ISO Involute Spline

Straight Cylindrical Involute Splines

Important note:
The information on this page is not detailed and has been obtained by reference to the relevant BS and Machinerys handbook.    Detail design should be completed using the relevant standards or quality reference sources.

Introduction

The notes and tables below relate to straight cylindrical involute splines in accordance with BS ISO 4156:2005-1 Straight cylindrical involute splines. Metric module, side fit.
It is emphasised that the splines identified are side fit with the centering based on the sides of the splines

Involute splines are the predominant form bacause they are stronger than straight sided splines and are easier to cut and the fit.  The external spline can be formed either by hobbing or by a gear shaper.    Internal splines are formed by broaching or by a gear shaper.    To control tolerances , the minimum efffective space width and the minimum major diameter of the internal spline are held to basic dimensions .    The external spline is varied to obtain the desired fit.

The very simplest method of initially selecting of involute spline based on a shaft dia is to arrive at an initial Pitch circle dia (D) and a module (m).     eg. a spline for a 50mm OD spline with say a module of 2mm would have a PCD (D) of 50 - 2 = 48mm. The number of teeth = D/m and as an whole number of teeth is required a value of D = 48mm is easily selected with number of teeth z =24 teeth.    The detailed manufacturing requirements are obtained using the various notes and tables in the relevant standard / detailed technical publications.    The notes and tables below provide outline information.

The fit of the spline ( interference , close, loose etc ) is primariy specified based on the ISO system of limits and fits as detailed in BS EN 20286-1/2. Refer to webpage ISO limits..

Shafts connections based on involute splines are suitable for transfering of high, cyclical and shock torsional moments.    Involute splines are used for fixed and for sliding connections of shafts with hubs..

The splined profile is shaped as involute toothing in the cross section, with nominal pressure angles of the profile 30°, 37.5° or 45°.

It is centered to the outer diameter or sides of the teeth. Centering to the diameter is more accurate. Centering to sides is more economical and is used much more frequently in practice.(see figure below )

The groove bottom can be flat or rounded.

BS ISO 4156 is based on the following modules.

For pressure angles of 30° and 37,5° the modules increments are   

0,5;  0,75;  1;   1,25;  1,5;  1,75;   2;    2,5;   3;  4; 5;  6; 8;   10

For pressure angle of 45° the module increments are    

0,25;  0,5;   0,75;  1;  1,25;   1,5;  1,75;  2;  2,5

Male Involute Spline

Advantages of the involute splines compared with keyways:

Lower pressures than couplings with keys, higher loading capacity of the coupling
Lower wear of sliding couplings
Suitable also for cyclical torsional moments
Easy assembly and disassembly of the coupling

Advantages of involute compared to straight splines

Higher number of teeth resulting in lower pressures and higher loading capacity of the coupling
More uniform distribution of forces along the perimeter
Option of fine adjustment of the hub on the shaft)
Stronger shaft the shaft, lower notch coefficient
Economical lot production using a hobbing method
High accuracy of production similarly as with accurate gears

Disadvantages of the Involute Splines

More complicated to engineer
Higher production costs than couplings with keys
Higher notch coefficient than couplings with keys
Difficult execution of alignment and perpendicularity of the coupling
non-parallelism of sides of the teeth causes additional radial forces in the coupling; these forces then try to open the hub

Standards

BS ISO 4156:2005_1: Straight cylindrical involute splines. Metric module, side fit. Generalities
BS ISO 4156:2005_2: Straight cylindrical involute splines. Metric module, side fit. Dimensions
BS ISO 4156:2005_3: Straight cylindrical involute splines. Metric module, side fit. Inspection

Designation of involute splines

An example set of designations for a mating spline pair with 32 teeth, 2,5 module, with a 30°pressure angle, a fillet root and a class 5 fit is...

Mating: INT/EXT 32z ° 2,5m ° 30R ° 5H/5f ISO 4156
Internal spline: INT 32z ° 2,5m ° 30R ° 5H ISO 4156
External spline: EXT 32z ° 2,5m ° 30R ° 5f ISO 4156

Pressure Angle	α	degree (o)
Pressure Angle At Pitch Diameter	αD	degree (o)
Number of teeth	z	-
Module	m	D / z
Pitch (Circular)	p	m.π
Basic Tooth thickness Circular	S	p/2
Basic space width Circular	E	p/2
Effective space width Circular (minimum)	EV min	p/2
Pitch diameter	D	m.z
Base diameter	Db	m° z °cos α D
Base pitch	pb	m° πα D
Fundamental deviation, external	es v	Tooth mod'n factor -Results from fit (h,f,e,d) see table below
Minimum major diameter, internal 30°, flat root 30°, fillet root 37,5°, fillet root 45°, fillet root	Dei min	m° (z + 1,5) m° (z + 1,8) m.° (z + 1,4) m.° (z + 1,2)
Maximum major diameter, internal	Dei max	D ei min + (T + λ ) / tan α D
Minimum form diameter, internal 30°, flat root &fillet 37,5°, fillet root 45°, fillet root	DFi min	m° (z +1)+2.cF m° (z +0,9)+2.cF m° (z +0,8)+2.cF)
Minimum minor diameter, internal	Di i min	DFe max + 2° cF
Maximum minor diameter, internal: m =< 0,75 0,75 < m < 2 m > 2	Di i max	Dii min +  IT 10 Dii min +  IT 11 Dii min +  IT 12
Maximum actual Space width class 4 class 5 class 6 class 7	Emax	EV min + (T + λ) EV min + (T + λ) EV min + (T + λ) EV min + (T + λ))
Minimum actual Space Width	Emin	Ev.min + λ
Maximum effective Space Width	EV max	Ev.min + TV
Maximum major dia external. 30°, flat root & fillet 37,5°, fillet root 45°, fillet root	Dee max	m° (z +1)+ esv  /  (tan αD ) m° (z + 0,9)+ esv  /  (tan αD ) m° (z + 0,8)+ esv   /  (tan αD )
Minimum major diameter, external: m =< 0,75 0,75 < m < 2 m = > 2	Dee min	Dee max - IT 10 Dee max - IT 11 Dee max - IT 12
Maximum form diameter	DFe max
Maximum minor diameter, external 30°, flat root 30°, fillet root 37,5°, fillet root 45°, fillet root	Die max	m° (z - 1,5)+ esv  /  (tan αD) m° (z -1,8)+ esv /  (tan αD) m.° (z - 1,4) + esv   /  (tan αD) m.° (z -1,2 ) + esv   /  (tan αD)
Minimum minor, external	Die min	D ie max - (T + λ )  / tan α D
Maximum effective tooth thickness	S V max	S + esV
Minimum actual tooth thickness class 4 class 5 class 6 class 7	Smin	SV max - (T + λ) SV max - (T + λ) SV max - (T + λ) SV max - (T + λ)
Maximum actual tooth thicknes	S max	Sv.max - λ
Minimum effective tooth thickness	S V min	Sv.max - TV
Total tolerance, Space width or tooth thickness	T + λ	See table below
Maximum effective clearance	C V max	Ev max - S v min
Minimum effective clearance	C V min	Ev min - S v max
Form tooth height	h s	See sketch and table for basic rack profile below

Note: The Form circle is the circle used to define the depth of involute profile control.
In the case of an external spline it is located near and above the minor diameter, and on an internal spline near and below the major diameter

Basic Rack Shape

Parameter	Pressure Angle
	30o		37,5o	45o
	Flat root	Fillet root
Major Space height	0,75.m	0,9.m	0,7.m	0,6. m
Major Tooth height	0,5.m	0,5.m	0,45.m	0,4. m
Form tooth height (hs	0,6.m	0,6.m	0,55.m	0,5. m
Minor tooth height	0,75.m	0,9.m	0,7.m	0,6. m
Root radius (ρFi )	0,2.m	0,4.m	0,3.m	0,25. m
Root radius (ρFe )	0,2.m	0,4.m	0,3.m	0,25. m
Form radius (cF)	0,1.m	0,1.m	0,1.m	0,1. m

Table showing tooth thickness modification (es_v   ) of external splines relative to basic tooth thickness

Important Note: These values are all reductions and so they are negative (-ve) values.

Pitch Dia. D (mm)	Selected fit class
	d	e	f	h
	Thickness reduction (esv) (mm)
= < 3	0,020	0,14	0,006	0
3 to 6(inc)	0,030	0,020	0,010	0
6 to 10(inc)	0,04	0,025	0,013	0
10 to 18(inc)	0,05	0,032	0,016	0
18 to 30(inc)	0,065	0,040	0,020	0
30 to 50(inc)	0,080	0,050	0,025	0
50 to 80(inc)	0,10	0,060	0,030	0
80 to 120(inc)	0,12	0,072	0,036	0

Table showing total Tolerance (T + λ) for Space width and Tooth thickness

The Total Tolerance (T + λ) is the sum of the Machining Tolerance (T) and the Effective variation (λ)

Spline Tolerance class(T + λ)	Formula for (T + λ)	formula for i1 & i2
4	10i1 + 40i2	i 1 = 0,001[0,453.Sqrt (D) +0,001D ] i 2 = 0,001[0,453.Sqrt (S +0,001 S ] Note:    i 1 applies for D < 500mm
5	16i1 + 64i2
6	25i1 + 100i2
7	40i1 + 160i2

The effective variation is the combined effect of the total index variation (F_p ), the positive profile variation (f_f ) and the tooth alignment variation ( F _β ) . These combined effect i for convenience calculated using the following equation

Table showing equations for (F_p ), ( f_f ) and ( F _β )

Spline Tolerance class	Fp	ff	F β
4	0,001.[ 2,5 Sqrt(M.z.β /2 ) + 6,3]	0,001 [1,6.m (1+0,012z) + 10]	0,01 [0,8.sqrt(g) + 4]
5	0,001.[ 3,55 Sqrt(M.z.β /2 ) + 9]	0,001 [2,5.m (1+0,012z) + 16]	0,01 [1,0.sqrt(g) + 5]
6	0,001.[ 5 Sqrt(M.z.β /2 ) + 12,5]	0,001 [4.m (1+0,012z) + 25]	0,01 [1,25.sqrt(g) + 6,3]
6	0,001.[ 7,1 Sqrt(M.z.β /2 ) + 18]	0,001 [6,3.m (1+0,012z) + 40]	0,01 [2,0.sqrt(g) + 10]

Table showing values for es_v   / (tan α_D ) for different Pressure angles and Fites

Pitch Dia. D (mm)	α = 30	α = 37,5	α = 45	α = 30	α = 37,5	α = 45	α = 30	α = 37,5	α = 45	α = All
	fit class = d			fit class = e			fit class = f			fit class = h
	esv    / (tan αD ) - (mm)
=< 3	0,035	0,026	0,020	0,024	0,018	0,014	0,010	0,008	0,006	0
3 to 6 inc.	0,035	0,039	0,030	0,035	0,026	0,020	0,017	0,013	0,010	0
6 to 10 inc.	0,052	0,052	0,040	0,043	0,033	0,025	0,023	0,017	0,013	0
10 to 18 inc.	0,069	0,065	0,050	0,055	0,042	0,032	0,028	0,021	0,016	0
18 to 30 inc.	0,087	0,085	0,065	0,069	0,052	0,040	0,035	0,026	0,020	0
30 to 50 inc.	0,113	0,104	0,080	0,087	0,065	0,050	0,043	0,033	0,025	0
50 to 80 inc.	0,139	0,130	0,100	0,104	0,078	0,060	0,052	0,039	0,030	0
80 to 120 inc.	0,173	0,156	0,120	0,125	0,094	0,072	0,062	0,047	0,036	0
180 to 250 inc.	0,208	0189	0,145	0,147	0,111	0,085	0,074	0,056	0,043	0
250 to 315 inc.	0,251	0,222	0,170	0,173	0,130	0,100	0,087	0,065	0,050	0
315 to 400 inc.	0,294	0,248	0,190	0,191	0,143	0,110	0,097	0,073	0,056	0