1.8 Rack shift of the gear
Undercut

When Number of teeth is below minimum as shownin Fig. 23, part of dedendum is no longer an Involutecurve but will look like a shape scooped out by cutter
tool.
Refer to drawing, when Involute curve shows thescooped out shape condition from Base circle (Toothtip side), it is called Undercut.
Gear with undercut has low strength of Dedendum
and provides bad influence to gear contact due toshortened Involute curve.
Calculation formula for minimum number of teeth (z)to prevent undercut is as follows,2z=sin 2α 0(α 0: Cutter pressure angle)

Basecircle

Rootcircle

Condition of Undercut generally appears when Number of teeth is 17 or less and pressure angle of gearis 20°. According to DIN standard, minimum Numberof teeth is 14 accepting slight Undercut which may
cause no serious influence.

(The Trochoid curve line on the right hand side is thecentre locus of roundness of cutter of rack tool. ( radius of roundness γ f =0.375m=7.5)

Profile Shifted Gear

It is possible to adjusting gear by item 2) to control
helix angle of Helical gear. However it is necessary toprovide thrust bearing in axis direction to countermeasure force (thrust force) occurring in Helical gear.
When design multi engagement between axes withdifferent gear ratio, items 2) is also useful (for example, speed reducer).
Generally, Positive profile shift (+) is the method ofgear fabrication where Reference pitch line of Rack
type cutter shifts x-times of module toward outerradius direction from Reference pitch. The Negative
profile shift (-) is that Reference pitch line of Rack
type cutter shifts x-times of module towards inner radius direction from Reference pitch. x.m is commonly
called the Amount of rack shift where x is called Rack
shift coefficient. (Please refer to Fig. 24).

(1) The Summary of Profile Shifted Gear
Using a rack tool (for example, hob cutter) to fabricate Profile Shifted Gear is to achieve the following
purposes.
1) Prevent condition of Undercut for gear with lessthan minimum Number of teeth.
2) When there is deviation or failure for centre distance, fabricate a modified gear to correct the fault
centre distance.
(3) Adjust distribution of Tooth thickness for gear pairto achieve equal gear strength.
4) Adjust to suitable contact ratio to lessen gear noiselevel and/or trapping of pump gear.
5) Take into consideration the wear of flank to adjustSpecific sliding. (Another theory states that Specificsliding and wear are not proportional.)

28

FixedFig. 23 Undercut

P
2

P=Pmtxm

AoGb

The Reference linefor Rack type cutter

GGa0
Fig. 24 Rack shift for Spur gear

Spur gear with full depth tooth(Rack shift coefficient x=0)

Positive (+) profile shifted gear
(Rack shift coefficient x=0.5)

Fig. 25 Profile shifted gear (Examples of Positive and Negative profile shifted gear, Number of teeth is 12z)

Note: (1) Adopted the old Standard term.

29

Limitation of Pointed tooth tip

When increase the positive amount of Rack shift, areaof top land is gets narrower and soon, Tooth profile
becomes sharp.
A sharp pointed Tooth profile has insufficient tooth
depth, thus Tooth tip of Mating gear may interferewith Root of tooth causing proper assembly andsmooth gear rotation to be impossible. Therefore
Rack shift of Top land exceeding zero is not advisable.
To calculate Top land ‘s’ of Spur gear by the following
formula,π1s = m( z + 2 + 2 x) • ­®( + 2 xtanα 0) • − (inv α a − invα 0)½¾z¯ 2¿
For easy reference, please refer to Table 14 for area offormed gear with Pressure angle 20°.

Fig. 26 Pointed tooth tip

Calculation for Rack shift coefficient.

y=(1) Rack shift coefficient to prevent Undercut.
Undercut is sure to occur when Number of teeth is 17
or below with Pressure angle 20°. Prevent Undercut
using theoretical Rack shift coefficient by following
calculation formula.
x=17 − z17(z: Practical number of teeth)

Practical number of teeth 14 z is available to use for
DIN standard, calculation formula of DIN is defined asfollows.
x=

14 − z17x=

17 − 10= 0.412
17(Please check for occurrence of sharp pointed tooth
top tip using Table 14.)
Practical rack shift coefficient is obtained by following
calculation.
x=

14 − 10= 0.235
17(2) Rack shift coefficient to adjust Centre distance
Below is the explanation using examples.
For example, calculate Rack shift coefficient for adjustable gear with Centre distance of 80.5mm (Properdistance is 80.0mm) with: Gear: Spur gear,Pressure angle:20°,
Module: 2.0mm,
Number of teeth for Pinion: 20z,
Number of teeth for Gear: 60z,
Centre distance modification coefficienty = (a' −a ) / m				30= (80.5 − 80) / 2= 0.25cos α w =cosα 0cos20°=2• y2 • 0.25 + 1+120 + 60z1 + z 2			 = 0.933856		 α w = 20.955894°
inv α w = tan α w − α w						= tan 20.955894° − 20.955894° • π / 180= 0.0172317

x1 + x 2 ·inv α w = 2 • tanα 0 • §¨¸ + invα 0 therefore© z1 + z 2 ¹(z: Practical number of teeth)

Theoretical Rack shift coefficient for Spur gear with
Number of teeth 10z with Pressure angle 20° is by following formula·z1 + z 2 § cosα 0− 1¸¨
2 © cos α w ¹ therefore

Sum of Rack shift coefficient§ inv α w − invα 0 ·x1 + x 2 = ¨¸ • ( z1 + z 2 )© 2 • tanα 0 ¹		

=0.0172317 - 0.0149044= 0.2557
2 • tan 20°

a’ : Actual centre distance (mm)a : Proper centre distance (mm)z1 : Number of teeth for Pinion
z2 : Number of teeth for Gear
α 0 : Pressure angle of Cutter (°)α w :pressure angle (°)
y : Centre distance increment coefficientx1 : Rack shift coefficient for Pinion
x2 : Rack shift coefficient for Gear
invα 0 : Functional involute for Cutter pressure angle

invα 0 = tanα 0 − α 0
inv 20° = 0.0149044(The lastα 0 is in Radian Unit)
You may provide the sum (0.2557) of this Rack shiftcoefficient to Pinion only or can divide between Gear
and Pinion.

Gear.
As for Rack shift to negative side, it is necessary tocheck for Undercut. If Undercut should occur, offset
the Negative rack shift coefficient to mating gear.
Refer to Table 14 to shown the area of formed gearwith pressure angle 20°.

(3) Guidelines for determining Rack shift coefficient.
Rack shift to positive side is mainly designed for Pinion. It is necessary to check that the calculated Rack
shift coefficient does not cause pointed tooth tip.
If design causes pointed tooth tip, reduce amount ofRack shift coefficient to Pinion and offset amount to2.5
A2.0A0 = 20°

Rack shift coeﬃcient x

Aʻ

1.5

Pointed tooth tip

1.0Top land 0.2m
Top land 0.4mLimitation of Theoreticaldedendum undercut
0.500ʻ0213456 7

10

Area of formed gear

2030

4050

70

No. of Teeth z(ZU)

- 0.5

B

Curved line A : Top land changed to 0.2・m by Rack shift coefficient and Number of teeth.
Curved line A’ : Top land changed to 0.4・m by Rack shift coefficient and Number of teeth.
Curved line B : R ack shift coefficient and Number of teeth for Limitation of Theoretical dedendum
undercut.
* For Helical gear, use horizontal axis in chart for Virtual number of teeth of spur gear Zv.

Zυ = Z / cos3 β
Table 14. Area of formed gear (pressure angle 20°)

Note (1) Adopted old gear terms.

31

The features of Tooth profile 05

Tooth profile of KG STOCK GEARS (Number of teethfrom 8z to 11z) has been adopted by type 05 in DIN
standard.
Tooth profile type 05 has its Rack shift coefficient
fixed to plus (+) 0.5. Adjust Addendum by shortening coefficient x module (κ .m) to fabricate smaller
Outside diameter, as the Bottom clearance have atendency to be narrow.
The calculation of Rack shift for Number of teeth
ranging from 8z to 11z for KG STOCK GEARS is as follows,Calculation formula for Working pressure angle α w is
as follows:
x1 + x 2 ·inv α w = 2 tanα §¨¸ + invα© z1 + z 2 ¹

Herebyz1=No. of teeth for Pinion
z2= No. of teeth for Gear
z1=Rack shift coefficient for Pinion
x2=Rack shift coefficient for Gear
α 0= Pressure angle (Cutter pressure angle)
inv= Involute function
invα =tan α -α(Refer to page 164-167 for the Involute function table)
Centre distance modification coefficient y is as follows:

y=·z1 + z 2 § cosα− 1¸¨2 © cos α w ¹

Centre distance ax is following formula:
z1 + z 2+ y ·¸ ma x = §¨
© 2¹

Herebym=module
Working pitch diameter d’1 and d’2 is by following formula:z1 ·d' 1 = 2 a x §¨¸© z1 + z 2 ¹z2 ·d' 2 = 2 a x §¨¸© z1 + z 2 ¹

32

Reference diameter d1 and d2 is by following formula:d 1 = z 1md 1 = z 2mTip (Outside) diameter dax is following formula:z +3− κ ·¸dax = 2m§¨
© 2¹Herebyκ =Truncation coefficientª§·ºκm = « x1 + x 2 − z1 + z 2 ¨ cosα − 1¸» m
2 © cos α w ¹¼¬
When Addendum of cutter is module 1.25, Bottom
clearance (minimum amount) is module 0.21.
The Centre distance for number of teeth 8z and 8z is
as follows,(Rack shift coefficient x=0.5)
a x / m = 8.7788mm

The centre distance for number of teeth 10z and 10z
is as follows.
(Rack shift coefficient x=0.5)
a x / m = 10.8043mm

The above calculations are for module 1.0. Example
for module is 2.0 with number of teeth 8z and 8z are
engaged, centre distance ax based on above ax /m =8.778mm is as follows:a x = 8.7788 × 2		 = 17.5576mm

Mating gear with other Number of teeth of KG STOCKGEARS is available.
With regards to the tooth profile of type 05 for theRack Shift Coefficient quoted by Gear Industry Volume No.54, “German Gear Standard” (DIN 3994 and3995)

The Centre distance between KG Rack shifted spur gear and KG STOCK GEARS
Usage of below comparison table: Where module is 1.0, calculate the centre distance ax multiply by module.
No. of teeth
No. of teeth
No. of teeth
8

No. of teeth

8

9

10

11

8

9

10

11

8.779

9.286

9.792

10.29845

26.97027.47127.97128.47246

27.47127.97128.47228.972

No. of teeth

99.286

9.792

10.29910.80410

9.792

10.29910.80411.31048

28.47228.97229.47329.97329.47329.97330.47330.97430.47330.97431.47431.97411

10.29910.80411.31011.81550

1210.43710.93911.441

11.94352

13

10.93911.441

11.94312.44554

31.47431.97432.47532.97514

11.44111.95312.44512.94655

31.97432.47532.97533.47515

11.94312.44512.94613.44856

32.47532.97533.47533.97616

12.44512.94613.44813.94958

33.47533.97634.47634.97617

12.94613.44813.94914.45160

34.47634.97635.47735.97718

13.44813.94914.45114.9526235.47735.97736.47736.97719

13.94914.45114.95215.4536436.477

36.97737.47837.97865

36.97737.47837.97838.47820

14.45114.95215.45315.95421

14.95215.45315.95416.45566

37.47837.97838.47838.97938.47838.97939.47939.97939.47939.97940.87940.9792215.45315.95416.45516.95668

23

15.95416.45516.95617.45770

24

16.45516.95617.45717.95872

40.47940.97941.48041.98025

16.95617.45717.95818.45975

41.98042.48042.98043.48026

17.45717.95818.45918.96080

44.48144.98145.48145.98127

17.95818.45918.96019.46184

49.48246.98247.48247.98228

18.45918.96019.46119.96285

46.98247.48247.98248.48229

18.96019.46119.962

20.46390

49.48349.98350.48350.98330

19.46119.962

20.46320.96395

51.98352.48352.98453.48496

52.48352.98453.48453.98432

20.46320.96321.46421.96534

21.46421.96522.46522.966100

54.48454.98455.48455.98510556.98557.48557.98558.48535

21.96522.46522.96623.46736

22.46522.96623.46723.967108

58.48558.98559.48559.98559.48559.98560.48560.98660.48560.98661.48661.98638

23.46723.96724.46824.96811040

24.46824.96825.46925.96911242

25.46925.96926.47026.970115

61.98662.48662.98663.48644

26.47026.97027.47127.971120

64.48664.98765.48765.987

Centre distance between KG-Rack shifted spur gear
and KG-Racka = h" +ah“m× z+ xm
2Herebya : Centre Distance (Distance from Datum of Rack
to Centre of KG-Spur gear)h’’ : Datum line of Rack (Refer to page 259)m : Module
x : Rack shift coefficientz : Number of teeth		
Module 1.0 and above		
For Number of teeth 8 to 11, x =0.5		
For Number of teeth 12 and above, x=033