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Table -1 shows the calculation steps. It will become a standard gearcalculation if x1 = x2 = 0.

If the center distance, a x, is given, x1 and x2 would be obtained from the inverse calcula-tion from item 4 to item 8 of Table -1. Theseinverse formulas are in Table -2 .

Pinion cutters are often used in cuttinginternal gears and external gears. The actualvalue of tooth depth and root diameter, aftercutting, will be slightly different from thecalculation. That is because the cutter has acoef cient of shifted pro le. In order to get acorrect tooth pro le, the coef cient of cuttershould be taken into consideration.

.2 Interference In Internal Gears

Three different types of interference canoccur with internal gears:

(a) Involute Interference(b) Trochoid Interference(c) Trimming Interference

(a) Involute Interference

This occurs between the dedendum ofthe external gear and the addendum of theinternal gear. It is prevalent when the numberof teeth of the external gear is small. Involuteinterference can be avoided by the conditionscited below:

z1 tan a2 1 ( -2) z2 tan w

where a2

is the pressure angle seen at a tip ofthe internal gear tooth. d b2 a2 = cos 1( ) ( -3) d a2and w is working pressure angle:

( z2 z1)mcos w = cos 1[ ] ( -4)2 a x Equation ( -3) is true only if the outsidediameter of the internal gear is bigger than thebase circle:

d a2 d b2 ( - )

3

20 16 24

0 0.5

0.060401

31.0937

0.389426

13.1683

48.000 72.000

45.105 67.658

52.673 79.010

3.000 1.500

6.75

54.000 69.000

40.500 82.500

x2 x12 tan ( ) + inv z2 z1Find from Involute FunctionTable

z2 z1 cos ( 1 )2 cos w z2 z1( + y ) m

2 zm

d cos d b cos w(1 + x1) m(1 x2) m

2.25 md 1 + 2 h a1d 2 2 h a2d a1 2 hd a2 + 2h

m

z1, z2 x1, x2

invww

y

a x

d

d b

d w

h a1 h a2 hd a1d a2d f 1d f 2

Module

Pressure Angle

Number of Teeth

Coef cient of Pro le Shift

Involute Function w

Working Pressure Angle

Center Distance Increment Factor

Center Distance

Pitch Diameter

Base Circle Diameter

Working Pitch Diameter

Addendum

Whole Depth

Outside Diameter

Root Diameter

Item Symbol FormulaNo.

1 2 34

5

6

7

8

9

10

11

12

13

14

15

Table 5-1 The Calculation of a Pro le Shifted Internal Gear and External Gear (1)

ExampleExternal InternalGear (1) Gear (2)

d b2d a2d 2d f2

O2

O1w

w a x

Fig. -1 The Meshing of Internal Gear and External Gear( = 20 , z1 = 16, z2 = 24, x1 = x2 = 0.5)

Table -2 The Calculation of Shifted Internal Gear and External Gear (2)

Item Symbol FormulaNo. Example

Center Distance

Center Distance Increment Factor

Working Pressure Angle

Difference of Coef cients ofPro le ShiftCoef cient of Pro le Shift

a x y

w

x2 x1

x1 , x2

a x z2 z1

m 2( z2 z1)cos cos 1[ ]2 y + z2 z1

( z2 z1 )(invw inv ) 2tan

1

2

3

4

5

13.1683

0.38943

31.0937

0.5

0 0.5

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This occurs in the radial direction in that it prevents pulling the gearsapart. Thus, the mesh must be assembled by sliding the gears together withan axial motion. It tends to happen when the numbers of teeth of the twogears are very close. Equation ( -9) indicates how to prevent this type ofinterference. z21 + inv a1 inv w ( 2 + inv a2 inv w) ( -9) z1Here 1 (cos a1 cos a2)2 1 = sin 1 1 ( z1 z2)2 ( -10 )

(cos a2 cos a1)2 1 2 = sin 1 ( z2 z1)

2

1

This type of interference can occur in the process of cutting an internalgear with a pinion cutter. Should that happen, there is danger of breakingthe tooling. Table -3a shows the limit for the pinion cutter to preventtrimming interference when cutting a standard internal gear, with pressureangle 20, and no pro le shift, i.e., xc = 0.

For a standard internal gear, where = 20, Equation ( - ) is validonly if the number of teeth is z2 > 34.

(b) Trochoid Interference

This refers to an interference occurring at the addendum of the externalgear and the dedendum of the internal gear during recess tooth action. Ittends to happen when the difference between the numbers of teeth of thetwo gears is small. Equation ( - ) presents the condition for avoidingtrochoidal interference. z1 1 + inv w inv a2 2 ( - ) z2Here r a22 r a12 a2 1 = cos 1( ) + inv a1 inv w 2 ar a1

( -7)

a 2 + r a2 2 r a1 2 2 = cos 1( )2 ar a2

where a1 is the pressure angle of the spur gear tooth tip: d b1 a1 = cos 1( ) ( -8) d a1

In the meshing of an external gear and a standard internal gear =20, trochoid interference is avoided if the difference of the number of teeth, z1 z2, is larger than 9.

(c) Trimming Interference

There will be an involute interference between the internal gear andthe pinion cutter if the number of teeth of the pinion cutter ranges from 15to 22 ( zc = 15 to 22). Table -3b shows the limit for a pro le shifted pinioncutter to prevent trimming interference while cutting a standard internal gear.The correction, xc , is the magnitude of shift which was assumed to be: xc =0.0075 zc + 0.05.

There will be an involute interference between the internal gear andthe pinion cutter if the number of teeth of the pinion cutter ranges from 15to 19 ( zc = 15 to 19).

.3 Internal Gear With Small Differences In Numbers Of Teeth

In the meshing of an internal gear and an external gear, if the differencein numbers of teeth of two gears is quite small, a pro le shifted gearcould prevent the interference. Table -4 is an example of how to preventinterference under the conditions of z2 = 50 and the difference of numbers of

teeth of two gears ranges from 1 to 8.All combinations above will not cause involute interference or trochoid

interference, but trimming interference is still there. In order to assemblesuccessfully, the external gear should be assembled by inserting in the axialdirection.

A pro le shifted internal gear and external gear, in which the differenceof numbers of teeth is small, belong to the eld of hypocyclic mechanism,which can produce a large reduction ratio in one step, such as 1/100. z2 z1Speed Ratio = ( -11 z1

Table -3a The Limit to Prevent an Internal Gear fromTrimming Interference ( = 20, xc = x2 = 0)

15

34

28

46

44

62

16

34

30

48

48

66

17

35

31

49

50

68

18

36

32

50

56

74

19

37

33

51

60

78

22

40

38

56

80

98

20

38

34

52

64

82

21

39

35

53

66

84

24

42

40

58

96

114

25

43

42

60

100

118

z c

z2

z c

z2

z c

z2

27

45

Table -3b The Limit to Prevent an Internal Gear fromTrimming Interference ( = 20, x2 = 0)

zc

xc

z2

zc

xc

z2

zc

xc

z2

15

0.1625

36

28

0.26

52

44

0.38

71

16

0.17

38

30

0.275

54

48

0.41

76

17

0.1775

39

31

0.2825

55

50

0.425

78

18

0.185

40

32

0.29

56

56

0.47

86

19

0.1925

41

33

0.2975

58

60

0.5

90

20

0.2

42

34

0.305

59

64

0.53

95

21

0.2075

43

35

0.3125

60

66

0.545

98

22

0.215

45

38

0.335

64

80

0.65

115

24

0.23

47

40

0.35

66

96

0.77

136

25

0.2375

48

42

0.365

68

100

0.8

141

27

0.252

50

z1

x1 z2

x2

w a

49

1.00

61.0605

0.971

1.105

48

0.60

46.0324

1.354

1.512

47

0.40

37.4155

1.775

1.726

46

0.30

32.4521

2.227

1.835

45

0.20

28.2019

2.666

1.933

44

0.11

24.5356

3.099

2.014

43

0.06

22.3755

3.557

2.053

42

0.01

20.3854

4.010

2.088

050

Table -4 The Meshing of Internal and External Gears of SmallDifference of Numbers of Teeth ( m = 1, = 20)

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In Figure -2 the gear train has a difference of numbers of teeth of only1; z1 = 30 and z2 = 31. This results in a reduction ratio of 1/30.

SECTION HELICAL GEARS

The helical gear differs from thespur gear in that its teeth are twistedalong a helical path in the axialdirection. It resembles the spur gearin the plane of rotation, but in the axialdirection it is as if there were a seriesof staggered spur gears. See Figure

-1. This design brings forth a numberof different features relative to the spurgear, two of the most important beingas follows:

1. Tooth strength is improvedbecause of the elongated helicalwraparound tooth base support.

2. Contact ratio is increased due to the axial tooth overlap. Helical gearsthus tend to have greater load carrying capacity than spur gears of thesame size. Spur gears, on the other hand, have a somewhat higheref ciency.

Helical gears are used in two forms:

1. Parallel shaft applications, which is the largest usage.2. Crossed-helicals (also called spiral or screw gears) for connecting

skew shafts, usually at right angles.

.1 Generation Of The Helical Tooth

The helical tooth form is involute in the plane of rotation and can bedeveloped in a manner similar to that of the spur gear. However, unlikethe spur gear which can be viewed essentially as two dimensional, thehelical gear must be portrayed in three dimensions to show changing axialfeatures.

Referring to Figure -2, there is a base cylinder from which a taut

plane is unwrapped, analogous to the unwinding taut string of the spurgear in Figure 2-2 . On the plane there is a straight line AB, which whenwrapped on the base cylinder has a helical trace A oBo. As the taut planeis unwrapped, any point on the line AB can be visualized as tracing aninvolute from the base cylinder. Thus, there is an in nite series of involutesgenerated by line AB, all alike, but displaced in phase along a helix on thebase cylinder.

Again, a concept analogous to the spur gear tooth development is toimagine the taut plane being wound from one base cylinder on to another asthe base cylinders rotate in opposite directions. The result is the generationof a pair of conjugate helical involutes. If a reverse direction of rotation isassumed and a second tangent plane is arranged so that it crosses the rst,a complete involute helicoid tooth is formed.

.2 Fundamentals Of Helical Teeth

In the plane of rotation, the helical gear tooth is involute and all of therelationships governing spur gears apply to the helical. However, the axialtwist of the teeth introduces a helix angle. Since the helix angle varies fromthe base of the tooth to the outside radius, the helix angle is de ned asthe angle between the tangent to the helicoidal tooth at the intersection ofthe pitch cylinder and the tooth pro le, and an element of the pitch cylinder.See Figure -3 .

The direction of the helical twist is designated as either left or right. Thedirection is de ned by the right-hand rule.

For helical gears, there are two related pitches one in the plane of

rotation and the other in a plane normal to thetooth. In addition, there is an axial pitch.

Referring to Figure -4, the two circularpitches are de ned and related as follows:

p n = p t cos = normal circular pitch( -1)

The normal circular pitch is less than thetransverse radial pitch, p t, in the plane of rota-tion; the ratio between the two being equal tothe cosine of the helix angle.

Consistent with this, the normal module isless than the t ransverse (radial) module.

The axial pitch of a helical gear, p x, is thedistance between corresponding points ofadjacent teeth measured parallel to the gear'saxis see Figure - . Axial pitch is related to

Fig. -1 Helical Gear

Elementof PitchCylinder(or gear'saxis)

Tangent to Helical Tooth

PitchCylinder

HelixAngle

Fig. 6-3 De nition of Helix Angle

Fig. -4 Relationshipof CircularPitches

p t

p n

p x

Fig. - Axial Pitch ofa Helical Gear

Twisted SolidInvolute

Taut Plane

Base Cylinder

B

B0A0

A

Fig. 6-2 Generation of the Helical Tooth Pro le

a x

Fig. -2 The Meshing of Internal Gear and External Gearin which the Numbers of Teeth Difference is 1

( z2 z1 = 1)