ISO 21771-1
(Main)Cylindrical involute gears and gear pairs — Part 1: Concepts and geometry
Cylindrical involute gears and gear pairs — Part 1: Concepts and geometry
This document specifies the geometric concepts and parameters for cylindrical gears with involute helicoid tooth flanks. Flank modifications are included. The formulas in this document apply to all pressure angles. It also covers the concepts and parameters for involute cylindrical gear pairs with parallel or crossed axes, and a constant gear ratio. Gear and mating gear in these gear pairs have the same basic rack tooth profile.
Roues et engrenages cylindriques en développante de cercle — Partie 1: Concepts et géométrie
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Cylindrical involute gears and
gear pairs —
Part 1:
Concepts and geometry
Roues et engrenages cylindriques à développante —
Partie 1: Concepts et géométrie
PROOF/ÉPREUVE
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© ISO 2024
All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may
be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting on
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Published in Switzerland
PROOF/ÉPREUVE
ii
Contents Page
Foreword .vii
1 Scope . 1
2 Normative references . 1
3 Terms, definitions, symbols, subscripts and units . 1
3.1 Terms and definitions .1
3.2 Symbols .2
3.3 Subscripts .9
3.4 Units and conventions .10
3.4.1 Units .10
3.4.2 Conventions .10
4 Individual cylindrical gears . 10
4.1 General .10
4.2 Concepts for an individual gear .11
4.2.1 Gear, cylindrical gear, external gear and internal gear .11
4.2.2 Tooth system, external teeth and internal teeth .11
4.2.3 Tooth and tooth space .11
4.2.4 Tooth system parameters .11
4.2.5 Number of teeth and sign of number of teeth.11
4.2.6 Tooth number . 12
4.2.7 Top land and bottom land . 12
4.2.8 Tooth flanks and flank sections . 12
4.3 Reference surfaces, datum lines and reference quantities . 13
4.3.1 Reference surface and datum face . 13
4.3.2 Reference rack . 13
4.3.3 Basic rack tooth profile for involute gear teeth .14
4.3.4 Reference cylinder, reference circle and reference diameter . 15
4.3.5 Gear axis . 15
4.3.6 Sections through a cylindrical gear . 15
4.3.7 Module .16
4.3.8 Diametral pitch .16
4.3.9 Facewidth . .16
4.3.10 Profile shift, profile shift coefficient and sign of profile shift .17
4.4 Involute helicoids .18
4.4.1 Generator of involute helicoids .18
4.4.2 Base cylinder, base circle and base diameter .19
4.4.3 Transverse pressure angle at a point and transverse pressure angle .19
4.4.4 Normal pressure angle at a point and normal pressure angle . 20
4.4.5 Involute function . 20
4.4.6 Lead .21
4.4.7 Helix angle and lead angle.21
4.4.8 Hand of helix . 22
4.4.9 Roll angle of the involute . 23
4.4.10 Radius of curvature of the involute and length of roll . 23
4.4.11 Length of involute profile, L . 23
yt
4.5 Angular pitch and pitches . 23
4.5.1 Angular pitch . 23
4.5.2 Pitches on the reference cylinder . 23
4.5.3 Pitches on any cylinder .24
4.5.4 Axial pitch . 25
4.5.5 Base pitch . 25
4.6 Diameters of gear teeth . 25
4.6.1 General . 25
4.6.2 V-cylinder and V-circle diameter . 26
4.6.3 y-cylinder and y-circle diameter . 26
PROOF/ÉPREUVE
iii
4.6.4 Tip alteration coefficient . 26
4.6.5 Tip cylinder, tip circle and tip diameter . 26
4.6.6 Profile control diameter, d . 26
Cf
4.6.7 Root cylinder, root circle and root diameter . 26
4.7 Gear tooth depth .27
4.7.1 Tooth depth .27
4.7.2 Addendum, working addendum, dedendum and working dedendum .27
4.8 Tooth thickness and space width .27
4.8.1 General .27
4.8.2 Transverse tooth thickness .27
4.8.3 Transverse tooth thickness half angle . 28
4.8.4 Transverse space width . 29
4.8.5 Transverse space width half angle . 29
4.8.6 Normal tooth thickness . 29
4.8.7 Normal space width . . 29
4.9 Radii of curvature of the tooth flanks in different planes . 30
4.10 Tilt angle of contact line .31
5 Parallel axis cylindrical gear pairs .32
5.1 General .32
5.2 Concepts for a gear pair .
...
ISO/DIS 21771-1.2:2024(Een)
Date: 2024-05- 1307
ISO/TC 60/SC 1/WG 4
Secretariat: SNV
Cylindrical involute gears and gear pairs — Part 1: Concepts and geometry
Roues et engrenages cylindriques à développante de cercle — Partie 1: Concepts et géométrie
Notes for ISO editor:
Figures 6, 14, 16, 22, 24, 25, 27, 28, 29, 34, 52, 53, 54, 59, 61, B.3, and
D.2 have been modified.
Throughout document, please keep the figure, the key, and the figure
title on the same page, either by adding blank space before the figure,
or moving the figure to a different part of the text.
Also, please keep a non-breaking space at the end of all subscripts. It is
very important to have this to insure readability of the variables.
Table of contents needs to be updated
ISO/DIS 21771-1.2:2024(E)
All rights reserved. Unless otherwise specified, or required in the context of its implementation, no
part of this publication may 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-1214 Vernier, Geneva
Phone: + 41 22 749 01 11
Email: copyright@iso.org
Website: www.iso.org
Published in Switzerland.
ii © ISO 2024 – All rights reserved
ISO/DIS 21771-1.2:2024(Een)
Contents
iii
ISO/DIS 21771-1.2:2024(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 document 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).
ISO draws attention to the possibility that the implementation of this document may involve the use of
(a) patent(s). ISO takes no position concerning the evidence, validity or applicability of any claimed
patent rights in respect thereof. As of the date of publication of this document, ISO had not received notice
of (a) patent(s) which may be required to implement this document. However, implementers are
cautioned that this may not represent the latest information, which may be obtained from the patent
database available at www.iso.org/patents. ISO shall not be held responsible for identifying any or all
such patent rights.
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 60, Gears, Subcommittee SC 1,
Nomenclature and wormgearing.
This first edition of ISO 21771-1 cancels and replaces ISO 21771:2007, which has been technically
revised.
The main changes are as follows:
[8]
— Thethe sign convention for internal gears used in the ISO 6336 series has been adopted. The
negative value for the number of teeth of an internal gear is applied to the diameters and centre
distance, so these dimensions of internal gears have negative values.;
— Flankflank direction has been renamed as hand of helix and sign (+/-) of helix angle is used.;
— A defintiona definition of normal surface has been added and this is used rather than normal plane.;
— Thethe annex on tooth thickness was removed because it is now addressed in anticipation of a part
ISO 21771-2 of this document which intends to address tooth thickness.
Additional material has been added to cover:
— calculation of form diameters for tooth tip corner radius and tooth root fillet radius in the transverse
plane for an involute cylindrical gear (Clauses 10, 11 and Annex B);
iv © ISO 2024 – All rights reserved
ISO/DIS 21771-1.2:2024(Een)
— calculation of the tooth tip corner radius for a specified form diameter and tip diameter of an involute
cylindrical gear;
— calculation of a radius tangent to the involutes of adjacent teeth at root or tip diameter (Annex A);
— generated tooth root fillet shape for individual involute cylindrical gears (Annex B);
— concepts and parameters for involute cylindrical gear pairs with crossed axes (Clause 6 and
Annex C);
— geometry of surfaces in contact (Annex D);
— projection of a transverse plane profile of a tooth onto another plane (Annex E);
— interface to ISO –10828 for involute worm gear geometry (Annex F).
A list of all parts in the ISO 21771 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.
v
International Standard ISO 21771-1:2024(en)
Cylindrical involute gears and gear pairs — Part 1: Concepts and
geometry
1 Scope
This document specifies the geometric concepts and parameters for cylindrical gears with involute
helicoid tooth flanks. Flank modifications are included. The formulae in this document apply to all
pressure angles.
It also covers the concepts and parameters for involute cylindrical gear pairs with parallel or crossed
axes, and a constant gear ratio. Gear and mating gear in these gear pairs have the same basic rack tooth
profile.
2 Normative references
There are no normative references in this document.
3 Terms, definitions, symbols, subscripts and units
3.1 Terms and definitions
NoFor the purposes of this document, the following terms and definitions are listed in this document.
apply.
ISO and IEC maintain terminology databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https://www.iso.org/obp
— IEC Electropedia: available at https://www.electropedia.org/
3.1.1
basic rack tooth profile for involute gear teeth
tooth profile of a normal section through the teeth of a basic rack which corresponds to an external spur
gear with number of teeth z = ∞ and diameter d = ∞
Note 1 to entry: The tooth of the basic rack tooth profile is bounded by the tip line at the top and by the parallel
root line at the bottom. The fillet between the straight part of the profile and the root line is a circular arc with a
radius equal to ρ .
fP
3.1.2
counterpart rack tooth profile
rack tooth profile symmetrical to the basic rack tooth profile about the datum line P-P and displaced by
half a pitch relative to it
3.1.3
nominal involute flank
ISO/DIS 21771-1.2:2024(E)
pure involute flank prior to any modifications
Note 1 to entry: See 4.4 for more information on a gear tooth involute flank.
3.1.4
tip alteration coefficient
k
change to the addendum relating to the standard addendum of one normal module, it is made non-
dimensional by dividing by the normal module
Note 1 to entry: Tip alteration is also known as addendum modification, tip shortening or truncation.
3.1.5
generating gear pair
generating tool (rack, hob, pinion-type cutter, or grinding wheel) and the gear being machined during
gear tooth machining process
3.2 Symbols
Table 1 provides all the symbols used in this document.
Table 1 — Symbols
Symbol Description Subclause
A pinion lower end point of meshing (near pinion root) 5.5.6.1
working centre distance of a cylindrical gear pair 5.3.3, 5.3.5,
a
w w
6.3.12
centre distance in the generating process with pinion-type cutter 11.2
a
w0 w0
B pinion lower point of single tooth contact (LPSTC) 5.5.6.1
b facewidth 4.3.9
bEa length of relief near tip 8.5.2
b length of relief near root 8.5.2
Ef
bF usable facewidth 4.3.9
bw active facewidth (the facewidth used) 5.5.9.2
C working pitch point 5.3.5
with subscript: amount of relief for modifications 8
C modification of the profile by function 8.6
ay
CEa amount of triangular end relief modification at tip 8.5.2
CEf amount of triangular end relief modification at root 8.5.2
C amount of transverse profile slope modification 8.3.3
Hα
CHβ amount of flank line slope modification 8.4.2
Ci,j amount of modification at point (i,j) 8.5.1
C amount of profile crowning (barrelling) 8.3.4
α
Cαa amount of tip relief 8.3.2
Cαf amount of root relief 8.3.2
2 © ISO 2024 – All rights reserved
Symbol Description Subclause
C amount of flank line crowning 8.4.3
β
CβI, CβlI amount of end relief 8.4.1
Cβy modification of the flank line by function 8.6
C modification of the flank surface by function 8.6
Σy
c tip clearance 5.3.8, 6.3.15
cF form over dimension 5.5.5
D pinion highest point of single tooth contact (HPSTC) 5.5.6.1
d reference diameter 4.3.4
da tip diameter 4.6.5
d tip diameter of pinion-type cutter 11.1
a0
db base diameter 4.4.2
db0 base diameter of the pinion-type cutter 11.1
d chamfering base diameter 8.2.2
bK
dCαa tip relief start diameter 8.3.2
dCαf root relief start diameter 8.3.2
d Profile control diameter 8.3.2
Cf
dEa diameter for start of triangular end relief at tooth tip 8.5.2
dEf diameter for start of triangular end relief at tooth root 8.5.2
d tip form diameter 9.7
Fa
dFf root form diameter 5.5.2, 9.7
df root diameter (nominal dimension) 4.6.7
d root diameter produced 9.6
fE
dNa active tip diameter 5.5.2, 6.4.3
dNf start of active profile diameter (SAP diameter, active root diameter) 5.5.2, 6.4.3
d V-circle diameter 4.6.2
v
dw working pitch diameter 5.3.5
dy Y-circle diameter 4.6.3
d reference diameter of pinion-type cutter 11.1.1.1
E pinion upper end point of meshing (near pinion tip) 5.5.6.1
Esni lower tooth thickness limit deviation for normal tooth thickness at reference 9.4
cylinder
Esns upper tooth thickness limit deviation for normal tooth thickness at reference 9.4
cylinder
en normal space width on the reference cylinder 4.8.7
eP space width of the basic rack tooth profile 4.3.3
e transverse space width on the reference cylinder 4.8.4
t
ISO/DIS 21771-1.2:2024(E)
Symbol Description Subclause
e normal space width on the y-cylinder 4.8.7
yn
eyt transverse space width on the y-cylinder 4.8.4
F force D.3
g length of addendum path of contact 5.5.6.2, 6.4.5.2
a
gf length of dedendum path of contact 5.5.6.2, 6.4.5.2
gα length of path of contact 5.5.6.2
g distance of a point Y from working pitch point C 5.7.5
αy
gβ overlap roll length (arc of contact) 5.5.9.4
h tooth depth (between tip line and root line) 4.7.1
h addendum from reference pitch circle 4.7.2
a
haP addendum of the basic rack tooth profile 4.3.2
haP0 addendum of the counterpart of the basic rack tooth profile 9.2.2
h addendum from working pitch circle 4.7.2
aw
ha0 addendum of pinion-type cutter 11.1.1.1
hFaP0 straight part of tip flank of tool-generating profile 9.2.2
h portion of the dedendum to root form line, of the basic rack tooth profile 4.3.2
FfP
hFfP0 portion of the dedendum to root form line of the counterpart rack tooth profile 9.2.1
hfP0 dedendum of the counterpart rack tooth profile and rack tool 9.2.2
h dedendum from reference circle 4.7.2
f
hfP dedendum of the basic rack tooth profile 4.3.3
hfw dedendum from working pitch circle 4.7.2
h height of the tip corner chamfering or tip corner rounding 8.2.2
K
hP tooth depth of basic rack tooth profile 4.3.2
hw working depth of teeth in a gear pair 5.3.7, 6.3.14
i transmission ratio of a gear pair 5.3.2
inv involute function (not a variable) 4.4.5
jbn normal base backlash 5.6.4, 6.5.2
j transverse backlash 5.6.2
bt
jr radial backlash 5.6.5
jt circumferential backlash at the reference circle 5.6.3, 6.5.3
j working normal backlash 6.5.2
wn
jwt circumferential backlash at the working pitch circle 5.6.3, 6.5.3
Kg sliding factor 5.7.6
K sliding factor at tooth tip 5.7.6
ga
Kgf sliding factor at tooth root 5.7.6
k tip alteration coefficient 3.51.3, 4.6.4,
5.3.9, 6.3.16
4 © ISO 2024 – All rights reserved
Symbol Description Subclause
L tip relief roll length 8.3.2
Ca
LCf root relief roll length 8.3.2
LCl, LClI length of end relief 8.4.1
L tip roll length of triangular end relief modification 8.5.2
Ea
LEf root roll length of triangular end relief modification 8.5.2
Ly length of roll to y-cylinder 4.4.10
L length of involute profile to y-cylinder 4.4.11
yt
lmax maximum length of a contact line 5.5.10
My a point on a tooth flank where radius of curvature is calculated 7.1
m module 4.3.7
mn normal module 4.3.7
mt transverse module 4.3.7
m axial module 4.3.7
x
N number of tooth or pitch 4.2.6
na rotational speed of driving gear (rpm) 5.3.2
n rotational speed of driven gear (rpm) 5.3.2
b
O centre of a circle 6.3.4
Pd diametral pitch 4.3.8
P normal diametral pitch 4.3.8
nd
pbn normal base pitch 4.5.5.1
pbt transverse base pitch 4.5.5.1
p normal base pitch on the path of contact 4.5.5.3
en
pet transverse base pitch on the path of contact 4.5.5.2
pn normal pitch on the reference cylinder 4.5.2.2
p transverse pitch on the reference cylinder 4.5.2.1
t
pwn normal pitch at the working diameter 6.3.6, 6.5.2
px axial pitch 4.5.4
p normal pitch on the y-cylinder 4.5.3
yn
pyt transverse pitch on the y-cylinder 4.5.3
pz lead 4.4.6
pr protuberance of the tool (as seen in ISO 6336-3) 9.2.1
q machining allowance on tooth flank 9.3
qFs magnitude (amount) of undercut in tran
...
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