant viscosity is
a) in the case of pure convection: =
T T
eff B
b) in the case of heat dissipation via the lubricant: T = T = 0,5 (T + T )
eff L en ex
At high peripheral speed, it is possible to select, instead of these mean values, a temperature which lies nearer to
the lubricant outlet temperature.
The values calculated for T and T shall be checked for their permissibility by comparison with the permissible
B ex
operational parameters T given in ISO 7902-3.
lim
In the sequence of calculation, at first only the operational data T or T are known, but not the effective
amb en
temperature T , which is required at the start of the calculation. The solution is obtained by first starting the
eff
calculation using an estimated temperature rise, i.e.
a) T - T = 20 K
B,0 amb
b) T - T = 20 K
ex,0 en
and the corresponding operating temperatures T . From the heat balance, corrected temperatures T or T are
eff B,1 ex,1
obtained, which, by averaging with the temperatures previously assumed (T or T ), are iteratively improved
B,0 ex,0
until the difference between the values with index 0 and 1 becomes negligibly small, for example 2 K. The condition
then attained corresponds to the steady condition. During the iterative steps, the influencing factors given in 6.7
shall be taken into account. As a rule, the iteration converges rapidly. It can also be replaced by graphical
interpolation in which, for calculating P and P or P ; several temperature differences are assumed. If the
th,f th,amb th,L
heat flows P = f(T ) or P = f(T ) are plotted, then the steady condition is given by the intersection of the
th,amb B th,L ex
two curves (see figure A.1).
6.5 Minimum lubricant film thickness and specific bearing load
The clearance gap, h, in a circular cylindrical journal bearing with the shaft offset is a function given by
hD=+05,cye1osj . . . (19)
()
eff
starting with j = j , in the widest clearance gap (see figure 1).
1
12
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SIST ISO 7902-1:2002
©
ISO
ISO 7902-1:1998(E)
The minimum lubricant film thickness
hD=−05,ye()1 . . . (20)
min eff
shall be compared with the permissible operational parameter h specified in ISO 7902-3.
lim
The specific bearing load
F
p = . . . (21)
DB
shall be compared with the permissible operational parameter p specified in ISO 7902-3.
lim
6.6 Operational conditions
Should the pla
...
NORME ISO
INTERNATIONALE 7902-1
Première édition
1998-04-01
Paliers lisses hydrodynamiques radiaux
fonctionnant en régime stabilisé — Paliers
circulaires cylindriques —
Partie 1:
Méthode de calcul
Hydrodynamic plain journal bearings under steady-state conditions —
Circular cylindrical bearings —
Part 1: Calculation procedure
A
Numéro de référence
ISO 7902-1:1998(F)
---------------------- Page: 1 ----------------------
ISO 7902-1:1998(F)
Avant-propos
L'ISO (Organisation internationale de normalisation) est une fédération
mondiale d'organismes nationaux de normalisation (comités membres de
l'ISO). L'élaboration des Normes internationales est en général confiée aux
comités techniques de l'ISO. Chaque comité membre intéressé par une
étude a le droit de faire partie du comité technique créé à cet effet. Les
organisations internationales, gouvernementales et non gouvernementales,
en liaison avec l'ISO participent également aux travaux. L'ISO collabore
étroitement avec la Commission électrotechnique internationale (CEI) en
ce qui concerne la normalisation électrotechnique.
Les projets de Normes internationales adoptés par les comités techniques
sont soumis aux comités membres pour vote. Leur publication comme
Normes internationales requiert l'approbation de 75 % au moins des
comités membres votants.
La Norme internationale ISO 7902-1 a été élaborée par le comité technique
ISO/TC 123, Paliers lisses, sous-comité SC 4, Méthodes de calcul des
paliers lisses.
L'ISO 7902 comprend les parties suivantes, présentées sous le titre
général Paliers lisses hydrodynamiques radiaux fonctionnant en régime
stabilisé — Paliers circulaires cylindriques:
— Partie 1: Méthode de calcul
— Partie 2: Fonctions utilisées pour le calcul
— Partie 3: Paramètres opérationnels admissibles
Les annexes A et B de la présente partie de l'ISO 7902 sont données
uniquement à titre d'information.
© ISO 1998
Droits de reproduction réservés. Sauf prescription différente, aucune partie de cette publi-
cation ne peut être reproduite ni utilisée sous quelque forme que ce soit et par aucun pro-
cédé, électronique ou mécanique, y compris la photocopie et les microfilms, sans l'accord
écrit de l'éditeur.
Organisation internationale de normalisation
Case postale 56 • CH-1211 Genève 20 • Suisse
Internet central@iso.ch
X.400 c=ch; a=400net; p=iso; o=isocs; s=central
Imprimé en Suisse
ii
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©
NORME INTERNATIONALE ISO ISO 7902-1:1998(F)
Paliers lisses hydrodynamiques radiaux fonctionnant en régime
stabilisé — Paliers circulaires cylindriques —
Partie 1:
Méthode de calcul
1 Domaine d'application
La présente partie de l'ISO 7902 prescrit une méthode de calcul des paliers lisses hydrodynamiques à lubrification
par huile dont l'arbre et les surfaces de frottement sont complètement séparés par une pellicule de lubrifiant.
Elle traite des paliers cylindriques circulaires à portée angulaire W, de 360°, 180°, 150°, 120° et 90° dont le segment
d'arc est mis en charge en son centre et dont le jeu circulaire est constant à quelques petites déformations
négligeables près résultant de la pression et de la température de la pellicule lubrifiante.
La méthode de calcul sert à optimaliser les cotes des paliers lisses utilisés dans les turbines, les générateurs, les
moteurs électriques, les engrenages, les trains de laminage, les pompes et autres machines. Elle est limitée aux
paliers en régime stabilisé, c'est-à-dire fonctionnant dans des conditions continues, sans variation de l'ampleur et
du sens de la mise en charge ni vitesses angulaires des parties tournantes. Elle reste applicable si le palier lisse
complet est soumis à une force constante tournant à n'importe quelle vitesse. Elle ne tient pas compte des mises
en charge dynamiques, c'est-à-dire qui varient en ampleur et sens avec le temps telles que peuvent en donner les
rotors rapides du fait des vibrations et de l'instabilité de fonctionnement.
2 Références normatives
Les normes suivantes contiennent des dispositions qui, par suite de la référence qui en est faite, constituent des
dispositions valables pour la présente partie de l'ISO 7902. Au moment de la publication, les éditions indiquées
étaient en vigueur. Toute norme est sujette à révision et les parties prenantes des accords fondés sur la présente
partie de l'ISO 7902 sont invitées à rechercher la possibilité d'appliquer les éditions les plus récentes des normes
indiquées ci-après. Les membres de la CEI et de l'ISO possèdent le registre des Normes internationales en vigueur
à un moment donné.
ISO 3448:1992, Lubrifiants liquides industriels — Classification ISO selon la viscosité.
ISO 7902-2:1998, Paliers lisses hydrodynamiques radiaux fonctionnant en régime stabilisé — Paliers circulaires
cylindriques — Partie 2: Fonctions utilisées pour le calcul.
ISO 7902-3:1998, Paliers lisses hydrodynamiques radiaux fonctionnant en régime stabilisé — Paliers circulaires
cylindriques — Partie 3: Paramètres opérationnels admissibles.
ISO 7904-2:1995, Paliers lisses — Symboles — Partie 2: Applications.
1
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ISO
ISO 7902-1:1998(F)
3 Base de calcul, hypothèses et conditions préalables
3.1 La base de calcul est toujours la solution numérique de l'équation différentielle de Reynolds appliquée à une
longueur finie de palier, compte tenu des conditions physiquement correctes aux limites engendrant les pressions:
¶ ¶p ¶ ¶p ¶h
33
h+ h =+6huu . . . (1)
()
JB
¶x ¶xz¶ ¶z ¶x
Les symboles sont donnés dans l'article 5.
Voir [1] à [3] et [11] à [14] de l'annexe B pour la dérivation de l'équation différentielle de Reynolds et [4] à [6] et [12]
et [13] pour la solution numérique.
3.2 Les hypothèses et conditions préalables idéales dont la validité a été confirmée de façon suffisante par
l'expérimentation et la pratique sont les suivantes:
a) Le lubrifiant est un fluide newtonien.
b) Tous les écoulements de lubrifiant sont laminaires.
c) Le lubrifiant adhère totalement aux surfaces de frottement.
d) Le lubrifiant est incompressible.
e) Dans la zone sous charge, la rainure de graissage est complètement remplie de lubrifiant. Le remplissage dans
la zone non chargée dépend du mode d'alimentation en lubrifiant.
f) Les effets dus à l'inertie, aux forces de la pesanteur et aux forces magnétiques du lubrifiant sont négligeables.
g) Les éléments formant la rainure de graissage sont rigides ou leur déformation est négligeable. Leur surface est
un cylindre circulaire idéal.
h) Les rayons de courbure des surfaces en mouvement les unes par rapport aux autres sont importants par
rapport à l'épaisseur de la pellicule de lubrifiant.
i) Dans le sens axial (axe z), l'épaisseur de la pellicule de lubrifiant demeure constante.
j) Les fluctuations de pression à l'intérieur de la pellicule de lubrifiant perpendiculairement à la surface du palier
(axe y) sont négligeables.
k) Il n'existe pas de mouvement normal à la surface du palier (axe y).
l) Le lubrifiant est isovisqueux dans la totalité de la rainure de graissage.
m) Le lubrifiant est introduit au niveau du coussinet de palier où la rainure de graissage est la plus large. La
pression d'alimentation en lubrifiant est négligeable par rapport aux pressions s'exerçant dans la pellicule
lubrifiante.
3.3 Les conditions aux limites engendrant les pressions dans la pellicule lubrifiante remplissent les critères de
continuité suivants:
— au niveau du bord d'attaque du profil de pression: p(j , z) = 0
1
— au niveau du bord du palier: p(j, z = – B/2) = 0
— au niveau du bord de fuite du profil des pressions: p[j (z), z] = 0
2
— et ¶p/¶j[j (z), z] = 0
2
2
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ISO
ISO 7902-1:1998(F)
Pour certains types ou dimensions de paliers, il est bon de spécifier les conditions aux limites.
Dans les paliers partiels, si l'inéquation suivante est satisfaite:
p
jp−−<()b
2
2
le bord de fuite du profil des pressions se trouvera à la sortie du palier:
pz(,jj==)0
2
3.4 On pratique l'intégration numérique de l'équation différentielle de Reynolds — éventuellement par
transformation de la pression comme le suggèrent [3], [11] et [12] — ou par transformation en équation différentielle
appliquée à une grille de points d'appui, ce qui donne un système d'équations linéaires. La précision de l'intégration
numérique dépend du nombre de points d'appui; il y a avantage à utiliser la grille à points non équidistants
suggérée en [6] et [13]. Après substitution dans l'équation des conditions aux limites au bord de fuite du profil des
pressions, l'intégration donne la répartition des pressions dans le sens axial sur la circonférence.
L'application du principe de similitude à la théorie des paliers lisses hydrodynamiques donne des grandeurs de
similitude sans dimensions pour les paramètres intéressants du type capacité de charge, comportement au
frottement, débit de lubrifiant, longueur relative du palier, etc. L'utilisation de ces grandeurs de similitude réduit le
nombre de solutions numériques à trouver à l'équation différentielle de Reynolds (voir ISO 7902-2). D'autres
solutions sont également valables si elles remplissent les conditions fixées dans l'ISO 7902-2 et ont une précision
numérique comparable.
3.5 L'ISO 7902-3 renferme les paramètres opérationnels admissibles sur lesquels les calculs doivent être orientés
pour garantir un fonctionnement correct des paliers lisses.
Dans des cas particuliers, il est permis de se mettre d'accord sur des paramètres différents.
4 Méthode de calcul
4.1 Par calcul on entend la détermination du fonctionnement correct à partir de paramètres de fonctionnement
réels (voir figure 1) comparables aux paramètres opérationnels. Les paramètres de fonctionnement définis dans
des conditions opératoires variables doivent donc se trouver dans les limites admissibles fixées pour les
paramètres opérationnels. Il faut donc déterminer toutes les conditions opératoires en fonctionnement continu.
4.2 L'absence d'usure n'est garantie que si le lubrifiant garantit une séparation complète des surfaces homologues
d'un palier. Un fonctionnement continu en régime mixte entraîne des défaillances. Il est inévitable de faire
fonctionner les paliers en régime mixte pendant de brèves périodes, par exemple au démarrage et à l'arrêt des
machines, et cela n'endommage en général pas les paliers, mais lorsque le palier est soumis à une lourde charge, il
peut s'avérer nécessaire de prévoir des dispositions hydrostatiques auxiliaires pour le démarrage et l'arrêt à faible
vitesse. L'aptitude au rodage et l'usure adaptée peuvent compenser les écarts par rapport à la surface géométrique
idéale lorsque ces phénomènes sont d'ampleur limitée en surface et dans le temps et n'entraînent pas de
surcharge. Dans certains cas, un rodage spécifique peut être intéressant, selon le matériau choisi.
4.3 Les limites de charge mécanique sont fonction de la résistance du matériau antifriction. De petites
déformations rémanentes sont admises si elles n'affectent pas le fonctionnement correct du palier.
4.4 Les limites de charge thermique sont fonction de la stabilité thermique du matériau antifriction mais aussi des
relations viscosité/température et de la tendance à la dégradation du lubrifiant.
3
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©
ISO
ISO 7902-1:1998(F)
Figure 1 — Abaque de calcul
4
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©
ISO
ISO 7902-1:1998(F)
4.5 Un calcul correct des paliers lisses suppose avant tout que l'on connaisse les conditions opératoires dans tous
les cas de fonctionnement continu. En pratique toutefois, il intervient fréquemment des phénomènes qui ne sont
pas connus au stade de la conception et ne sont pas toujours prévisibles. Aussi est-il recommandé de prévoir une
certaine marge de sécurité entre les paramètres réels de fonctionnement et les paramètres opérationnels permis.
Parmi les phénomènes perturbateurs, on peut citer, par exemple:
— les forces parasites (balourd, vibrations, etc.);
— écarts de forme géométrique (tolérances d'usinage, écarts de montage, etc.);
— contamination du lubrifiant par la poussière, l'eau, l'air, etc.;
— corrosion, érosion électrique, etc.
D'autres grandeurs d'influence sont indiquées en 6.7.
4.6 Le nombre de Reynolds sert à vérifier que l'ISO 7902-2, dans laquelle un écoulement laminaire est nécessaire
dans la rainure de graissage, est bien applicable:
C C
RR,,eff eff
rU pDN
J J
D
22
Re== < 41,3 . . . (2)
h v C
R,eff
Dans les paliers lisses, dont Re > 41,/3 DC (en raison par exemple d'une vitesse périphérique élevée), il faut
R,eff
s'attendre à des coefficients de perte élevés et à des températures importantes dans le palier. La capacité de
charge peut également augmenter. Les paliers à écoulement turbulent ne peuvent pas être calculés à l'aide de la
présente partie de l'ISO 7902.
4.7 Le calcul des paliers lisses tient compte des facteurs suivants (fonction des dimensions du palier et des
données opérationnelles que l'on connaît):
— rapport entre la capacité de charge et l'épaisseur de la pellicule lubrifiante;
— la puissance de frottement;
— le débit de lubrifiant;
— le bilan thermique.
Ces facteurs sont tous interdépendants.
La solution est recherchée par une méthode itérative dont la procédure est illustrée à la figure 1.
Pour optimaliser les divers paramètres, on peut les faire varier et également modifier la suite des opérations.
5 Symboles et unités
Voir figure 2 et tableau 1.
Épaisseur minimale de la pellicule de lubrifiant, h :
min
DD−
J
h = −=eD05, ye1−
()
min
2
5
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©
ISO
ISO 7902-1:1998(F)
où l'excentricité relative, e, est donnée par
e
e =
DD−
J
2
Si
p
jp−−b<
()
2
2
alors
hD=+05,cye1osj
()
min 2
Figure 2 — Illustration des symboles
6
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©
ISO
ISO 7902-1:1998(F)
Tableau 1 — Symboles et leurs désignations
Symbole Désignation Unité
2
A Surface émettant de la chaleur (logement) m
b Largeur de la rainure de graissage m
G
B Largeur nominale du palier m
Capacité de chaleur massique du lubrifiant J/(kg K)
c �
C Jeu nominal du palier m
C Jeu radial utile du palier m
R,eff
d Diamètre du trou de graissage m
L
D Diamètre nominal du palier (diamètre intérieur) m
D Diamètre nominal de l'arbre m
J
D Valeur maximale de D m
J,max J
D Valeur minimale de D m
J,min J
D Valeur maximale de D m
max
D Valeur minimale de D m
min
e Excentricité entre l'axe d'un arbre et l'axe du palier m
E Module d'élasticité 1
f Coefficient de frottement 1
F Force d'appui (charge nominale) N
F Force de frottement dans la zone sous charge du film de lubrifiant N
f
F′ Force de frottement dans la zone non chargée du film de lubrifiant N
f
G Module de cisaillement 1
h Épaisseur locale du film de lubrifiant m
h Épaisseur minimale admissible du film de lubrifiant m
lim
h Épaisseur minimale du film de lubrifiant m
min
h Ondulation de la surface de frottement m
wav
h Ondulation utile de la surface de frottement m
wav,eff
h Ondulation utile maximale admissible de la surface de frottement m
wav,eff,lim
2
k Coefficient de transmission thermique extérieure (aire de référence A) W/(m �K)
A
l Longueur de la rainure de graissage m
G
Longueur de la poche à huile m
l
P
L Longueur du logement de palier à angle droit de l'axe m
H
N Fréquence de rotation du palier m/s
B
N Fréquence de rotation de la force d'appui m/s
F
N Fréquence de rotation de l'arbre m/s
J
p Pression locale du film de lubrifiant Pa
Charge spécifique Pa
p
p Pression d'alimentation en lubrifiant Pa
en
p Pression admissible du film de lubrifiant Pa
lim
p Charge spécifique admissible sur le palier Pa
lim
P Puissance de frottement W
f
P Débit thermique W
th
P Débit thermique à l'ambiante W
th,amb
P Débit thermique en fonction de la puissance de frottement W
th,f
P Débit thermique dans le lubrifiant W
th,L
3
Q Débit de lubrifiant m /s
3
Q Débit de lubrifiant à l'entrée de l'ouverture m /s
1
3
Q Débit de lubrifiant à la sortie de l'ouverture m /s
2
3
Q Débit de lubrifiant dû à la pression hydrodynamique m /s
3
*
Q Paramètre de débit de lubrifiant dû à la pression hydrodynamique 1
3
7
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ISO
ISO 7902-1:1998(F)
Symbole Désignation Unité
3
Q Débit de lubrifiant en fonction de la pression d'alimentation m /s
p
Q* Paramètre de débit de lubrifiant en fonction de la pression d'alimentation 1
p
Rz Hauteur moyenne de crête à creux de la surface de frottement du palier m
B
Rz Hauteur moyenne de crête à creux de la surface correspondante de l'arbre m
J
Re Nombre de Reynolds 1
So Nombre de Sommerfeld 1
T Température ambiante °C
amb
T Température du palier °C
B
T Température initiale supposée du palier °C
B,0
Température calculée du palier résultant d'une itération °C
T
B,1
T Température du lubrifiant à l'entrée du palier °C
en
T Température du lubrifiant à la sortie du palier °C
ex
T Température initiale supposée du lubrifiant à la sortie du palier °C
ex,0
T Température calculée du lubrifiant à la sortie du palier °C
ex,1
T Température de l'arbre °C
J
T Température maximale admissible du palier °C
lim
T Température moyenne du lubrifiant °C
L
U Vitesse périphérique du palier m/s
B
U Vitesse périphérique de l'arbre m/s
J
V Vitesse de l'air de ventilation m/s
a
x Coordonnée parallèle à la surface de frottement, dans le sens circonférentiel m
y Coordonnée perpendiculaire à la surface de frottement m
z Coordonnée parallèle à la surface de frottement, dans le sens axial m
-1
a Coefficient de dilatation thermique linéaire du palier K
l,B
-1
a Coefficient de dilatation thermique linéaire de l'arbre K
l,J
Angle d'assiette (position angulaire de l'excentricité de l'arbre par rapport à la direction °
b
d'application de la charge)
d Défaut d'alignement angulaire de l'arbre rad
J
e Excentricité relative 1
h Viscosité dynamique du lubrifiant Pa�s
h Viscosité dynamique effective du lubrifiant Pa�s
eff
n Viscosité cinématique du lubrifiant Pa�s
x Coefficient de résistance à la rotation dans la zone sous charge du film de lubrifiant 1
x¢ Coefficient de résistance à la rotation dans la zone non chargée du film de lubrifiant 1
x Coefficient de résistance à la rotation dans la zone de la rainure circulaire de graissage 1
G
x Coefficient de résistance à la rotation dans la zone de la poche à huile 1
P
3
r Masse volumique du lubrifiant kg/m
j Coordonnée angulaire dans le sens circonférentiel rad
j Coordonnée angulaire du bord d'attaque rad
1
j Coordonnée angulaire du bord de fuite rad
2
Jeu relatif du palier 1
y
y
Jeu moyen relatif du palier 1
y Jeu relatif utile du palier 1
eff
y Valeur maximale de y 1
max
y Valeur minimale de y 1
min
-1
w Vitesse angulaire du palier s
B
-1
w Vitesse angulaire hydrostatique s
h
-1
w Vitesse angulaire de l'arbre s
J
W Portée angulaire du segment de palier °
W Portée angulaire de la rainure de graissage °
G
W Portée angulaire de la poche d'huile °
P
8
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ISO 7902-1:1998(F)
6 Définition des symboles
6.1 Capacité de charge
Le paramètre caractéristique de la capacité de charge est le nombre de Sommerfeld, , nombre sans dimension:
So
2
Fy
B
eff
So== So e,,W . . . (3)
DBhw D
eff h
Les valeurs de So sont données dans l'ISO 7902-2 en fonction de l'excentricité relative e, de la longueur relative de
palier B/D et du battement angulaire du segment de palier W. Les variables w , h et y tiennent compte des
h eff eff
effets thermiques, de la vitesse angulaire de l'arbre et du palier, et de la force dans le palier (voir 6.4 et 6.7).
L'excentricité relative e décrit, avec l'angle d'inclinaison b (voir ISO 7902-2), la valeur et la position de l'épaisseur
minimale de la pellicule de lubrifiant. Dans un palier complet (W = 360 °C), l'huile doit être introduite là où le jeu est
le plus grand ou, compte tenu du sens de rotation, juste avant. C'est la raison pour laquelle il est utile de connaître
l'angle d'inclinaison b.
6.2 Perte par frottement
Le frottement engendré dans un palier lisse hydrodynamique par la contrainte de cisaillage visqueux est caractérisé
par le coefficient de frottement f = F /F et par les caractéristiques non dimensionnelles dérivées de perte par
f
frottement x et f/y .
eff
Fy
feff
x= . . . (4)
DBhw
eff h
f x
= . . . (5)
y So
eff
Ces formules sont applicables si l'on tient compte de la perte par frottement uniquement dans la partie sous charge
de la pellicule de lubrifiant.
S'il est nécessaire de calculer la perte par frottement dans la partie sous charge et dans la partie non chargée, les
valeurs de
f
fF,, x,
f
y
eff
sont remplacées par
f′
fF′′,, x′,
f
y
eff
dans les équations (4) et (5), ce qui revient à dire que tout le jeu est rempli de lubrifiant.
Les valeurs de f/y et f¢/y sont données dans l'ISO 7902-2 pour diverses valeurs de e, B/D et W.
eff eff
L'ISO 7902-2 donne également les valeurs approchées des équations dérivées de [15] qui servent à déterminer la
perte par frottement dans le palier due aux poches à huile et rainures de graissage.
Le frottement ou la chaleur produite dans le palier s'exprime sous la forme
PP==fF . . . (6)
fth,f
Pf′ = ′F . . . (7)
f
9
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ISO 7902-1:1998(F)
6.3 Débit de lubrifiant
Le lubrifiant introduit dans le palier forme une pellicule qui sépare les surfaces de frottement. L'augmentation de
pression dans cette pellicule chasse le lubrifiant vers les extrémités du palier.
La proportion de débit de lubrifiant résultant de cette augmentation de pression hydrodynamique est
Q
3
3
QD= y wQ * . . . (8)
3 eff h3
où QQ**= (eW,BD, ) est donné dans l'ISO 7902-2.
31
3
Il se produit également un écoulement de lubrifiant dans le sens circulaire par la rainure de graissage qui va en
s'élargissant. Si la charge augmente et que la rainure est petite, cette proportion du débit de lubrifiant peut être
négligée.
La pression d'alimentation en lubrifiant p chasse également le lubrifiant vers les extrémités du palier. La
en
proportion Q de débit de lubrifiant résultant de la pression d'alimentation est
p
3
3
Dpy
en
eff
Q= Q * . . . (9)
pp
h
eff
où QQ**= (eW,,BD ) est donné dans l'ISO 7902-2.
pp
6.3.1 Les éléments utilisés pour la lubrification sont les trous de graissage, les rainures de graissage et les poches
à huile. Il convient que la pression d'alimentation en lubrifiant p soit sensiblement inférieure à la charge spécifique
en
du palier p, pour éviter d'introduire des charges hydrostatiques supplémentaires. En règle générale, p se situe
en
entre 0,05 MPa et 0,2 MPa. La profondeur des rainures de graissage et poches à huile est beaucoup plus grande
que le jeu du palier.
6.3.2 Les rainures de graissage sont des éléments conçus pour diffuser le lubrifiant dans le sens circonférentiel.
Les encoches usinées dans la surface de frottement courent sur la périphérie et sont étroites dans le sens axial. Si
les rainures se situent au voisinage d'une augmentation de pression, la pression se répartit entre deux «crêtes»
indépendantes, ce qui réduit de façon notable la capacité de charge (voir figure 3). Dans ce cas, on effectue le
calcul sur une charge partagée par moitié entre chaque demi-coussinet. Toutefois, en raison de l'augmentation de
la pression hydrodynamique, Q , on ne tiendra compte que de la moitié du débit pour équilibrer les pertes
3
thermiques (voir 6.4) puisque le retour dans la rainure de graissage ne joue aucun rôle dans la dissipation de la
chaleur. Il est plus avantageux pour un palier complet de placer la rainure de graissage dans une région non
chargée car la totalité de l'écoulement de lubrifiant Q est prise en compte dans le bilan thermique.
p
6.3.3 Les poches à huile sont des éléments conçus pour diffuser le lubrifiant sur toute la longueur du palier. Les
encoches usinées dans la surface de frottement sont orientées dans le sens axial et il convient qu'elles prennent
aussi peu de place que possible dans le sens circonférentiel. Il est recommandé que la longueur relative des
poches remplisse la condition b /B < 0,7. Bien que le débit d'huile augmente avec leur nombre, l'huile sortant des
p
petites nervures rétrécies aux extrémités du palier ne joue aucun rôle dans la dissipation de chaleur. C'est encore
plus vrai si les nervures sont orientées dans le sens axial. Les paliers complets (W = 360°) ont une poche de
graissage dans le sens opposé de la charge et deux poches de graissage perpendiculaires à la charge. Le débit de
lubrifiant aidant à dissiper la chaleur de frottement due aux cisaillements, même dans la partie non chargée du
palier, les poches à bulle doivent être prises en compte totalement dans le bilan thermique. Dans les segments
partiels (W < 360°), le débit de lubrifiant dû à la pression d'alimentation des poches à huile à l'entrée ou à la sortie
du segment ne contribue presque pas à la dissipation de chaleur car les poches se rétrécissent à peine aux
extrémités de segment et la majeure partie du lubrifiant sort directement.
Si le lubrifiant remplit la partie sous charge du palier alors qu'il n'y en a pas dans la partie non chargée, on ne tient
compte que du débit de lubrifiant dans la partie sous charge dans la dissipation de chaleur.
L'influence du type et de la disposition des éléments utilisés pour la lubrification sur le débit de lubrifiant est traitée
dans l'ISO 7902-2.
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Le débit global de lubrifiant s'exprime sous la forme
QQ= . . . (10)
3
si le lubrifiant ne remplit que la partie sous charge du palier, et
QQ=+Q . . . (11)
3 p
si le lubrifiant remplit tout le jeu circulaire y compris la partie non chargée, soit 2p.
Légende
1 Trou de graissage
2 Rainure de graissage
Figure 3
6.4 Bilan thermique
L'état thermique d'un palier lisse résulte de son bilan thermique. Le débit thermique, P , résultant des frottements
th,f
dans les paliers, P, se dissipe dans l'environnement par le logement de palier et le lubrifiant qui en sort. En
f
pratique, il existe deux modes de dissipation de la chaleur dont l'un domine. Si l'on néglige l'autre, on s'assure d'une
marge de sécurité supplémentaire au stade de la conception. On peut faire les hypothèses suivantes:
a) Dans les paliers lubrifiés sans pression (par exemple bague de graissage), la chaleur se dissipe principalement
par convection vers l'environnement: P = P
th,f th,amb
b) Dans les paliers lubrifiés sous pression, la chaleur se dissipe principalement à travers le lubrifiant: =
P P
th,f th,L
6.4.1 Dissipation par convection
Le phénomène a lieu par convection thermique dans le logement de palier et par rayonnement et convection de la
surface du logement vers l'extérieur. Le processus complexe du transfert de chaleur peut se résumer comme suit:
Pk=−ATT . . . (12)
()
th,amb A B amb
où
22
k=⋅15Wm K à 20Wm⋅K
() ()
A
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ou, si l'on ventile le logement du palier avec de l'air circulant à une vitesse V > 1,2 m/s
a
k =+712 V . . . (13)
A a
(Voir [3] et [14])
Si l'on ne connaît pas exactement la surface réchauffée A du logement de palier, on peut utiliser la formule
approchée suivante:
— pour les logements cylindriques
p
2 2
AD=−2 D+pDB . . . (14)
() HH
H
4
— pour les chaises paliers
H
AH=+p B . . . (15)
H
2
— pour les paliers de socles de machine
AD=()15 à 20B . . . (16)
où
B est la longueur du logement axial;
H
D est le diamètre extérieur du logement;
H
H est la hauteur totale de la chaise palier.
6.4.2 Dissipation par le lubrifiant
Si la lubrification se fait sous pression, la dissipation de chaleur se fait par le lubrifiant:
Pc=−rQT T . . . (17)
()
th,L ex en
Pour les lubrifiants minéraux la capacité thermique volumique est donnée par
6 3
rc=×18,J10 m⋅K . . . (18)
()
Du bilan thermique, il découle que
P = P pour les paliers lubrifiés sans pression,
th,f th,amb
P = P pour les paliers lubrifiés sous pression,
th,f th,L
ce qui donne la température du palier T (voir [15]) et la température de sortie du lubrifiant T (voir [15]). La
B ex
température effective de la pellicule de lubrifiant compte tenu de la viscosité est
a) dans le cas d'une convection pure: T = T
eff B
b) dans le cas d'une dissipation par le lubrifiant: T = T = 0,5 (T + T )
eff L en ex
Aux vitesses périphériques élevées, il est possible de choisir, au lieu de ces valeurs moyennes, une température
plus voisine de la température de sortie du lubrifiant.
Les valeurs calculées de T et T doivent être vérifiées en fonction des paramètres opérationnels admis T
B ex lim
donnés dans l'ISO 7902-3.
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Dans la suite des calculs, on ne connaît d'abord que les données opérationnelles T et T et non la température
amb en
effective T nécessaire dès le début. La solution est de commencer avec l'augmentation de température estimée,
eff
soit
a) = 20 K
T - T
B,0 amb
b) T - T = 20 K
ex,0 en
et les températures de fonctionnement correspondantes T . Puis, à partir du bilan thermique, de calculer les
eff
températures corrigées T ou T qui sont améliorées par intégrations successives avec les hypothèses (T ou
B,1 ex,1 B,0
T ) jusqu'à ce que la différence entre les valeurs indicées 0 et 1 demeure négligeable (par exemple 2 K). La
ex,0
condition ainsi obtenue correspond à un fonctionnement stabilisé. Pendant l'itération, il convient de tenir compte des
facteurs indiqués en 6.7. En règle générale, l'itération converge rapidement. On peut la remplacer par une
interpolation graphique supposant plusieurs différences de température pour calculer P et P ou P . Si l'on
th,f th,amb th,L
trace les courbes des flux thermiques P = f(T ) ou P = f(T ), la position immobile est donnée par
th,amb B th,L ex
l'intersection des deux courbes (voir figure A.1).
6.5 Épaisseur minimale de la pellicule de lubrifiant et cha
...
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