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ELEMENTS OF METRIC GEAR TECHNOLOGY
T-150
SECTION 17 STRENGTH AND DURABILITY OF GEARS
The strength of gears is generally expressed in terms of bending strength and surface durability. These are independent criteria which can have differing criticalness, although usually both are important. Discussions in this section are based upon equations published in the literature of the Japanese Gear Manufacturer Association (JGMA). Reference is made to the following JGMA specifications:
Specifications of JGMA:
JGMA 401-01 Bending Strength Formula of Spur Gears and Helical Gears JGMA 402-01 Surface Durability Formula of Spur Gears and Helical Gears JGMA 403-01 Bending Strength Formula of Bevel Gears JGMA 404-01 Surface Durability Formula of Bevel Gears JGMA 405-01 The Strength Formula of Worm Gears
Generally, bending strength and durability specifications are applied to spur and helical gears (including double helical and internal gears) used in industrial machines in the following range:
Module: m 1.5 to 25 mm Pitch Diameter: d 25 to 3200 mm Tangential Speed: v less than 25 m/sec Rotating Speed: n less than 3600 rpm Conversion Formulas: Power, Torque and Force
Gear strength and durability relate to the power and forces to be transmitted. Thus, the equations that relate tangential force at the pitch circle, Ft (kgf), power, P (kw), and torque, T (kgf m) are basic to the calculations. The relations are as follows:
102 P 1.95 x 106 P 2000 T Ft = = = (17-1) v dw n dw
Ft v 10 6 P = = Ft dwn (17-2) 102 1.95
Ft dw 974 P T = = (17-3) 2000 n
where: v : Tangential Speed of Working Pitch Circle (m/sec) dwn v = 19100
dw : Working Pitch Diameter (mm) n : Rotating Speed (rpm)
17.1 Bending Strength Of Spur And Helical Gears
In order to confirm an acceptable safe bending strength, it is necessary to analyze the applied tangential force at the working pitch circle, Ft , vs. allowable force, Ft lim. This is stated as:
Ft Ft lim (17-4)
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ELEMENTS OF METRIC GEAR TECHNOLOGY
T-151
It should be noted that the greatest bending stress is at the root of the flank or base of the dedendum. Thus, it can be stated:
sF = actual stress on dedendum at root sF lim = allowable stress
Then Equation (17-4) becomes Equation (17-5)
sF sF lim (17-5)
Equation (17-6) presents the calculation of Ft lim:
mnb K LK FX
1 Ft lim = sF lim () (kgf) (17-6) YFYeYb KV KO SF Equation (17-6) can be converted into stress by Equation (17-7):
YFYeYb KV KO sF = Ft () SF (kgf/mm2) (17-7) mnb KLKFX
17.1.1 Determination of Factors in the Bending Strength Equation If the gears in a pair have different blank widths, let the wider one be bw and the narrower one be bs . And if: bw bs mn, bw and bs can be put directly into Equation (17-6). bw bs > mn, the wider one would be changed to bs + mn and the narrower one, bs , would be unchanged.
17.1.2 Tooth Profile Factor, YF
The factor YF is obtainable from Figure 17-1 based on the equivalent number of teeth, zv, and coefficient of profile shift, x, if the gear has a standard tooth profile with 20 pressure angle, per JIS B 1701. The theoretical limit of undercut is shown. Also, for profile shifted gears the limit of too narrow (sharp) a tooth top land is given. For internal gears, obtain the factor by considering the equivalent racks.
17.1.3 Load Distribution Factor, Ye
Load distribution factor is the reciprocal of radial contact ratio.
1 Ye = (17-8) ea
Table 17-1 shows the radial contact ratio of a standard spur gear.
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ELEMENTS OF METRIC GEAR TECHNOLOGY
T-152
Fig. 17-1 Chart of Tooth Profile Factor, YF
Normal STD PressureAngle an = 20Addendum ha = 1.00 mnDedendum hf = 1.25 mnCorner Radius of Cutterg = 0.375 mn
Theo
retica
l Und
ercut
Limit
Narrow Tooth Top Limit
Toot
h Pr
ofile
Fac
tor,
Y F (
Form
Fac
tor)
3.8
3.7
3.6
3.5
3.4
3.3
3.2
3.1
3.0
2.9
2.8
2.7
2.6
2.5
2.4
2.3
2.2
2.1
2.0
1.9
1.8
3.5
3.4
3.3
3.2
3.1
3.0
2.9
2.8
2.7
2.6
2.5
2.4
2.3
2.2
2.1
2.0
1.9
1.810 11 12 13 14 15 16 17 18 19 20 25 30 35 40 45 50 60 80 100 200 400
X = 0.5
X = 0.4
X = 0.3
X = 0.2X = 0.1
X = 0X = 0.1
X = 0.2X = 0.3
X = 0.4X = 0.5X = 0.6
X = 0.7X = 0.8
X = 0.9X = 1.0
zEquivalent Spur Gear Number of Teeth zv = cos3 b
Metric
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ELEMENTS OF METRIC GEAR TECHNOLOGY
T-153
120
1.87
1
1.92
6
110
1.86
3
1.86
7
1.99
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95 1.84
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0
1.85
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1.85
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80 1.82
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1.83
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1.83
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1.83
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1.83
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1.84
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1.84
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1.90
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60 1.78
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1.79
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1.79
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1.80
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1.80
5
1.80
9
1.81
3
1.81
6
1.81
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1.82
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1.82
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1.88
3
50 1.75
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1.76
3
1.77
0
1.77
6
1.78
1
1.78
6
1.79
0
1.79
4
1.79
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1.80
1
1.80
4
1.80
9
1.81
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1.86
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55 1.77
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1.77
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1.78
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1.78
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1.79
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1.79
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1.80
2
1.80
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1.80
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1.81
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1.81
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1.82
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1.87
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45 1.73
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1.74
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1.75
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1.76
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1.77
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1.77
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1.78
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1.84
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30 1.65
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1.67
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1.68
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1.69
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1.70
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1.73
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1.74
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1.74
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1.75
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1.81
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1.72
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1.72
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1.73
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1.74
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1.74
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1.75
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1.76
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1.76
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1.76
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1.77
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1.77
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1.77
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1.83
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12 1.42
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1.45
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1.48
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1.51
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1.53
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1.56
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1.64
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1.70
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1.51
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1.56
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1.58
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1.59
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1.60
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1.62
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1.63
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1.63
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1.64
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1.64
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1.65
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1.65
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1.66
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1.66
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1.66
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1.67
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1.67
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1.73
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20 1.55
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1.58
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1.60
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1.62
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1.63
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1.64
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1.65