4.2 Method of measurement with Over balls or Rollers
For spur gear, putting Over balls or Rollers to Spacewidth. External gear is measured by outside dimension of Overball or Rollers. Internal gear is measured by inner dimension of Over balls or Rollers.
Use method of Over balls or Rollers for Helical gear. Measurement for Internal gear, this method has advantages overothers.
Odd number of teethdm

Even number of teethdmExternalgear
dp

dm

dm

dp

Internalgear

dp

dp*Note for gear with small and odd number of teeth
and large helix angle. We recommend using methodof Over balls or Rollers due to unstable pins causing
inaccurate measurement

Fig. 2 Method of Over balls or Rollers

Diameter for Over balls or Rollers

Refer to Fig. 3 to find diameter of Over balls or Rollers. Obtain the nearest available diameter from the graph as standard for Over balls or Rollers to measure the gear.

A = 20°
1.0

Example (1)When conditions of
m=2,α=20º, z=17 and x=+0.8
calculation is by following formula,Example (2)When conditions of
mn=4,αn=20º, z=19, β=26º42’ and xn=+0.4calculation is by following formula,

3.0

0.8

0.6
2.5
0.4

Rack shift coeﬃcient x

Diameter of Over balls or Rollers dpmm

3.5

0.2

2.0

0

1.5

- 0.2

- 0.4
00.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

Number of teeth z

Fig. 3 Graph to find suitable diameter of Over balls or Rollers (for module mn=1.0)
Increase proportionately by module used. Number of teeth of Helical gear is Virtual number of teeth of Spur gear.

63

Dimensions of Over balls or Rollers forSpur gear.

For even number of teeth, calculation is by following
formuladm=zmcosα+ dp
cosφ

For odd number of teeth, calculation is by following
formuladm=zmcosα90°cos+ dpcosφz

For invφ , calculation is by following formulainvφ =π − invα · + 2 xtanα
dp− §¨¸zmcosα © 2 zz
¹Herebydm : Over balls or Rollers dimension(mm)z : Number of teeth
x : Rack shift coefficient
φ : Pressure angle (°) at pin centre
dp : Diameter of Practical Over balls or Rollers (mm)m : Module (mm)α : Reference pressure angle(°)

Example 1, even number of teeth
When module m=2.0, Number of teeth z=30, Reference pressure angle α =20° and Rack shift coefficient
x= 0.15. Calculations of Over balls or Rollers dimensions is as follows,① Over balls or Rollers dimension dp
Refer to Fig. 3, d p=1.73 multiply by module 2.0 =3.46(mm)Use nearest available pin size dp=3.5(mm) instead of3.46(mm)② Pressure angle φ at contact point between flankand Over balls or Rollers
invφ =				π − inv 20° ·3.5− §¨¸30 • 2 • cos20° © 2 • 30¹+2 • 0.15 • tan 20°30			 = 0.0282613 (inv20° = 0.0149044)		
φ = 24.5388° (See page 164 to 167 for Involute function charts)③ Over balls or Rollers dimension dm
dm =30 • 2 • cos20°+ 3.5 = 65.48 (mm)cos 24.5388°

Example 2, odd number of teeth
Follow example 1 for calculation method, Number ofteeth is changed to 29 (Other data remains the same)

dpA)

F (z+2xFzcosAdm① Over balls or Rollers dimension dp dp=3.5(mm)② Pressure angle (°) at pin centre
invφ =				FinvFFig. 4 Dimension of Over balls or Rollers for Spur gear

+2 • 0.15 • tan 20°
29			 = 0.0287218		
φ = 24.6645° (24°39' 52" )③ Over balls or Rollers dimension dm
dm =64π − inv 20° ·3.5− §¨¸29 • 2 • cos20° © 2 • 29
¹90°
29 • 2 • cos20°• cos+ 3.5 = 63.39 (mm)29
cos24.6645°

Dimension of Over balls or Rollers for Internal gear

Calculation for even number of teeth is by following
formuladm=zmcosα− dpcosφCalculation for odd number of teeth is by following
formuladm=zmcosα90°cos− dpcosφz

For invφ , calculation is by following formulaπ + invα · + 2 xtanα − dpinvφ = §¨¸zzmcosα© 2z¹Example 1, even number of teeth
When module m=1.0, Number of teeth z=80, Reference pressure angleα ＝ 20° and Rack shift coefficient
x= 0.12. Calculations for dimensions of Over balls orRollers is as follows,① Over balls or Rollers diameter dp
Refer to Fig. 3, dp=1.68 multiply by module 1.0 = 1.68(mm)Use nearest available pin size dp=1.70(mm) instead of1.68(mm)② Pressure angle (°) at pin centre

π + inv 20° · −1.7invφ = §¨¸© 2 • 80
¹ 80 • 1 • cos20°						dpinvFPz

2 • 0.12 • tan 20°80= 0.0130174

φ = 19.145° (19°8' 42" )③ Over balls or Rollers dimension dmdm =Base circle

dbdm

Fx2+80 • 1 • cos 20°− 1.7 = 77.88 (mm)cos19.145°

Example 2, odd number of teeth
Number of teeth for calculation example 1 is changed
to 81 (other data remains the same).
① Over balls or Rollers dimension dp dp=1.7(mm)② Pressure angle (°) at pin centre

Fig. 5 Over balls or Rollers dimension forInternal gearπ + inv 20° · −1.7invφ = §¨¸© 2 • 81
¹ 80 • 1 • cos 20°							

+2 • 0.12 • tan 20°81= 0.0130407

φ = 19.156° (19°9' 22" )③ Over balls or Rollers dimension dmdm =90°81 • 1 • cos20°• cos− 1.7 = 78.86 (mm)81
cos19.156°

65

Over balls or Rollers for Straight tooth rackd m = h" +dp §1 · πm
¨1 +¸−2 © sinα ¹ 4 tanĮHerebyh’’ : Datum line (mm) is from Rack base to Reference line (1).
Helical rack is the same as straight rack at normal section. The above formula can be used. For calculationof Pressure angleα and module m, use α n and mn at
normal section.

Example,When module m=3.0, Reference pressure angle α =20°
and Datum line h”=32. Calculations of Over balls or Rollers dimensions is as follows,① Over balls or Rollers diameter dp
Refer to Fig. 3, dp=1.68 multiply by module 3.0 = 5.04(mm)Use nearest available pin size dp=5.0(mm) instead of5.04(mm)② Over balls or Rollers dimension dmd m = 32 +		π •35 §1 ·• ¨1 +¸−
2 © sin 20° ¹ 4 tan 20°= 35.34 (mm)Pm
12 Pmh"dm

Pm
4tanAdp

Fig. 6 Over balls or Rollers dimension for Straight rack

Note: (1) Adopted the old Standard term.

66

Over balls or Rollers dimension for Helicalgear③ Transverse module mtmt =Calculation for even number of teeth is by following
formula④ Transverse pressure angle α tz mt cos α t+ dp
cosφ

dm=Calculation for odd number of teeth is by following
formuladm=z mt cos α t90°cos+ dp
cosφz		

⑤ Pressure angle (°) at pin centre

π − inv α t · + 2 xn tan α ndp− §¨¸z mn cos α n © 2 zz
¹Herebymn : Normal module (mm)α n : Normal pressure angle(°)
xn : Normal rack shift coefficient
mt : Transverse module
α t : Transverse pressure angle(°)

			

3.5
π − inv 20.646896° ·− §¨¸36 • 2 • cos 20° © 2 • 36¹+2 • 0.05 • tan 20°
36⑥ Over balls or Rollers dimension dm
dm =36 • 2.07055 • 20.646896°+ 3.5 = 79.72 (mm)cos23.77°

Example 2, odd number of teeth
Number of teeth for calculation example 1 is changed
to 35 (other data remains the same).

)
F(z+2xnFzcosAcdm

invφ =		= 0.025562 (inv20.646896° = 0.0164533)
φ = 23.77° ( 23°46'12" See page 164 to 167 for Involute function charts)		
φ = 23		.77° (23°46'12" See page 164 to 167 for Involute function charts)

dp

AtF

§ tan α n ·§ tan 20° ·¸ = tan -1 ¨¸βcos© cos15° ¹©
¹= 20.646896° (20°38' 48" )α t = tan -1 ¨

For inv φ , calculation is by following formulainvφ =mn
2== 2.07055 (mm)cosβ cos15°

invF① (1)Virtual Number of teeth of Spur gear zυ

zυ =z
35
== 38.84 ѳ 39
cos3 β cos315°② Over balls or Rollers diameter dp
Refer to Fig. 3, dp=1.7 multiply by module 2.0 = 3.4 (mm)Use nearest available pin size dp=3.5(mm) instead of 3.4(mm)③ Transverse module mt

Fig. 7 Over balls or Rollers dimension for Helical gear(shown in axis section)

mt=2.07055(mm)Calculations is the same as above in even number of teeth part ③

Example 1, even number of teeth
When module m=2.0, Number of teeth z=36, helix
angle β =15°, Normal pressure angleα n ＝ 20° andNormal rack shift coefficient xn= 0.05. Calculations ofOver balls or Rollers dimensions is as follows,④ Transverse pressure angle α t
α t =20.646896° (20°38’48”)
Calculations is the same as Example 1, even number of teeth part ④① (1)Virtual Number of teeth of Spur gear zυ
zυ =z
36== 39.94 ѳ 40
cos3 β cos315°

Note: (1) Adopted the old Standard term.
② Over balls or Rollers diameter dp
Refer to Fig. 3, dp=1.7 multiply by module 2.0 = 3.4(mm)Use nearest available pin size dp=3.5(mm) instead of3.4(mm)⑤ Pressure angle at pin centre (°)
invφ =			

π − inv 20.646896° ·3.5− §¨¸35 • 2 • cos20° © 2 • 35¹+2 • 0.05 • tan 20°
35		= 0.025822 (inv20.646896° = 0.0164533)		 φ = 23.8465° (23°50' 47" See page 164 to 167 for Involute function charts		 ° (23°50' 47" See page 164 to 167 for Involute function charts)
φ = 23.8465⑥ Over balls or Rollers dimension dm
dm =		35 • 2.07055 • cos20.646896
90°
• cos
cos23.8465°
35+3.5 = 77.57 (mm)67Over balls or Rollers dimension for Wormgear

To obtain Over balls or Rollers dimension for Worm
gear, introduce following methods,1) Substituted three wire method from thread screw
used for measurement.
2) Use same calculation method of Rack for Wormgear.
3) Use same calculation method of Helical gear forWorm gear. However, only formula 3) is introduced.

­ 13 ½d m = d (1 + A) + dp + Ae2d ®
+ e 2 ¾ − A2 e 4 d2(1A)8+¿¯
A=px =px·1 §¨ dp − cos γ b ¸dsin γ b ©
2¹π mncosȖe=z px
cot γ bπdExample
When module mn=2.0, Number of thread z=1, Pitch
diameter of Worm gear d = 31, Normal pressure angle
α n=20° and Reference cylinder lead angle γ = 3°42’(3.7°). Calculations of Over balls or Rollers dimensions ofWorm gear is as follows.
① Over balls or Rollers diameter dp
Refer to Number of teeth(10 to ∞) in Fig. 3, dp=1.68
and multiply by module 2.0 = 3.36 (mm)Use nearest available pin size dp=3.4 (mm) instead of3.36(mm)② Transverse pressure angle α t§ tanγ ·¸© cos α t ¹

α t = tan -1 §¨ tan 20° ·¸ = 79.9459°

γ b = tan -1 ¨© sin 3.7° ¹§ tan α n ·¸© sinȖ ¹

α t = tan -1 ¨

③ Base cylinder lead angle γ bHerebyd : Pitch diameter of Worm gear (mm)z : Number of thread of Worm gearpx : Axial pitch of Worm gear (mm)γ : Reference cylinder lead angle (°)γ b : Base cylinder lead angle (°)§ tan α n ·§ tan 20° ·¸ = tan -1 ¨¸© cos15° ¹© cosβ ¹

α t = tan -1 ¨

④ Axial pitchpx =ʌ•2= 6.2963
cos3.7°⑤AdpA=		1§ 3.4 − 6.2963 cos20.3256° ·¨¸
231sin 20.3256° ©
¹= 0.04159

dm⑥ee=

1 • 6.2963
cot 20.3256°
π • 31		 = 0.17453⑦ Over balls or Rollers dimensiond m = 35.71 (mm) (Substitution method omitted)

Fig. 8 Over balls or Rollers dimension of Worm gear68