ISO/TS 31657-1:2020

TECHNICAL ISO/TS
SPECIFICATION 31657-1
First edition
2020-06
Plain bearings — Hydrodynamic plain
journal bearings under steady-state
conditions —
Part 1:
Calculation of multi-lobed and tilting
pad journal bearings
Reference number
©
ISO 2020
© ISO 2020
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ii © ISO 2020 – All rights reserved

Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols and units . 1
5 General principles, assumptions and preconditions . 7
6 Calculation method . 9
6.1 General . 9
6.2 Load carrying capacity .11
6.3 Frictional power.11
6.4 Lubricant flow rate .12
6.5 Heat balance .13
6.6 Maximum lubricant film temperature .14
6.7 Maximum lubricant film pressure .15
6.8 Operating states .15
6.9 Further influencing parameters .15
6.10 Stiffness and damping coefficients .16
7 Figures .18
Annex A (informative) Calculation examples .23
Bibliography .37
Foreword
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This document was prepared by Technical Committee ISO/TC 123, Plain bearings, Subcommittee SC 8,
Calculation methods for plain bearings and their applications.
A list of all parts in the ISO/TS 31657 series can be found on the ISO website.
Any feedback or questions on this document should be directed to the user’s national standards body. A
complete listing of these bodies can be found at www .iso .org/ members .html.
iv © ISO 2020 – All rights reserved

Introduction
The aim of this document is the operationally-safe design of plain journal bearings for medium or high
journal circumferential velocities, U , up to approximately 90 m/s by applying a calculation method for
J
oil-lubricated hydrodynamic plain bearings with complete separation of journal and bearing sliding
surfaces by a lubricating film.
For low circumferential velocities up to approximately 30 m/s usually circular cylindrical bearings
are applied. For these bearings a similar calculation method is given in ISO 7902-1, ISO 7902-2 and
ISO 7902-3.
Based on practical experience the calculation procedure is usable for application cases where specific
bearing load times circumferential speed, pU⋅ , does not exceed approximately 200 MPa·m/s.
J
This document discusses multi-lobed journal bearings with two, three and four equal, symmetrical
sliding surfaces, which are separated by laterally-closed lubrication pockets, and symmetrically-loaded
tilting-pad journal bearings with four and five pads. Here, the curvature radii, R , of the sliding
B
surfaces are usually chosen larger than half the bearing diameter, D, so that an increased bearing
clearance results at the pad ends.
The calculation method described here can also be used for other gap forms, for example asymmetrical
multi-lobed journal bearings like offset-halves bearings, pressure-dam bearings or other tilting-pad
journal bearing designs, if the numerical solutions of the basic formulas are available for these designs.
TECHNICAL SPECIFICATION ISO/TS 31657-1:2020(E)
Plain bearings — Hydrodynamic plain journal bearings
under steady-state conditions —
Part 1:
Calculation of multi-lobed and tilting pad journal bearings
1 Scope
This document specifies the general principles, assumptions and preconditions for the calculation of
multi-lobed and tilting-pad journal bearings by means of an easy-to-use calculation procedure based on
numerous simplifying assumptions. For a reliable evaluation of the results of this calculation method,
it is indispensable to consider the physical implications of these assumptions as well as practical
experiences for instance from temperature measurements carried out on real machinery under
typical operating conditions. Applied in this sense, this document presents a simple way to predict
the approximate performance of plain journal bearings for those unable to access more complex and
accurate calculation techniques.
The calculation method serves for the design and optimisation of plain bearings, for example in
turbines, compressors, generators, electric motors, gears and pumps. It is restricted to steady-state
operation, i.e. in continuous operating states the load according to size and direction and the angular
velocity of the rotor are constant.
Unsteady operating states are not recorded. The stiffness and damping coefficients of the plain journal
bearings required for the linear vibration and stability investigations are indicated in ISO/TS 31657-2
and ISO/TS 31657-3.
2 Normative references
There are no normative references in this document.
3 Terms and definitions
No terms and definitions are listed in this document.
ISO and IEC maintain terminological databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at http:// www .electropedia .org/
4 Symbols and units
Table 1 contains the symbols used in the ISO 31657 series.
Table 1 — Symbols and units
Symbol Description Unit
B Bearing width m
B
*
*
Relative bearing width, width ratio as given by: B = 1
B
D
Table 1 (continued)
Symbol Description Unit
b Width of lubricant pocket m
P
b
* P
*
b 1
Relative width of lubricant pocket, as given by: b =
P
P
B
Bearing radial clearance, as given by: CR=−R
C m
RJ
R
C
Effective radial bearing clearance m
R,eff
c Stiffness coefficient of lubricant film (i,k = 1,2) N/m
ik
Non-dimensional stiffness coefficient of lubricant film, as given by:
* ψ
c 1
eff
*
ik
c = ⋅=ci(),,k 12
ik ik
2⋅⋅Bηω⋅
eff
J/(kg
c
Specific heat capacity (p = constant)
p
K)
D Nominal bearing diameter (inside diameter of journal bearing) m
D Maximum value of D m
max
D
Minimum value of D m
min
D
Journal diameter (diameter of the shaft section located inside of a journal bearing) m
J
D Maximum value of D
m
Jm, ax J
D Minimum value of D
m
Jm, in J
d
Damping coefficient of lubricant film (i,k = 1,2) N s/m
ik
Non-dimensional damping coefficient of lubricant film, as given by:
* 3
ψ 1
d
ik eff
*
d = ⋅⋅ω di(),,k=12
ik ik
2⋅⋅Bηω⋅
eff
e Eccentricity (distance between journal and bearing axis) m
Eccentricity of the bearing sliding surfaces (pads) of a multi-lobed or tilting-pad journal
e m
B
bearing
f Bearing force, bearing load, nominal bearing load, load-carrying capacity N
ΔF Component of additional dynamic force in x-direction N
x
ΔF
Component of additional dynamic force in y-direction N
y
Component of additional dynamic force parameter in x-direction, as given by:
* ΔF ⋅ψ
ΔF x
eff
*
x
ΔF =
x
BD⋅⋅ηω⋅
eff
Component of additional dynamic force parameter in y-direction, as given by:
*
ΔF ⋅ψ
ΔF 1
y eff
y
*
ΔF =
y
BD⋅⋅ηω⋅
eff
Friction force, as given by: Ff=⋅F
F
N
f
f
f
*
*
Friction force parameter, as given by: F =⋅So
F 1
f
f
ψ
eff
F
Bearing force at transition to mixed friction N
tr
F Coefficient of friction 1
f
Journal deflection m
J
h(φ) Local lubricant film thickness m
2 © ISO 2020 – All rights reserved

Table 1 (continued)
Symbol Description Unit
h()ϕ
*
*
Relative local lubricant film thickness, as given by: h ()ϕ =
h ()ϕ
C
R
h
Minimum admissible lubricant film thickness at transition to mixed friction m
lim,tr
Minimum admissible relative lubricant film thickness at transition to mixed friction, as
h
*
lim,tr
h 1
*
lim,tr
given by: h =
lim,tr
C
R,eff
h
Minimum lubricant film thickness, minimum gap m
min
h
min
*
*
Minimum relative lubricant film thickness, minimum relative gap, as given by: h =
h 1
min
min
C
R,eff
h
Minimum lubricant film thickness at transition to mixed friction m
min,tr
Minimum relative lubricant film thickness at transition to mixed friction, as given by:
h
*
min,tr
h 1
*
min,tr
h =
min,tr
C
R,eff
h ϕ
() Local gap at ε=0 , gap function m
h ()ϕ
*
Relative local gap at ε=0 , profile function, as given by: h ()ϕ =
*
h ()ϕ 1
C
0 R
h
Maximum gap at ε=0 m
0,max
h
0,max
*
Maximum relative gap at ε=0 , gap ratio, as given by: h =
*
0,max
h 1
C
0,max
R
Profile factor (relative difference between lobe or pad bore radius and journal radius), as
ΔR
K
B 1
P
given by: K ==
P
Cm1−
R
K
Effective profile factor 1
P,eff
K
Profile factor at 20 °C 1
P,20
M Mixing factor 1
m Preload factor, preload of bearing or pad sliding surface 1
−1
N Rotational speed (rotational frequency) of the rotor (revolutions per time unit) s
−1
N
Critical speed (critical rotational frequency) s
cr
Rotational speed (rotational frequency) at the stability speed limit of the rotor supported
−1
N
s
lim
by plain bearings
−1
N
Resonance speed (resonance rotational frequency) of the rotor supported by plain bearings s
rsn
Rotational speed (rotational frequency) at transition to mixed friction, transition rotational
−1
N s
tr
speed, transition rotational frequency
O Centreline of plain bearing 1
B
O
Centreline of sliding surface No. i 1
i
O
Centreline of journal 1
J
P Frictional power, as given by: PF=⋅U
W
f ff J
P
Heat flow via the lubricant W
th,L
p Lubricant film pressure, local lubricant film pressure Pa
Table 1 (continued)
Symbol Description Unit
F
p Pa
Specific bearing load, as given by: p=
BD⋅
p Lubricant supply pressure Pa
en
p ⋅ψ
en
eff
* *
p Lubricant supply pressure parameter, as given by: p = 1
en en
ηω⋅
eff
p Maximum admissible lubricant film pressure Pa
lim
p
Maximum admissible specific bearing load at transition to mixed friction Pa
lim,tr
p Maximum lubricant film pressure Pa
max
p
max
* *
p Maximum lubricant film pressure parameter, as given by: p = 1
max max
p
F
tr
p Pa
Specific bearing load at transition to mixed friction, as given by: p =
tr
tr
BD⋅
Lubricant flow rate, as given by: QQ=+Q
Q m /s
3 p
Q Minimum admissible lubricant flow rate m /s
lim
Q
Lubricant flow rate due to supply pressure m /s
p
Q
p
* *
Q
Lubricant flow rate parameter due to supply pressure, as given by: Q = 1
p p
*
pQ⋅
en 0
3 3
Q
Reference value of Q, as given by: QR=⋅ωψ⋅ m /s
0 eff
Q Lubricant flow rate at the entrance into the lubrication gap (circumferential direction) m /s
Lubricant flow rate at the exit of the lubrication gap (circumferential direction), as
Q
m /s
given by: QQ=−Q
21 3
Lubricant flow rate parameter at the exit of the lubrication gap (circumferential direction),
Q
*
Q 2 1
*
2 as given by: Q =
Q
Q Lubricant flow rate due to hydrodynamic pressure build-up (side flow rate) m /s
Lubricant flow rate parameter due to hydrodynamic pressure build-up (side flow parame-
* Q
3 1
Q *
ter), as given by: Q =
Q
D
R m
Journal bearing inside radius, as given by: R=
R Lobe or pad bore radius of a multi-lobed or tilting-pad journal bearing m
B
Difference between lobe or pad bore radius and journal radius, as given by: ΔRR=−R
ΔR
m
BB J
B
Journal radius (radius of the shaft section located inside of a journal bearing), as given by:
R D
m
J
J
R =
J
R
Surface finish ten-point average of bearing sliding surface m
z,B
R
Surface finish ten-point average of journal sliding surface m
z,J
ρω⋅⋅RC⋅
R,eff
Re Reynolds number, as given by: Re= 1
η
eff
Re
Critical Reynolds number 1
cr
4 © ISO 2020 – All rights reserved

Table 1 (continued)
Symbol Description Unit
F⋅ψ
eff
So Sommerfeld number, as given by: So= 1
BD⋅⋅ηω⋅
eff
So Sommerfeld number at transition to mixed friction 1
tr
S Displacement amplitude of the rotor (mechanical oscillation) m
T Temperature °C
Heating of lubricant between bearing entrance and exit, as given by: ΔTT=−T
ΔT K
ex en
ΔT Maximum admissible heating of lubricant between bearing entrance and exit K
lim
T
Bearing temperature °C
B
T Effective temperature of lubricant film °C
eff
T
Lubricant temperature at the bearing entrance °C
en
T Lubricant temperature at the bearing exit °C
ex
T
Journal temperature °C
J
T Maximum admissible bearing temperature °C
lim
T
Maximum temperature of lubricant film °C
max
Difference between maximum temperature of lubricant film and lubricant temperature in
ΔT
K
max
the lubricant pocket, as given by: ΔTT=−T
maxmax 1
Non-dimensional difference between maximum temperature of lubricant film and lubricant
* ρψ⋅⋅c
ΔT p eff 1
*
max
temperature in the lubricant pocket, as given by: ΔΔT = ⋅ T
max max
pf⋅
T Lubricant temperature at the entrance into the lubrication gap (circumferential direction) °C
Difference between lubricant temperature at the entrance into the lubrication gap and
ΔT
K
lubricant temperature at the bearing entrance, as given by: ΔTT=−T
11 en
T
Lubricant temperature at pressure profile trailing edge (circumferential direction) °C
Difference between lubricant temperature at pressure profile trailing edge and lubricant
ΔT K
temperature at the entrance into the lubrication gap, as given by: ΔTT=−T
22 1
t Time s
Circumferential speed of the journal, sliding velocity
U
m/s
J
UR=⋅ω
JJ
U
Circumferential speed at transition to mixed friction m/s
tr
U
Minimum admissible circumferential speed at transition to mixed friction m/s
lim,tr
u Velocity component in the φ-direction m/s
u
Average velocity component in the φ-direction m/s
w Velocity component in the z- direction m/s
w Average velocity component in the z-direction m/s
x Coordinate of journal radial motion, normal to direction of load m
x
*
*
Relative coordinate of journal radial motion, normal to direction of load, as given by: x =
x
C
R
Coordinate normal to sliding surface (across the lubricant film, in the radial direction);
y m
coordinate of journal radial motion, in direction of load
y
*
*
Relative coordinate of journal radial motion, in direction of load, as given by: y =
y
C
R
Table 1 (continued)
Symbol Description Unit
y Coordinate normal to sliding surface (across the lubricant film) m
h
Z Number of sliding surfaces (pads), number of pockets per bearing 1
Coordinate parallel to the sliding surface, normal to direction of motion (normal to circum-
z m
ferential direction, in the axial direction)
−1
α
Linear thermal expansion coefficient of bearing material K
lB,
−1
α
Linear thermal expansion coefficient of journal material K
l ,J
β Attitude angle (angular position of journal eccentricity related to the direction of load) °
β
Angle between direction of load and position of minimum lubricant film thickness °
h,min
δ
Journal misalignment angle (angular deviation of journal) °
J
e
Relative eccentricity: ε=
ε 1
C
R,eff
η Dynamic viscosity of the lubricant Pa s
η
Effective dynamic viscosity in the lubricant film Pa s
eff
ρ Density of the lubricant kg/m
φ Angular coordinate in circumferential direction °
ϕ
Angular coordinate of pivot position of pad (tilting-pad bearing) °
F
ϕ Angular coordinate of lubricant pocket centreline °
P
Angular coordinate of bearing sliding surface (segment or pad) centreline at multi-lobed or
ϕ °
tilting-pad journal bearings (with non-tilted pads), see Figure 1, a)
ϕ
Angular coordinate at the entrance into the gap °
ϕ
Angular coordinate at the end of the hydrodynamic pressure build-up °
ϕ
Angular coordinate at the exit of the gap °
C
R
ψ ‰
Relative bearing clearance, as given by: ψ=
R
Tolerance of ψ, as given by: Δψ=−ψψ
Δψ ‰
maxmin
ψ
Effective relative bearing clearance ‰
eff
ψ
Maximum value of ψ ‰
max
ψ
Minimum value of ψ ‰
min
Δψ
Thermal change of ψ ‰
th
ψ
Relative bearing clearance at 20 °C ‰
Angular span of bearing sliding surface (segment or pad), as given by: Ω=−ϕϕ
Ω °
Angular distance between leading edge and pivot position of pad (tilting-pad bearing), as
Ω
°
F
given by: Ω =−ϕϕ
FF 1
Relative angular distance between leading edge and pivot position of pad (tilting-pad
*
Ω 1
*
F
bearing), as given by: ΩΩ= /Ω
FF
360°
Ω
°
Angular span of lubricant pocket, as given by: ΩΩ= −
P
P
Z
−1
Angular speed of the rotor, as given by: ωπ=⋅2 ⋅N
ω s
−1
ω
Angular speed at transition to mixed friction s
tr
6 © ISO 2020 – All rights reserved

5 General principles, assumptions and preconditions
The bearing bore form of multi-lobed journal bearings [see Figure 1, a)] and tilting-pad journal bearings
h ϕ
()
*
[with non-tilted pads according to Figure 1, b)] is described by the profile function h ()ϕ = in the
C
R
e
case of a centric journal position ε ==0 . The angle φ is counted, starting from the load direction, in
C
R
the journal rotational direction.
Formula (1) applies to the shell segment or pad i with the angular length Ω =−ϕϕ :
ii31,,i
ΔΔR R
 
B B
*
h ()ϕϕ=+ −11⋅−cos( ϕ ),i= ,.,Ζ (1)
 
00,i ,i
C C
 
R R
with the profile factor
RR−
ΔR e
B J
B B
K == =+1 ,
P
C C C
R R R
minimum clearance
DD−
J
CR=−R =
R J
and the lubricant film thickness ratio as given by Formula (2):
h
0,max
* *
hh()ϕ == (2)
00Pi,,max
C
R
Here the position of the sliding surface (segment or pad) axis (curvature centre "point") of the shell
segment or pad i is uniquely described by the sliding surface eccentricity e and the associated angle
B
coordinate ϕ .
0,i
*
In the case of cylindrical bearings, K = 1 and h ϕ =1 .
()
P
NOTE Instead of the profile factor, K the "preload factor", m, is frequently used internationally; the
P,
following relation exists between both variables:
K =
P
1−m
In the case of an eccentric position of the journal (ε, β), Formula (3) applies to the lubricant film
thickness, h(φ), of the multi-lobed journal bearings [(see Figure 1, c)]:
**
hC()ϕϕ=⋅hC()=⋅[(h ϕϕ)c−⋅εβos()− ] (3)
RR 0
In the case of tilting-pad journal bearings [see Figure 1, d)], the individual pads automatically adjust
themselves (optimally) so that the lubricant film force F passes through the supporting pad pivot,
i
[9]
respectively . For a more precise calculation of tilting-pad journal bearings, the elasticities in the pad
support and the elastic and thermal deformations of the pads shall be considered.
The pressure formation in the lubrication gaps is basically calculated with the numerical solutions of
the Reynolds differential equation for a finite bearing width:
1 ∂  ∂p  ∂ ∂p ∂h
 
⋅ h ⋅ + h ⋅ =⋅6ηω⋅⋅ (4)
 
 
∂ϕϕ∂ ∂z ∂z ∂ϕ
R  
 
J
with ωπ=⋅2 ⋅N angular speed of the rotor.
For derivation of the Reynolds differential equation, reference is made to Reference [8], for the
numerical solution to Reference [9].
When solving Formula (4), the following idealising assumptions and preconditions are made, whose
[10]
permissibility shall be estimated according to Clause 6, if necessary .
a) The lubricant corresponds to a Newtonian fluid.
b) All flow processes of the lubricant are laminar.
c) The lubricant adheres fully to the sliding surfaces.
d) The lubricant is incompressible.
e) At the leading edge of the segment or pad, the lubrication gap is completely filled with lubricant.
f) Inertia effects, gravitation and magnetic forces of the lubricant are negligible.
g) The components forming the lubrication gap are rigid or their deformation is negligible; the
surfaces of the journal and bearing bore are ideal circular cylinders or cylindrical segments.
h) The curvature radii of the surfaces moving relative to one another are large in comparison to the
lubricant film thicknesses.
i) The lubricant film thickness in an axial direction (z coordinate) is constant.
j) Pressure changes in the lubricant film normal to the sliding surfaces (in the lubricant film thickness
direction) are negligible.
k) A movement normal to the sliding surfaces (in the lubricant film thickness direction) is not
considered here, in contrast to 6.10.
l) The lubricant film is isoviscous in the entire lubrication gap.
m) The lubricant is supplied at the leading edge of the segments or pads respectively; the level of the
supply pressure is negligible compared to the lubricant film pressures themselves.
The boundary conditions for the lubricant film pressure build-up satisfy the continuity condition.
The following applies respectively to the individual segments or pads (see Figures 2 and 4):
— at the lateral bearing edge  pzϕ,/=±B 20= ;
()
— in the lubrication pocket and on the sealing land pz()ϕ, =0 ;
∂p
— at the pressure profile trailing edge pz[(ϕ ),z][= ϕ ()zz,]=0 ;
∂ϕ
∂p ∂p
— at the beginning of cavitation area pz[(ϕ ),z][= ϕϕ()zz,]= [(zz), ]=0 ;
∂ϕ ∂z
— at the end of cavitation area  pz[(ϕ ),z]=0 .
[15][16]
The cavitation theory according to Jakobsson, Floberg and Olsson is used in the cavitation area
and on its edge for fulfilment of the continuity condition.
8 © ISO 2020 – All rights reserved

The numerical integration of the Reynolds differential equation is done using the transformation of the
pressure proposed in Reference [9] by conversion into a difference formula, which is applied to a grid of
nodal points and which leads to a system of linear formulas.
After specifying the boundary conditions, the integration yields the pressure profile in the
circumferential and axial direction.
The maximum lubricant film temperature is calculated using the numerical solution of the energy
equation averaged by integration with respect to the lubricant film thickness, h
h 22
 
   
u ∂T ∂T η 1 ∂u ∂w
 
⋅ +⋅w = ⋅⋅ + ⋅dy (5)
   
h
∫
R ∂ϕ ∂zcρ⋅ h ∂y ∂y
 
   
J ph h
00 
[9][13][14]
for the two-dimensional temperature distribution T(φ, z) .
This includes
h h
u =⋅ ud⋅=yw, ⋅⋅wdy
h h
∫∫
h h
the flow rates averaged over the lubricant film thickness h in the circumferential and axial direction.
When deriving the energy equation, Formula (5), it is also assumed besides the above preconditions
that no heat is dissipated from the lubrication gap by thermal conduction (adiabatic calculation).
When solving Formula (5) the following boundary conditions apply (see Figure 4):
— at the entrance gap Tz(,ϕ )=T ;
∂T
— in the axial bearing centre (,ϕ z==00) .
∂z
The numerical integration of Formula (5) is carried out similar to the solution of the Reynolds
differential equation, Formula (4), using a suitable difference formula and yields for the specified
boundary conditions the temperature distribution in the circumferential and axial direction.
The application of the similarity principle in the hydrodynamic plain bearing theory leads to
dimensionless similarity variables for the interesting characteristic values (such as load-carrying
capacity, frictional power, lubricant flow rate and relative bearing width). Use of the similarity variables
reduces the number of necessary numerical solutions of the Reynolds differential equation, Formula (4),
and the energy equation, Formula (5), which are summarised in ISO/TS 31657-2 and ISO/TS 31657-3.
As a rule, other solutions can also be used, insofar as they satisfy the conditions indicated in this
document and a corresponding numerical accuracy.
ISO/TS 31657-4 contains operational guide values for checking the calculation results, in order to
ensure the functionality of the plain bearings.
In special use cases, operational guide values different from ISO/TS 31657-4 can be agreed.
6 Calculation method
6.1 General
Calculation refers to the mathematical determination of the functional capability based on operational
characteristic values (see Figure 5), which are to be compared with permissible operational parameter
values. The operational characteristic values determined in different operating states shall be
permissible with their permissible operational parameter values. All continuous operating states shall
be examined for this.
The safety against wear is ensured when a complete separation of the sliding partners is attained by the
lubricant. Continuous operation in the mixed friction area leads to premature functional incapability.
Short-term operation in the mixed friction area, for example when starting up and running down of
machines with slide bearings, is unavoidable and does not usually lead to bearing damage. At high
loads, a hydrostatic jacking can be required during slow start-up or run-down. Running-in and adapting
wear for compensating the surface form deviations from the ideal form are permissible as long as these
occur with local and time restrictions and without signs of overload.
The limits of the mechanical load are given by the strength of the bearing material. Minor plastic
deformations are permissible as long as they do not impair the functional capability of the plain bearing.
The limits of the thermal load result from the high-temperature strength of the bearing material, but
also from the viscosity temperature dependence and the tendency of the lubricant to age.
The calculation of the functional capability of plain bearings presupposes that the operating conditions
are known for all continuous operating states. In practice, however, additional disturbing influences
frequently occur, which are still unknown during the design and which are also not always accessible
to a mathematical approach. It is therefore recommended to work with a corresponding safety interval
between the operational characteristic values and the permissible operating parameter values.
Disturbing influences are for example:
— disturbing forces (e.g. imbalances, vibrations);
— form deviations from the ideal geometry (e.g. operational deformations, production tolerances,
assembly deviations);
— lubricant impurities due to solid, liquid and gaseous foreign bodies;
— corrosion, electro-erosion.
Information on some further influencing variables is given in 6.9.
The applicability of this document, in which a laminar flow in the lubrication gap is presupposed, shall
[11][12]
be checked by the Reynolds number :
ρω⋅⋅RC⋅
41,3
R,eff
Re= ≤≈Re (6)
cr
η
K ⋅ψ
eff
P eff
C
R,eff
with ψ = effective relative bearing clearance.
eff
R
In the case of plain bearings with Re > Re (e.g. due to high circumferential velocities), higher power
cr
[10][11][12]
losses shall be expected. The load carrying capacity can rise .
Bearings with turbulent flow can only be calculated approximately according to this document.
The plain bearing calculation grasps the following, based on the known bearing dimensions and
operating data:
— the relation between bearing load carrying capacity and lubricant film thickness;
— the frictional power;
— the lubricant flow rate;
— the heat balance;
— the maximum lubricant film temperature and the maximum lubricant film pressure;
all these interacting with one another. The solution happens in an iterative process whose sequence is
summarised in the calculation flow chart according to Figure 5.
10 © ISO 2020 – All rights reserved

A parameter variation can be performed for the optimisation of individual parameters. It is possible to
modify the calculation procedure
6.2 Load carrying capacity
Characteristic for the load carrying capacity is the (non-dimensional) Sommerfeld number
F⋅ψ
eff
So= (7)
BD⋅⋅ηω⋅
eff
e B
*
whose dependence on the relative eccentricity ε = , the relative bearing width B = and the
C D
R,eff
*
profile function h ϕ is indicated in ISO/TS 31657-2 and ISO/TS 31657-3. The state variables η , ψ
()
eff eff
consider thermal influences (see 6.5 and 6.9).
*
From this, with the attitude angle β [So, B*, h ()ϕ ] according to ISO/TS 31657-2 and ISO/TS 31657-3
the components of the static bearing flexibility ε cos β, ε sin β depending on the static load parameter So
can be determined.
In the case of multi-lobed and tilting-pad journal bearings, the static displacement e and minimum
lubricant film thickness h add up vectorially to the radial journal mobility (see Figure 3). The
min
h
min
* **
dependence indicated in ISO/TS 31657-2 and ISO/TS 31657-3 hS oB,,h ϕ = yields
()
min 0
 
C
R,eff
h
lim,tr
*
through comparison with the permissible operational parameter valueh = the load carrying
lim,tr
C
R,eff
capacity at the transition to mixed friction.
6.3 Frictional power
The losses due to frictional power in a hydrodynamic plain bearing are determined by the dimensionless
*
friction force, F , (or the friction coefficient, f )
f
f
*
F =⋅So (8)
f
ψ
eff
*
whose dependence on So, B* and h ()ϕ is indicated in ISO/TS 31657-2 and ISO/TS 31657-3. Here it is
assumed that the lubricant supply pressure, p , remains very low and, in cavitation areas, the friction
en
[15][16]
force has a linear dependence on calculated degree of filling .
The friction power in the bearing or the heat flow caused by it is
PF=⋅U (9)
ff J
with Ff=⋅F friction force and UR=⋅ω circumferential speed of the journal.
f JJ
NOTE Particularly in the lubricant pockets of multi-lobed journal bearings and between the pads of tilting-
pad journal bearings already at moderate circumferential speeds, turbulent flow can occur, leading to increasing
power losses in these areas not taken into account in the calculation procedure described in this document.
In addition, depending on the geometrical design of the lubricant supply and lubricant scraper elements (if
applicable), churning losses can arise in the cavities between the tilting-pads. The extent of these power losses
not considered in this calculation method depend also on the filling degree of the cavities (see 6.5).
6.4 Lubricant flow rate
The lubricant supplied to the bearing via lubrication pockets forms a load-carrying lubricant film for
separating the sliding surfaces. The pressure formation in the lubricant film forces the lubricant out
from the sides of the bearing.
[9][13]
This is the fraction, Q , of lubricant flow rate due to inherent pressure formation :
*
QQ=⋅Q (10)
30 3
D
with QU=⋅C ⋅=R ⋅⋅ωψ reference lubricant flow rate
0 J,R effeff
* *
and Q = f [So, B*, h ()ϕ ] according to ISO/TS 31657-2 and ISO/TS 31657-3.
3 0
The lubricant supply pressure, p , at the inlet into the bearing also forces lubricant out from the sides
en
[13]
of the plain bearing. This is the fraction, Q , of lubricant flow rate due to supply pressure :
p
**
QQ=⋅pQ⋅ (11)
pp0en
* *
with Q = f [So, B*, h ()ϕ ] according to ISO/TS 31657-2
p 0
p ⋅ψ
en eff
*
and p = as lubricant supply pressure parameter.
en
ηω⋅
eff
The lubricant supply pressure, p is normally between 0,05 and 0,2 MPa (above the ambient pressure).
en,
In the case of tilting-pad journal bearings, the lubricant flow rate Q is normally set by throttling the
p
supply or discharge flow, or via corresponding nozzles (in case of injection lubrication).
The total lubricant flow rate is:
QQ=+Q (12)
3 p
For the (later) calculation of the lubricant temperature in the pockets, the lubricant flow rate, Q , which
enters in the circumference direction through the narrowest lubrication gap into the divergent gap, is
also required:
*
QQ=−QQ=⋅Q (13)
21 30 2
* *
with Q = f [So, B*, h ()ϕ ] according to ISO/TS 31657-2 and ISO/TS 31657-3.
2 0
Lubrication pockets as defined in this document are design elements for the distribution of the lubricant
over the bearing width. The recesses machined into the sliding surfaces of the bearing extend in the
axial direction and should be as short as possible in the circumferential direction.
*
Relative pocket widths should be bb=≤/,B 08 .
PP
Although greater values increase the oil flow rate, the oil escaping at the narrow throttling lands at the
sides does not take part in the heat dissipation. This applies more so if the side lands have axial grooves.
*
For the calculation of the flow rate fraction Q a relative pocket width of b = 0,8 is presupposed in this
p
P
document. The effect of the lubricant inertia forces is not considered here.
The depth of the lubrication pockets is significantly greater than the bearing clearance.
12 © ISO 2020 – All rights reserved

6.5 Heat balance
The thermal state of the plain bearing results from the heat balance.
The heat flow resulting from the frictional power, P , in the bearing is dissipated to the surroundings
f
via the bearing housing and via the lubricant escaping from the bearing.
Pressure-lubricated multi-lobed and tilting-pad journal bearings (forced lubrication) primarily
dissipate the heat via the lubricant (recooling):
P = P (14)
f th,L
By neglecting the convective heat dissipation via the bearing housing, an additional safety results with
the design. Formula (15) applies for the heat dissipation by the lubricant:
Pc=⋅ρρ⋅⋅QT −Tc=⋅ ⋅⋅QTΔ (15)
()
th,eLp xen p
In the case of mineral lubricants, the volume-specific heat capacity is:
6 3
ρ⋅=c ()17,,…18 ⋅⋅10 J/(m K)
p
In practice, with regard to the lubricant service life and/or the available cooling capacity of the
lubricating system, the heating of lubricant, ΔT, frequently has to be limited to a certain extent, ΔT ,
lim
(e.g. 20 … 25 K) by increasing the total lubricant flow rate, Q, appropriately as shown in Formula (16):
P
f
Q = (16)
lim
ρΔ⋅⋅cT
p lim
Mixing processes in the lubrication pockets:
As a multi-lobed and tilting-pad journal bearing comprises several pads, it is necessary to consider not
only the lubricant flow rate of an individual pad, but that of the complete bearing and hence also the
reciprocal effect of the individual lubricant flow rate fractions. The lubricant escaping at the end of
the segments or pads is mixed with freshly supplied lubricant in the following oil pocket. This means
that the lubricant temperature, T , at the entrance of the lubrication gap is higher than that of lubricant
freshly supplied with the temperature, T (see Figure 4).
en
To simplify, the same temperature, T , and, when calculating it, for all segments or pads, an averaged oil
heating to the temperature, T , is presupposed for all oil pockets. When determining the temperature
difference as shown in Formula (17)
ΔTT=−T (17)
11 en
an empirical factor must be introduced, as a purely theoretical treatment of this mixing problem has
not yet led to satisfactory results.
For adaptation to the experience gained up to the present (see Reference [14]), it is possible via a heat
balance at the lubrication pockets (see Figure 4) to introduce a mixing factor, M, as follows:
Q
ΔΔT = ⋅ T (18)
1 2
MQ⋅+()1−⋅MQ
The limiting values are considered for explanation of the mixing factor. A mixing factor M = 0 means no
mixing in the lubrication pockets, i.e., the lubricant flow rate leaving the lubrication gaps Q fully enters
into the following lubrication gaps. As a result, a high lubricant flow rate, Q, would be ineffective, as the
majority of this freshly supplied lubricant would flow axially out of the lubrication pockets, without
affecting the operational characteristic values. A mixing factor M = 1 means “complete” mixing in the
lubrication pockets. M = 0,4 to 0,6 can be used as an empirical value. It is influenced by the engineering
design to an extent that cannot be indicated more precisely. Particularly in the case of tilting-pad
bearings, it is possible to increase the mixing factor (e.g. up to 0,75) by using directed lubrication
methods combined with unrestricted drain instead of a flooded design with fixed or floating seals on
the bearing sides. In addition, such designs are suitable to reduce the churning losses in the cavities
between the tilting-pads (see 6.3).
The averaged oil heating ∆T at the pressure profile trailing edge, given in Formula (19)
ΔTT=−T (19)
22 1
can be calculated from the frictional power P in the lubrication gap via Formula (14):
f
P
f
ΔT = (20)
Q
 
ρ⋅⋅cQ +
 
p 2
 
As a result, the average pocket temperature T and the effective lubricant film temperature T can be
1 eff
determined:
TT=+ΔT ,
11en
(21)
TT+ ΔT
12 2
T = =+TTΔ +
effen 1
In the calculation sequence, only the lubricant film supply temperature T is initially known, but not
en
the effective lubricant film temperature, T , which is required at the beginning of the calculation.
eff
The temperatures resulting from the heat balance are improved iteratively by averaging with the
temperatures previously forming the basis until the difference between the results of two iteration
steps in succession becomes negligibly small, e.g. <1 K. The state then attained corresponds to the
steady state. The iteration usually converges rapidly (see Annex A).
Additionally, in the case of tilting-pad journal bearings, the lubricant flow rate, Q , to be set for ensuring
p
the functional capability (see 6.4) is generally not initially known in the calculation sequence. In such
cases, it is therefore recommended setting in a first step Q = 0 and checking the functional capability
p
of the bearing (i.e. in particular compliance with the permissible operational parameter values for the
minimum lubricant film thickness, h , and the maximum lubricant film temperature, T ) for this. If
min max
necessary, the lubricant flow rate, Q , can be determined in a second step by targeted specification of a
p
value Q > 0 so that the functional capability of the bearing is ensured (see A.2).
p
6.6 Maximum lubricant film temperature
The now known average pocket temperature, T , allows the maximum lubricant film temperature, T
1 max,
of the bearing to be determined from the energy equation, Formula (5) , (see Figure 4), as shown in
Formula (22):
TT=+ΔT (22)
maxm1 ax
with
*
F
pf⋅ p f
* *
ΔΔT = ⋅=T ⋅⋅ΔT , (23)
maxmax max
ρψ⋅⋅c ρ⋅c So
ppeff
* *
ΔT = f [So, B*, h ()ϕ ] according to ISO/TS 31657-2 and ISO/TS 31657-3,
max 0
14 © ISO 2020 – All rights reserved

ISO/T
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