Straight cylindrical involute splines — Metric module, side fit — Generalities, dimensions and inspection

Provides data and guidance for the design, manufacture and inspection of the involute splines. Limiting dimensions, tolerances and manufacturing errors are defined and tabulated. Linear and angular dimensions are expressed in mm and degrees, respectively.

Cannelures cylindriques droites à flancs en développante — Module métrique, à centrage sur flancs — Généralités, dimensions et vérification

Ravni utori z evolventnimi boki na valjih - Metrski modul, bočno prileganje - Splošno, mere in kontrola

General Information

Status
Withdrawn
Publication Date
30-Apr-1981
Withdrawal Date
30-Apr-1981
Current Stage
9599 - Withdrawal of International Standard
Start Date
21-Sep-2005
Completion Date
21-Sep-2005

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International Standard

INTERNATIONAL ORGANIZATION FOR STANDARDIZATION.MEIK,4YHAPO~HAR OPl-AHM3Al.&lR fl0 CTAH~APTM3ALWWORGANISATION INTERNATIONALE DE NORMALISATION

Straight cylindrical involute splines - Metric module,
side fit - Generalities, dimensions and inspection

Cannelures cylindriques droites à flancs en développante - Module métrique, à centrage sur flancs - Généralités, dimensions et

vérification
First edition - 1981-05-01
U DC 621 n 24.44 Ref. No. ISO4156-1981 (E)

dimensions, dimensional tolerances, limits, definitions, symbols, formulae (mathemathics), design.

Descriptors : splines,
Price based on 138 pages
---------------------- Page: 1 ----------------------
Foreword

ISO (the International Organization for Standardization) is a worldwide federation of

national standards institutes IISO member bodies). The work of developing Inter-
national Standards is carried out through ISO technical committees. Every member

body interested in a subject for which a technical committee has been set up 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.

Draft International Standards adopted by the technical committees are circulated to

the member bodies for approval before their acceptance as International Standards by

the ISO Council.
International Standard ISO 4156 was developed by Technical Committee ISO/TC 32,
Splines and serrations, and was circulated to the member bodies in October 1978.
It has been approved by the member bodies of the following countries :
Australia India Sweden
Austria
Italy Turkey
Belgium Japan United Kingdom
Chile Korea, Rep. of USA
Finland South Africa, Rep. of Y ugoslavia
France Spain

The member bodies of the fol lowing countries expressed disapproval of the document

on technical grounds :
Czechoslovakia
Germany, F. R.
0 International Organization for Standardization, 1981
Printed in Switzerland
---------------------- Page: 2 ----------------------
Contents
Page
Section one : Generalities
.........................................................
1 Scope and field of application
............................................
2 Terms and definitions relating to splines

3 Symbols ....................................................................................

4 Pressure angle (standard) ..............................................................

Type of fit ..................................................................................

...................... 8
6 Space width and tooth thickness, total tolerance (T + A.)
...........................................................
7 Basic rack profiles for spline
.................................................
8 Machining tolerances and variations
...............................................................
9 Effect of spline variations
.........................................
10 Effective and actual dimensions for splines
Il Use of effective and actual dimensions for space width and tooth

thickness ...................................................................................

12 Drawing data ..............................................................................

Section two : Dimensions

. . . . . . . . . . ..I..............................................................

13 Introduction.. . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14 Dimensions and tolerances
Tables
30° pressure angle
38-65

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- flat root
66-93

- fillet root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...*.

94-121

37,5O pressure angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122-139

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45O pressure angle
Section three : Inspectiod)
1) In preparation : Will be the subject of an addendum.
---------------------- Page: 3 ----------------------
ISO 4X56-1981 (E)
INTERNATIONAL STANDARD
lute splines - Metric module,
Straight cylindric
side fit - Generalities, dimensions and inspection
: Generalities
Section one
2.6 fillet root spline : A spline having a tooth or space pro-
1 Scope and field of application
file in which the opposing involute flanks are connected to the
root circle (Dei or Di, diameter) by a single fillet.
This International Standard provides data and guidance for the
design, manufacture and inspection of straight (non-helical)
cylindrical involute splines with side fit. It establishes a
2.7 flat root spline : A spline having a tooth or space profile
specification based on the module within the range 0,25 to 10
in which each of the opposing involute flanks are connected to
inclusive, relating to nominal pressure angles of 30°, 37,5O and
the root circle (Dei or Di, diameter) by a fillet.
45O. (For electronic data processing purposes, the form of ex-
“37,5”’ has been adopted instead of 37O 30’.)
pression
module, m : The ratio of the circular pitch, expressed in
2.8
millimetres, to the number n: (or the ratio of the pitch diameter,
Limiting dimensions, tolerances, manufacturing errors and
expressed in millimetres, to the number of teeth).
their effects on the fit between connecting co-axial spline
elements are defined and tabulated. Linear dimensions are ex-
2.9 pitch circle : The reference circle from which all normal
pressed in millimetres and angular dimensions in degrees.
spline dimensions are derived, and the circle on which the
specified pressure angle has its nominal value.
2 Terms and definitions relating to splines
2.10 pitch diameter, D : The diameter of the pitch circle, in
millimetres, equal to the number of teeth multiplied by the
module.
2.1 spline joint : Connecting, co-axial elements that
transmit torque through the simultaneous engagement of

equally spaced teeth situated around the periphery of a cylin- 2.11 pitch point : The intersection of the spline tooth profile

with the pitch circle.
drical external member with similar spaced mating spaces
situated around the inner surface of the related cylindrical inter-
nal member.
A length of arc of the pitch circle
2.12 circular pitch, p :
between two consecutive pitch points of left- (or right-) hand
flanks, which has a normal value of the number n: multiplied by
One member of spline joint having
2.2 involute spline :
the module.
teeth or spaces that have involute flank profiles.
The acute angle between a radial
2.13 pressure angle, a :
: A spline formed on the inner surface of
2.3 interna1 spline
line passing through any point on a tooth flank and the tangent
a cylinder.
plane to the flank at that point.
2.14 standard pressure angle, an : The pressure angle at
2.4 external spline : A spline formed on the outer surface of
the specified pitch point.
a cylinder.
2.15 base circle : The circle from which involute spline
2.5 fillet : The concave surface of the tooth or space con-
tooth profiles are generated.
necting the involute flank and the root circle. This curved sur-
face as generated varies and cannot be properly specified by a
: The diameter of the base circle.
radius of any given value. 2.16 base diameter, D,
---------------------- Page: 4 ----------------------
ISO 4156-1981 (E)

: The arc length of the base circle be- 2.30 effective clearance, cv Hooseness or interference) :

2.17 base pitch, Pb
The effective space width of the interna1 spline minus the effec-
tween two consecutive corresponding flanks.
tive tooth thickness of the mating external spline.
2.18 major circle : The circle formed by the outermost sur-
2.31 theoretical clearance, c (looseness or interference) :
face of the spline. It is the outside circle (tooth tip circle) of the
The actual space width of an interna) spline minus the actual
external spline or the root circle of the interna) spline.
tooth thickness of the mating external spline. It does not define
the fit between mating members, because of the effect of varia-
2.19 major diameter, Dec, Dei : The diameter of the major
tions.
2.32 form clearance, CF : The radial depth of involute pro-
2.20 minor circle : The circle formed by the innermost sur-
file beyond the depth of engagement with the mating part. It
face of the spline. It is the root circle of the external spline or
allows eccentricity of the minor circle (internal), of the major
the inside circle (tooth tip circle) of the interna1 spline.
circle (external) and of their respective pitch circles.
Dii : The diameter of the minor
2.21 minor diameter, Di,,
2.33 total index variation : Amount of absolute values of
circle.
the two greatest actual (or practically measured) positive and
negative variations from the theoretical spacing.
2.22 form circle : The circle which establishes the deepest
points of involute form control of the tooth profile. This circle
2.34 total profile variation : Amount of absolute values of
along with the tooth tip circle (or start of chamfer circle) deter-
the two greatest positive and negative variations, from the
mines the limits of tooth profile requiring control. It is located
theoretical tooth profiles, measured normal to flanks.
near and below the major circle on the interna1 spline and near
and above the minor circle on the external spline.
2.35 total lead variation : Amount of absolute values of the
two greatest opposite direction variations, from the theoretical
2.23 form diameter, &,, DFi : The diameter of the form
direction (parallel to the datum axis), also including parallelism
circle.
and alignment variations (see figure IL
NOTE - Straight (non-helical) splines have an infinite lead.
: The radial distance from the
2.24 depth of engagement
minor circle of the interna1 spline to the major circle of the ex-
ternal spline, minus corner clearance and/or chamfer depth.
2.25 basic (circular) space width or tooth thickness at
the pitch diameter, E or S : For 30°, 37,5’ and 45O pressure
Datum axis
angle splines, half the circular pitch.
2.26 actual space width : The practically measured circular
space width, on the pitch circle, of any single space width
a) Lead variation
within the limit values E,,,, and E,i,..
Datum axis -
2.27 effective space width, EV : For an interna1 spline,
equal to the circular tooth thickness on the pitch circle of an im-
aginary Perfect external spline which would fit the interna1
spline without looseness or interference, considering engage-
ment of the entire axial length of the splined assembly. The
b) Parallelism variation
minimum effective space width (EV min., always equal to E) of
Effective spline
the interna1 spline is always basic, as shown in table 1. Fit varia-
tions may be obtained by adjusting the tooth thickness of the
Datum axis
external spline.
-YY
2.28 actual tooth thickness : The practically measured cir-
cular tooth thickness, on the pitch circle, of any single tooth
C) Alignment variation
within the limit values S,,,. and S,i,..
2.29 effective tooth thickness, S, : For an external spline,
equal to the circular space width on the pitch circle of an im-
Figure 1 - Lead variations
aginary Perfect interna1 spline which would fit the external
spline without looseness or interference, considering engage-
ment of the entire axial length of the splined assembly. Fit
variations are obtained by adjusting this value S,.
---------------------- Page: 5 ----------------------
ISO 41!56-1981 (E)
parallelism variation : The variation of parallelism of a
2.36 Db [DB1 = base diameter
single spline tooth to any other single spline tooth (see
dei = pin contact diameter, interna1 spline
figure IL
d,, = pin contact diameter, external spline
DFe [DFEI = form diameter, external spline
2.37 alignment variation : The variation of the effective
spline , axis with respect to the reference axis (sec figure 1).
DFi [DFII = form diameter, interna1 spline
Dii [DII] = minor diameter, interna1 spline
2.38 out-of-roundness : The variation of the spline from a
Dec [DEEI = major diameter, external spline
true circular configuration.
Die [DIEI = minor diameter, external spline
2.39 effective variation : The accumulated effect of the
D,i [DEI1 = major diameter, interna1 spline
spline variations on the fit with the mating part.
DRe [DREI = diameter of measuring pin for external spline
D,i [DRI] = diameter of measuring pin for interna1 spline
2.40 variation allowance, A : The permissible effective
Â. = variation allowance
variation.
inv a = involute a ( = tan a - a)
2.41 machining tolerance, T : The permissible variation in
Ke [KEI = approximation factor for external spline
actual space width or actual tooth thickness.
Ki [KI] = approximation factor for interna1 spline
= spline length
2.42 total tolerance, (T + A) : The machining tolerance
plus the variation allowance. The total tolerance on an interna1
= active spline length
spline is the difference between the minimum effective space
= length of engagement
width and the maximum actual space width; on an external
T = machining tolerance
spline, it is the difference between the maximum effective tooth
thickness and the minimum actual tooth thickness.
IMRe [MRE] = measurement over two pins, external spline
h/lRi [MRI] = measurement between two pins, interna1 spline
2.43 length of engagement, g, : The axial length of con-
W = measurement over k teeth, external spline
tact between mating splines.
2 = number of teeth
2.44 active spline length, g, : The maximum axial spline m = module
length in contact (when working) with the mating spline. On
Pb = base pitch
sliding splines, the active length exceeds the length of engage-
p = circular pitch
ment.
@Fe = fillet radius of the basic rack, external spline
2.6 basic dimension : A numerical value to describe the
@Fi = fillet radius of the basic rack, interna1 spline
theoretically exact size, shape or location of a feature. It is the
E = basic space width, circular
basis from which permissible variations are established by
E = actual maximum space width, circular
max
tolerances.
= actual minimum space width, circular
min
2.46 auxiliary dimension : A dimension, without tolerance,
E,, EV1 = effective space width, circular
given for information purposes only, for the determination of
S= basic tooth thickness, circular
the useful production and control dimensions.
S = actual maximum tooth thickness, circular
max
S = actual minimum tooth thickness, circular
min
3 Symbols
= effective tooth thickness, circular
s, WI
a = pressure angle
3.1 General symbols
= standard pressure angle
The general symbols used to designate the various spline terms
a,i = pressure angle at pin contact diameter, interna) spline
and dimensions are given below (sec figures 10, 11, 12, 13, 14
a = pressure angle at pin contact diameter, external
and 15).
spline
NOTE - In electronic data processing (EDP), it is not always possible
ai = pressure angle at pin centre, interna1 spline
to present symbols in their theoretically correct form because of limita-
a, = pressure angle at pin centre, external spline
tions of connected printing equipment. For this reason, some alter-
native symbols for EDP usage are given in brackets below (for exam-
= pressure angle at form diameter, external spline
aFe
ple, the symbol Db for base diameter may be printed as DB).
aFi = pressure angle at form diameter, interna1 spline
c, = effective clearance (looseness or interference)
k-j,--h-f- e and d = fundamental deviation on the
cF = form clearance
external spline = c, min
D = pitch diameter H = fundamental deviation on the interna1 spline
---------------------- Page: 6 ----------------------
ISO 41!56-1981 (El

Tables 1 and 2 give the basic dimensions and fundamental formulae, a graphical presentation of which is given by figure 2.

Table 1 - Theoretical dimensions for spllines
Dimensions in millimetres
Basic space width or tooth thickness Base pitch
Circular
Tooth Module at pitch diameter E or S
pitch
-- -- --1
o!D 30” * m
&D 30” QD 37,5O o!D 45” &D 30” a!D 37,5O &D 45O
10 31,416 15,708 15,708 27,207 0 24,923 9
25,133 12,566 12,566 21,765 6 19,939 2
18,850 9,425 9,425 16,324 2 14,954 4
5 15,708 7,854 7,854 13,603 5 12,462 0
4 12,566 6,283 6,283 10,882 8 9,969 6
3 9,425 4,712 4,712 8,162 1 7,477 2
5,553 6
7,854 3,927 3,927 3,927 6,801 7 6,231 0
2,5
2 6,283 3,142 3,142 3,142 5,441 4 4,984 8 4,442 9
1,75 5,498 2,749 2,749 2,749 4,761 2 4,361 7 3,887 5
4,712 2,356 2,356 2,356 4,081 0 3,738 6 3,332 2
1,5
1,25 3,927 1,963 1,963 1,963 3,400 9 3,115 5 2,776 8
3,142 1,571 1,571 1,571 2,720 7 2,492 4 2,221 4
0,75 2,356
1,178 1 ,178 1 ,178 2,040 5 1,869 3 1,666 1
1,571 0,785 0,785 0,785 1,360 4 1,246 2 1,110 7
Or5
0,25 0,785 0,393 0,555 4

* For illustration purposes : relative tooth sizes for various spline modules for pressure angle aiD = 30”.

---------------------- Page: 7 ----------------------
ISO 4156-‘l981 (E)
Table 2 - ForanuBae for dimensions and tolerances for all fit classes
Formul a
Pitch diameter D mZ
Base diameter m z Cos O”D
Circular pitch Km
Base pitch x m Cos aiD
Effective Upper deviation, external
resulting from deviation allowance (fundamental)
k-j,--h-f-eandd
Minimum diameter, interna1
30°, flat root D
m(Z+ 1,5)
ei min
30”, fi I let root D
m(Z+ 1,8)
ei min
37,5 O, fillet root D m(Z+ 1,4)
ei min
45’, fillet root D
m(Z+ 1,2)
ei min
Maximum major diameter, interna1 D D
t (T t h)/tan aD (sec note 1 )
ei max ei min
Minimum form diameter, interna1
30°, flat root and fillet root m(zt 1) + 2cF
DFi min
37,5O, fi I let root m(z t o,g) t 2 cF
DFi min
45” , fi I let root dz + 0,8) + 2 CF
DFi min
Minimum diameter, interna1
DFe max f 2 CF (Set? Ilote 2)
Dii min
Maximum minor diameter, interna1
m < 0,75
+ tol. H 10
Dii max Dii min
0,75 < m < 2
+ tol. H 11
Dii max Dii min
m>2
+ toi. H 12
Dii max Dii min
Basic space width and E and
0,5x m
Minimum effective space width E
v min
Maximum actual space width
class 4 E E v min + (T + A) (sec note 3)
max
class 5 E E v min + (T + Al (sec note 3)
max
class 6 E E
v min + (T + h) (sec note 3)
max
class 7 E E
v min + (T + A) (sec note 3)
max
Minimum actual space width E E
v min + h (sec 8.2)
min
Maximum effective space width E E
max - h (see 8.2)
v max
Maximum major diameter, external
30°, flat root and fillet root D
m(z t 1) t esv/tan o!D (sec note 4)
ee max
37,5’, fillet root D
m(z t o,g) t es,/tan aD (sec note 4)
ee max
45”) fillet root
D m(Z + 0,8) + es,ltan QD (sec note 4)
ee max
Minimum major diameter, external
m < 0,75 D - tol. h 10
ee min ee max
0,75 < m < 2 D .
ee min
ma2 D
ee min
22)
Maximum form diameter, external
DFe max
Maximum minor diameter, external
(Sec note 6)
30”, flat root
m (z - 1 ,5) t es&n aD
Die max
30”, fillet root
m(Z- 1,8) + es,/tan &D
Die max
37,5’, fillet root
m (z - 1 $1 t esv/tan Q!D
Die max
45”, fil let root
m (z - 1 ,2) f es&an o!D
Die max
---------------------- Page: 8 ----------------------
ISO 4156-1981 (E)
Table 2 - Formulae for dimensions and tolerances for all fit classes (conchded)
Basic tooth thickness
Maximum effective tooth thickness
Minimum actual tooth thickness
class 4 v max - (T + A) (sec note 3)
class 5
v max - (T f A) (sec note 3)
class 6 v max - (T + h) (sec note 3)
class 7 v max - (T + A) (sec note 3)
Maximum actual tooth thickness
Minimum effective tooth thickness
Total tolerance, space width or tooth thickness
Form clearance
Pin diameter, interna1 spline -
Pin diameter, external spline
Measurement between pins
Measurement over pins
Change factor, interna1 see section three, “1 nspection”
Change factor, external see section three, “Inspection”
NOTES
1 (T + h) for class 7 - see clause 6.
2 For all classes of fit, always take the &e max
value corresponding to the H/h fit.
3 See clause 6 and section two - tables of dimensions.
4 Take a null Upper deviation value for j, and k fundamental deviations.

5 See section two : “Dimensions”, and section three : “Inspection”, concerning &oice of pins.

6 For h, see figures 3,4,5 and 6.
---------------------- Page: 9 ----------------------
ISO 4156-1981 (E)
Piece
E effective
actual
max
(actual space width)
min -
vmax -
l- .
basic
v min
-0,5 n m-
(effective space width) z
basic :
a\1 max
(effective tooth thickness)
vmin -
max -
min
(actual tooth thickness)
Figure 2 - Graphical representation of tables 1 and 2
---------------------- Page: 10 ----------------------
ISO 4156-1981 (El
3.2 Subscripts classes as follows :

The following subscripts (see also the note in 3.1) are used as Spline fit class Effective interfeaence

part of the above general symbols to designate relative con-
H/k cVmax = (T + A)
ditions or locations :
T+A
deviation allowance j, = -
MS CV max =
= minor or interna1 (in this last case, in the last position)
ilIl
c 2 )
= major or external (in this last case, in the last position)
CE1 Effective looseness
,[Bl = at base
H/h Cv min = deviation allowance h = zero
= at contact point
H/f CV min = deviation allowance f
= pertaining to form diameter
&FI
deviation allowance e
H/e Cv min =
= effective
“[VI
H/d Cv min = deviation allowance d
= active
The deviation allowances (fundamental) k - j, - h - f - e
pertaining to gauges
,Ml =
and d are the standard deviations selected from ISO/R 286 /SO
- of engagement System of limits and fits - Part 7 : General, tolerances and
deviations, which are applied to the external spline. A pre-
n = standard
scribed maximum effective interference or minimum effective
looseness is obtained, allowing the fitting by adjusting from the
zero line the maximum effective and minimum actual limit
values of tooth thickness by the amount of the deviation
4 Pressure angle (standard)
allowance (see 8.72). The spline dimensions in the spline
tables of this International Standard are given for spline fit class
Standard pressure angles included in this International Stan-
H/h, C, min = zero.
dard for involute splines are 30°, 37,5’ and 45’.
6 Space width and tooth thickness,
total tolerance (T + A)
5 Type of fit
Tolerance classes
This International Standard deals with only one type of fit, the
This International Standard includes four classes of total
side fit, for 30°, 37,5O and 45O pressure angle splines. Formulae
tolerance ( T + A) on space width and tooth thickness selected

for the dimensions and tolerances for these splines are shown from a combination of tolerance units (i) in ISO/R 286. The

in table 2.
tolerance classes are indicated below, with corresponding com-
bination of tolerance units (i). For the calculation of T and Â,
sec clause 8. The values of Â. are given in tables 3 to 6 of
5.1 Side fit
clause 8.
In this fit, the mating members contact on the sides of the teeth
Tolerance unit
Spline tolerance
only. Major and minor diameters are clearance dimensions. The
class (il
tooth sides act as drivers and centralize mating splines.
4 = (10 i* + 40 i”“)
5 = (16 i* + 64 i”“)
5.2 Spline fit classes
6 = (25 i* + 100 i”“)

This International Standard provides the side fit in six spline fit 7 = (40 i* + 160 i”“)

Tolerance based on pitch diameter
i = 0,45 $D + 0,001 D for-D Q 5OOmm
i = 0,004D + 2,l forD > 5OOmm
Tolerance based on space width or tooth thickness
i =
0,45 $?m + 0,001 E (or SI
where
D is the pitch diameter in millimetres;
E is the basic space width in millimetres;
S is the basic tooth thickness in millimetres.

the resultant ( T + A) in micrometres. For ( T + Â) in millimetres, multiply the result by 0,001.

For the calculation of tolerance units (i), only the above indicated formulae (notes *, **) are to be taken into consideration.

---------------------- Page: 11 ----------------------
ISO 41564981 (El
volute spline of infinitely large diameter on a plane at right
It should to be noted that total values ( T + A) may always be

subtracted from the limit values of space width and tooth angles to the tooth surfaces, the profile of which is used as the

basis for defining the standard tooth dimensions of a system of
thickness given in section 2 and are usable even if the chosen
involute splines.
fit class is other than H/h.
NOTE - Below are listed the combinations of tolerance qualities IT
corresponding to the combinations of tolerance units (i) indicated
above. Those combinations of tolerance qualities IT are given only to
7.2 The reference line is a straight line crossing the profile of
indicate the conception principle of the tolerance system regarding its

eventual extension, and only to rediscover the corresponding com- the basic rack, with reference to which the tooth dimensions

binations of tolerance units (il.
are specif ied.
Spline tolerance Tolerance quality
class W-I
4 = (IT 6* + IT 9”“)
7.3 The profile of the basic rack for the standard pressure
angle splines is represented in the following figures :
5 = (IT 7* + IT 10”“)
6 = (IT 8* + IT II**)
figure 3 : 30° flat root spline, for modules 0,5 to 10;
7 = (IT 9* + IT 12”“)
figure 4 : 30° fillet root spline, for modules 0,5 to 10;
7 Basic rack profiles for spline
figure 5 : 37,5O fillet root spline, for modules 0,5 to 10;

7.1 The basic rack is a section of the tooth surface of an in- figure 6 : 45’ fillet root spline, for modules 0,25 to 2,5.

* Tolerance based on pitch diameter.
Tolerance based on space width and tooth thickness.
---------------------- Page: 12 ----------------------
60 41564981 (El
0,5nm A 0,5n:m
=0,2m* /
I -Pl=i
I nternal spline
External spline \
i Reference line
Figure 3 - Profile of the basic rack for 30° flat root spline
External spline
Reference line
Figure 4 - Profile of the basic rack for 30’ fillet root spline
* and ** See next page.
---------------------- Page: 13 ----------------------
ISO 4156-1981 0
= 0,3m”
PFi
Reference line
Figure 5 - Profile of the basic rack for 37,5’ fillet root spline
Interna1 spline
Figure 6 - Profile of the basic rack for 45O fillet root spline
NOTE concerning figures 3 to 6 :

For interna1 splines (hub), the form diameter, obtained by generating from the basic rack, is always greater than the form diameter shown in the

tables of dimensions (sec section two), which corresponds in all fit cases to the major maximum diameter of the shaft (with Upper deviation - es, -

zero) increased to diametral form clearance (2 c,-) equal to 0,2 module.

** For external splines (shafts), cF is obtained by generation from the basic rack @Fe maX

1 and for H/h fit (sec note 2 under table 2).

The size cF indicated is in fact a deviation permitting the form clearance cF given in table 2 to be obtained and is equal to 0,l m.

---------------------- Page: 14 ----------------------
ISO 4156-1981 (E)

7.4 The minimum radial form clearance, CF, shown in the 8 Machining tolerances and variations

basic rack profiles is equal to 0,l m for all pressure angle
8.1 Variation allowance (il)
splines.
The variation allowance, being the addition of the total index
The form clearance is modified for fits other than H/h.
variation, total profile variation and total lead variation, has an
effect on the effective fit of an involute spline. The effect of
these individual spline variations on the fit is less than their
total, because areas of more than minimum clearance cari have
form, lead, or index errors without changing the fit. It is also
7.5 External splines may be produced by generating with a
unlikely that these errors would occur in their maximum
pinion-type shaper cutter or with a hob, or by cutting with no
amounts simultaneously on the same spline. For this reason,
generating motion using a tool formed to the contour of a tooth
total index variation, profile variation and total lead variation are
space. External splines are also made by cold forming, and in
added together statistically and 60 % of this total is taken to
these cases are usually of the fillet root design. Interna1 splines
determine the effect that these variations have on spline fit. On
are usually produced by broaching, by form cutting, or by
this basis, the variation allowance becomes as follows :
generating with a shaper cutter. Even when full tip radius tools
L = 0,6 [(total index variation)* + (totale profile variation)*
are used, each of these cutting methods produces a fillet con-
+ (total lead variation)*] 1’2
tour with individual characteristics. Generated spline fillets are
curves related to the prolate epicycloid for external splines and
The above variation allowance is based on a length of engage-
the prolate hypocycloid for interna1 splines. These fillets have a
ment equal to one-half the pitch diameter. Adjustment on the
minimum radius of curvature at the point where the fillet is
variation allowance may be required for a greater length of
tangent to the external spline minor diameter circle or the inter-
engagement.
nal spline major diameter circle, and a rapidly increasing radius
of curvature up to the point where the fil
...

SLOVENSKI STANDARD
SIST ISO 4156:2000
01-januar-2000
5DYQLXWRUL]HYROYHQWQLPLERNLQDYDOMLK0HWUVNLPRGXOERþQRSULOHJDQMH
6SORãQRPHUHLQNRQWUROD

Straight cylindrical involute splines -- Metric module, side fit -- Generalities, dimensions

and inspection

Cannelures cylindriques droites à flancs en développante -- Module métrique, à centrage

sur flancs -- Généralités, dimensions et vérification
Ta slovenski standard je istoveten z: ISO 4156:1981
ICS:
21.120.30 Mozniki, utori za moznike, Keys and keyways, splines
razcepke
SIST ISO 4156:2000 en

2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

---------------------- Page: 1 ----------------------
SIST ISO 4156:2000
---------------------- Page: 2 ----------------------
SIST ISO 4156:2000
International Standard

INTERNATIONAL ORGANIZATION FOR STANDARDIZATION.MEIK,4YHAPO~HAR OPl-AHM3Al.&lR fl0 CTAH~APTM3ALWWORGANISATION INTERNATIONALE DE NORMALISATION

Straight cylindrical involute splines - Metric module,
side fit - Generalities, dimensions and inspection

Cannelures cylindriques droites à flancs en développante - Module métrique, à centrage sur flancs - Généralités, dimensions et

vérification
First edition - 1981-05-01
U DC 621 n 24.44 Ref. No. ISO4156-1981 (E)

dimensions, dimensional tolerances, limits, definitions, symbols, formulae (mathemathics), design.

Descriptors : splines,
Price based on 138 pages
---------------------- Page: 3 ----------------------
SIST ISO 4156:2000
Foreword

ISO (the International Organization for Standardization) is a worldwide federation of

national standards institutes IISO member bodies). The work of developing Inter-
national Standards is carried out through ISO technical committees. Every member

body interested in a subject for which a technical committee has been set up 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.

Draft International Standards adopted by the technical committees are circulated to

the member bodies for approval before their acceptance as International Standards by

the ISO Council.
International Standard ISO 4156 was developed by Technical Committee ISO/TC 32,
Splines and serrations, and was circulated to the member bodies in October 1978.
It has been approved by the member bodies of the following countries :
Australia India Sweden
Austria
Italy Turkey
Belgium Japan United Kingdom
Chile Korea, Rep. of USA
Finland South Africa, Rep. of Y ugoslavia
France Spain

The member bodies of the fol lowing countries expressed disapproval of the document

on technical grounds :
Czechoslovakia
Germany, F. R.
0 International Organization for Standardization, 1981
Printed in Switzerland
---------------------- Page: 4 ----------------------
SIST ISO 4156:2000
Contents
Page
Section one : Generalities
.........................................................
1 Scope and field of application
............................................
2 Terms and definitions relating to splines

3 Symbols ....................................................................................

4 Pressure angle (standard) ..............................................................

Type of fit ..................................................................................

...................... 8
6 Space width and tooth thickness, total tolerance (T + A.)
...........................................................
7 Basic rack profiles for spline
.................................................
8 Machining tolerances and variations
...............................................................
9 Effect of spline variations
.........................................
10 Effective and actual dimensions for splines
Il Use of effective and actual dimensions for space width and tooth

thickness ...................................................................................

12 Drawing data ..............................................................................

Section two : Dimensions

. . . . . . . . . . ..I..............................................................

13 Introduction.. . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14 Dimensions and tolerances
Tables
30° pressure angle
38-65

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- flat root
66-93

- fillet root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...*.

94-121

37,5O pressure angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122-139

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45O pressure angle
Section three : Inspectiod)
1) In preparation : Will be the subject of an addendum.
---------------------- Page: 5 ----------------------
SIST ISO 4156:2000
---------------------- Page: 6 ----------------------
SIST ISO 4156:2000
ISO 4X56-1981 (E)
INTERNATIONAL STANDARD
lute splines - Metric module,
Straight cylindric
side fit - Generalities, dimensions and inspection
: Generalities
Section one
2.6 fillet root spline : A spline having a tooth or space pro-
1 Scope and field of application
file in which the opposing involute flanks are connected to the
root circle (Dei or Di, diameter) by a single fillet.
This International Standard provides data and guidance for the
design, manufacture and inspection of straight (non-helical)
cylindrical involute splines with side fit. It establishes a
2.7 flat root spline : A spline having a tooth or space profile
specification based on the module within the range 0,25 to 10
in which each of the opposing involute flanks are connected to
inclusive, relating to nominal pressure angles of 30°, 37,5O and
the root circle (Dei or Di, diameter) by a fillet.
45O. (For electronic data processing purposes, the form of ex-
“37,5”’ has been adopted instead of 37O 30’.)
pression
module, m : The ratio of the circular pitch, expressed in
2.8
millimetres, to the number n: (or the ratio of the pitch diameter,
Limiting dimensions, tolerances, manufacturing errors and
expressed in millimetres, to the number of teeth).
their effects on the fit between connecting co-axial spline
elements are defined and tabulated. Linear dimensions are ex-
2.9 pitch circle : The reference circle from which all normal
pressed in millimetres and angular dimensions in degrees.
spline dimensions are derived, and the circle on which the
specified pressure angle has its nominal value.
2 Terms and definitions relating to splines
2.10 pitch diameter, D : The diameter of the pitch circle, in
millimetres, equal to the number of teeth multiplied by the
module.
2.1 spline joint : Connecting, co-axial elements that
transmit torque through the simultaneous engagement of

equally spaced teeth situated around the periphery of a cylin- 2.11 pitch point : The intersection of the spline tooth profile

with the pitch circle.
drical external member with similar spaced mating spaces
situated around the inner surface of the related cylindrical inter-
nal member.
A length of arc of the pitch circle
2.12 circular pitch, p :
between two consecutive pitch points of left- (or right-) hand
flanks, which has a normal value of the number n: multiplied by
One member of spline joint having
2.2 involute spline :
the module.
teeth or spaces that have involute flank profiles.
The acute angle between a radial
2.13 pressure angle, a :
: A spline formed on the inner surface of
2.3 interna1 spline
line passing through any point on a tooth flank and the tangent
a cylinder.
plane to the flank at that point.
2.14 standard pressure angle, an : The pressure angle at
2.4 external spline : A spline formed on the outer surface of
the specified pitch point.
a cylinder.
2.15 base circle : The circle from which involute spline
2.5 fillet : The concave surface of the tooth or space con-
tooth profiles are generated.
necting the involute flank and the root circle. This curved sur-
face as generated varies and cannot be properly specified by a
: The diameter of the base circle.
radius of any given value. 2.16 base diameter, D,
---------------------- Page: 7 ----------------------
SIST ISO 4156:2000
ISO 4156-1981 (E)

: The arc length of the base circle be- 2.30 effective clearance, cv Hooseness or interference) :

2.17 base pitch, Pb
The effective space width of the interna1 spline minus the effec-
tween two consecutive corresponding flanks.
tive tooth thickness of the mating external spline.
2.18 major circle : The circle formed by the outermost sur-
2.31 theoretical clearance, c (looseness or interference) :
face of the spline. It is the outside circle (tooth tip circle) of the
The actual space width of an interna) spline minus the actual
external spline or the root circle of the interna) spline.
tooth thickness of the mating external spline. It does not define
the fit between mating members, because of the effect of varia-
2.19 major diameter, Dec, Dei : The diameter of the major
tions.
2.32 form clearance, CF : The radial depth of involute pro-
2.20 minor circle : The circle formed by the innermost sur-
file beyond the depth of engagement with the mating part. It
face of the spline. It is the root circle of the external spline or
allows eccentricity of the minor circle (internal), of the major
the inside circle (tooth tip circle) of the interna1 spline.
circle (external) and of their respective pitch circles.
Dii : The diameter of the minor
2.21 minor diameter, Di,,
2.33 total index variation : Amount of absolute values of
circle.
the two greatest actual (or practically measured) positive and
negative variations from the theoretical spacing.
2.22 form circle : The circle which establishes the deepest
points of involute form control of the tooth profile. This circle
2.34 total profile variation : Amount of absolute values of
along with the tooth tip circle (or start of chamfer circle) deter-
the two greatest positive and negative variations, from the
mines the limits of tooth profile requiring control. It is located
theoretical tooth profiles, measured normal to flanks.
near and below the major circle on the interna1 spline and near
and above the minor circle on the external spline.
2.35 total lead variation : Amount of absolute values of the
two greatest opposite direction variations, from the theoretical
2.23 form diameter, &,, DFi : The diameter of the form
direction (parallel to the datum axis), also including parallelism
circle.
and alignment variations (see figure IL
NOTE - Straight (non-helical) splines have an infinite lead.
: The radial distance from the
2.24 depth of engagement
minor circle of the interna1 spline to the major circle of the ex-
ternal spline, minus corner clearance and/or chamfer depth.
2.25 basic (circular) space width or tooth thickness at
the pitch diameter, E or S : For 30°, 37,5’ and 45O pressure
Datum axis
angle splines, half the circular pitch.
2.26 actual space width : The practically measured circular
space width, on the pitch circle, of any single space width
a) Lead variation
within the limit values E,,,, and E,i,..
Datum axis -
2.27 effective space width, EV : For an interna1 spline,
equal to the circular tooth thickness on the pitch circle of an im-
aginary Perfect external spline which would fit the interna1
spline without looseness or interference, considering engage-
ment of the entire axial length of the splined assembly. The
b) Parallelism variation
minimum effective space width (EV min., always equal to E) of
Effective spline
the interna1 spline is always basic, as shown in table 1. Fit varia-
tions may be obtained by adjusting the tooth thickness of the
Datum axis
external spline.
-YY
2.28 actual tooth thickness : The practically measured cir-
cular tooth thickness, on the pitch circle, of any single tooth
C) Alignment variation
within the limit values S,,,. and S,i,..
2.29 effective tooth thickness, S, : For an external spline,
equal to the circular space width on the pitch circle of an im-
Figure 1 - Lead variations
aginary Perfect interna1 spline which would fit the external
spline without looseness or interference, considering engage-
ment of the entire axial length of the splined assembly. Fit
variations are obtained by adjusting this value S,.
---------------------- Page: 8 ----------------------
SIST ISO 4156:2000
ISO 41!56-1981 (E)
parallelism variation : The variation of parallelism of a
2.36 Db [DB1 = base diameter
single spline tooth to any other single spline tooth (see
dei = pin contact diameter, interna1 spline
figure IL
d,, = pin contact diameter, external spline
DFe [DFEI = form diameter, external spline
2.37 alignment variation : The variation of the effective
spline , axis with respect to the reference axis (sec figure 1).
DFi [DFII = form diameter, interna1 spline
Dii [DII] = minor diameter, interna1 spline
2.38 out-of-roundness : The variation of the spline from a
Dec [DEEI = major diameter, external spline
true circular configuration.
Die [DIEI = minor diameter, external spline
2.39 effective variation : The accumulated effect of the
D,i [DEI1 = major diameter, interna1 spline
spline variations on the fit with the mating part.
DRe [DREI = diameter of measuring pin for external spline
D,i [DRI] = diameter of measuring pin for interna1 spline
2.40 variation allowance, A : The permissible effective
Â. = variation allowance
variation.
inv a = involute a ( = tan a - a)
2.41 machining tolerance, T : The permissible variation in
Ke [KEI = approximation factor for external spline
actual space width or actual tooth thickness.
Ki [KI] = approximation factor for interna1 spline
= spline length
2.42 total tolerance, (T + A) : The machining tolerance
plus the variation allowance. The total tolerance on an interna1
= active spline length
spline is the difference between the minimum effective space
= length of engagement
width and the maximum actual space width; on an external
T = machining tolerance
spline, it is the difference between the maximum effective tooth
thickness and the minimum actual tooth thickness.
IMRe [MRE] = measurement over two pins, external spline
h/lRi [MRI] = measurement between two pins, interna1 spline
2.43 length of engagement, g, : The axial length of con-
W = measurement over k teeth, external spline
tact between mating splines.
2 = number of teeth
2.44 active spline length, g, : The maximum axial spline m = module
length in contact (when working) with the mating spline. On
Pb = base pitch
sliding splines, the active length exceeds the length of engage-
p = circular pitch
ment.
@Fe = fillet radius of the basic rack, external spline
2.6 basic dimension : A numerical value to describe the
@Fi = fillet radius of the basic rack, interna1 spline
theoretically exact size, shape or location of a feature. It is the
E = basic space width, circular
basis from which permissible variations are established by
E = actual maximum space width, circular
max
tolerances.
= actual minimum space width, circular
min
2.46 auxiliary dimension : A dimension, without tolerance,
E,, EV1 = effective space width, circular
given for information purposes only, for the determination of
S= basic tooth thickness, circular
the useful production and control dimensions.
S = actual maximum tooth thickness, circular
max
S = actual minimum tooth thickness, circular
min
3 Symbols
= effective tooth thickness, circular
s, WI
a = pressure angle
3.1 General symbols
= standard pressure angle
The general symbols used to designate the various spline terms
a,i = pressure angle at pin contact diameter, interna) spline
and dimensions are given below (sec figures 10, 11, 12, 13, 14
a = pressure angle at pin contact diameter, external
and 15).
spline
NOTE - In electronic data processing (EDP), it is not always possible
ai = pressure angle at pin centre, interna1 spline
to present symbols in their theoretically correct form because of limita-
a, = pressure angle at pin centre, external spline
tions of connected printing equipment. For this reason, some alter-
native symbols for EDP usage are given in brackets below (for exam-
= pressure angle at form diameter, external spline
aFe
ple, the symbol Db for base diameter may be printed as DB).
aFi = pressure angle at form diameter, interna1 spline
c, = effective clearance (looseness or interference)
k-j,--h-f- e and d = fundamental deviation on the
cF = form clearance
external spline = c, min
D = pitch diameter H = fundamental deviation on the interna1 spline
---------------------- Page: 9 ----------------------
SIST ISO 4156:2000
ISO 41!56-1981 (El

Tables 1 and 2 give the basic dimensions and fundamental formulae, a graphical presentation of which is given by figure 2.

Table 1 - Theoretical dimensions for spllines
Dimensions in millimetres
Basic space width or tooth thickness Base pitch
Circular
Tooth Module at pitch diameter E or S
pitch
-- -- --1
o!D 30” * m
&D 30” QD 37,5O o!D 45” &D 30” a!D 37,5O &D 45O
10 31,416 15,708 15,708 27,207 0 24,923 9
25,133 12,566 12,566 21,765 6 19,939 2
18,850 9,425 9,425 16,324 2 14,954 4
5 15,708 7,854 7,854 13,603 5 12,462 0
4 12,566 6,283 6,283 10,882 8 9,969 6
3 9,425 4,712 4,712 8,162 1 7,477 2
5,553 6
7,854 3,927 3,927 3,927 6,801 7 6,231 0
2,5
2 6,283 3,142 3,142 3,142 5,441 4 4,984 8 4,442 9
1,75 5,498 2,749 2,749 2,749 4,761 2 4,361 7 3,887 5
4,712 2,356 2,356 2,356 4,081 0 3,738 6 3,332 2
1,5
1,25 3,927 1,963 1,963 1,963 3,400 9 3,115 5 2,776 8
3,142 1,571 1,571 1,571 2,720 7 2,492 4 2,221 4
0,75 2,356
1,178 1 ,178 1 ,178 2,040 5 1,869 3 1,666 1
1,571 0,785 0,785 0,785 1,360 4 1,246 2 1,110 7
Or5
0,25 0,785 0,393 0,555 4

* For illustration purposes : relative tooth sizes for various spline modules for pressure angle aiD = 30”.

---------------------- Page: 10 ----------------------
SIST ISO 4156:2000
ISO 4156-‘l981 (E)
Table 2 - ForanuBae for dimensions and tolerances for all fit classes
Formul a
Pitch diameter D mZ
Base diameter m z Cos O”D
Circular pitch Km
Base pitch x m Cos aiD
Effective Upper deviation, external
resulting from deviation allowance (fundamental)
k-j,--h-f-eandd
Minimum diameter, interna1
30°, flat root D
m(Z+ 1,5)
ei min
30”, fi I let root D
m(Z+ 1,8)
ei min
37,5 O, fillet root D m(Z+ 1,4)
ei min
45’, fillet root D
m(Z+ 1,2)
ei min
Maximum major diameter, interna1 D D
t (T t h)/tan aD (sec note 1 )
ei max ei min
Minimum form diameter, interna1
30°, flat root and fillet root m(zt 1) + 2cF
DFi min
37,5O, fi I let root m(z t o,g) t 2 cF
DFi min
45” , fi I let root dz + 0,8) + 2 CF
DFi min
Minimum diameter, interna1
DFe max f 2 CF (Set? Ilote 2)
Dii min
Maximum minor diameter, interna1
m < 0,75
+ tol. H 10
Dii max Dii min
0,75 < m < 2
+ tol. H 11
Dii max Dii min
m>2
+ toi. H 12
Dii max Dii min
Basic space width and E and
0,5x m
Minimum effective space width E
v min
Maximum actual space width
class 4 E E v min + (T + A) (sec note 3)
max
class 5 E E v min + (T + Al (sec note 3)
max
class 6 E E
v min + (T + h) (sec note 3)
max
class 7 E E
v min + (T + A) (sec note 3)
max
Minimum actual space width E E
v min + h (sec 8.2)
min
Maximum effective space width E E
max - h (see 8.2)
v max
Maximum major diameter, external
30°, flat root and fillet root D
m(z t 1) t esv/tan o!D (sec note 4)
ee max
37,5’, fillet root D
m(z t o,g) t es,/tan aD (sec note 4)
ee max
45”) fillet root
D m(Z + 0,8) + es,ltan QD (sec note 4)
ee max
Minimum major diameter, external
m < 0,75 D - tol. h 10
ee min ee max
0,75 < m < 2 D .
ee min
ma2 D
ee min
22)
Maximum form diameter, external
DFe max
Maximum minor diameter, external
(Sec note 6)
30”, flat root
m (z - 1 ,5) t es&n aD
Die max
30”, fillet root
m(Z- 1,8) + es,/tan &D
Die max
37,5’, fillet root
m (z - 1 $1 t esv/tan Q!D
Die max
45”, fil let root
m (z - 1 ,2) f es&an o!D
Die max
---------------------- Page: 11 ----------------------
SIST ISO 4156:2000
ISO 4156-1981 (E)
Table 2 - Formulae for dimensions and tolerances for all fit classes (conchded)
Basic tooth thickness
Maximum effective tooth thickness
Minimum actual tooth thickness
class 4 v max - (T + A) (sec note 3)
class 5
v max - (T f A) (sec note 3)
class 6 v max - (T + h) (sec note 3)
class 7 v max - (T + A) (sec note 3)
Maximum actual tooth thickness
Minimum effective tooth thickness
Total tolerance, space width or tooth thickness
Form clearance
Pin diameter, interna1 spline -
Pin diameter, external spline
Measurement between pins
Measurement over pins
Change factor, interna1 see section three, “1 nspection”
Change factor, external see section three, “Inspection”
NOTES
1 (T + h) for class 7 - see clause 6.
2 For all classes of fit, always take the &e max
value corresponding to the H/h fit.
3 See clause 6 and section two - tables of dimensions.
4 Take a null Upper deviation value for j, and k fundamental deviations.

5 See section two : “Dimensions”, and section three : “Inspection”, concerning &oice of pins.

6 For h, see figures 3,4,5 and 6.
---------------------- Page: 12 ----------------------
SIST ISO 4156:2000
ISO 4156-1981 (E)
Piece
E effective
actual
max
(actual space width)
min -
vmax -
l- .
basic
v min
-0,5 n m-
(effective space width) z
basic :
a\1 max
(effective tooth thickness)
vmin -
max -
min
(actual tooth thickness)
Figure 2 - Graphical representation of tables 1 and 2
---------------------- Page: 13 ----------------------
SIST ISO 4156:2000
ISO 4156-1981 (El
3.2 Subscripts classes as follows :

The following subscripts (see also the note in 3.1) are used as Spline fit class Effective interfeaence

part of the above general symbols to designate relative con-
H/k cVmax = (T + A)
ditions or locations :
T+A
deviation allowance j, = -
MS CV max =
= minor or interna1 (in this last case, in the last position)
ilIl
c 2 )
= major or external (in this last case, in the last position)
CE1 Effective looseness
,[Bl = at base
H/h Cv min = deviation allowance h = zero
= at contact point
H/f CV min = deviation allowance f
= pertaining to form diameter
&FI
deviation allowance e
H/e Cv min =
= effective
“[VI
H/d Cv min = deviation allowance d
= active
The deviation allowances (fundamental) k - j, - h - f - e
pertaining to gauges
,Ml =
and d are the standard deviations selected from ISO/R 286 /SO
- of engagement System of limits and fits - Part 7 : General, tolerances and
deviations, which are applied to the external spline. A pre-
n = standard
scribed maximum effective interference or minimum effective
looseness is obtained, allowing the fitting by adjusting from the
zero line the maximum effective and minimum actual limit
values of tooth thickness by the amount of the deviation
4 Pressure angle (standard)
allowance (see 8.72). The spline dimensions in the spline
tables of this International Standard are given for spline fit class
Standard pressure angles included in this International Stan-
H/h, C, min = zero.
dard for involute splines are 30°, 37,5’ and 45’.
6 Space width and tooth thickness,
total tolerance (T + A)
5 Type of fit
Tolerance classes
This International Standard deals with only one type of fit, the
This International Standard includes four classes of total
side fit, for 30°, 37,5O and 45O pressure angle splines. Formulae
tolerance ( T + A) on space width and tooth thickness selected

for the dimensions and tolerances for these splines are shown from a combination of tolerance units (i) in ISO/R 286. The

in table 2.
tolerance classes are indicated below, with corresponding com-
bination of tolerance units (i). For the calculation of T and Â,
sec clause 8. The values of Â. are given in tables 3 to 6 of
5.1 Side fit
clause 8.
In this fit, the mating members contact on the sides of the teeth
Tolerance unit
Spline tolerance
only. Major and minor diameters are clearance dimensions. The
class (il
tooth sides act as drivers and centralize mating splines.
4 = (10 i* + 40 i”“)
5 = (16 i* + 64 i”“)
5.2 Spline fit classes
6 = (25 i* + 100 i”“)

This International Standard provides the side fit in six spline fit 7 = (40 i* + 160 i”“)

Tolerance based on pitch diameter
i = 0,45 $D + 0,001 D for-D Q 5OOmm
i = 0,004D + 2,l forD > 5OOmm
Tolerance based on space width or tooth thickness
i =
0,45 $?m + 0,001 E (or SI
where
D is the pitch diameter in millimetres;
E is the basic space width in millimetres;
S is the basic tooth thickness in millimetres.

the resultant ( T + A) in micrometres. For ( T + Â) in millimetres, multiply the result by 0,001.

For the calculation of tolerance units (i), only the above indicated formulae (notes *, **) are to be taken into consideration.

---------------------- Page: 14 ----------------------
SIST ISO 4156:2000
ISO 41564981 (El
volute spline of infinitely large diameter on a plane at right
It should to be noted that total values ( T + A) may always be

subtracted from the limit values of space width and tooth angles to the tooth surfaces, the profile of which is used as the

basis for defining the standard tooth dimensions of a system of
thickness given in section 2 and are usable even if the chosen
involute splines.
fit class is other than H/h.
NOTE - Below are listed the combinations of tolerance qualities IT
corresponding to the combinations of tolerance units (i) indicated
above. Those combinations of tolerance qualities IT are given only to
7.2 The reference line is a straight line crossing the profile of
indicate the conception principle of the tolerance system regarding its

eventual extension, and only to rediscover the corresponding com- the basic rack, with reference to which the tooth dimensions

binations of tolerance units (il.
are specif ied.
Spline tolerance Tolerance quality
class W-I
4 = (IT 6* + IT 9”“)
7.3 The profile of the basic rack for the standard pressure
angle splines is represented in the following figures :
5 = (IT 7* + IT 10”“)
6 = (IT 8* + IT II**)
figure 3 : 30° flat root spline, for modules 0,5 to 10;
7 = (IT 9* + IT 12”“)
figure 4 : 30° fillet root spline, for modules 0,5 to 10;
7 Basic rack profiles for spline
figure 5 : 37,5O fillet root spline, for modules 0,5 to 10;

7.1 The basic rack is a section of the tooth surface of an in- figure 6 : 45’ fillet root spline, for modules 0,25 to 2,5.

* Tolerance based on pitch diameter.
Tolerance based on space width and tooth thickness.
---------------------- Page: 15 ----------------------
SIST ISO 4156:2000
60 41564981 (El
0,5nm A 0,5n:m
=0,2m* /
I -Pl=i
I nternal spline
External spline \
i Reference line
Figure 3 - Profile of the basic rack for 30° flat root spline
External spline
Reference line
Figure 4 - Profile of the basic rack for 30’ fillet root spline
* and ** See next page.
---------------------- Page: 16 ----------------------
SIST ISO 4156:2000
ISO 4156-1981 0
= 0,3m”
PFi
Reference line
Figure 5 - Profile of the basic rack for 37,5’ fillet root spline
Interna1 spline
Figure 6 - Profile of the basic rack for 45O fillet root spline
NOTE concerning figures 3 to 6 :

For interna1 splines (hub), the form diameter, obtained by generating from the basic rack, is always greater than the form diameter shown in the

tables of dimensions (sec section two), which corresponds in all fit cases to the major maximum diameter of the shaft (with Upper deviation - es, -

zero) increased to diametral form clearance (2 c,-) equal to 0,2 module.

** For external splines (shafts), cF is obtained by generation from the basic rack @Fe maX

1 and for H/h fit (sec note 2 under table 2).

The size cF indicated is in fact a deviation permitting the form clearance cF given in table 2 to be obtained and is equal to 0,l m.

---------------------- Page: 17 ----------------------
SIST ISO 4156:2000
ISO 4156-1981 (E)

7.4 The minimum radial form clearance, CF, shown in the 8 Machining tolerances and variations

basic rack profiles is equal to 0,l m for all pressure angle
8.1 Variation allowance (il)
splines.
The variation allowance, being the addition of the total index
The form clearance is modified for fits other than H/h.
variation, total profile variation and total lead variation, has an
effect on the effective fit of an involute spline. The effect of
these individual spline variations on the fit is less than their
total, because areas of more than minimum clearance cari have
form, lead, or index errors without changing the fit. It is also
7.5 External splines may be produced by generating with a
unlikely that these errors would occur in their maximum
pinion-type shaper cutter or with a hob, or by cutting with no
amounts simultaneously on the same spline. For this reason,
generating motion using a tool formed to the contour of a tooth
total index variation, profile variation and total lead variation are
space. External splines are also made by cold f
...

Norme internationale

INTERNATIONAL ORGANIZATION FOR STANDARDIZATlON.ME~YHAPOjjHAR OPTAHM3Ai&lfl IlO CTAH~APTIl3A~WWORGANlSATlON INTERNATIONALE DE NORMALISATION

Cannelures cylindriques droites 8 flancs en
Module metrique, 9 centrage sur flancs -
dheloppante -
Gh&aMs, dimensions et verification

Straight cylindrical involute spines - Metric module, side fit - Generalities, dimensions and inspection

Premihe edition - 1981-05-01
CDU 621324.44 Ref. no : ISO4156-1981 (F)

Descripteurs : cannelure, dimension, tolkrance de dimension, limit& dbfinition, symbole, formule, conception

Prix base sur 138 pages
---------------------- Page: 1 ----------------------
Avant-propos

L ’ISO (Organisation internationale de normalisation) est une federation mondiale

d ’organismes nationaux de normalisation (comites membres de I ’ISO). L ’elaboration

des Normes internationales est confide aux comites techniques de I ’ISO. Chaque

comite membre interesse par une etude a le droit de faire partie du comite technique

correspondant. Les organisations internationales, gouvernementales et non gouverne-

mentales, en liaison avec I ’ISO, participent egalement aux travaux.

Les projets de Normes internationales adopt& par les comites techniques sont soumis

aux comites membres pour approbation, avant leur acceptation comme Normes inter-
nationales par le Conseil de I ’ISO.

La Norme internationale IS0 4156 a ete elaboree par le comite technique ISO/TC 32,

Cannelures et dentelures, et a ete soumise aux comites membres en octobre 1978.
Les comites membres des pays suivants l ’ont approuvee :
Afrique du Sud, Rep. d’ Espagne Royaume-Uni
Australie Finlande Suede
Autriche France Turquie
Belgique lnde USA
Chili ltalie Yougoslavie
Co&e, Rep. de Japon

Les comites membres des pays suivants I ’ont desapprouvee pour des raisons techni-

ques :
Allemagne, R. F.
Tchecoslovaquie
@ Organisation internationale de normalisation, 1981 0
imprime en Suisse
---------------------- Page: 2 ----------------------
Sommaire
Page
Section un : Gh5raMs

1 Objet et domaine d ’application ........................................................

................................................ 1
2 Termes et definitions des cannelures

3 Symboles ..................................................................................

4 Angle de pression (normali&) .........................................................

.........................................................................
5 Type de montage

6 Tolkrance totale sur I ’intervalle et I ’epaisseur (T + ;1) ............................

7 Profil de la cr~maillhre de reference pour les cannelures .........................

8 Tolhances d ’usinage et &arts ........................................................

9 Effet des erreurs de forme des cannelures ..........................................

Dimensions effectives et dimensions reelles des cannelures ....................
11 Usage des dimensions effectives et des dimensions rbelles d ’intervalle et

................................................................................ 28

d ’kpaisseur

12 Indications sur les dessins ..............................................................

Section deux : Dimensions

13 Introduction ...............................................................................

14 Dimensions et tolhrances ...............................................................

Tableaux
Angle de pression 30°

- fond plat ................................................................................ 38-65

66-93

- plein rayon .............................................................................

94-121

Angle de pression 37,5O plein rayon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122-139

Angle de pression 45’ plein rayon
Section trois : Vhificationl)
1) En preparation, fera I ’objet d ’un additif.
. . .
III
---------------------- Page: 3 ----------------------
NORME INTERNATIONALE IS0 4X6-1981 (F)
Cannelures cylindriques droites 5 flancs en
dhveloppante - Module mbrique, S centrage sur flancs -
Gh&aIit6s, dimensions et vhification
Section un : G&&alit&

1 Objet et domaine d ’application 2.6 cannelure ti plein rayon : Cannelure ayant un profil de

dent ou d ’entredent dont les flancs opposes en developpante

La presente Norme internationale fournit les donnees et indica- sont raccordes au cercle de pied (de diametre D,i ou Die) par

tions necessaires a la conception, a la fabrication et a la verifica- une seule surface de raccordement.

tion des cannelures cylindriques droites (non helicoi ’dales) a
flancs en developpante, 8 centrage sur flancs. Elle fixe des speci-
2.7 cannelure G fond plat : Cannelure ayant un profil de
fications, basees sur le module dans la gamme 0,25 8 10 inclus,
dent ou d ’entredent dont chacun des flancs anti-homologues
concernant les angles nominaux de pression de 30°, 37,5O et
est raccorde au cercle de pied (de diametre D,i ou Die) par une
45O. (Pour des raisons pratiques d ’exploitation par ordinateur,
surface de raccordement particuliere.
le terme ((37,5O)) a ete adopt6 a la place de 37O 30 ’.)
module, m : Quotient du pas circulaire, exprime en milli-
Les dimensions limites, tolerances, erreurs de fabrication et 2.8
metres, par le nombre n: IOU quotient du diametre primitif,
leurs effets sur I ’ajustement entre les elements d ’accouplement
exprime en millimetres, par le nombre de dents).
cannel& coaxiaux sont definis et present& dans des tableaux.
Les dimensions Iineaires sont exprimees en millimetres et les
dimensions angulaires en degres.
2.9 cercle primitif : cercle 8 partir duquel sont etablies tou-
tes les dimensions courantes des cannelures et au niveau
duquel I ’angle de pression specific a sa valeur nominale.
2 Termes et definitions des cannelures
2.10 diam8tre primitif, D : Diametre du cercle primitif en

2. ‘l assemblage par cannelures : Accouplement d ’ele- millimetres egal au nombre de dents multiplie par le module.

ments coaxiaux transmettant un couple par engagement simul-
tane de dents egalement espacees sur le pourtour d ’un element
2.11 point primitif : Intersection d ’un profil de dent de can-
externe cylindrique dans les entredents correspondants espa-
nelure avec le cercle primitif.
ces de facon identique sur la surface interne de I ’element cylin-
drique creux associe.
2.12 pas primitif, p : Longueur d ’art du cercle primitif entre
deux points primitifs de deux flancs homologues consecutifs,
2.2 cannelure en dkveloppante : Un des elements
qui a comme valeur nominale le nombre 7t multiplie par le
d ’accouplement par cannelures dont les dents ou les intervalles
module.
ont des flancs a profil en developpante de cercle.
2.13 angle de pression, a : Angle aigu forme par une ligne
2.3 cannelure interne : Cannelure formee sur la surface
radiale passant par un point quelconque d ’un flanc de dent, et
interne d ’un cylindre.
le plan tangent au flanc en ce point.
2.4 cannelure externe : Cannelure formee sur la surface
2.14 angle de pression normalis6, an : Angle de pression
externe d ’un cylindre.
au point primitif specific.
2.5 surface de raccordement (flanc de raccord) : Sur-
2.15 cercle de base : Cercle fictif a partir duquel est genere
face concave de la dent ou de I ’entredent au cercle de pied.
le profil de la cannelure en developpante de cercle.
Cette surface incurvee varie suivant la facon dont elle est gene-
ree et ne peut etre correctement specifiee par aucun rayon de
valeur donnee. 2.16 diam&re de base, Db : Diametre du cercle de base.
---------------------- Page: 4 ----------------------
IS0 4156-1981 (F)

2.17 pas de base, Pb : Longueur d ’art du cercle de base s ’ajusterait sans jeu ni serrage dans une cannelure exterieure

sur toute la longueur axiale de I ’assemblage cannel& Les diffe-
entre deux flancs homologues consecutifs.
rents ajustements s ’obtiennent en faisant varier cette valeur S,.
2.18 cercIe majeur : Cercle forme par la surface la plus a

I ’exterieur d ’une cannelure. C ’est le cercle exterieur (cercle de 2.30 jeu ou serrage effectif, c, : Difference entre I ’inter-

valle effectif d ’une cannelure interne et I ’epaisseur effective de
tete des dents) d ’une cannelure externe ou le cercle de pied
la cannelure externe conjuguee.
d ’une cannelure interne.

2.19 diametre majeur, Dee, D,i : Diametre du cercle 2.31 jeu ou serrage theorique, c : Difference entre I ’inter-

valle reel d ’une cannelure interne et I ’epaisseur reelle de la can-
majeur.
nelure externe conjuguee; elle ne definit pas I ’ajustement entre
les deux elements en raison de l ’influence des defauts.
2.20 cercle mineur : Cercle forme par la surface la plus a
I ’interieur d ’une cannelure. C ’est le cercle de pied d ’une canne-

lure externe ou le cercle interieur (cercle de tete des dents) 2.32 securite de forme, cF : Hauteur radiale du profil de

d ’une cannelure interne. developpante au-dela de la hauteur de contact avec la piece

conjuguee. Elle autorise un defaut d ’excentricite du cercle
mineur (interieur), du cercle majeur (exterieur) et de leurs cer-
2.21 diametre mineur, Die, Dii : Diametre du cercle mineur.
cles primitifs respectifs.
2.22 cercle de forme : Cercle qui definit les points les plus
2.33 erreur totale de division : Somme des valeurs abso-
bas du controle de la forme de la developpante du profil des
lues des deux plus grands &arts reels (ou pratiquement mesu-
dents. Ce cercle et le cercle de t&e des dents (ou le cercle de
res) des flancs homologues de dents, de signe oppose, par rap-
debut du chanfrein) definissent les limites du profil des dents
port a I ’ecartement theorique.
exigeant un controle. II se situe a proximite et au-dessous du
cercle majeur pour les cannelures internes et a proximite et au-
2.34 erreur totale de profil : Somme des valeurs absolues
dessus du cercle mineur pour les cannelures externes.
des deux plus grands &arts de signe oppose, par rapport au
profil theorique des dents, mesures suivant la normale aux
2.23 diametre de forme, II,,, D,i : Diametre du cercle de
flancs.
forme.
2.35 erreur totale de pas helicoidal : Somme des valeurs
2.24 hauteur de contact : Distance radiale entre le cercle
absolues des deux plus grands &arts de direction des flancs,
mineur d ’une cannelure interne et le cercle majeur d ’une canne-
de signe oppose, par rapport a la direction theorique (parallele a
lure externe diminuee du degagement d ’angle et/au de la hau-
I ’axe de reference). Cette erreur inclue egalement les erreurs de
teur de chanfrein.
parallelisme et de distorsion (voir figure 1).
NOTE - Les cannelures droites (non h6licoi ’dales) ont un pas hblicoi ’dal
2.25 intervalle ou epaisseur circulaire theorique au pri-
infini.
mitif, E ou S : Pour des cannelures a angles de pression de
30°, 37,5O ou 45O, egal 9 la moitie du pas primitif de reference.
Axe des dents
Axe de
intervalle reel : lntervalle circulaire pratiquement
2.26 r6f6rence
mesure, circulaire au cercle primitif, d ’un entredent quelconque
compris dans les valeurs limites Emax. et Emin .
a) Erreur de pas h6lico ’idaI
2.27 intervalle effectif, Ev : Pour une cannelure interne,
egal a I ’epaisseur circulaire de dent au cercle primitif d ’une can-
Axe des dents
Axe de
nelure externe imaginaire parfaite, qui s ’ajusterait sans jeu ni
r6f6rence
serrage dans une cannelure interne sur toute la longueur axiale
de I ’assemblage cannel6 L ’intervalle effectif minimal (E, min.1
toujours egal a El de la cannelure interne est toujours I ’element
de base comme le montre le tableau 1. Des variations d ’ajuste-
b) Erreur de paralklisme
ment ne peuvent etre obtenues qu ’en jouant sur I ’epaisseur de
,-Axe effectif
dent de la cannelure externe.
de la cannelure
Axe de
r6fhrence
2.28 epaisseur reelle :
ipaisseur circulaire, pratiquement
mesuree, circulaire au cercle primitif, d ’une dent quelconque
comprise dans les valeurs limites S,,,. et Smin .
c) Erreur de distorsion
2.29 Bpaisseur effective, S, : Pour la dent d ’une cannelure
externe, egale a I ’intervalle circulaire au cercle primitif de

I ’entredent d ’une cannelure interne imaginaire parfaite qui Figure 1 - Erreurs de pas helicoidal

---------------------- Page: 5 ----------------------
IS0 4156-1981 (F)

2.36 erreur de paralklisme : Defaut de parallelisme d ’une d,, = diametre de contact des piges de controle, cannelure

externe
dent de cannelure par rapport a une autre (voir figure 1).
&, [DFE] = diametre de forme, cannelure externe
2.37 erreur de distorsion : Ecart de I ’axe effectif de la can-
DFi [DFI] = diametre de forme, cannelure interne
nelure par rapport a son axe de reference (voir figure 1).
Dii [DII] = diametre mineur, cannelure interne
Dee [DEE] = diametre majeur, cannelure externe
2.38 faux-rond : Ecart de la cannelure par rapport a une
Die [DIE] = diametre mineur, cannelure externe
forme circulaire exacte.
D,i [DEI] = diametre majeur, cannelure interne
D,, [DRE] = diametre de la pige de controle pour cannelure
2.39 erreur effective :
Effet cumule des defauts de la can-
externe
nelure sur son montage avec la piece qui lui est conjuguee.
DRi [DRI] = diametre de la pige de controle pour cannelure
interne
2.40 erreur globale de forme, A : Erreur effective admissi-
il = erreur globale de forme
ble de forme.
inv a = involute a ( = tan a - a)
Ke [KE] = facteur d ’approximation pour cannelure externe
2.41 tolkance d ’usinage, T : Erreur admissible sur I ’inter-
valle ou I ’epaisseur reels.
Ki [KI] = facteur d ’approximation pour cannelure interne
g = longueur cannelee
2.42 tokance totale, (T + A) : Somme de la tolerance
longueur cannelee utile
&J =
d ’usinage et de l ’erreur globale de forme. La tolerance totale sur
g, = longueur en prise
une cannelure interne est la difference entre I ’intervalle effectif

minimal et I ’intervalle reel maximal; sur une cannelure externe, T = tolerance d ’usinage

c ’est la difference entre I ’epaisseur effective maximale et
MRe [MRE] = mesure sur deux piges de controle, cannelure
I ’epaisseur reelle minimale.
externe
MRi [MRI] = mesure entre deux piges de controle, cannelure
2.43 longueur en prise, g, : Longueur axiale de contact
interne
entre cannelures conjuguees.
W= mesure sur ire dents, cannelure externe
2 = nombre de dents
2.44 longueur cannel&e utile, g, : Longueur axiale maxi-
m = module
male de cannelure en contact, lors du fonctionnement, avec la
cannelure conjuguee. Pour les cannelures glissantes, la lon- pb = pas de base
gueur utile est superieure a la longueur en prise.
p = pas primitif
= rayon de raccordement de la cremaillere de reference,
@Fe
2.6 dimension thborique : Valeur numerique theorique
cannelure externe
definissant les dimensions, la forme ou I ’emplacement exacts
@Fi = rayon de raccordement de la cremaillere de reference,
d ’un element. C ’est a partir de cette valeur que sont etablis les
cannelure interne
&arts admissibles sous forme de tolerances.
E = intervalle circulaire theorique
E = intervalle circulaire reel maximal
2.46 dimension auxiliaire : Dimension sans indication de max
tolerance, utilisee 8 titre d ’information uniquement, en vue de E
intervalle circulaire reel minimal
min =
determiner des dimensions utiles a la fabrication ou au con-
E, [EVI = intervalle circulaire effectif
trole.
= epaisseur circulaire theorique
S = epaisseur circulaire reelle maximale
max
3 Symboles
S = epaisseur circulaire reelle minimale
min
s, WI = epaisseur circulaire effective
3.1 Symboles ghnhaux
a = angle de pression
Les symboles generaux, utilises pour designer les divers ele-
aD = angle de pression normalise
ments et tolerances des cannelures, sont donnes ci-dessous
aci = angle de pression aux points de contact des piges de
(voir figures 10, 11, 12, 13, 14 et 15) :
controle, cannelure interne
NOTE - Dans le traitement electronique de I ’information, il n ’est pas
= angle de pression aux points de contact des piges de
toujours possible de presenter les symboles sous leur forme theorique-
controle, cannelure externe
ment correcte, a cause des limitations du materiel imprimant utilise.
ai = angle de pression au diametre passant par les centres des
Pour cette raison, certains symboles sont donnes ci-dessous, en
piges de controle, cannelure interne
variante, entre crochets, pour utilisation dans le traitement electroni-
(2, = angle de pression au diametre passant par lescentresdes
que de I ’information (par exemple, le symbole Db pour le diametre de
piges de controle, cannelure externe
base peut etre imprime sous la forme DB).
angle de pression au diametre de forme, cannelure
aFe =
= jeu effecif ou serrage effectif
externe
securite de forme
CF =
= angle de pression au diametre de forme, cannelure
aFi
D= diametre primitif
interne
Db [DB] = diametre de base
k-j,- h - f - e et d = &art fondamental sur cannelure
externe = c, min
dci = diametre de contact des piges de controle, cannelure
interne H = &art fondamental sur cannelure interne
---------------------- Page: 6 ----------------------
IS0 4156-1981 (F)

Les tableaux 1 et 2 donnent les dimensions theoriques et formules de base, dont une repksentation graphique est donnee 6 la figure 2.

Tableau 1 - Dimensions theoriques des cannelures
Dimensions en millim6tres
lntervalle ou bpaisseur thborique Pas de base
Pas
Module au primitif E ou S
Denture cyo 30” * prim itif
&D30° aD 37,5O aD45" aD30O "0 37,5' &D45O
d-3 ,I 31,416 15,708 15,708 - 27,207 0 24,923 9 -
25,133 12,566 12,566 - 21,765 6 19,939 2 -
: 8
18,850 16,324 2 14,9544 -
9,425 9,425 -
L 6
12,462 0 -
15,708 7,854 7,854 13,603 5
12,566 6,283 6,283 - 10,882 8 9,969 6 -
r 4
\ 3 9,425 4,712 4,712 - 7,477 2 -
8,162 1
1 2 ‘5 7,854 3,927 3,927 3,927 6,801 7 6,231 0 5,553 6
6,283 3,142 3,142 5,441 4 4,442 9
3,142 4,984 8
P--L 2
I,75 5,498 2,749 2,749 2,749 4,761 2 4,361 7 3,887 5
4,712 3,332 2
1 I,5 2,356 2,356 2,356 4,081 0 3,738 6
1,25 3,927
1,963 1,963 1,963 3,400 9 3,115 5 2,7768
1 3,142 1,571 1,571 1,571 2,720 7 2,492 4 2,221 4
0 ‘75 2,356 1,178 1,178
1,178 2,0405 1,869 3 I,6661
1,571
0,785 0,785 0,785 I,3604 1,246 2 I,1107
m Ot5
- - -
0,25 0,785 - 0,393 0,555 4

* k titre d ’illustration : grandeurs relatives des divers modules de cannelures pour angle de pression &D = 30".

---------------------- Page: 7 ----------------------
IS0 4156-1981 (F)

Tableau 2 - Formules de calcul des dimensions et de leurs tolhrances pour toutes classes d ’ajustement

Formule
Terme Symbole
D mZ
Diametre primitif
m z cos “0
Diametre de base
Pas prim itif P
n m cos a!D
Pas de base
resultant des &arts (fondamentaux) k -j, - h -f -
icart superieur effectif, externe
es,
e et d
Diametre majeur minimal, interne
30 ‘, fond plat m(Z+ I,51
ei min
30 ”, plein rayon m(Z+ 1,8)
ei min
37,5 ’, plein rayon D m(Z+ 1,4)
ei min
D m(Z + 1,2)
45 ”, plein rayon
ei min
D + (T + &)/tan a!D (voir note 1)
Diametre majeur maximal, interne D
ei max ei min
Diametre de forme minimal, interne
m(z + 1) + 2 cF
30 ”, fond plat et plein rayon
DFi min
37,5 ’, plein rayon m(z + 0,9) f 2 cF
DFi min
45 ”, plein rayon m(z + 0,8) + 2 CF
DFi min
Diametre mineur minimal, interne DF, max + 2 CF (voir note 2)
Dii min
Diametre mineur maximal, interne
m < 0,75 + tol. H 10
Dii max Dii min
0,75 < m < 2 + tol. H 11
Dii max Dii min
m>2 + tot. H 12
Dii min
Dii max
Intervalle theorique et E et
0,5n m
Intervalle effectif minimal E
v min I
I ntervalle reel maximal
v min + (T+ h),voir note 3
classe 4 E E
max
v min + (T + A), voir note 3
classe 5 E E
max
classe 6 E
E v min + (T + A), voir note 3
max
v min + (T + A), voir note 3
classe 7 E E
max
Intervalle reel minimal E v min + A (voir 8.2)
min
Intervalle effectif maximal E E max - h (voir 8.2)
v max
Diametre majeur maximal, externe
30 ”, fond plat et plein rayon D m(z + 1 ) + es&an aD (voir note 4)
ee max
37,5 ’, plein rayon D m(z -k 0,9) + es&m @D (voir note 4)
ee max
45 ”, plein rayon D m(Z + 0,8) + es&an &D (voir note 4)
ee max
Diametre majeur minimal, externe
D Deemax-ttol. h 10
m < 0,75
ee min
0 ‘75 < 177 < 2 D D tot. h 11
ee min ee max -
m>2 D D tol. h 12
ee min ee max -
0,5 es, 2
h,- -
tan a!D
Diametre de forme maximal, externe Db)* +
DFe max
sin aD
(Voir note 6)
Diametre mineur maximal, externe
30 ”, fond plat
m(z- 1 ,5) + es&an ct!D
Die max
30 ”, plein rayon m (Z - I,81 + es&an QD
Die max
37,5O, plein rayon m(z- 1 ,4) + es,/tan aD
Die max
45 ”, plein rayon m(z- 1,2) -k es&an aD
Die max
---------------------- Page: 8 ----------------------
IS0 4156-1981 (F)
Tableau 2 -

Formules de calcul des dimensions et de leurs tolhrances pour toutes classes d ’ajustement (fin)

Terme Symbole Formule
Diametre mineur minimal, externe - (T + h)/tan ci!D (voir note 1 )
Die max
Die min
Epaisseur theorique S 0,5n: m
ipaisseur effective maximale S + es,
v max
ipaisseur reelle, minimale
classe 4 S S v max - (T + A) (voir note 3)
min
classe 5 S S v max - (T + A) (voir note 3)
min
classe 6 S S v max - (T + A) (voir note 3)
min
classe 7 S S v max - (T + h) (voir note 3)
min
ipaisseur reelle maximale S S --A
max v max
Epaisseur effective minimale S S
v min min + h
Tolerance totale sur intervalle ou sur epaisseur (T+ A) (voir chapitre 6)
Securitk de forme
0,l m
Diametre de pige, cannelure interne Voir note 5
DRi
Diametre de pige, cannelure externe Voir note 5
DRC?
Mesure entre piges Voir note 5
MRi
Mesure sur piges Voir note 5
w=k
Voir section trois, W&-if ication ))
Facteur de correction, interne
Facteur de correction, externe Voir section trois, Werification ))
Kc?
NOTES
1 (T + A) pour classe 7, voir chapitre 6.
2 Quelle que soit la classe d ’ajustement, prendre toujours la valeur DFe max
correspondant a I ’ajustement H/h.
3 Voir chapitre 6, et section deux : Dimensions.

4 Prendre une valeur d ”ecart superieur nulle pour les &arts fondamentaux j, et k.

5 Voir section deux : Dimensions, et section trois :
~~Verificatiorw , concernant le choix des piges.
6 Pour h,, voir figures 3,4, 5 et 6.
---------------------- Page: 9 ----------------------
IS0 4156-1981 (F)
Pike
r 1
E reel effectif
max
(intervalle Gel)
min -
vmax -
E thgorique
v min
-0,5 n m-
(intervalle effectif) .
S thgorique
v max
(Gpaisseur effective)
v min
max
C ”f
l-u
J!-
min
(bpaisseur reel le)
Figure 2 - ReprQsentation graphique des tableaux 1 et 2
---------------------- Page: 10 ----------------------
IS0 41564981 (F)
3.2 Indices tement de centrage sur flancs des cannelures :
Les indices suivants (voir egalement la note en 3.1) sont utilises
Classe
Serrage effectif
comme partie des symboles generaux ci-dessus pour designer d ’ajustement
des conditions ou des positions relatives.
H/k cVmax = (T + A)
i[l] = mineur ou interne (dans ce dernier cas, en derniere
= &art fondamental j, =
position) MS % max
J E] = majeur ou externe (dans ce dernier cas, en derniere
Jeu effectif
position 1
H/h Cv min = &art fondamental h = zero
,[B] = de base
H/f Cv min = &art fondamental f
= diametre aux poins de contact
H/e Cv min = &art fondamental e
,[F] = concernant le diametre de forme
H/d Cv min = &art fondamental d
= effectif
“WI
= utile
w Les &arts fondamentaux k - js - h - f - e et d sont les
&arts-types, figurant dans l ’ISO/R 286 Systkme /SO de to/h-
,[R] = relatif aux calibres de controle
rances et d ’ajustemen ts - Partie 7 : G&&alit&, tokrances et
= en prise
&carts, qui s ’appliquent a la cannelure externe. On obtient un
serrage effectif maximal ou un jeu effectif minimal prescrit, per-
D = normalise
mettant le montage, en decalant a partir de la ligne zero les
valeurs limites effective maximale et reelle minimale de I ’epais-
4 Angle de pression (normalisb)
seur (voir 8.7.2). Les dimensions de cannelures don&es dans
les tableaux de la presente Norme internationale correspondent
Les angles de pression normalises, consider& dans la presente
a la classe d ’ajustement H/h, c, min = zero.
Norme internationale pour les cannelures en developpante,
sont 30°, 37,5O et 45 ’.
6 Tolhrance totale sur I ’intervalle et
I ’bpaisseur (T + A)
5 Type de montage
Classes de tolhances
La presente Norme internationale ne traite que d ’un seul type

de montage, le ((centrage sur flancs)) pour les cannelures a La presente Norme internationale considere quatre classes de

angles de pression de 30°, 37,5’ et 45 ’. Les formules de calcul tolerances totales ( T + A) sur I ’intervalle et sur I ’epaisseur choi-

sies parmi une combinaison d ’unites de tolerances (9 de
des dimensions et des tolerances de ces cannelures sont don-
nees dans le tableau 2. l ’ISO/R 286. Les classes de tolerances sont indiquees
ci-dessous, avec les combinaisons correspondantes d ’unites de
tolerances (i). Pour le calcul de T et de A, se reporter au
5.1 Centrage sur flancs
chapitre 8. Les valeurs de A sont donnees dans les tableaux 3
a 6 du chapitre 8.
Avec ce type de montage, les elements conjugues ne sont en
contact que sur les flancs des dents. Les dimensions caracteris-
Classe de tolhanck Unit& de tokance
tiques du jeu sont les diametres majeur et mineur. Les flancs
d ’une cannelure (i)
des dents servent d ’entralnement et centrent les cannelures
4 = (10 i* + 40 i ”“)
conjuguees.
5 = (16 i* + 64 i**)
5.2 Classes d ’ajustement des cannelures
6 = (25 i* + 100 i ”“)

La presente Norme internationale consider-e six classes d ’ajus- 7 = (40 i* + 160 i ”“)

* Tolerance basee sur le diametre primitif
i = 0,45 $E+ 0,001 D pour D Q 500 mm
i = 0,004 D + 2,l pour D > 500 mm
** Tolerance basee sur I ’intervalle ou I ’epaisseur theorique
i = 0,45 $m + 0,001 E (OU S)
Oti
D est le diametre primitif, en millimetres;
E est I ’intervalle theorique, en millimetres;
S est I ’epaisseur theorique, en millimetres.

La resultante ( T + A) est en micrometres; pour ( T + A) en millimetres, multiplier le resultat par 0,001.

Pour le calcul des unites de tolerances ii), seules les formules indiquees ci-dessus (notes *, **) sont 5 prendre en consideration

---------------------- Page: 11 ----------------------
IS0 4156-1981 (F)
II y a lieu de noter que les valeurs totales ( T + A) peuvent tou-
des dents d ’une cannelure en dkveloppante, de diamhtre infini

jours &re dbduites des valeurs limites d ’epaisseur et d ’inter- dans un plan perpendiculaire aux surfaces des dents, dont le

valle don&es dans la section 2 et sont utilisables mQme si la profil sert de base de definition des dimensions normalisees des

classe d ’ajustement choisie est diffhrente de la classe
dents d ’un ensemble cannel6 en d&eloppante.
H/h.
On trouvera ci-dessous les combinaisons de qualites de tole-
NOTE -
rances IT correspondant aux combinaisons d ’unites de tolerances (i)
7.2 La Iigne de rbfhence est une droite coupant le profil de la
indiquees ci-dessus. Ces combinaisons de qualites de tolerances IT ne
cr~maillhe de refhrence et par rapport 2 laquelle sont spkifiGes
sont don&es que pour indiquer le principe de conception du systeme
les dimensions des dents.
de tolerances en vue de son extension eventuelle et uniquement pour
retrouver les combinaisons correspondantes d ’unites de tolerances
ii).
7.3 Le profil de la cremaill&e de ref&-ence des cannelures 5
Classe de tolkrance Qualit& de tolbrance
angle de pression normalis est represent6 aux figures suivan-
d ’une cannelure
(IT)
tes :
4 = (IT 6* + IT 9 ”“)
figure 3 : Cannelure 5 fond plat, angle de pression 30°, pour
5 = (IT 7* + IT IO ”“)
modules 0,5 6 10;
6 = (IT 8* + IT 11 ”“)
7 = (IT 9* + IT 12 ”“)
figure 4 : Cannelures 3 plein rayon, angle de pression 30°, pour
modules 0,5 5 10;
figure 5 : Cannelure 3 plein rayon, angle de pression 37,5 ’, pour
7 Profil de la cr6maiIlhe de rbfbrence pour
modules 0,5 & IO;
les cannelures
figure 6 : Cannelures 2 plein rayon, angle de pression 45O, pour
7.1 La cr~maillhre de ref&ence est une section de la surface
modules 0,25 5 2,5.
* Tolerance basee sur le diametre primitif.
** Tolerance basee sur I ’intervalle ou I ’epaisseur theorique.
---------------------- Page: 12 ----------------------
IS0 4156-1981 IF)
/ \
FE ’
Ef I 0’
1 I t&/////k \ Pannnll fil
vcIIIII~lyre externe
d \
1 . . I ! \ ‘A# ;A
/ \\Y
lirnmy
i,,,, de rbfhence W

Figure 3 - Profil de la cr6maiII&re de refbrence pour cannelures 6 fond plat - 30°

0,5 n m
CY-
PFi = 0,4m*
& / / -
k------r /
Ai / /I I ‘/AI
’ Y/ ‘/I t
I//
y--f
-2iZ4, Cannelure interne
$I!W(,,, gg$K$& 51
-~ --
co,
o), 0
: ~~~~~~~~n~~~~
r-, :,-I “$$I!\
= 0,4 m
ipFe

Figure 4 - Profil de la cr6maillhre de rkfbrence pour cannelures ZI plein rayon - 30’

+ et ** Voir page suivante.
---------------------- Page: 13 ----------------------
IS0 4156-1981 (F)
Cannelure interne
‘\ .
\ \ \ \ ‘\
<-- k
\ ‘\ ’
= 0,3 m
-PFe
’ I

Figure 5 - Profil de la crQmaill&e de kfhrence pour cannelures 6 plein rayon - 37,5O

Cannelu re interne
igne de rdfbrence
Figure 6 -
Profil de la crtSmaill&re de rhftkence pour cannelures 8 plein rayon - 45O
NOTE concernant les figures 3 a 6 :

* Pour les cannelures internes (moyeu), le diametre de forme, obtenu par generation a partir de la cremaillere de reference, est toujours plus grand

que le diametre de forme present4 dans les tables de dimensions (voir section deux), qui correspond, dans t

...

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