SIST ISO 21771:2008

INTERNATIONAL ISO
STANDARD 21771
First edition
2007-09-01
Gears — Cylindrical involute gears and
gear pairs — Concepts and geometry
Engrenages — Roues et engrenages cylindriques à développante —
Concepts et géométrie
Reference number
©
ISO 2007
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ii © ISO 2007 – All rights reserved

Contents Page
Foreword. iv
1 Scope . 1
2 Normative references . 1
3 Symbols, subscripts and units. 2
3.1 Symbols . 2
3.2 Subscripts . 8
3.3 Units . 9
4 Individual cylindrical gears. 10
4.1 Concepts for an individual gear . 10
4.2 Reference surfaces, datum lines and reference quantities. 12
4.3 Involute helicoids. 17
4.4 Angular pitch and pitches. 21
4.5 Diameters of gear teeth. 24
4.6 Gear tooth height. 24
4.7 Tooth thickness, space width. 25
5 Cylindrical gear pairs . 27
5.1 Concepts for a gear pair . 27
5.2 Mating quantities . 28
5.3 Calculation of the sum of the profile shift coefficients. 31
5.4 Tooth engagement. 32
5.5 Backlash . 40
5.6 Sliding conditions at the tooth flanks. 41
6 Tooth flank modifications . 44
6.1 Tooth flank modifications which restrict the usable flank . 44
6.2 Transverse profile modifications . 45
6.3 Flank line (helix) modifications . 48
6.4 Flank face modifications. 49
6.5 Descriptions of modifications by functions. 51
7 Geometrical limits. 52
7.1 Counterpart rack tooth profile. 53
7.2 Machining allowance. 54
7.3 Deviations in tooth thickness. 55
7.4 Generating profile shift, generating profile shift coefficient. 56
7.5 Generated root diameter . 57
7.6 Usable area of the tooth flank, tip and root form diameter . 57
7.7 Undercut . 59
7.8 Overcut . 59
7.9 Minimum tooth thickness at the tip circle of a gear. 59
Annex A (informative) Calculations related to tooth thickness. 60
Bibliography . 83

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.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. 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.
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.
ISO 21771 was prepared by Technical Committee ISO/TC 60, Gears, Subcommittee SC 1, Nomenclature and
wormgearing.
This first edition of ISO 21771 cancels and replaces ISO/TR 4467:1982, of which it constitutes a technical
revision.
iv © ISO 2007 – All rights reserved

INTERNATIONAL STANDARD ISO 21771:2007(E)

Gears — Cylindrical involute gears and gear pairs — Concepts
and geometry
1 Scope
This International Standard specifies the geometric concepts and parameters for cylindrical gears with involute
helicoid tooth flanks. Flank modifications are included.
It also covers the concepts and parameters for cylindrical gear pairs with parallel axes and a constant gear
ratio, which consist of cylindrical gears according to it. Gear and mating gear in these gear pairs have the
same basic rack tooth profile.
The equations given are not restricted to the pressure angle,α = 20°.
P
The standard is structured as follows.
⎯ Listing of symbols and nomenclature for a unique description of gears and gear pairs (see Clause 3).
⎯ Equations and explanations of the relevant values for defining a cylindrical gear and its tooth system. The
equations for determination of the nominal values for zero-deviation gear description parameters are
stated for radial tooth dimensions (gear tooth heights), the distance between flanks of the same hand, the
distance between flanks of opposite hand, as well as the tooth flank characterizing parameters (see
Clause 4).
⎯ Equations and explanations of the relevant values for defining cylindrical gear pairs. The equations for the
essential parameters characterizing the engagement conditions of the unloaded gear pair are listed (see
Clause 5).
⎯ Equations and suggestions for desired flank modifications (see Clause 6).
⎯ Concepts and recommendations needed for a unique geometrical definition of the intended results from
manufacture (Clause 7).
⎯ Equations for determination of the nominal values or the limiting values for the most used inspection
methods for tooth thickness (see Annex A).
2 Normative references
The following referenced documents are indispensable for the application of this document. For dated
references, only the edition cited applies. For undated references, the latest edition of the referenced
document (including any amendments) applies.
ISO 53:1998, Cylindrical gears for general and heavy engineering — Standard basic rack tooth profile
ISO 1328-1:1995, Cylindrical gears — ISO system of accuracy — Part 1: Definitions and allowable values of
deviations relevant to corresponding flanks of gear teeth
ISO 1328-2:1997, Cylindrical gears — ISO system of accuracy — Part 2: Definitions and allowable values of
deviations relevant to radial composite deviations and runout information
3 Symbols, subscripts and units
3.1 Symbols
Symbol Description Used in
a centre distance of a cylindrical gear pair 5.2.3
w
a centre distance in the generating gear unit 7.5
a centre distance for tooth flank engagement A.8
L
b facewidth 4.2.8
b usable facewidth 4.2.8
F
b contact line overlap (for measuring base tangent length) A.2.1
M
b active facewidth (the facewidth used) 5.4.7.2
w
c
tip clearance 5.2.7
c form over dimension 5.4.4
F
d
reference diameter 4.2.4
d tip diameter 4.5.3
a
d tip diameter of tool 7.5
a0
d tip diameter of overcut cylindrical gears A.9
aM
d base diameter 4.3.10
b
d base diameter of the pinion-type cutter 7.6
b0
d root diameter (nominal dimension) 4.5.4
f
d
root diameter produced 7.5
fE
d root diameter of the pinion-type cutter A.9
f0
d V-circle diameter 4.5.1
v
d working pitch diameter 5.2.5
w
d
Y-circle diameter 4.3.3
y
d tip form diameter 7.6
Fa
d tip form diameter of the pinion-type cutter 7.6
Fa0
d root form diameter 7.6
Ff
d diameter of circle through centre of ball A.5
K
d
diameter of a measuring circle A.2.1
M
d active tip diameter 5.4.1
Na
d start of active profile diameter (SAP diameter, active root diameter) 5.4.1
Nf
e space width on the reference cylinder 4.7.3
t
e
space width on the Y-cylinder 4.7.3
yt
2 © ISO 2007 – All rights reserved

Symbol Description Used in
e space width of the standard basic rack tooth profile 4.2.3
P
g length of addendum path of contact 5.4.5.2
a
g length of dedendum path of contact 5.4.5.2
f
g length of path of contact 5.4.5.2
α
g distance of a point Y from pitch point C 5.6.1
αy
g arc of contact 5.4.7.4
β
h tooth depth (between tip line and root line) 4.6.1
h addendum 4.6.2
a
h
addendum of the standard basic rack tooth profile Figure 4
aP
h addendum of the tool standard basic rack tooth profile 7.5
aP0
h dedendum 4.6.2
f
h dedendum of the standard basic rack tooth profile Figure 4
fP
h dedendum of the tool standard basic rack tooth profile A.9
fP0
h working depth of teeth in a gear pair 5.2.6
w
h depth of dedendum form of the standard basic rack tooth profile Figure 4
FfP
h radius of the tip corner chamfering or tip corner rounding 6.1.2
K
h tooth depth of standard basic rack tooth profile Figure 4
P
inv involute function 4.3.9
j contact backlash 5.5
bn
j
radial backlash 5.5
r
j circumferential backlash at the reference circle 5.5.2
t
j circumferential backlash at the pitch circle 5.5
wt
k number of teeth, spaces or pitches in a span (e.g. number of teeth A.2.1
spanned)
k
addendum modification coefficient 4.5.2
l path of engagement 5.4.8
max
∑l sum of path of contact 5.4.8
m normal module 4.2.7
n
m transverse module 4.2.7
t
m axial module 4.2.7
x
n number of revolutions of driving gear (rpm) 5.2.2
a
n number of revolutions of driven gear (rpm) 5.2.2
b
p
pitch, pitch on the reference cylinder Figure 4
Symbol Description Used in
p normal pitch on the base cylinder 4.4.5
bn
p transverse pitch on the base cylinder 4.4.5.1
bt
p normal base pitch on the path of contact 4.4.5.2
en
p transverse base pitch on the path of contact 4.4.5.1
et
p normal pitch 4.4.2.2
n
p
transverse pitch 4.4.2.1
t
p axial pitch 4.4.4
x
p normal pitch on the Y-cylinder 4.4.3
yn
p transverse pitch on the Y-cylinder 4.4.3
yt
p lead 4.3.2
z
q machining allowance on tooth flank 7.2
q undercut Figure 24
Fs
s
residual tooth thickness at tip with tip corner chamfering or tip corner 6.1.2
aK
rounding
s normal tooth thickness on the base circle A.2.2
bn
s normal tooth thickness on the reference circle 4.7.5
n
s minimum normal tooth thickness on the reference circle 7.3
ni
s
maximum normal tooth thickness on the reference circle 7.3
ns
s transverse tooth thickness on the reference circle 4.7.1
t
s normal tooth thickness on the Y-cylinder 4.7.5
yn
s transverse tooth thickness on the Y-cylinder 4.7.1
yt
s
tooth thickness of the standard basic rack tooth profile 4.2.3
P
u gear ratio 5.2.1
v sliding speed 5.6.1
g
v sliding speed at the addendum 5.6.1
ga
v sliding speed at the dedendum 5.6.1
gf
v normal speed 5.6.1
n
x profile shift coefficient 4.2.9
x generating profile shift coefficient 7.4
E
x generating profile shift coefficient at undercut limit 7.7
Emin
x profile shift coefficient of master gear A.8
L
z number of teeth 4.1.5
z number of teeth of driving gear 5.2.2
a
4 © ISO 2007 – All rights reserved

Symbol Description Used in
z number of teeth of driven gear 5.2.2
b
z number of teeth of master gear A.8
L
z number of teeth of pinion-type cutter 7.6
A starting point of meshing 5.4.3
B starting point of single tooth contact on driving gear 5.4.5.1
C pitch point, depth of relief for modifications 5.4.3
C modification of the profile 6.5
ay
C modification of the flank line 6.5
βy
C modification of the flank surface 6.5
Σy
C amount of tip relief 6.2.1
αa
C amount of root relief 6.2.1
αf
C tip amount of triangular end relief modification 6.4.2
Ea
C root amount of triangular end relief modification 6.4.2
Ef
C amount of modification at point (i,j) 6.4.1
i,j
C amount of transverse profile slope modification 6.2.2
Hα
C amount of profile crowning (barrelling) 6.2.3
α
C , C amount of end relief 6.3.1
βI βlI
C amount of flank line crowning 6.3.3
β
C amount of flank line slope modification 6.3.2
Hβ
D
measuring ball or measuring cylinder diameter A.5
M
D end point of single tooth contact point on driving gear 5.4.5.1
E end point of meshing 5.4.3
E normal tooth thickness deviation limit (or allowance) A.9
sn
E
lower deviation limit for tooth thickness 7.3
sni
E upper deviation limit for tooth thickness 7.3
sns
K sliding factor 5.6.2
g
K sliding factor at tooth tip 5.6.2
ga
K sliding factor at tooth root 5.6.2
gf
L roll length 6.2
AE
L tip relief roll length 6.2.1
Ca
L root relief roll length 6.2.1
Cf
L , L length of end relief 6.3.1
Cl ClI
L
tip roll length of triangular end relief modification 6.4.2
Ea
Symbol Description Used in
L root roll length of triangular end relief modification 6.4.2
Ef
M dimension over balls A.7
dK
M dimension over cylinders A.7.1
dZ
M radial single-ball dimension A.5
rK
M radial single-cylinder dimension A.6
rZ
N
number of tooth or pitch 4.1.6
O centre of a circle Figure 10
S twist of the transverse profile 6.4.3
α
S twist of the flank line 6.4.3
β
T tooth thickness tolerance Figure 37
sn
T contact point of tangent (lines of engagement) at base circle Figure 10
U involute point of origin 4.3.7
W base tangent length over k measured teeth or measured spaces A.2.1
k
Y any point on a tooth flank or involute 4.3.5
normal pressure angle 4.3.6
α
n
α transverse pressure angle 4.3.5
t
working transverse pressure angle of gear pair 5.2.4
α
wt
α working transverse pressure angle in the generating gear unit 7.6
wt0
normal pressure angle at the Y-cylinder 4.3.6
α
yn
transverse pressure angle at the Y-cylinder 4.3.5
α
yt
pressure angle at root form circle 7.6
α
Ff
α pressure angle at circle through centre of ball A.5
K
transverse pressure angle at a point at circle through centre of ball A.5
α
Kt
α transverse pressure angle at a point at measuring circle A.5
Mt
pressure angle of the standard basic rack tooth profile 4.3.6
α
P
pressure angle of the tool basic rack tooth profile 7
α
P0
working transverse pressure angle for double-flank engagement A.8
α
L
transverse pressure angle at the V-cylinder A.5
α
vt
helix angle 4.3.3
β
base helix angle 4.3.3
β
b
helix angle at Y-cylinder 4.3.3
β
y
angle of rocking for span measurement A.2.1
δ
w
6 © ISO 2007 – All rights reserved

Symbol Description Used in
lead angle at reference cylinder 4.3.3
γ
γ lead angle at Y-cylinder 4.3.3
y
transverse contact ratio 5.4.7.1
ε
α
ε overlap ratio 5.4.7.3
β
total contact ratio 5.4.7.5
ε
γ
specific sliding 5.6.3
ζ
ζ specific sliding at end points of path of contact 5.6.3
f
space width half angle at reference circle 4.7.4
η
base space width half angle 4.7.4
η
b
η space width half angle at Y-circle 4.7.4
y
rolling angle of the involute at point Y 4.3.7
ξ
y
ξ rolling angle at tip form circle of pinion-type cutter 7.6
Fa0
rolling angle at root form circle 7.6
ξ
Ff
ξ rolling angle at active tip circle 5.4.1
Na
rolling angle at active root circle 5.4.1
ξ
Nf
ρ root radius on the standard basic rack tooth profile Figure 4
fP
radius of curvature of the involute at point Y 4.3.8
ρ
y
angular pitch 4.4.2
τ
ϕ backlash angle 5.5.2
j
transverse angle of transmission 5.4.7.1
ϕ
α
overlap angle 5.4.7.3
ϕ
β
total angle of transmission 5.4.7.5
ϕ
γ
tooth thickness half angle at reference circle 4.7.2
ψ
base tooth thickness half angle 4.7.2
ψ
b
tooth thickness half angle at Y-circle 4.7.2
ψ
y
angular velocity of driving gear 5.2.2
ω
a
ω angular velocity of driven gear 5.2.2
b
sum of profile shift coefficients 5.3
∑x
sum of profile shift coefficient, non-zero backlash 5.3
∑x
E
3.2 Subscripts
Subscript Description Used
b
in
a
—
a for quantities associated with the tip of a tooth or for the driving gear 5.2.2
b for quantities associated with the base cylinder 4.3.10
b for quantities associated with the driven gear 5.2.2
e for quantities associated with the plane of action
f for quantities associated with the root
g for “sliding”
i for the lower limit in the case of deviations
k for a number of teeth, spaces, pitches or spans
l for “left-hand”
m for a mean value
max for a maximum value
min for a minimum value
n for quantities in a normal section 4.2.6.2
r for “right-hand”
s relating to “tooth thickness”, for the upper limit in the case of deviations
t for quantities in a transverse section 4.2.6.1
v for quantities associated with the V-cylinder 4.5.1
w for quantities associated with the pitch cylinder and working values of a
gear pair
x for quantities in an axial section 4.2.6.3
y for values at a point Y (on the Y-cylinder)
E relating to “generating” (e.g. quantities generated on the cylindrical gear) or
“generator”
F for quantities determining form circles and maximum usable flank area
K for quantities resulting from corner chamfering or for ball dimensions
L for designating a master gear
L for designating left flanks 4.1.8.2
M for designating a measured value
N for active circles
P for quantities of the standard basic rack tooth profile
P0 for quantities of the tool standard basic rack tooth profile
R for designating right flanks 4.1.8.2
V for working side, for rough gear cutting
W for measuring base tangent length
8 © ISO 2007 – All rights reserved

Subscript Description Used
b
in
Z for quantities associated with cylinder dimensions
α
for quantities associated with contact
β for quantities associated with a tooth trace
γ
for total contact ratio
for “sum”
∑
0 for quantities associated with the generating tool or the generating gear unit 7
1 for quantities associated with the pinion (smaller gear) of a gear pair 5.1.3
2 for quantities associated with the wheel (larger gear) or internal gear, used 5.1.3
for designating a coefficient relating to the module
I for locating face 4.2.1
II for the face opposite the locating face 4.2.1
a
No subscript designates quantities associated with the reference cylinder.
b
Used with the symbols listed in 3.1 or as additions.

3.3 Units
The quantities dealt with in this International Standard are to be stated in the following units:
⎯ modules, lengths and linear dimensions in millimetres (mm);
⎯ angles which are to be used in equations in radians (rad);
⎯ angles which can be used for entries or to display results in degrees (°);
⎯ angular velocity in radians per second (rad/s).
NOTE The notation |z|, denotes the absolute value, which is always positive, e.g. |−50| = +50. The expression
z
is used to extract the sign of the tooth number and is convenient for programming. In particular, it is used often to
z
determine the appropriate sign for an element of an expression; the result is 1 for external gears and −1 for internal gears.
4 Individual cylindrical gears
In this clause, the geometry of gear teeth is described using a generation process based on zero backlash
engagement with a basic rack. The relationships are valid for any basic rack, but the standard basic rack
(see ISO 53) is used for illustration. The standard basic rack tooth profile of the tooth system has straight
flanks. Its datum line is the straight line on which the nominal dimensions of tooth thickness and space width
are defined as equal to half the pitch. The standard basic rack tooth profile has the same pressure angles for
the left and right flanks and the addendum plus bottom clearance equal to the dedendum. The helix angles for
all the tooth flanks of a gear have the same nominal value.
4.1 Concepts for an individual gear
4.1.1 Gear, cylindrical gear, external gear, internal gear
A gear is a rotationally symmetrical object (gear blank) with a tooth system worked into the rim. A cylindrical
gear is a gear with a cylindrical reference surface. A distinction is made between external and internal gears
according to the radial arrangement of the teeth in each case. The tips of the teeth point outwards in an
external gear and inwards in an internal gear.
4.1.2 Tooth system, external teeth, and internal teeth
The tooth system refers to all the teeth and space widths around the rim of a gear. As in 4.1.1, a distinction is
made between internal and external gear teeth.
4.1.3 Tooth and space
A tooth is a geometrical element on the gearwheel body that enables the transmission of force and motion.
The form and dimensions of the teeth and the distance between consecutive teeth are defined by the tooth
system parameters. The space is the gap between two consecutive teeth.
4.1.4 Tooth system parameters
The nominal dimensions of involute cylindrical gear teeth are uniquely determined by the diameter of the
reference cylinder, the associated basic rack and its position in relation to the reference circle. The nominal
dimensions are defined by the following parameters, which are independent of each other:
⎯ number of teeth, z;
⎯ standard basic rack tooth profile;
⎯ normal module, m ;
n
⎯ helix angle, β, and flank direction;
⎯ profile shift coefficient, x;
⎯ tip diameter, d ;
a
⎯ facewidth, b.
4.1.5 Number of teeth and sign of number of teeth
The number of teeth around the rim of the gearwheel is denoted by z.
The number of teeth, z, of an external cylindrical gear must be taken as a positive value in the following
equations while the number of teeth, z, in an internal cylindrical gear is to be taken as a negative value.
In the case of segments, the number of teeth, z, used in calculations is the number that there would be on the
whole circumference.
10 © ISO 2007 – All rights reserved

4.1.6 Tooth number
When numbering teeth, the designations tooth 1, tooth 2, etc. are to be defined on a transverse surface
(datum face) viewed in an agreed direction so that the teeth are numbered in ascending order (moving in a
clockwise direction). If the letter N is used to denote a reference tooth, the next tooth in the direction of
counting is denoted by N + 1 and the previous tooth going in the opposite direction by N − 1. Tooth No. z is
followed by tooth 1 in the direction of counting, see Figure 1.

Figure 1 — Numbering of teeth and spaces on datum face

4.1.7 Top land and bottom land
4.1.7.1 Top land
The top land of a tooth is the outermost (innermost in the case of internal gears) periphery of the tooth
concentric to the reference cylinder, see Figure 2.
4.1.7.2 Bottom land
The bottom land is the innermost (outermost in the case of internal gears) periphery of the space width
concentric to the reference cylinder, see Figure 2.
4.1.8 Tooth flanks and flank sections
4.1.8.1 Tooth flank
Tooth flanks are those parts of the surface of a tooth that are located between the top land and the bottom
land, see Figure 2.
4.1.8.2 Right flank, left flank
The right flank (or left flank) is the tooth flank that an observer sees on the right-hand (or left-hand) side when
viewing the datum face of a tooth when it is pointing upwards. This definition applies to both external and
internal gears, see Figure 2.
Right flank parameters are indicated by the subscript R and left flank parameters by the subscript L.
a) Internal tooth b) External tooth
Key
1 top land
2 addendum flank
3 reference cylinder
4 dedendum flank
5 bottom land
6 datum face
Figure 2 — Top land, bottom land and tooth flank with division (internal and external teeth)

4.1.8.3 Addendum flank, dedendum flank
The addendum flank (or dedendum flank) is that part of a tooth flank that is located between the reference
cylinder and the top land (or the bottom land), see Figure 2.
4.1.8.4 Usable flank
The usable flank is that part of a tooth flank that can be used to engage with a mating flank. On a cylindrical
gear, it is part of the involute helicoid including any flank modifications.
4.2 Reference surfaces, datum lines and reference quantities
4.2.1 Reference surface, datum surface, datum face
The reference surface of the teeth is an imaginary surface to which the geometrical parameters relate. In the
case of cylindrical gears, the reference surface is termed the reference cylinder.
The agreed front of the gear (usually used for text or suitably marked) is used as the datum face. Parameters
which relate to the datum face are denoted by the subscript I while parameters which relate to the opposite
face are denoted by the subscript II.
4.2.2 Reference rack
The reference rack is the rack that can be produced using the same gear-cutting tool, gear-cutting method
and pitch point (pitch axis) as the actual cylindrical gear. It is characterized by its profile, the direction of its
teeth in relation to the pitch axis of the generating gear unit, tip plane, root plane and facewidth, see Figure 3.
12 © ISO 2007 – All rights reserved

Key
1 transverse section
2 facewidth
3 normal section
4 standard basic rack tooth profile
Figure 3 — Concepts and parameters relating to reference rack

4.2.3 Basic rack tooth profile for involute gear teeth
The basic rack tooth profile is defined in a normal section. The flanks of the basic rack tooth profile of involute
teeth are straight lines. Tooth thickness, s , and space width, e , on the datum line of the basic rack (P-P) in
P P
the reference plane are equal, see Figure 4.
The standard basic rack tooth profile for involute teeth is standardized in ISO 53.

Key
1 basic rack profile
2 datum line
3 root line
4 tip line
Figure 4 — Terms and parameters relating to basic rack tooth profile in normal section
4.2.4 Reference cylinder, reference circle, reference diameter
The reference cylinder is the reference surface for the cylindrical gear teeth. Its axis coincides with the axis of
the gear (gear axis). The reference circle is the intersection of the reference cylinder with a transverse plane
section. The reference diameter, d, is determined by
z m
n
dz==m (1)
t
cosβ
4.2.5 Gear axis
The axis of a gear (gear axis) is the axis that is defined by the geometrical axis of the support surfaces.
4.2.6 Sections through a cylindrical gear
4.2.6.1 Transverse section, transverse profile
The sectioning of cylindrical gear teeth by a plane perpendicular to the gear axis yields a transverse section.
For helical gears, quantities in the transverse section are denoted by the subscript t. The intersection of a
tooth with a transverse plane is termed the transverse profile.
4.2.6.2 Normal section, normal profile
The sectioning of involute helical gear teeth by a surface perpendicular to the flank lines of the involute
helicoid yields a normal surface. The normal surface is curved three-dimensionally.
Quantities on the normal surface are denoted by the subscript n. The intersection of a tooth with a normal
surface is termed the normal profile.
4.2.6.3 Axial section, axial profile
The sectioning of cylindrical gear teeth by a plane containing the gear axis yields an axial section.
Quantities in the axial section are denoted by the subscript x. The intersection of a tooth with an axial plane is
termed the axial profile.
4.2.6.4 Cylindrical section, flank lines
The flank lines are lines of intersection of the right and left flanks with a cylinder that has an axis which
coincides with the gear axis. Hence, right and left flank lines are to be distinguished.
The reference flank line (tooth trace) is the line of intersection of the flank with the reference cylinder. The
base flank line is the line of intersection of the involute flank — possibly imagined as extended — with the
base cylinder. The base flank line is a helix on the base cylinder. The origin of the involute helicoid is a base
flank line. The tip flank line is the line of intersection of the involute flank — possibly imagined as extended —
with the tip cylinder.
The flank lines are helices in the case of helical gear teeth and straight lines in the case of spur gear teeth.
4.2.7 Module
The module of a basic rack is found as the pitch of the rack divided by the number π (see Figure 4). The
normal module, m , of the cylindrical gear is found as the module of the standard basic rack tooth profile
n
(module series ISO 54).
14 © ISO 2007 – All rights reserved

For a helical gear, the transverse module, m , is found as
t
m
n
m = (2)
t
cos β
and the axial module, m , as
x
m
mm
nn t
m== = (3)
x
sin β cosγβtan
For a spur gear, the module is m = m = m .
t n
4.2.8 Facewidth
The facewidth, b, is the length of the toothed part of the cylindrical gear measured in the axial direction on the
V-cylinder. (See 4.5.1.)
The usable facewidth, b , is the distance between two transverse sections that contain the fully developed
F
height of the tooth flank. (See Figure 5.)

Key
1 developed view of V-cylinder
Figure 5 — Facewidth b, usable facewidth b
F
4.2.9 Profile shift, profile shift coefficient and sign of profile shift
The profile shift, xm , for involute gear teeth is the displacement of the basic rack datum line from the
n
reference cylinder. The magnitude of the profile shift can be made non-dimensional by dividing by the normal
module, and it is then expressed by the profile shift coefficient, x. Positive profile shift increases the tooth
thickness on the reference cylinder. (See Figure 6.)

Key
measurement along an arc
P–P datum line of basic rack
1 external gear
2 internal gear
Figure 6 — Profile shift for external and internal gear teeth
16 © ISO 2007 – All rights reserved

4.3 Involute helicoids
4.3.1 Generator of involute helicoids
In developing the base cylinder surface as a plane, a flank line on the base cylinder describes an involute
helicoid. The straight line inclined to the axial line in the developed surface (base cylinder tangential plane) is
the generator of the involute helicoid, see Figure 7.

Key
1 developed axial line
2 involute helicoid
3 involute
4 base helix
5 base cylinder axial line
6 base cylinder
7 developed base cylinder envelope
8 involute
9 straight line generator
Figure 7 — Base cylinder with generator and involute helicoids
4.3.2 Lead
The lead, p , is the distance between successive intersections of an axial line with the involute helicoid, see
z
Figure 8. The lead is independent of the diameter of the cylinder.
zmππz m
nt
p== = zp (4)
zx
sinββtan
β + γ = 90°
Key
1 normal plane
2 reference cylinder envelope line, gear axis
3 reference trace
4 projection of the gear axis
Figure 8 — Lead triangle, lead, helix angle, lead angle
4.3.3 Helix angle, lead angle
The helix angle, β, is the angle between a tangent to a reference helix and the reference cylinder envelope
line through the tangent contact point. In special cases, the helix angle, β , of right flanks may differ from the
R
helix angle, β , of left flanks; however, all equations are based on equal helix angles.
L
The relationship between β and the base helix angle, β (helix angle on the base cylinder), is found from
b
Equations (5) to (7):
tanββ= tan cosα (5)
t
b
sinββ= sin cosα (6)
n
b
sinα
cosααsin
yn
nn
cos β==cosβα= = cos tanα+ cosβ (7)
nn
b
cosααsin sinα
tt yt
18 © ISO 2007 – All rights reserved

On a cylinder with arbitrary diameter, d , the helix angle, β , is found from Equations (8) to (10):
y y
dd
cosα tan β
yy
tb
tanββ==tan tanβ = tanβ = (8)
y
b
ddcosα cosα
yt b yt
cosα sin β
n b
sinββ==sin (9)
y
cosα cosα
yn yn
cosα cos β
tanα
yn yt b
cos β== (10)
y
tanααcos
yn
yt
The lead angle, γ, is the angle at which the normal plane crosses the gear axis, see Figure 8. It is also the
angle between a tangent to a reference helix (reference flank line) and the transverse section through the
tangent contact point:
γ =°90− β (11)
yy
For spur gears, β = 0° and γ = 90°.
4.3.4 Flank direction
The flank direction is right-handed if the flank line describes a right-hand helix and left-handed if the flank line
describes a left-hand helix. (See Figure 9.)

a) Right-hand teeth
b) Left-hand teeth
Figure 9 — Direction of helix
4.3.5 Transverse pressure angle at a point, transverse pressure angle
In a transverse section, the tangent to the involute at the arbitrary point Y is inclined to the radius to that point
by the transverse pressure angle, α :
yt
d
d
b
cosα == cosα (12)
yt t
dd
yy
(See Figure 10.)
The transverse pressure angle, α , is the acute angle between the tangent to the involutes at their point of
t
intersection with the reference circle and the radius through this point of intersection. It is expressed by
d
b
cosα = (13)
t
d
Figure 10 — Parameters relating to involute
4.3.6 Normal pressure angle at a point, normal pressure angle
In the normal section of the involute helicoid, the tangent to this section at an arbitrary point Y is inclined to the
radius through Y by the normal pressure angle at that point, α . The corresponding angle of inclination at the
yn
reference cylinder is the normal pressure angle, α ; this is equal to the pressure angle, α , of the standard
n P
basic rack tooth profile.
tanα = tanαβcos (14)
n
t
tanα = tanαβcos (15)
yn y
yt
For a spur gear, α = α and α = α .
n yn
t yt
20 © ISO 2007 – All rights reserved

4.3.7 Roll angle of the involute
The angle at the centre over the base circle arc from the origin, U, of the involute to the contact point, T, of the
tangent from point Y to the base circle is the roll angle, ξ , of the involute, see Figure 10. The base circle arc,
y
UT, is equal to the tangent portion, YT, hence
ξ = tanα (16)
y
yt
4.3.8 Radius of curvature of the involute, length of roll
The tangent portion, YT, is the radius of curvature, ρ , of the involute at point Y and at the same time the
y
length of roll, L , belonging to point Y, i.e. the developed base circle arc from the involute origin, U. In the
y
triangle OTY it is the side opposite the transverse pressure angle, α , at the centre of the circle O:
yt
dd−
dd
y
zz z
b
bb
L==ρξ= tanα = (17)
yy y
yt
zz22 z 2
(See Figure 10.)
4.3.9 Involute function
The angular difference, ξ − α , is termed the involute function of angle α and is denoted by inv α (to be
yyt yt yt
read as “involute α ”):
yt
invαξ=−αα= tan −α (18)
y
yt yt yt yt
(See Figure 10.)
4.3.10 Base cylinder, base circle, base diameter
The base cylinder is that cylinder coaxial with the gear axis that is determinative for the generation of the
involute helicoids, see Figure 10. Quantities associated with the base cylinder are denoted by the subscript b.
The base circle is the intersection of the base cylinder with a plane of transverse section. The involutes from
the base circle form the transverse profiles of the gearing. The base diameter, d , is given by
b
zm cosα zm
n n
t
dd==cosααzm cos= = (19)
b tt t
2 2
cos β
tanαβ+ cos
n
cosα
n
dz= m (20)
n
b
cos β
b
4.4 Angular pitch and pitches
4.4.1 Angular pitch
The angular pitch, τ, is that angle laying in transverse sections that result from the dividing of the complete
periphery of a circle into z equal parts.
2 p
2π
yt
τ== in radians (21)
z d
y
τ = in degrees (22)
z
4.4.2 Pitches on the reference cylinder
4.4.2.1 Transverse pitch
The (reference cylinder) transverse pitch, p , is the length of the reference circle arc between two successive
t
equal-handed tooth flanks (right or left flanks):
πm
ddπ
n
p== τ= =πm (23)
tt
cos β 2 z
(See Figure 11.)
Key
measurement along an arc
Figure 11 — Diameter, angular pitch, transverse pitches on helical cylindrical gear
4.4.2.2 Normal pitch
The (reference cylinder) normal pitch, p , is the length of the helix arc between two successive equal-handed
n
tooth flanks (right or left flanks) on the reference cylinder in the normal section of the gear:
pm=π =p cos β (24)
nn
t
(See Figure 12.)
4.4.3 Pitches on any cylinder
It is necessary to distinguish between the transverse pitch, p , and the normal pitch, p , on a cylinder of
yt yn
any diameter, d , (Y-cylinder):
y
ddπ d
yy y
p==τ = p (25)
yt t
2 zd
pp= cos β (26)
yn y
yt
22 © ISO 2007 – All rights reserved

4.4.4 Axial pitch
The axial pitch, p , of a helical gear is the portion of a generation line of a cylinder concentric with the gear
x
axis between two successive equal-handed tooth flanks (right or left flanks), see Figure 12. The axial pitch is
independent of the diameter of the cylinder. Axial pitch does not apply to spur gears. It is expressed by
p
p
p
πm
πm
yt yn
nz t
pm==π= = = = (27)
x
x
sin β z tanββtan sinβ
yy
Figure 12 — Geometrical relations between transverse, normal
and axial pitch in a developed view of a reference cylinder
4.4.5 Base pitch
The distance between successive equal-handed tooth flanks (right or left flanks) on the developed base
cylinder tangential plane is the base pitch.
⎯ Transverse base pitch:
ddπd
bbb
(28)
p==ταppcos = cosα= =p
bt tt yt yt t
2 zd
⎯ Normal base pitch:
pp==cosαβp cos (29)
nn
bn bt b
4.4.5.1 Transverse base pitch on the path of contact
The transverse base pitch on path of contact, p , is the distance between two parallel tangents in a
et
transverse section which contact two successive equal-handed tooth flanks:
p = p (30)
et bt
(See Figure 11.)
4.4.5.2 Normal base pitch on the plane of contact
The normal base pitch, p , is the distance between two parallel tangential planes which contact two
en
successive equal-handed tooth flanks:
p = p (31)
en bn
4.5 Diameters of gear
...