Chapter 10 Calculation for Gear strength10.1 Calculation of strength for Spur and Helical gears
There are calculations for Tooth bending strength(hereby called Bending strength), Surface durability
and Scoring when considering gear strength. These
are from ISO, JGMA, AGMA, DIN, BS and JSME.
KG had developed and marketed KG-CALMET foreasy searching of suitable KG STOCK GEARS by entering gear data, tooth strength (Bending strength andSurface durability), profile generation, condition ofengagement, Number of teeth and transfer torque.
Now to introduce the selected calculation formula for
Bending strength and Surface durability from formula
extracted from JGMA (Japan Gear Manufacture Association Standard) as follows.

Calculation formula of Bending strength for Spur and Helical gears JGMA 401-01 (1974).
Calculation formula of Surface durability for Spur and Helical gears JGMA 402-01 (1975).
1. Application range (common)

1.1 This standard is applied to Spur, Helical, Double
helical and Internal gears that uses general industrial machinery transfer power.
Module: 1.5 to 25.0 mm
Reference pitch diameter : 25 to 3,200 mm
Circumferential velocity : Below 25m/s
Revolving velocity: Below 3,600 m-1
Tooth profile of Spur and normal type of Helicalgears as stipulated in JIS B 1701 (Involute tooth
profile and dimensions). Also applicable to gears
with Normal reference pressure angle of 22.5°and
25°
Accuracy: A c c u r a c y c l a s s e s 1
to 6 stipulated in JIS
B 1702 (Accuracy for
Spur and Helical gear).

1.2.1 T his standard stipulates calculation for Bending
allowable load and when determining designated dimension based on Tooth root bending
stress.
1.2.2 T his standard stipulates calculation for Toothsurface allowable load for a gear with designated dimension and for calculating specifications
based on flank stress.

2 Definition

2.1 Bending strength
Bending allowable load for gear is Allowable tangential load on the Reference pitch circle based on
Allowable tooth root bending stress of gears whentransferring power during operation.

2.2 Surface durability
Surface durability is stipulated as capacity of load it
can withstand and still provide necessary strengthand enough safety for gear against progressive pitting. Therefore, meaning of Allowable flank load is Allowable tangential load on the Reference pitch circle
determined in accordance to Surface durability of its
gears when transferring power during operations.

3. Basic formula (common)

In regards to calculating Gear strength, the conversion formulas related to calculating Tangential loadon Reference pitch circle, Nominal power and Nominal torque are as follows.
3.1 Nominal tangential load on Reference pitch circle
Ft(kgf)Ft =102 P 1.95 × 106 P=. ..................................................... (1)ȣdnHerebyP : Nominal power (kW)υ : Circumferential velocity (m/s) on the Reference
pitch circle
d : Reference pitch diameter (mm)n : Revolving velocity (min-1)ȣ=dn19100 ................................................................................ (2)

Or
Ft =

2000T
. ........................................................................... (3)dHerebyT : Nominal torque (kgf・m)
3.2 Nominal power (kW)P=

F t ȣ 10−6=
F t dn . ........................................................... (4)102 1.95

123

3.3 Nominal torque (kgf・m)T=

Ft d................................................................................. (5)
2000

Or
T=

974 P
................................................................................ (6)n4. Calculation formula for Strength

4.1 Bending strength
Nominal tangential load on the Reference pitch circle
is necessary as reference for calculating Bending
strength. Therefore, Nominal tangential load on the
Reference pitch circle should be equal or below Allowable tangential load on the Reference pitch circle,
which is derived from calculating Allowable toothroot bending stress. Therefore,Ft ҇ Ftlim ................................................................................... (7)HerebyFt: N ominal tangential load on the Reference
pitch circle (kgf)
Ftlim : Calculate Allowable tangential load (kgf) on
the Reference pitch circle by selecting thesmaller value from either pinion or gear.
On the other hand, Tooth root stress calculated from
Nominal tangential load on the Reference pitch circle
should be equal or below Allowable tooth root bending stress.
Thereforeσ F ҇ ıFlim .................................................................................. (8)HerebyσF: Dedendum stress calculated from Nominal
tangential load on Reference pitch circle(kgf/mm2)
σ Flim : A llowable tooth root bending stress (kgf/mm2)
4.1.1 Calculation for Allowable tangential load on the
Reference pitch circle is as follow.
Ftlim = ıFlim

mn b § KLKFX · 1¨¸YFYİYȕ © KVKO ¹ SF ............................................. (9)Herebymn : Normal module (mm)b : Facewidth (mm)YF : Form factor
Yε : Load distribution factor
Yβ : Helix angle factor
KL : Life factor
KFX : Dimension factor for Tooth root stress
Kv : Dynamic factor
K0 : Overload factor
SF : Safety factor for Tooth root bending damage
4.1.2 Calculation for Tooth root bending stress is asfollow.
ı F = Ft

124YFYİYȕ § KVKO ·¨¸ SF . ............................................... (10)mn b © KLKFX ¹

4.2 Calculation for Surface durability
Nominal tangential load on the Reference pitch
circle is necessary as reference for calculating Surface
strength. Therefore, Nominal tangential load on the
Reference pitch circle should be equal or below Allowable tangential load on the Reference pitch circle,
which is derived from calculating Allowable Hertz
stress. Therefore,Ft ҇ Ftlim ................................................................................ (11)Hereby Ft : Nominal tangential load on the Reference pitch circle (kgf)
Ftlim : Calculate Allowable tangential load (kgf)
on the Reference pitch circle by selecting the smaller value (kgf ) from either
pinion or gear.
On the other hand, Hertz stress from Nominal tangential load on the Reference pitch circle should be equal
or below Allowable hertz stress.
Thereforeσ H ҇ ıHlim .............................................................................. (12)Hereby		σH: Hertz stress calculated from Nominal tangential load on Reference pitch circle (kgf/mm2)
σ Hlim : Allowable hertz stress ((kgf/mm2)
4.2.1 Calculation for Allowable tangential load on the
Reference pitch circle is as follow.
2u § KHLZLZRZVZWZHX ·¨¸ ×
u ± 1 © ZHZM Z ε Zβ ¹11
‫ޓ‬KHβKVKO SH 2 . ........................................................ (13)2

Ft lim = ıH lim d 1bH+/- : ‘+’ indicate the engagement with both External gears. ‘-‘ for engagement with External andInternal gears.
Herebyd1 : Reference pitch diameter for pinion (mm)bH : Effective facewidth for Surface durability (mm)u : Gear ratio
ZH : Zone factor
ZM : Elasticity factor
Zε : Contact ratio factor
Zβ : Helix angle factor
KHL : Life factor for Surface durability
ZL : Lubricating oil factor
ZR : Roughness factor
ZV : Lubricating speed factor
ZW : Work hardening factor
KHX : Dimension factor for Surface durability
KHβ : Face load factor for Contact stress
KV : Dynamic factor
KO : Overload factor
SH : Safety factor for Surface durability

ZHZM Z ε Zβ
Ft u ± 1×
d 1 bH u KHLZLZRZVZWKHX. ............................................................ (14)KHβKVKO SH

σH =+/- : ‘+’ indicate the engagement with both External gears. ‘-‘ for engagement with External andInternal gears.

5. Calculation formula for types of factor

5.1	How to obtain the types of factor using the calculation formula of Bending strength.
The following stipulates types of factor from calculation formula of Bending strength in previous paragraph.
5.1.1 Facewidth b
When Facewidths differs, assume wider Facewidth tobe bw and smaller Facewidth to be bs. bw - bs ≦ mn, use
actual Facewidth for calculations.
When bw - bs ＞ mn, bs is used in formula bs + mn to calculation of Facewidth.
5.1.2 Form factor YF
Refer to Fig. 1 to find Form factor.
For Virtual number of teeth of spur gear for Helicalgear, use following calculation formula.
zυ =z
cos3 β.......................................................................... (15)For Form factor for Tooth profile excluding Fig. 1
please refer to this original standard.
5.1.3 Load distribution factor Yε
Calculating Load distribution factor using following
formula.
Yε =

1

εα

. ............................................................................... (16)Herebyε α : Transverse contact ratio
Calculation formulas of Transverse contact ratio are as
follows,
Spur gear

ra12 − rb12 + ra 2 2 − rb 2 2 − asin α ω ......
??????
(17): εα =
mπ cosα 0

β : Reference pitch cylindrical helix angle (°)β b : Base cylinder helix angle (°)
Subscript
1 : Pinion
2 : Gear
Remark 1. Table 1 shows the Transverse contact ratio
ε α for Standard spur gear with Reference
pressure angle 20°.
Remark 2. Use following formula to calculate approximate value of Yε for Helical gear.
cos 2 β b

εα

=

1

ε α n .................................................................... (20)

However, obtain Transverse contact ratio ε α n for Virtual spur gear from Table 1 by using Virtual numberof teeth of spur gear zv1 and zv2.
5.1.4 Helix angle factor Yβ
Calculate helix angle factor using following formula.

β

For 0° ≦ β ≦ 30° : Yβ = 1 − 120 ................................... (21): Yβ = 0.75 . ....................................... (22)For β ≧ 30°β

Fig. 2 Helix angle factor

1.00
Helix angle factor YB

4.2.2 Calculation for Hertz stress is as follows.

0.90
0.80
0.75

0

10

20

30

40

Helix angle B(º)

5.1.5 Life factor KL
Refer to Table 2 to obtain Life factor.
Table 2. Life factor KL
Number of repeated

Hardness (1)(2) Hardness (2)HB120 - 220 Above HB221

Carburizing
gear

Below 10,000

1.4

1.5

Approx. 100,000

1.2

1.4

1.5
1.5

Approx. 106

1.1

1.1

1.1

Above 107

1.0

1.0

1.0

2
2
2
2
Helical gear : ε α = ra1 − rb1 + ra 2 − rb 2 − asin α ωt ...... (18)mt π cos α t
cos 2 β b = 1 − sin 2 β • cos 2 α n . ........................................... (19)Note (1) Steel casted gears to use this Table
Note (2) Core hardness is used for Induction hardened
gear.

HerebyMeaning of repeated rotations is number of repeat
during life span of gears. If uncertain, KL =1.0.

γ a : Tip (Outside) radius (mm)γ b : Base radius (mm)a : Centre distance (mm)α w : Working pressure angle (°)α wt : Transverse contact pressure angle (°)α : Reference pressure angle (°)α n : Normal pressure angle (°)α t : Transverse reference pressure angle (°)

125

Fig. 1 Graph for Form factor (Part 1 in Table 3)
3.8
3.7

Normal reference pressure angle An = 20º
Addendum hk = 1.0 mn

3.6Dedendum hƒ = 1.25 mn
Gorge of Tool tip G = 0.375 mn

3.5

3.5

3.4

3.4

3.3

3.3

3.2

3.2

3.1

3.1

2.9

2.9

4

-0.5

-0.

2.7

2

2.7

2.8

3

-0.

Lim

n

tio

ita

3.0

u

-0.

2.8

o

t

re

eo

h
ft

l
ica

e
nd

x=

Form factor YF

ut

rc

3.0

.1

-0
x=
0

2.6

0.1

2.6

0.2

2.5

2.5

0.3

2.4

2.4

0.4

2.3

2.3

x = 0.5

2.2

2.2

0.6

2.1

2.1

0.7

2.0

Lim

ita

tio

no

1.9
1.8
10

11

2.0

0.8

fP

oin

ted

12

0.9

too

th

tip

13

1.9

x = 1.014

15

16 17 18 19 202530

Virtual number of teeth of spur gear Zv

126

35 40 45 50 60

80 100 200 400

1.8

d

127

19

20

21

22

24

26

εα =

25

28

30

32

34

36

38

40

ra12 − rb12 + ra 2 2 − rb 2 2 − a • sin α ωmπ cos α

42

45

22

=

48

55

1.722

58

a

a+b+c

60

65

75

1.737

80

90

100

110

120

130

140

150

160

170

180

190

200

Calculate cos2β b = 0.900286478 from formula 21
Therefore,
εα = 1.73554 x 0.90286478 = 1.567 ( theoretical value is 1.568).

c = 22 and 110 is 1.722

119.1 − 110
a = 0.004 ×= 0.00364½°
a½+ b + c
119.1 − 110
a = 0.004 120
× − 110= 0.00364¾ =°1.73554
23.8120
− 22
b+c
− 110
b = 0.011 ×= 0.0099 ° ¾ =a1+.73554
23.8 − 22
b = 0.011 ×24 − 22
= 0.0099 ¿ °24 −ε22¿
α between
Transverse contact ratio

Firstly calculate Zv1 = 23.8, Zv2 = 119.1 from formula (18), secondly
calculate εα = 1.722 to compare between engagements of 22 and 110
where the difference is 0.004.
Obtain εα = 1.726 to compare between engagements of 22 and 120
where the difference is 0.004.
Obtain εα = 1.737 to compare between engagements of 24 and 120
where the difference is 0.011.

Example: Calculate εα of Helical gear with z1 = 20, z2 = 100, β = 19°22”

Calculation of transverse contact ratio for Helical gear

200 1.713 1.721 1.728 1.734 1.741 1.746 1.757 1.762 1.766 1.775 1.783 1.790 1.796 1.802 1.808 1.813 1.817 1.824 1.830 1.833 1.837 1.841 1.846 1.848 1.854 1.860 1.865 1.869 1.876 1.882 1.887 1.892 1.895 1.899 1.902 1.904 1.906 1.908 1.910 1.912

190 1.712 1.719 1.726 1.733 1.739 1.745 1.755 1.760 1.765 1.773 1.781 1.788 1.795 1.800 1.806 1.811 1.816 1.822 1.828 1.832 1.835 1.840 1.844 1.847 1.853 1.858 1.863 1.867 1.874 1.881 1.886 1.890 1.894 1.897 1.900 1.902 1.905 1.907 1.909

180 1.710 1.717 1.724 1.731 1.737 1.743 1.753 1.758 1.763 1.771 1.779 1.786 1.793 1.799 1.804 1.809 1.814 1.820 1.826 1.830 1.833 1.838 1.842 1.845 1.851 1.856 1.861 1.865 1.873 1.879 1.884 1.888 1.892 1.895 1.898 1.901 1.903 1.905

170 1.708 1.715 1.722 1.729 1.735 1.741 1.751 1.756 1.761 1.769 1.777 1.784 1.791 1.797 1.802 1.807 1.812 1.818 1.824 1.828 1.831 1.836 1.840 1.843 1.849 1.854 1.859 1.863 1.871 1.877 1.882 1.886 1.890 1.893 1.896 1.898 1.901

160 1.706 1.713 1.720 1.727 1.733 1.738 1.749 1.754 1.759 1.767 1.775 1.782 1.788 1.794 1.800 1.805 1.810 1.816 1.822 1.825 1.829 1.834 1.838 1.840 1.847 1.852 1.857 1.861 1.868 1.874 1.879 1.884 1.888 1.891 1.894 1.896

150 1.703 1.710 1.717 1.724 1.730 1.736 1.747 1.751 1.756 1.765 1.772 1.779 1.786 1.792 1.797 1.802 1.807 1.813 1.819 1.823 1.826 1.831 1.835 1.838 1.844 1.849 1.854 1.858 1.866 1.872 1.877 1.881 1.885 1.888 1.891

140 1.700 1.708 1.715 1.721 1.727 1.733 1.744 1.749 1.753 1.762 1.769 1.777 1.783 1.789 1.794 1.799 1.804 1.811 1.816 1.820 1.823 1.828 1.832 1.835 1.841 1.847 1.851 1.856 1.863 1.869 1.874 1.878 1.882 1.885

130 1.697 1.704 1.711 1.718 1.724 1.730 1.740 1.745 1.750 1.758 1.766 1.773 1.780 1.786 1.791 1.796 1.801 1.807 1.813 1.817 1.820 1.825 1.829 1.832 1.838 1.843 1.848 1.852 1.860 1.866 1.871 1.875 1.879

120 1.693 1.701 1.708 1.714 1.720 1.726 1.737 1.742 1.746 1.755 1.762 1.770 1.776 1.782 1.787 1.792 1.797 1.804 1.809 1.813 1.816 1.821 1.825 1.828 1.834 1.840 1.844 1.849 1.856 1.862 1.867 1.871

110 1.689 1.696 1.703 1.710 1.716 1.722 1.732 1.737 1.742 1.750 1.758 1.765 1.772 1.778 1.783 1.788 1.793 1.799 1.805 1.809 1.812 1.817 1.821 1.824 1.830 1.835 1.840 1.844 1.852 1.858 1.863

100 1.684 1.691 1.698 1.705 1.711 1.717 1.727 1.732 1.737 1.745 1.753 1.760 1.767 1.773 1.778 1.783 1.788 1.794 1.800 1.804 1.807 1.812 1.816 1.819 1.825 1.830 1.835 1.839 1.846 1.853

90 1.678 1.685 1.692 1.699 1.705 1.711 1.721 1.726 1.731 1.739 1.747 1.754 1.760 1.766 1.772 1.777 1.782 1.788 1.794 1.798 1.801 1.806 1.810 1.813 1.819 1.824 1.829 1.833 1.840

80 1.670 1.678 1.685 1.691 1.697 1.703 1.714 1.719 1.723 1.732 1.740 1.747 1.753 1.759 1.765 1.770 1.774 1.781 1.787 1.790 1.794 1.798 1.803 1.805 1.811 1.817 1.821 1.826

75 1.666 1.673 1.680 1.687 1.693 1.699 1.710 1.714 1.719 1.728 1.735 1.742 1.749 1.755 1.760 1.765 1.770 1.777 1.782 1.786 1.789 1.794 1.798 1.801 1.807 1.812 1.817

zv2

1.726

b

70

70 1.661 1.669 1.676 1.682 1.688 1.694 1.705 1.710 1.714 1.723 1.731 1.738 1.744 1.750 1.756 1.761 1.765 1.772 1.778 1.781 1.785 1.789 1.794 1.796 1.802 1.808

65 1.656 1.663 1.670 1.677 1.683 1.689 1.699 1.704 1.709 1.717 1.725 1.732 1.739 1.745 1.750 1.755 1.760 1.766 1.772 1.776 1.779 1.784 1.788 1.791 1.797

60 1.650 1.657 1.664 1.671 1.677 1.683 1.693 1.698 1.703 1.711 1.719 1.726 1.733 1.739 1.744 1.749 1.754 1.760 1.766 1.770 1.773 1.778 1.782 1.785

58 1.647 1.655 1.662 1.668 1.674 1.680 1.691 1.696 1.700 1.709 1.716 1.724 1.730 1.736 1.741 1.746 1.751 1.758 1.763 1.767 1.770 1.775 1.779

55 1.643 1.650 1.657 1.664 1.670 1.676 1.686 1.691 1.696 1.704 1.712 1.719 1.726 1.732 1.737 1.742 1.747 1.753 1.759 1.763 1.766 1.771

52 1.638 1.646 1.653 1.659 1.665 1.671 1.682 1.687 1.691 1.700 1.707 1.715 1.721 1.727 1.732 1.737 1.742 1.749 1.754 1.758 1.761

50 1.635 1.642 1.649 1.656 1.662 1.668 1.678 1.683 1.688 1.696 1.704 1.711 1.718 1.724 1.729 1.734 1.739 1.745 1.751 1.755

c

52

24

110

23.8

εα

50

cos 2 β b

48 1.631 1.639 1.646 1.652 1.658 1.664 1.675 1.680 1.684 1.693 1.701 1.708 1.714 1.720 1.725 1.731 1.735 1.742 1.748

45 1.625 1.633 1.640 1.646 1.652 1.658 1.669 1.674 1.678 1.687 1.695 1.702 1.708 1.714 1.720 1.725 1.729 1.736

42 1.619 1.626 1.633 1.640 1.646 1.652 1.662 1.667 1.672 1.680 1.688 1.695 1.702 1.708 1.713 1.718 1.723

40 1.614 1.622 1.629 1.635 1.641 1.647 1.658 1.663 1.667 1.676 1.684 1.691 1.697 1.703 1.708 1.714

38 1.609 1.617 1.624 1.630 1.636 1.642 1.653 1.658 1.662 1.671 1.678 1.686 1.692 1.698 1.703

36 1.604 1.611 1.618 1.625 1.631 1.637 1.647 1.652 1.657 1.665 1.673 1.680 1.687 1.692

34 1.598 1.605 1.612 1.619 1.625 1.631 1.641 1.646 1.651 1.659 1.667 1.674 1.681

32 1.591 1.599 1.606 1.612 1.618 1.624 1.635 1.640 1.644 1.653 1.661 1.668

30 1.584 1.592 1.599 1.605 1.611 1.617 1.628 1.633 1.637 1.646 1.654

28 1.576 1.584 1.591 1.597 1.604 1.609 1.620 1.625 1.629 1.638

26 1.568 1.575 1.582 1.589 1.595 1.601 1.611 1.616 1.621

25 1.563 1.571 1.578 1.584 1.590 1.596 1.607 1.612

24 1.558 1.566 1.573 1.579 1.586 1.591 1.602

22 1.548 1.555 1.562 1.569 1.575 1.581

21 1.542 1.549 1.556 1.563 1.569

20 1.536 1.543 1.550 1.557

19 1.530 1.537 1.544

18 1.522 1.529

18

zv1

17

17 1.515

Table 1. Transverse contact ratio ε α for Standard spur gear

11
9.1
12
0

5.1.6 Dimension factor KFX for Tooth root stress
With increased Tooth profile, Bending strength isinfluenced. At the moment, due to insufficient data
Dimension factor will be 1.0.

5.1.7 Dynamic factor Kv
Obtain Dynamic factor from Table 3 using gear accuracy and Circumferential speed on the Reference
pitch circle.

Table 3. Dynamic factor Kv
System of accuracy from JIS B 1702
Tooth profile
NormalModified
1
1
2
2
3
3
4
4
5
6
-

Above 1.0 to
below 3.0
1.0
1.1
1.2
1.3
1.4
1.5

Below 1
1.0
1.0
1.0
1.1
1.2

Circumferential speed on the Reference pitch circle (m/s)
Above 3.0 to
Above 5.0 to
Above 8.0 to
Above 12.0 to
below 5.0
below 8.0
below 12.0
below 18.0
1.0
1.0
1.1
1.2
1.05
1.1
1.2
1.3
1.15
1.2
1.3
1.5
1.3
1.4
1.5
1.4
1.5
1.5
-

5.1.9 S afety factor S F for damage from Tooth root

5.1.8 Overload factor K0
Obtain Overload factor using following formula.
Actual tangential load
KO =Nominal tangential load ( Ft ) ............................... (23)

Use Table 4 to obtain Actual tangential load if uncertain of value.
Table 4. Overload factor K0
Impact from motor side

Impact from load
Flat load

Flat load (Electric, turbine, hydraulic motors)
Light impact (Multi cylinder engine)
Average impact(Single
cylinder engine)

Above 18.0 to
below 25.0
1.3
1.5
-

Average impact Heavy impact

1.0

1.25

1.75

1.25

1.5

2.0

1.5

1.75

2.25

Note:	If the impact from load is unknown, refer toTable 5.

bending
Fixed value of Safety factor for damage from Toothroot bending is difficult to be determined due to various internal and external factors. Minimum factor of
1.2 is necessary.
5.1.10 Allowable Tooth root bending stress σ Flim
Refer to Tables 9 and 10 for Allowable tooth root
bending stress for gear with fixed load direction. Forintermediate Hardness values in the tables shown, it
is our recommendation to use interpolation values.
When load direction is bi-directional, value of Allowable tooth root bending stress σ Flim will be 2/3 ofvalues in the table. For exmple, an idler gear or gear
which alternates bi-directionally and for equal loads
on either right or left teeth.
Value of hardness or core hardness uses centre ofTooth root.

Table 5. Classification of load for Driven machine
Name of Driven machine

Range

Name of Driven machine

Range

Name of Driven machine

Range

Agitator

M

ElevatorU

Petroleum refinery machinery

M

BlowerU

ExtruderU

Paper mill machinery

M

Brewing and Distillation apparatusU

Fan (electric fan)U

Timber mill machinery

H

VehiclesM

Fan (for industries)

M

Pump

M

ClarifierU

FeederM

Rubber machinery (medium load)

M

Sorting MachineM

Feeder (to and fro motion)

H

Rubber machinery (heavy load)

H

Ceramics industry machine (medium load)

M

Food machinery

M

Water treatment machine (light load)

U

Ceramics industry machine (heavy load)

H

Hammer millH

Water treatment machine (medium load)M

Compressor

M

Hoist

M

Screen (fluid)

U

Conveyer (uniform load)

U

Machine tools (main drive)M

Screen (gravel)

M

Conveyer (uniform load / heavy load)

M

Machine tools (supplementary drive)U

Sugar plant machinery

M

Crane

U

Metalwork machinery

H

Textile machinery

M

Crusher

H

Rotary millM

Iron mill machinery (hot rolling)

H

Dredger (Medium load)

M

Tumbler

H

Iron mill machinery (cold rolling)U

Dredger (heavy load)

H

Mixer

M

Note U: Uniform load, M: Medium impact, H: Heavy impact

128

5.2.2 Zone factor ZH
Calculation of Zone factor is as follows.

5.2 How to obtain each factor based on calculationfor Surface durability
Factors using calculation formulas based on Surface
durability as mentioned above is defined below.

2 cos β b cos α wt
1=
cos 2 α t sin α wt
cos α tZH =2 cos β b .................
(24)tan α wt

Herebyβ b : Base cylinder helix angle (°)α wt : Transverse contact pressure angle (°)α t : Transverse reference pressure angle (°)

5.2.1 Effective facewidth for Surface durability bH (mm)Obtain Effective facewidth for Surface durability from(a) and (b).
(a) For different Facewidth between pinion and gear,
select the narrower Facewidth as Effective facewidth.
(b) F or Facewidth with end relief at both ends, Effective facewidth is the narrower of the Facewidth
deducted by such end relief areas.

(a) O btain Zone factor from Fig. 3 with Normal reference pressure angle of 20° defined in JIS.

Fig. 3 Zone factor ZH (Normal reference pressure angle α n = 20°)
3.0

x2) /
(z1±

2.8

(x1±

5

.01

-0

2.9

z2)=

-0

- 0.

.01

02

2.7
-0

.00

2.6

- 0.0

5

025

0

2.5
Zone factor ZH+ 0.0025+ 0.005

2.4+ 0.012.3+ 0.01

5+ 0.022.2+ 0.025+ 0.032.1+ 0.04+ 0.052.0+ 0.06+ 0.071.9+ 0.08+ 0.09
+ 0.1

1.8
1.7

1.6
1.5

0°

5°

10°

15°

20°

25°

30°

35°

40°

45°

Reference pitch cylindrical helix angle B

129

In Fig. 3, x : Rack shift coefficient (Normal rack shiftcoefficient for Helical gear and Superscript) 1 is Pinion and 2 is Gear.)		z : Number of teeth		
β : Reference pitch cylindrical helix angle (°)

Herebyν : Poisson’s ratio
E : Modulus of direct elasticity (Young’s modulus)(kgf/mm2)
For ZM, refer to Table 6 for combinations of main gear
materials.

(b) Factors from above formula and figure are defined
as follows.
β b = tan −1 ( tan ȕ cos Įt ) ....................................................... (25)5.2.4 Contact ratio factor ZεObtain Contact ratio factor using following formula(refer to Fig 4).
Spur gear : Z İ = 1.0 . ....................................................... (29)x1 ± x 2 ·inv α wt = 2tan α n §¨
¸ + inv α t .................................. (26)© z1 ± z 2 ¹
α t = tan −1 (tan Į n / cosβ ) .................................................... (27)Fig. 4 Contact ratio factor(c) Zone factor is based upon Curvature radius of flankat Pitch point. Therefore this factor is used for calculating Allowable load for flank. Due to Relative
curvature radius at the worst load point is slightly
smaller than that at Pitch point, such Zone factor
cannot be use. These are Spur gear or Helical gearwith extremely small Overlap ratio (ε β ＜ about 0.5)

0.2

0.95

0.

8

0.85

1

1
2
2
......................................... (28)
−
1
ν
1
§
+ 1 − ν 2 ·¸
π¨
E
E
1
2
©
¹

0.6

≧

5.2.3 Elasticity factor ZM
Calculation of Elasticity factor is as follows.

0.90

EB

ZE Contact ratio factor

0.4

with below minimum number of teeth (z ≦ about
23) and small Rack shift coefficient. For such cases,
please refer to 4.2.2 to check for Hertz stress at theworst load point.

ZM =EB = 0

1.00

0.80

0.75

0.70
1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

Transverse contact ratio EA

Table 6. Elasticity factor ZM
Gear
Materials

Structural steel

Casting steel

Vocabularies

∗(1)

SC

Mating gear

Modulus of
direct elasticity
E
kgf/mm2

Poisson's ratio
ν

21000

20500

Spheroidal
graphite iron

FCD

17600

Gray iron casting

FC

12000

0.3

Materials

Vocabularies

Structural steel

*(1)

21000

60.6

Casting steel

SC

20500

60.2

Spheroidal
graphite iron

FCD

17600

57.9

Gray iron casting

FC

12000

51.7

Casting steel

SC

20500

Spheroidal
graphite iron

FCD

17600

Gray iron casting

FC

12000

51.5

Spheroidal
graphite iron

FCD

17600

55.5

Gray iron

FC

12000

50.0

Gray iron casting

FC

12000

45.8

Note(1) ∗Structural steel to be S ～ C, SNC, SNCM, SCr, SCM.

130

Elasticity factor
ZM(kgf/mm2) 0.5

Modulus of
direct elasticity
E
kgf/mm2

Poisson's ratio
ν

0.3

59.9
57.6

Herebyε α : Transverse contact ratio (refer to Clause 5.1.3
and Reference 1)
ε β : Overlap ratio
5.2.5 Helix angle factor Zβ
Helix angle factor for Surface durability is difficult toaccurately stipulate due to insufficient data. Calculation formula will beZβ =1.0.................................................................................... (32)5.2.6 Life factor for Surface durability KHL
Obtain Life factor for Surface durability from Table 7
Table 7. Life factor of Surface durability
Number of repeatedLife factor for Surface durability

Below 10,000

1.5

About 100,000

1.3

About 106

1.15

Above 107

1.0

Remark 1. ‘Repeated’ is number of times of engaged
rotation during life span.
Remark 2. Normally idler gear makes 2 engagements
per rotation. However for engagements
between different flanks for 1 rotation, it
should be counted as 1 engagement.
Remark 3. For reversible rotation or similar conditions,
number of rotation is from larger load applied to either flank.

Lubricating oil factor ZL

If number of times is uncountable, life factor to be
KHL = 1.0................................................................................ (33)5.2.7 Lubricating oil factor ZL
For the 2 types of gear stated below, obtain Lubricating oil factor from Fig. 5 based on Kinematic viscosity(cSt) at 50°C.
Fig. 5 Lubricating oil factor
1.2

Thermal reﬁned gear

1.1
1.0

Rmaxm =

Rmax1 + Rmax 2 3 100( µ m) ..................................... (34)a2(1) Thermal refined gear (1): Use solid line in Fig. 6.
(2) Surface hardened gear: Use broken line in Fig. 6.
Refer to 5.2.7 for Note (1) and Remark
Fig. 6 Roughness factor

1.1
1.0

Surface hardened gear

0.9
0.8
0.7

Thermal reﬁned gear

1

2

3

4 5 6
7 8 9 10 11 12 13 14
Average roughness Rmax m (Mm)

5.2.9 Lubricating speed factor ZV
Find Lubricating speed factor based on maximum
height of profile roughness of flank Rmaxm(μ m) from
Fig. 7 using either pinion or gears(1) Thermal refined gear (1): Use solid line in Fig. 7.
(2) Surface hardened gear: Use broken line in Fig. 7.
Refer to 5.2.7 for Note (1) and Remark

1.2

Fig. 7 Lubricating speed factor
Thermal reﬁned gear

1.1
1.0

Surface hardened gear

0.9
0.8
0.5

1

2
4
6 8 1020 25 (40) (60)
Velocity on the Pitch circle N(m/s)

5.2.10 Work hardening factor ZW
Hardness ratio factor is applied to engagement between gear and pinion(1) which is hardened ground.
Calculation for Work hardening factor Zw is as follow.
(Refer to Fig. 8)
ZW = 1.2 −Surface hardened gear

0.9
0.8

Find Roughness factor based on average roughnessof flank Rmaxm(μ m) from Fig. 6 for 2 types of gears. Use
the following formula to obtain the average of maximum height of profile roughness of flank Rmaxm from
Rmax1, Rmax2 and centre distance a(mm). (Meaning of
Rmax1, Rmax2 is Maximum height if profile roughness offlank inclusive of the effects of warm up and test run.)

Roughness factor ZR

1: in case Z İ = εα εβ ＞ 1....................... (31)		

5.2.8 Roughness factor ZR

Lubricating speed factor ZVεβHelical gear : in case Z İ = 1 − εβ + εα εβ ≦ 1.......... (30)

HB2 − 130
. ....................................................... (35)1700

Hereby0100
200
Kinematic viscosity (cSt) at 50ºC

300(1) Thermal refined gear (1): Use solid line in Fig. 5.
(2) Surface hardened gear: Use broken line in Fig. 5.
Note (1)	Thermal refined gear includes gear with
quenching, tempering and normalizing.
Remark:	Casting steel gear is equivalent to thermal
refined gear.

: Hardness of gear flank (indicated by Brinell
hardness)
However
Gear with conditions that cannot match above (35)
and 130 ≦ HB2 ≦ 470, Pinion to be
ZW =1.0................................................................................... (36)
HB2

131

Work hardening factor ZW

1.20

Table 8. Face load factor for contact stressFig. 8 Work hardening factor

Supporting method

1.15

b
d1

1.10
1.05
1.00
130

200

300

400

470

Flank hardness for gear HB2

Note (1)	Flank roughness of pinion is RMAX1 ≦ 6μ m
when engaged with stipulated gear.
5.2.11 Dimension factor KHX for Surface durability
If Tooth profile and gear size increases, Surface durability also increases but has a tendency to increase
disproportionately. Due to insufficient data at the
moment Dimension factor
KHX = 1.0............................................................................... (37)5.2.12 Face load factor for contact stress KHβ
Obtain Face load factor for contact stress for Surface
durability using following formula.
(a) I f unable to estimate tooth contact conditions
when load is applied to gear. Obtain Tooth traceload distribution factor from ratio (b/d1) between
Facewidth b and Reference diameter d1 of pinion
and from method of gear support from Table 8.
(b) S atisfactory tooth contact when load is applied togear.
Tooth trace load distribution factor KHβ for Surface
durability depends on level of modification compared to used load (reference value). When calculating modifications on Tooth trace for following cases,
analyse all causes that influence Tooth bearing whenload is applied. Apply modifications of Proper Tooth
trace for gear, Helix angle, Axial parallelism. Warm up
and test run is performed and confirm Tooth bearing
is secured during operation.
KHβ =1.0 ～ 1.2..................................................................... (38)5.2.13 Dynamic factor KV (common)
Obtain Dynamic factor based on gear accuracy andCircumferential speed on the Reference pitch circle
from 5.1.7 of Table 3.
5.2.14 Overload factor Ko (common)
Obtain the overload factor from 5.1.8 - Table 4.
5.2.15 Safety factor for flank damage (Pitting) SH
A minimum Safety factor for flank damage (Pitting)value of 1.15 is necessary even though it is difficult tofind fixed value of internal and external factors.

132

Support on both end
Bearing is on
Balanced to
one side andboth bearings stiffness of axis
is increased.

Bearing is on
one side andless stiffnessof axis.

Unbalanced
support

0.2

1.0

1.0

1.1

1.2

0.4

1.0

1.1

1.3

1.45

0.6

1.05

1.2

1.5

1.65

0.8

1.1

1.3

1.7

1.85

1.0

1.2

1.45

1.85

2.0

1.2

1.3

1.6

2.0

2.15

1.4

1.4

1.8

2.1

-

1.6

1.5

2.05

2.2

-

1.8

1.8

-

-

-

2.0

2.1

-

-

-

Remark 1.	b is Effective facewidth for Spur and Helicalgears. For Double helical gear, b is lengthof facewidth inclusive of cutter groove at
centre of gear.
Remark 2.	Tooth contact has to be satisfactory without load.
Remark 3.	Inapplicable to Idler gear and pinion (Idler)
engaged with gears.
5.2.16 Allowable hertz stress σ Hlim
Refer to Tables 9 ～ 12 to find the Allowable hertz
stress. For values not listed, use interpolation. Meaning of flank’s hardness is hardness near Pitch circle.

Table 9. Gear without surface hardening
Hardness of flankMaterials (Arrow marks are for references only)

HB

SC37
SC42
SC46
SC49
SCC3

Casting steel

S25C

S35C

Carbon steel for structural
use with Normalizing

S43C
S48C
S53C
S58C

S35C

Carbon steel for structural
use with Quenching andTempering

S43C

S48C
S53C
S58C

SMn443
Alloy steel for structural
use with Carburizing,
Quenching andTempering

HV

SNC836
SCM435
SCM440

SNCM439

120
130
140
150
160
170
180
190
200
210
220
230
240
250
160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
310
320
330
340
350
220
230
240
250
260
270
280
290
300
310
320
330
340
350
360
370
380
390
400

126
136
147
157
167
178
189
200
210
221
231
242
252
263
167
178
189
200
210
221
231
242
252
263
273
284
295
305
316
327
337
347
358
369
231
242
252
263
273
284
295
305
316
327
337
347
358
369
380
391
402
413
424

Lower limit oftensile strength
kgf/mm2(reference)
37
42
46
49
55
60
39
42
45
48
51
55
58
61
64
68
71
74
77
81
51
55
58
61
64
68
71
74
77
81
84
87
90
93
97
100
103
106
110
113
71
74
77
81
84
87
90
93
97
100
103
106
110
113
117
121
126
130
135

σ Flimkgf/mm2

σ Hlimkgf/mm2

10.4
12.0
13.2
14.2
15.8
17.2
13.8
14.8
15.8
16.5
17.6
18.4
19.0
19.5
20
20.5
21
21.5
22
22.5
18.2
19.4
20.2
21
22
23
23.5
24
24.5
25
25.5
26
26
26.5

34
35
36
37
39
40
41.5
42.5
44
45
46.5
47.5
49
50
51.5
52.5
54
55
56.5
57.5
51
52.5
54
55.5
57
58.5
60
61
62,5
64
65.5
67
68.5
70
71
72.5
74
75.5
77
78.5
70
71.5
73
74.5
76
77.5
79
81
82.5
84
85.5
87
88.5
90
92
93.5
95
96.5
98

25
26
27.5
28.5
29.5
31
32
33
34
35
36.5
37.5
39
40
41

133

Table 9. Gear with High frequency induction hardening (continued)
Materials (Arrow marks are for references only)

S43C

Conditions of Heat treatment
before High-frequency
induction hardening
Normalizing

S48C
Carbon steel for
structural use
S48C

Induction hardening andTempering

S43C

SCM440Alloy steel for
structural use

SMn443
SNCM439

Induction hardening andtempering

SCM435
SNC836

Normalizing

Carbon steel for
structural use

S43C
S48C

Induction hardening andTempering

Alloy steel for
structural use

SMn443
SCM435
SCM440
SNC836
SNCM439

Induction hardening andTemperingCore hardness
HB

HV

160
180
220
240
200
210
220
230
240
250
230
240
250
260
270
280
290
300
310
320

167
189
231
252
210
221
231
242
252
263
242
252
263
273
284
295
305
316
327
337

Flank hardness (1)HV

σ Flim(2)kgf/mm2

Above 550〃〃〃
Above 550〃〃〃〃〃
Above 550
〃
〃
〃
〃
〃
〃
〃
〃
〃
420
440
460
480
500
520
540
560
580
Above 600
500
520
540
560
580
600
620
640
660
Above 680
500
520
540
560
580
600
620
640
660
Above 680

21
21
21.5
22
23
23.5
24
24.5
25
25
27
28
29
30
31
32
33
34
35
36.5

σ Hlimkgf/mm2

77
80
82
85
87
90
92
93.5
95
96
96
99
101
103
105
106.5
107.5
108.5
109
109.5
109
112
115
117
119
121
123
124
125
126

Note(1) When flank hardness is low, use σ Flim value which is equivalent to gear without hardened surface.
Note(2)	When gear has defects such as quenching cracks, insufficient hardening depth and uneven hardness, precaution is necessary as values of σ Flim may become significantly lower compared with Tables 9 and 10.
		Values in Tables 9 and 10 are shown for full quenching at bottomland. Assuming insufficient quenching atbottomland, value will be 75% from Table 9 and 10.

134

Table 10. Gear with case hardening

Carbon steel formachinestructural use

Core hardness (1)
HB
HV
140
147
150
157
160
167
170
178
180
189
190
200

Effective carburizing
depth (2)Materials (Arrow marks are references only)

Flank hardness
HV

σ Hlimkgf/mm2

580
600
620
640
660
680
700
720
740
760
780
800

S15C
S15CK
Relatively shallow depth (A)SCM415σ Flim(2)kgf/mm2
18.2
19.6
21
22
23
24

220
230
240
250
260
270
280
290
300
310
320
330
340
350
360
370

SCM415

SCM420

SNC815
SNCM420

231
242
252
263
273
284
295
305
316
327
337
347
358
369
380
390

115
117
118
119
120
120
120
119
118
117
115
113
34
36
38
39
41
42.5
44
45
46
47
48
49
50
51
51.5
52

580
600
620
640
660
680
700
720
740
760
780
800
580
600
620
640
660
680
700
720
740
760
780
800

Alloy steel formachinestructural use
Relatively shallow depth (B)SCM415(21)SCM420(22)SNC415(21)SNC815(22)SNCM420(23)Relatively deeper than above
B

131
134
137
138
138
138
138
137
136
134
132
130
156
160
164
166
166
166
164
161
158
154
150
146

Note (1) Relatively shallow effective case depth refers to below A and relatively deeper depth refers to B or more.
Meaning of Effective case depth is hardness of up to HV513 (HRC50). Depth for Ground gear is after process.
ModuleEffective depth

1.5

2

3

4

5

6

8

10

15

20

25

A

0.2

0.2

0.3

0.4

0.5

0.6

0.7

0.9

1.2

1.5

1.8

B

0.3

0.3

0.5

0.7

0.8

0.9

1.1

1.4

2.0

2.5

3.4

Remark: Especially in engagement between gears, we recommend providing bigger Safety factor SH., starting pointof Maximum inner shear-stress force at inner gear tooth from surface pressure of flank is deeper than thedepth of Case hardening which affects effectiveness of Carburizing depth.

135

Table 12. Nitro-carburizing (1)Table 11. Nitriding gear (1)Flank hardness(reference)

Material
Nitriding
steel

SACM
645 andothersσ Hlim kgf/mm2

Above HV 650

Normal120

Sustained period of
Nitriding treatment

130 140

Material

Carbon steel andAlloy steel for
structural use

Note (1)	Applicable to gear with proper Nitriding
depth and hardened surface for improving
Surface durability. We recommend providing a larger safety factor than usual when
Surface hardness is lower than above table.
Starting point of Maximum shear-stressforce at inner gear tooth is deeper thandepth of Nitriding.

Alloy steel for structural
use without
Nitriding steel

Nitriding steel SACM645

Above
HV650

Above
HV650

Below 10

10 - 20

Above 20

100

90

80

4

110

100

90

6

120

110

100

0

Gear ratio
1

60

2

50

3
4

Core hardness

σ FlimHB

HV

kgf/mm2

220

231

30

240

252

33

260

273

36

280

295

38

300

316

40

320

337

42

340

358

44

360

380

46

220

231

32

240

252

35

260

273

38

280

295

41

6

300

316

44

5

Note (1)	Applicable to gear with proper Nitriding
depth for improving Surface durability.
However Nitriding layer is extremely thinfrom Nitro-carburizing, use σ f lim value of thegear without hardened surface.

136

Relative curvature radius (mm)(2)

Fig. 9 Relative curvature radius
Relative curvature radius (mm)Material

2

σ Hlim kgf/mm2

Note (1)	Applicable to Salt bath and Gas Nitro-carburizing gears.
(2) Use Fig. 9 to obtain Relative curvature radius
Remark. Use properly adjusted gear material for core.

Table 13. Nitriding gear (1)Flankhardness(reference)

Nitriding
period (h)

20
20

5
6

30

20
15

10
10

40

10
8
7

80

100

150

200
300
400
Centre distance A (mm)500 600 700 800