Plain bearings — Hydrodynamic plain journal bearings under steady-state conditions — Part 3: Functions for calculation of tilting pad journal bearings

This document specifies the characteristic values for selected tilting-pad journal bearings with four or five centrally or eccentrically supported tilting pads and with angular spans of pad sliding surfaces of Ω = 80°, 60° and 45°. The functions plotted and listed in table form below are required for the operationally safe design of hydrodynamic tilting-pad journal bearings according to ISO/TS 31657-1. They are based on the presumptions and boundary conditions indicated there and only apply to stationary operating states. The symbols used are explained in ISO/TS 31657-1; calculation examples are also included there. The calculation method described in ISO/TS 31657-1 can also be used for other tilting-pad journal bearing designs, if the numerical solutions of the basic equations are available in the same manner for these designs.

titre manque — Partie 3: titre manque

General Information

Status

Published

Publication Date

23-Jun-2020

ICS

21.100.10 - Plain bearings

Technical Committee

ISO/TC 123/SC 8 - Calculation methods for plain bearings and their applications

Drafting Committee

ISO/TC 123/SC 8 - Calculation methods for plain bearings and their applications

Current Stage

6060 - International Standard published

Start Date

28-Apr-2020

Completion Date

24-Jun-2020

Ref Project

Buy Standard

ISO/TS 31657-3:2020 - Plain bearings -- Hydrodynamic plain journal bearings under steady-state conditions

Technical specification

ISO/TS 31657-3:2020 - Plain bearings -- Hydrodynamic plain journal bearings under steady-state conditions

English language

73 pages

sale 15% off

Preview

sale 15% off

Preview

Technical specification

ISO/TS 31657-3:2020 - Plain bearings -- Hydrodynamic plain journal bearings under steady-state conditions

English language

73 pages

sale 15% off

Preview

sale 15% off

Preview

Standards Content (sample)

TECHNICAL ISO/TS
SPECIFICATION 31657-3
First edition
2020-06
Plain bearings — Hydrodynamic plain
journal bearings under steady-state
conditions —
Part 3:
Functions for calculation of tilting pad
journal bearings
Reference number
ISO/TS 31657-3:2020(E)
ISO 2020
---------------------- Page: 1 ----------------------
ISO/TS 31657-3:2020(E)
COPYRIGHT PROTECTED DOCUMENT
© ISO 2020

be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting

on the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address

below or ISO’s member body in the country of the requester.
ISO copyright office
CP 401 • Ch. de Blandonnet 8
CH-1214 Vernier, Geneva
Phone: +41 22 749 01 11
Email: copyright@iso.org
Website: www.iso.org
Published in Switzerland
ii © ISO 2020 – All rights reserved
---------------------- Page: 2 ----------------------
ISO/TS 31657-3:2020(E)
Contents Page

Foreword ........................................................................................................................................................................................................................................iv

1 Scope ................................................................................................................................................................................................................................. 1

2 Normative references ...................................................................................................................................................................................... 1

3 Terms and definitions ..................................................................................................................................................................................... 1

4 Functions for calculation of multi-lobed journal bearings ...................................................................................... 1

4.1 General ........................................................................................................................................................................................................... 1

4.2 Tilting-pad journal bearings with four tilting pads ................................................................................................. 4

4.3 Tilting-pad journal bearings with five tilting pads ...............................................................................................39

Annex A (informative) Characteristic curves for tilting-pad bearings with five pads ....................................62

Bibliography .............................................................................................................................................................................................................................73

© ISO 2020 – All rights reserved iii
---------------------- Page: 3 ----------------------
ISO/TS 31657-3:2020(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.

The procedures used to develop this document and those intended for its further maintenance are

described in the ISO/IEC Directives, Part 1. In particular, the different approval criteria needed for the

different types of ISO documents should be noted. This document was drafted in accordance with the

editorial rules of the ISO/IEC Directives, Part 2 (see www .iso .org/ directives).

Attention is drawn to the possibility that some of the elements of this document may be the subject of

patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of

any patent rights identified during the development of the document will be in the Introduction and/or

on the ISO list of patent declarations received (see www .iso .org/ patents).

Any trade name used in this document is information given for the convenience of users and does not

constitute an endorsement.

For an explanation of the voluntary nature of standards, the meaning of ISO specific terms and

expressions related to conformity assessment, as well as information about ISO's adherence to the

World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT), see www .iso .org/

iso/ foreword .html.

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
---------------------- Page: 4 ----------------------
TECHNICAL SPECIFICATION ISO/TS 31657-3:2020(E)
Plain bearings — Hydrodynamic plain journal bearings
under steady-state conditions —
Part 3:
Functions for calculation of tilting pad journal bearings
1 Scope

This document specifies the characteristic values for selected tilting-pad journal bearings with four or

five centrally or eccentrically supported tilting pads and with angular spans of pad sliding surfaces of

Ω = 80°, 60° and 45°.

The functions plotted and listed in table form below are required for the operationally safe design

of hydrodynamic tilting-pad journal bearings according to ISO/TS 31657-1. They are based on the

presumptions and boundary conditions indicated there and only apply to stationary operating states.

The symbols used are explained in ISO/TS 31657-1; calculation examples are also included there.

The calculation method described in ISO/TS 31657-1 can also be used for other tilting-pad journal

bearing designs, if the numerical solutions of the basic equations are available in the same manner for

these designs.
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 Functions for calculation of multi-lobed journal bearings
4.1 General

The characteristic values for symmetrically loaded tilting-pad journal bearings with four and five

centrally (relative angular distance between leading edge and pivot position of pad Ω = 0,5) or

eccentrically (Ω = 0,6) supported tilting pads are indicated below in table form.

The characteristic values were calculated for the geometrical parameters summarised in Figure 1

(angular spans of pad sliding surface Ω, angular coordinates of pivot position of pad ϕ , profile factors

F ,1
© ISO 2020 – All rights reserved 1
---------------------- Page: 5 ----------------------
ISO/TS 31657-3:2020(E)
* *

K , bearing width ratios B ) in the operating range 0,02 ≤ h ≤ 1. The following (dimensionless)

P min
characteristic values are individually listed in table form:
a) Static characteristic values
Sommerfeld number:
F⋅ψ
eff
So=
BD⋅⋅ηω⋅
eff
Relative eccentricity:
ε =
R,eff
Attitude angle β in ° .
Product of maximum lubricant film pressure parameter and Sommerfeld number:
p ⋅ψ
max eff
pS⋅=o
max
ηω⋅
eff
Minimum relative lubricant film thickness:
min
h =
min
R,eff
Friction force parameter:
F =⋅So
eff
Lubricant flow rate parameter due to hydrodynamic pressure build-up:
Q =
Lubricant flow rate parameter due to supply pressure:
Q =
PQ⋅
en 0
Lubricant flow rate parameter at the exit of the lubricant gap:
Q =

Non-dimensional difference between maximum temperature of lubricant film and lubricant

temperature in the lubricant pocket:
ρψ⋅⋅c
p eff
ΔΔT = ⋅ T
max max
pf⋅
b) Dynamic characteristic values
Non-dimensional lubricant film stiffness coefficients:
2 © ISO 2020 – All rights reserved
---------------------- Page: 6 ----------------------
ISO/TS 31657-3:2020(E)
eff
c = ⋅c (i, k = 1, 2)
ik, ik,
2⋅⋅Bηω⋅
eff
Non-dimensional lubricant film damping coefficients:
eff
d = ⋅⋅ω d (i, k = 1, 2)
ik, ik,
2⋅⋅Bηω⋅
eff

For some selected tilting-pad journal bearings (Ω = 0,6, Ω = 45°, ϕ = 36°), these characteristic

F F ,1

values are shown graphically as a function of the Sommerfeld number So and the profile factor K

in Annex A, Figures A.1 to A.11.
Figure 1 — Geometrical parameters of selected tilting-pad journal bearings
* *
(Ω = 0,5 and Ω = 0,6)
F F
© ISO 2020 – All rights reserved 3
---------------------- Page: 7 ----------------------
ISO/TS 31657-3:2020(E)
4.2 Tilting-pad journal bearings with four tilting pads

Characteristic values for tilting-pad journal bearings with four centrally or eccentrically supported

tilting pads and angular spans of pad sliding surface of Ω = 80° or 60° are contained in Tables 1 to 34.

4 © ISO 2020 – All rights reserved
---------------------- Page: 8 ----------------------
ISO/TS 31657-3:2020(E)
© ISO 2020 – All rights reserved 5

Table 1 — Characteristic values of a tilting-pad bearing with four centrally (Ω = 0,5) supported tilting pads (Ω = 80°, φ = 45°, K = 2,

F,1 P
B* = 0,5)
* * * *
* * * * * * * * * * *
So ε β [°] F
pS⋅ o h Q Q Q ΔT c c c c d d d d
max min 3 P 2 max 11 12 21 22 11 12 21 22

0,000 0,000 0,00 0,278 0,829 2,785 0,909 0,000 1,730 1,02 0,283 0,000 0,000 0,283 0,335 0,000 0,000 0,335

0,021 0,050 0,00 0,308 0,796 2,789 0,909 0,000 1,730 1,10 0,286 0,000 0,000 0,286 0,337 0,000 0,000 0,337

0,042 0,100 0,00 0,342 0,763 2,800 0,907 0,000 1,730 1,18 0,295 0,000 0,000 0,295 0,342 0,000 0,000 0,342

0,086 0,200 0,00 0,426 0,698 2,846 0,902 0,000 1,732 1,36 0,330 0,000 0,000 0,330 0,362 0,000 0,000 0,362

0,136 0,300 0,00 0,539 0,632 2,925 0,893 0,000 1,735 1,58 0,397 0,000 0,000 0,397 0,399 0,000 0,000 0,399

0,197 0,400 0,00 0,692 0,567 3,043 0,881 0,000 1,739 1,84 0,505 0,000 0,000 0,505 0,456 0,000 0,000 0,456

0,273 0,500 0,00 0,907 0,503 3,210 0,863 0,000 1,744 2,15 0,678 0,000 0,000 0,678 0,546 0,000 0,000 0,546

0,374 0,600 0,00 1,216 0,440 3,436 0,841 0,000 1,752 2,51 0,954 0,000 0,000 0,954 0,686 0,000 0,000 0,686

0,512 0,700 0,00 1,674 0,378 3,741 0,812 0,000 1,760 2,93 1,403 0,000 0,000 1,403 0,905 0,000 0,000 0,905

0,713 0,800 0,00 2,392 0,317 4,157 0,776 0,000 1,773 3,51 2,167 0,000 0,000 2,167 1,253 0,000 0,000 1,253

1,023 0,900 0,00 3,587 0,259 4,735 0,732 0,000 1,788 4,34 3,562 0,000 0,000 3,562 1,841 0,000 0,000 1,841

1,245 0,950 0,00 4,501 0,230 5,111 0,707 0,000 1,796 4,88 4,707 0,000 0,000 4,707 2,299 0,000 0,000 2,299

1,541 1,000 0,00 5,768 0,202 5,568 0,679 0,000 1,806 5,65 6,389 0,000 0,000 6,389 2,931 0,000 0,000 2,931

1,943 1,050 0,00 7,584 0,174 6,134 0,648 0,000 1,816 6,49 8,951 0,000 0,000 8,951 3,848 0,000 0,000 3,848

2,512 1,100 0,00 10,311 0,147 6,850 0,613 0,000 1,829 7,76 13,077 0,000 0,000 13,077 5,241 0,000 0,000 5,241

3,358 1,150 0,00 14,651 0,121 7,787 0,575 0,000 1,842 9,64 20,191 0,000 0,000 20,191 7,500 0,000 0,000 7,500

4,708 1,200 0,00 22,167 0,095 9,074 0,532 0,000 1,857 12,41 33,852 0,000 0,000 33,852 11,308 0,000 0,000 11,308

7,086 1,250 0,00 36,774 0,070 10,962 0,482 0,000 1,874 16,96 63,748 0,000 0,000 63,748 19,069 0,000 0,000 19,069

12,031 1,300 0,00 71,724 0,046 14,101 0,422 0,000 1,895 26,41 148,095 0,000 0,000 148,095 37,650 0,000 0,000 37,650

26,371 1,350 0,00 197,100 0,024 20,706 0,346 0,000 1,923 51,75 535,824 0,000 0,000 535,824 107,479 0,000 0,000 107,478

58,756 1,380 0,00 571,372 0,012 31,042 0,284 0,000 1,945 109,29 2 141,057 0,000 0,000 2 141,055 306,671 0,000 0,000 306,670

---------------------- Page: 9 ----------------------
ISO/TS 31657-3:2020(E)
6 © ISO 2020 – All rights reserved

Table 2 — Characteristic values of a tilting-pad bearing with four eccentrically (Ω = 0,6) supported tilting pads (Ω = 80°, φ = 45°, K = 2,

F,1 P
B* = 0,5)
* * * * * * * * * * * * * *
So ε β [°] F
pS⋅ o h Q Q Q ΔT c c c c d d d d
max min 3 P 2 max 11 12 21 22 11 12 21 22

0,000 0,000 0,00 0,443 0,664 2,789 1,572 0,000 1,542 1,07 0,473 0,000 0,000 0,473 0,459 0,000 0,000 0,459

0,025 0,050 0,00 0,481 0,643 2,792 1,572 0,000 1,542 1,14 0,477 0,000 0,000 0,477 0,461 0,000 0,000 0,461

0,051 0,100 0,00 0,522 0,622 2,803 1,570 0,000 1,544 1,23 0,486 0,000 0,000 0,486 0,468 0,000 0,000 0,468

0,106 0,200 0,00 0,623 0,582 2,844 1,567 0,000 1,545 1,42 0,528 0,000 0,000 0,528 0,496 0,000 0,000 0,496

0,166 0,300 0,00 0,752 0,540 2,916 1,561 0,000 1,546 1,64 0,605 0,000 0,000 0,605 0,545 0,000 0,000 0,545

0,234 0,400 0,00 0,925 0,496 3,016 1,544 0,000 1,553 1,91 0,728 0,000 0,000 0,728 0,621 0,000 0,000 0,621

0,320 0,500 0,00 1,159 0,450 3,164 1,533 0,000 1,556 2,25 0,920 0,000 0,000 0,920 0,735 0,000 0,000 0,735

0,440 0,600 0,00 1,491 0,402 3,381 1,527 0,000 1,556 2,65 1,222 0,000 0,000 1,222 0,902 0,000 0,000 0,902

0,595 0,700 0,00 1,982 0,352 3,666 1,512 0,000 1,560 3,14 1,707 0,000 0,000 1,707 1,150 0,000 0,000 1,150

0,819 0,800 0,00 2,751 0,300 4,066 1,494 0,000 1,564 3,77 2,533 0,000 0,000 2,533 1,529 0,000 0,000 1,529

1,164 0,900 0,00 4,056 0,246 4,649 1,474 0,000 1,567 4,81 4,052 0,000 0,000 4,052 2,134 0,000 0,000 2,134

1,412 0,950 0,00 5,059 0,219 5,042 1,463 0,000 1,568 5,45 5,296 0,000 0,000 5,296 2,623 0,000 0,000 2,623

1,737 1,000 0,00 6,449 0,192 5,527 1,449 0,000 1,571 6,18 7,103 0,000 0,000 7,103 3,316 0,000 0,000 3,316

2,175 1,050 0,00 8,424 0,166 6,132 1,434 0,000 1,573 7,14 9,848 0,000 0,000 9,848 4,311 0,000 0,000 4,311

2,791 1,100 0,00 11,376 0,140 6,905 1,416 0,000 1,576 8,33 14,237 0,000 0,000 14,237 5,801 0,000 0,000 5,801

3,700 1,150 0,00 16,040 0,115 7,920 1,395 0,000 1,580 10,23 21,778 0,000 0,000 21,778 8,167 0,000 0,000 8,167

5,133 1,200 0,00 24,049 0,091 9,305 1,372 0,000 1,583 13,00 36,003 0,000 0,000 36,003 12,305 0,000 0,000 12,305

7,634 1,250 0,00 39,539 0,067 11,327 1,343 0,000 1,588 17,56 67,340 0,000 0,000 67,340 20,207 0,000 0,000 20,207

12,774 1,300 0,00 76,100 0,045 14,616 1,308 0,000 1,594 27,05 154,291 0,000 0,000 154,291 39,597 0,000 0,000 39,597

27,474 1,350 0,00 205,321 0,024 21,403 1,264 0,000 1,601 52,69 555,053 0,000 0,000 555,052 109,234 0,000 0,000 109,234

60,147 1,380 0,00 584,387 0,012 31,725 1,228 0,000 1,607 107,13 2 152,788 0,000 0,000 2 152,785 321,212 0,000 0,000 321,212

---------------------- Page: 10 ----------------------
ISO/TS 31657-3:2020(E)
© ISO 2020 – All rights reserved 7

Table 3 — Characteristic values of a tilting-pad bearing with four centrally (Ω = 0,5) supported tilting pads (Ω = 80°, φ = 45°, K = 3,

F,1 P
B* = 0,5)
* * * *
* * * * * * * * * * *
So ε β [°] F
pS⋅ o h Q Q Q ΔT c c c c d d d d
max min 3 P 2 max 11 12 21 22 11 12 21 22

0,000 0,000 0,00 0,373 0,782 2,821 1,349 0,000 1,664 1,02 0,342 0,000 0,000 0,342 0,257 0,000 0,000 0,257

0,021 0,050 0,00 0,409 0,750 2,825 1,349 0,000 1,664 1,10 0,344 0,000 0,000 0,344 0,258 0,000 0,000 0,258

0,043 0,100 0,00 0,450 0,718 2,836 1,347 0,000 1,664 1,17 0,353 0,000 0,000 0,353 0,262 0,000 0,000 0,262

0,088 0,200 0,00 0,548 0,655 2,881 1,342 0,000 1,666 1,35 0,388 0,000 0,000 0,388 0,279 0,000 0,000 0,279

0,139 0,300 0,00 0,677 0,593 2,958 1,332 0,000 1,668 1,54 0,452 0,000 0,000 0,452 0,309 0,000 0,000 0,309

0,198 0,400 0,00 0,848 0,532 3,072 1,320 0,000 1,671 1,76 0,554 0,000 0,000 0,554 0,355 0,000 0,000 0,355

0,270 0,500 0,00 1,082 0,472 3,229 1,303 0,000 1,675 2,04 0,711 0,000 0,000 0,711 0,424 0,000 0,000 0,424

0,363 0,600 0,00 1,411 0,413 3,439 1,283 0,000 1,680 2,40 0,954 0,000 0,000 0,954 0,525 0,000 0,000 0,525

0,487 0,700 0,00 1,892 0,355 3,719 1,258 0,000 1,686 2,86 1,341 0,000 0,000 1,341 0,679 0,000 0,000 0,679

0,664 0,800 0,00 2,627 0,299 4,093 1,229 0,000 1,693 3,45 1,983 0,000 0,000 1,983 0,921 0,000 0,000 0,921

0,929 0,900 0,00 3,826 0,244 4,603 1,195 0,000 1,701 4,32 3,128 0,000 0,000 3,128 1,324 0,000 0,000 1,324

1,116 0,950 0,00 4,729 0,217 4,932 1,176 0,000 1,706 4,96 4,054 0,000 0,000 4,054 1,625 0,000 0,000 1,625

1,359 1,000 0,00 5,960 0,190 5,328 1,155 0,000 1,711 5,69 5,385 0,000 0,000 5,385 2,052 0,000 0,000 2,052

1,686 1,050 0,00 7,703 0,164 5,814 1,133 0,000 1,716 6,67 7,390 0,000 0,000 7,390 2,658 0,000 0,000 2,658

2,141 1,100 0,00 10,283 0,139 6,428 1,108 0,000 1,722 8,03 10,579 0,000 0,000 10,579 3,553 0,000 0,000 3,553

2,808 1,150 0,00 14,332 0,114 7,230 1,081 0,000 1,729 10,02 16,050 0,000 0,000 16,050 4,930 0,000 0,000 4,930

3,851 1,200 0,00 21,191 0,090 8,323 1,051 0,000 1,737 13,13 26,193 0,000 0,000 26,193 7,403 0,000 0,000 7,403

5,655 1,250 0,00 34,402 0,066 9,932 1,017 0,000 1,745 18,22 48,319 0,000 0,000 48,319 12,108 0,000 0,000 12,108

9,329 1,300 0,00 65,317 0,044 12,599 0,979 0,000 1,754 28,06 109,751 0,000 0,000 109,751 23,143 0,000 0,000 23,143

19,717 1,350 0,00 174,369 0,023 18,175 0,933 0,000 1,764 54,67 386,812 0,000 0,000 386,812 63,618 0,000 0,000 63,617

42,609 1,380 0,00 493,091 0,012 26,726 0,896 0,000 1,772 110,73 1 476,149 0,000 0,000 1 476,147 185,825 0,000 0,000 185,825

---------------------- Page: 11 ----------------------
ISO/TS 31657-3:2020(E)
8 © ISO 2020 – All rights reserved

Table 4 — Characteristic values of a tilting-pad bearing with four eccentrically (Ω = 0,6) supported tilting pads (Ω = 80°, φ = 45°, K = 3,

F,1 P
B* = 0,5)
* * * * * * * * * * * * * *
So ε β [°] F
pS⋅ o h Q Q Q ΔT c c c c d d d d
max min 3 P 2 max 11 12 21 22 11 12 21 22

0,000 0,000 0,00 0,482 0,695 2,767 2,044 0,000 1,530 1,07 0,448 0,000 0,000 0,448 0,345 0,000 0,000 0,345

0,024 0,050 0,00 0,521 0,671 2,770 2,044 0,000 1,530 1,15 0,450 0,000 0,000 0,450 0,347 0,000 0,000 0,347

0,049 0,100 0,00 0,565 0,648 2,780 2,043 0,000 1,530 1,24 0,459 0,000 0,000 0,459 0,351 0,000 0,000 0,351

0,100 0,200 0,00 0,671 0,599 2,822 2,041 0,000 1,530 1,43 0,497 0,000 0,000 0,497 0,370 0,000 0,000 0,370

0,157 0,300 0,00 0,809 0,549 2,895 2,037 0,000 1,530 1,66 0,564 0,000 0,000 0,564 0,402 0,000 0,000 0,402

0,223 0,400 0,00 0,993 0,498 3,004 2,032 0,000 1,531 1,90 0,673 0,000 0,000 0,673 0,451 0,000 0,000 0,451

0,304 0,500 0,00 1,243 0,446 3,158 2,025 0,000 1,530 2,19 0,841 0,000 0,000 0,841 0,523 0,000 0,000 0,523

0,408 0,600 0,00 1,603 0,392 3,371 2,018 0,000 1,529 2,63 1,105 0,000 0,000 1,105 0,626 0,000 0,000 0,626

0,547 0,700 0,00 2,130 0,338 3,664 2,009 0,000 1,528 3,16 1,524 0,000 0,000 1,524 0,786 0,000 0,000 0,786

0,742 0,800 0,00 2,937 0,284 4,063 1,995 0,000 1,528 3,83 2,215 0,000 0,000 2,215 1,048 0,000 0,000 1,048

1,031 0,900 0,00 4,239 0,232 4,614 1,975 0,000 1,528 4,74 3,438 0,000 0,000 3,438 1,482 0,000 0,000 1,482

1,233 0,950 0,00 5,214 0,207 4,970 1,962 0,000 1,529 5,32 4,420 0,000 0,000 4,420 1,805 0,000 0,000 1,805

1,495 1,000 0,00 6,535 0,182 5,400 1,949 0,000 1,530 6,06 5,828 0,000 0,000 5,828 2,256 0,000 0,000 2,256

1,844 1,050 0,00 8,399 0,157 5,927 1,932 0,000 1,531 7,05 7,933 0,000 0,000 7,933 2,902 0,000 0,000 2,902

2,327 1,100 0,00 11,149 0,133 6,589 1,914 0,000 1,533 8,41 11,266 0,000 0,000 11,266 3,846 0,000 0,000 3,846

3,029 1,150 0,00 15,432 0,109 7,446 1,892 0,000 1,535 10,41 16,930 0,000 0,000 16,930 5,320 0,000 0,000 5,320

4,119 1,200 0,00 22,653 0,086 8,603 1,867 0,000 1,538 13,28 27,451 0,000 0,000 27,451 7,877 0,000 0,000 7,876

5,988 1,250 0,00 36,430 0,064 10,283 1,837 0,000 1,541 18,27 50,254 0,000 0,000 50,254 12,726 0,000 0,000 12,726

9,759 1,300 0,00 68,399 0,043 13,015 1,801 0,000 1,545 27,87 112,309 0,000 0,000 112,309 24,491 0,000 0,000 24,491

20,316 1,350 0,00 179,949 0,023 18,692 1,755 0,000 1,550 55,61 397,952 0,000 0,000 397,952 64,243 0,000 0,000 64,243

43,289 1,380 0,00 498,175 0,012 27,286 1,715 0,000 1,555 112,94 1 521,926 0,000 0,000 1 521,923 182,322 0,000 0,000 182,322

---------------------- Page: 12 ----------------------
ISO/TS 31657-3:2020(E)
© ISO 2020 – All rights reserved 9

Table 5 — Characteristic values of a tilting-pad bearing with four centrally (Ω * = 0,5) supported tilting pads (Ω = 80°, φ = 45°, K = 5,

F F,1 P
B* = 0,5)
* * * * * * * * * * * * * *
So ε β [°] pS⋅ o F Q ΔT
h Q Q c c c c d d d d
max f 3 max
min P 2 11 12 21 22 11 12 21 22

0,000 0,000 0,00 0,463 0,722 2,837 2,019 0,000 1,582 0,97 0,348 0,000 0,000 0,348 0,184 0,000 0,000 0,184

0,019 0,050 0,00 0,502 0,693 2,840 2,018 0,000 1,582 1,03 0,350 0,000 0,000 0,350 0,186 0,000 0,000 0,186

0,039 0,100 0,00 0,546 0,663 2,849 2,017 0,000 1,582 1,12 0,357 0,000 0,000 0,357 0,188 0,000 0,000 0,188

0,079 0,200 0,00 0,652 0,606 2,887 2,012 0,000 1,584 1,27 0,384 0,000 0,000 0,384 0,198 0,000 0,000 0,198

0,124 0,300 0,00 0,787 0,549 2,952 2,003 0,000 1,586 1,48 0,435 0,000 0,000 0,435 0,216 0,000 0,000 0,216

0,175 0,400 0,00 0,965 0,493 3,047 1,990 0,000 1,589 1,70 0,515 0,000 0,000 0,515 0,245 0,000 0,000 0,245

0,236 0,500 0,00 1,204 0,437 3,177 1,974 0,000 1,593 2,02 0,638 0,000 0,000 0,638 0,287 0,000 0,000 0,287

0,313 0,600 0,00 1,532 0,383 3,350 1,953 0,000 1,597 2,39 0,825 0,000 0,000 0,825 0,349 0,000 0,000 0,349

0,414 0,700 0,00 2,004 0,329 3,580 1,928 0,000 1,603 2,88 1,117 0,000 0,000 1,117 0,441 0,000 0,000 0,441

0,552 0,800 0,00 2,710 0,277 3,885 1,898 0,000 1,610 3,56 1,592 0,000 0,000 1,592 0,584 0,000 0,000 0,584

0,755 0,900 0,00 3,835 0,226 4,300 1,863 0,000 1,618 4,50 2,423 0,000 0,000 2,423 0,816 0,000 0,000 0,816

0,895 0,950 0,00 4,666 0,201 4,567 1,843 0,000 1,623 5,19 3,085 0,000 0,000 3,085 0,988 0,000 0,000 0,988

1,075 1,000 0,00 5,785 0,177 4,888 1,822 0,000 1,628 5,97 4,025 0,000 0,000 4,025 1,228 0,000 0,000 1,228

1,313 1,050 0,00 7,356 0,153 5,286 1,799 0,000 1,634 7,14 5,441 0,000 0,000 5,441 1,549 0,000 0,000 1,549

1,639 1,100 0,00 9,647 0,129 5,783 1,773 0,000 1,640 8,59 7,634 0,000 0,000 7,634 2,043 0,000 0,000 2,043

2,110 1,150 0,00 13,188 0,106 6,433 1,746 0,000 1,646 10,68 11,328 0,000 0,000 11,328 2,807 0,000 0,000 2,807

2,833 1,200 0,00 19,100 0,084 7,323 1,715 0,000 1,653 13,92 18,208 0,000 0,000 18,208 4,077 0,000 0,000 4,077

4,063 1,250 0,00 30,378 0,062 8,632 1,679 0,000 1,661 19,50 32,932 0,000 0,000 32,932 6,504 0,000 0,000 6,504

6,516 1,300 0,00 56,299 0,041 10,786 1,639 0,000 1,669 30,08 73,179 0,000 0,000 73,179 12,099 0,000 0,000 12,099

12,626 1,350 0,00 146,038 0,022 14,877 1,584 0,000 1,680 59,35 246,790 0,000 0,000 246,790 32,334 0,000 0,000 32,334

27,843 1,380 0,00 402,270 0,011 21,987 1,547 0,000 1,686 116,92 941,968 0,000 0,000 941,967 90,403 0,000 0,000 90,403

---------------------- Page: 13 ----------------------
ISO/TS 31657-3:2020(E)
10 © ISO 2020 – All rights reserved

Table 6 — Characteristic values of a tilting-pad bearing with four eccentrically (Ω = 0,6) supported tilting pads (Ω = 80°, φ = 45°, K = 5,

F,1 P
B* = 0,5)
* * * * * * * * * * * * * *
So ε β [°] F
pS⋅ o h Q Q Q ΔT c c c c d d d d
max min 3 P 2 max 11 12 21 22 11 12 21 22

0,000 0,000 0,00 0,527 0,684 2,755 2,901 0,000 1,505 1,06 0,395 0,000 0,000 0,395 0,219 0,000 0,000 0,219

0,021 0,050 0,00 0,570 0,657 2,758 2,901 0,000 1,505 1,14 0,398 0,000 0,000 0,398 0,219 0,000 0,000 0,219

0,043 0,100 0,00 0,618 0,630 2,769 2,901 0,000 1,505 1,23 0,405 0,000 0,000 0,405 0,222 0,000 0,000 0,222

0,088 0,200 0,00 0,734 0,576 2,813 2,899 0,000 1,503 1,42 0,435 0,000 0,000 0,435 0,233 0,000 0,000 0,233

0,137 0,300 0,00 0,884 0,522 2,887 2,896 0,000 1,502 1,64 0,490 0,000 0,000 0,490 0,253 0,000 0,000 0,253

0,193 0,400 0,00 1,078 0,469 2,995 2,889 0,000 1,500 1,91 0,577 0,000 0,000 0,577 0,282 0,000 0,000 0,282

0,260 0,500 0,00 1,337 0,417 3,141 2,879 0,000 1,500 2,21 0,707 0,000 0,000 0,707 0,329 0,000 0,000 0,329

0,344 0,600 0,00 1,693 0,365 3,335 2,865 0,000 1,499 2,59 0,906 0,000 0,000 0,906 0,395 0,000 0,000 0,395

0,452 0,700 0,00 2,201 0,315 3,590 2,847 0,000 1,499 3,10 1,214 0,000 0,000 1,214 0,493 0,000 0,000 0,493

0,601 0,800 0,00 2,956 0,265 3,926 2,824 0,000 1,500 3,73 1,713 0,000 0,000 1,713 0,646 0,000 0,000 0,646

0,816 0,900 0,00 4,156 0,217 4,380 2,796 0,000 1,501 4,74 2,585 0,000 0,000 2,585 0,887 0,000 0,000 0,887

0,964 0,950 0,00 5,037 0,193 4,668 2,780 0,000 1,502 5,44 3,273 0,000 0,000 3,273 1,068 0,000 0,000 1,068

1,153 1,000 0,00 6,221 0,170 5,013 2,761 0,000 1,504 6,23 4,259 0,000 0,000 4,259 1,309 0,000 0,000 1,309

1,401 1,050 0,00 7,864 0,147 5,432 2,740 0,000 1,505 7,26 5,693 0,000 0,000 5,693 1,670 0,000 0,000 1,670

1,741 1,100 0,00 10,277 0,125 5,960 2,717 0,000 1,507 8,87 7,999 0,000 0,000 7,999 2,150 0,000 0,000 2,150

2,227 1,150 0,00 13,952 0,103 6,637 2,690 0,000 1,509 10,98 11,797 0,000 0,000 11,797 2,942 0,000 0,000 2,942

2,971 1,200 0,00 20,108 0,081 7,553 2,659 0,000 1,513 14,25 18,798 0,000 0,000 18,798 4,276 0,000 0,000 4,276

4,227 1,250 0,00 31,629 0,061 8,883 2,622 0,000 1,516 19,39 33,785 0,000 0,000 33,785 6,773 0,000 0,000 6,773

6,713 1,300 0,00 58,147 0,041 11,053 2,579 0,000 1,520 29,68 74,071 0,000 0,000 74,071 12,692 0,000 0,000 12,692

13,511 1,350 0,00 148,407 0,022 15,499 2,520 0,000 1,526 56,63 249,869 0,000 0,000 249,868 33,964 0,000 0,000 33,964

27,982 1,380 0,00 403,229 0,011 22,181 2,472 0,000 1,529 113,12 929,177 0,000 0,000 929,176 95,057 0,000 0,000 95,057

---------------------- Page: 14 ----------------------
ISO/TS 31657-3:2020(E)
© ISO 2020 – All rights reserved 11

Table 7 — Characteristic values of a tilting-pad bearing with four centrally (Ω = 0,5) supported tilting pads (Ω = 80°, φ = 45°, K = 2,

F,1 P
B* = 0,75)
* * * *
* * * * * * * * * * *
So ε β [°] F
pS⋅ o h Q Q Q ΔT c c c c d d d d
max min 3 P 2 max 11 12 21 22 11 12 21 22

0,000 0,000 0,00 0,385 0,847 2,804 0,941 0,000 2,708 1,02 0,381 0,000 0,000 0,381 0,499 0,000 0,000 0,499

0,028 0,050 0,00 0,425 0,814 2,807 0,941 0,000 2,708 1,09 0,385 0,000 0,000 0,385 0,502 0,000 0,000 0,502

0,057 0,100 0,00 0,469 0,780 2,819 0,939 0,000 2,708 1,18 0,396 0,000 0,000 0,396 0,509 0,000 0,000 0,509

0,118 0,200 0,00 0,579 0,714 2,865 0,933 0,000 2,709 1,37 0,442 0,000 0,000 0,442 0,537 0,000 0,000 0,537

0,186 0,300 0,00 0,722 0,648 2,946 0,924 0,000 2,712 1,60 0,526 0,000 0,000 0,526 0,588 0,000 0,000 0,588

0,268 0,400 0,00 0,916 0,581 3,067 0,910 0,000 2,715 1,87 0,663 0,000 0,000 0,663 0,668 0,000 0,000 0,668

0,369 0,500 0,00 1,184 0,515 3,238 0,891 0,000 2,720 2,22 0,878 0,000 0,000 0,878 0,785 0,000 0,000 0,785

0,501 0,600 0,00 1,566 0,450 3,474 0,870 0,000 2,724 2,64 1,221 0,000 0,000 1,221 0,967 0,000 0,000 0,967

0,681 0,700 0,00 2,124 0,386 3,794 0,840 0,000 2,732 3,12 1,777 0,000 0,000 1,777 1,255 0,000 0,000 1,255

0,938 0,800 0,00 2,979 0,323 4,231 0,803 0,000 2,742 3,78 2,709 0,000 0,000 2,709 1,710 0,000 0,000 1,710

1,326 0,900 0,00 4,373 0,262 4,840 0,757 0,000 2,756 4,73 4,381 0,000 0,000 4,381 2,457 0,000 0,000 2,457

1,601 0,950 0,00 5,421 0,233 5,238 0,731 0,000 2,765 5,36 5,737 0,000 0,000 5,737 3,029 0,000 0,000 3,029

1,961 1,000 0,00 6,853 0,204 5,723 0,702 0,000 2,775 6,16 7,700 0,000 0,000 7,700 3,825 0,000 0,000 3,825

2,445 1,050 0,00 8,880 0,175 6,323 0,669 0,000 2,786 7,23 10,678 0,000 0,000 10,678 4,942 0,000 0,000 4,942

3,121 1,100 0,00 11,878 0,148 7,082 0,633 0,000 2,799 8,60 15,410 0,000 0,000 15,410 6,615 0,000 0,000 6,615

4,112 1,150 0,00 16,578 0,121 8,073 0,593 0,000 2,814 10,58 23,452 0,000 0,000 23,452 9,314 0,000 0,000 9,314

5,669 1,200 0,00 24,584 0,095 9,430 0,547 0,000 2,832 13,68 38,646 0,000 0,000 38,646 13,883 0,000 0,000 13,883

8,368 1,250 0,00 39,980 0,069 11,425 0,495 0,000 2,852 19,08 71,831 0,000 0,000 71,831 22,600 0,000 0,000 22,600

13,879 1,300 0,00 76,266 0,046 14,707 0,432 0,000 2,877 29,20 163,463 0,000 0,000 163,463 43,668 0,000 0,000 43,668

27,847 1,350 0,00 206,016 0,024 20,802 0,348 0,000 2,913 60,47 571,280 0,000 0,000 571,279 119,574 0,000 0,000 119,574

64,154 1,380 0,00 591,907 0,012 32,223 0,288 0,000 2,937 118,29 2 276,467 0,000 0,000 2 276,464 332,605 0,000 0,000 332,605

---------------------- Page: 15 ----------------------
ISO/TS 31657-3:2020(E)
12 © ISO 2020 – All rights reserved

Table 8 — Characteristic values of a tilting-pad bearing with four eccentrically (Ω = 0,6) supported tilting pads (Ω = 80°, φ = 45°, K = 2,

F,1 P
B* = 0,75)
* * * * * * * * * * * * * *
So ε β [°] F
pS⋅ o h Q Q Q ΔT c c c c d d d d
max min 3 P 2 max 11 12 21 22 11 12 21 22

0,000 0,000 0,00 0,661 0,648 2,923 1,786 0,000 2,381 1,08 0,692 0,000 0,000 0,692 0,707 0,000 0,000 0,707

0,037 0,050 0,00 0,712 0,628 2,926 1,786 0,000 2,381 1,15 0,697 0,000 0,000 0,697 0,710 0,000 0,000 0,710

0,074 0,100 0,00 0,770 0,609 2,937 1,785 0,000 2,381 1,24 0,710 0,000 0,000 0,710 0,720 0,000 0,000 0,720

0,151 0,200 0,00 0,906 0,571 2,980 1,782 0,000 2,382 1,43 0,766 0,000 0,000 0,766 0,759 0,000 0,000 0,759

0,237 0,300 0,00 1,080 0,532 3,056 1,776 0,000 2,383 1,65 0,866 0,000 0,000 0,866 0,829 0,000 0,000 0,829

0,337 0,400 0,00 1,307 0,491 3,168 1,769 0,000 2,385 1,93 1,027 0,000 0,000 1,027 0,938 0,000 0,000 0,938

0,458 0,500 0,00 1,610 0,447 3,325 1,759 0,000 2,388 2,27 1,275 0,000 0,000 1,275 1,099 0,000 0,000 1,099

0,612 0,600 0,00 2,031 0,401 3,542 1,748 0,000 2,389 2,70 1,656 0,000 0,000 1,656 1,332 0,000 0,000 1,332

0,818 0,700 0,00 2,637 0,352 3,838 1,733 0,000 2,393 3,27 2,262 0,000 0,000 2,262 1,674 0,000 0,000 1,674

1,108 0,800 0,00 3,562 0,301 4,255
...

TECHNICAL ISO/TS
SPECIFICATION 31657-3
First edition
Plain bearings — Hydrodynamic plain
journal bearings under steady-state
conditions —
Part 3:
Functions for calculation of tilting pad
journal bearings
PROOF/ÉPREUVE
Reference number
ISO/TS 31657-3:2020(E)
ISO 2020
---------------------- Page: 1 ----------------------
ISO/TS 31657-3:2020(E)
COPYRIGHT PROTECTED DOCUMENT
© ISO 2020

be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting

on the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address

below or ISO’s member body in the country of the requester.
ISO copyright office
CP 401 • Ch. de Blandonnet 8
CH-1214 Vernier, Geneva
Phone: +41 22 749 01 11
Fax: +41 22 749 09 47
Email: copyright@iso.org
Website: www.iso.org
Published in Switzerland
ii PROOF/ÉPREUVE © ISO 2020 – All rights reserved
---------------------- Page: 2 ----------------------
ISO/TS 31657-3:2020(E)
Contents Page

4 Functions for calculation of multi-lobed journal bearings ...................................................................................... 1

4.2 Tilting-pad journal bearings with four tilting pads ................................................................................................. 4

4.3 Tilting-pad journal bearings with five tilting pads ...............................................................................................39

Annex A (informative) Characteristic curves for tilting-pad bearings with five pads ....................................62

© ISO 2020 – All rights reserved PROOF/ÉPREUVE iii
---------------------- Page: 3 ----------------------
ISO/TS 31657-3:2020(E)
Foreword