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Elements of Metric Gear Technology

17.2 Surface Strength Of Spur And Helical Gears
The following equations can be applied to both spur and helical gears, including double helical and internal gears, used in power transmission. The general range of application is:

Module: m, 1.5 to 25 mm
Pitch Circle: d, 25 to 3200 mm
Linear Speed: v, less than 25 m/sec
Rotating Speed: n, less than 3600 rpm

17.2.1 Conversion Formulas
To rate gears, the required transmitted power and torques must be converted to tooth forces. The same conversion formulas, Equations (17-1), (17-2) and (17-3), of SECTION 17 are applicable to surface strength calculations.

17.2.2 Surface Strength Equations
As stated in SECTION 17.1, the tangential force, Ft, is not to exceed the allowable tangential force, Ft lim. The same is true for the allowable Hertz surface stress, σH lim. The Hertz stress σH is calculated from the tangential force, Ft. For an acceptable design, it must be less than the allowable Hertz stress σH lim. That is:
Equation 17-12

The tangential force, σH lim, in kgf, at the standard pitch circle, can be calculated from Equation (17-13).

Equation 17-13

The Hertz stress σH (kgf/mm²) is calculated from Equation (17-14), where u is the ratio of numbers of teeth in the gear pair.

Equation 17-14

The "+" symbol in Equations (17-13) and (17-14) applies to two external gears in mesh, whereas the "–" symbol is used for an internal gear and an external gear mesh. For the case of a rack and gear, the quantity u/(u ± 1) becomes 1.

17.2.3 Determination Of Factors In The Surface Strength Equations


17.2.3.A Effective Tooth Width, bH (mm)
The narrower face width of the meshed gear pair is assumed to be the effective width for surface strength. However, if there are tooth modifications, such as chamfer, tip relief or crowning, an appropriate amount should be subtracted to obtain the effective tooth width.

17.2.3.B Zone Factor, ZH
The zone factor is defined as:
Equation 17-15

where: βb = tan-1 (tan β cos αt)

Figure 17-2 Figure 17-2
The zone factors are presented in Figure 17-2 for tooth profiles per JIS B 1701, specified in terms of profile shift coefficients x1 and x2, numbers of teeth z1 and z2 and helix angle β.

The "+" symbol in Figure 17-2 applies to external gear meshes, whereas the "–" is used for internal gear and external gear meshes.

17.2.3.C Material Factor, ZM
Equation 17-16

where:
Table 17-9 Table 17-9
ν = Poisson's Ratio, and E = Young's Modulus

Table 17-9 contains several combinations of material and their material factor.

17.2.4 Contact Ratio Factor, Zε
This factor is fixed at 1.0 for spur gears.
For helical gear meshes, Zε is calculated as follows:
Equation 17-17

17.2.5 Helix Angle Factor, Zβ
This is a difficult parameter to evaluate. Therefore, it is assumed to be 1.0 unless better information is available.

Equation 17-18

17.2.6 Life Factor, KHL
Table 17-10 Table 17-10
This factor reflects the number of repetitious stress cycles. Generally, it is taken as 1.0. Also, when the number of cycles is unknown, it is assumed to be 1.0. When the number of stress cycles is below 10 million, the values of Table 17-10 can be applied.

Figure 17-3 Figure 17-3
17.2.7 Lubricant Factor, ZL
The lubricant factor is based upon the lubricant's kinematic viscosity at 50°C. See Figure 17-3.

Figure 17-4 Figure 17-4
17.2.8 Surface Roughness Factor, ZR
This factor is obtained from Figure 17-4 on the basis of the average roughness Rmaxm (µm). The average roughness is calculated by Equation (17-19) using the surface roughness values of the pinion and gear, Rmax1 and Rmax2, and the center distance, a, in mm.

Equation 17-19

Figure 17-5 Figure 17-5
17.2.9 Sliding Speed Factor, ZV
This factor relates to the linear speed of the pitch line. See Figure 17-5.

17.2.10 Hardness Ratio Factor, ZW
The hardness ratio factor applies only to the gear that is in mesh with a pinion which is quenched and ground. The ratio is calculated by Equation (17-20).
Equation 17-20

where: HB2 = Brinell hardness of gear range: 130 ≤ HB2 ≤ 470
If a gear is out of this range, the ZW is assumed to be 1.0.

17.2.11 Dimension Factor, KHX
Because the conditions affecting this parameter are often unknown, the factor is usually set at 1.0.
Equation 17-21

Table 17-11 Table 17-11
17.2.12 Tooth Flank Load Distribution Factor, KH β
(a) When tooth contact under load is not predictable: This case relates the ratio of the gear face width to the pitch diameter, the shaft bearing mounting positions, and the shaft sturdiness. See Table 17-11. This attempts to take into account the case where the tooth contact under load is not good or known.

(b) When tooth contact under load is good: In this case, the shafts are rugged and the bearings are in good close proximity to the gears, resulting in good contact over the full width and working depth of the tooth flanks. Then the factor is in a narrow range, as specified below:
Equation 17-22

17.2.13 Dynamic Load Factor, KV
Dynamic load factor is obtainable from Table 17-3 according to the gear's precision grade and pitch line linear speed.

Table 17-3 Table 17-3
Table 17-4 Table 17-4
Table 17-12 Table 17-12
Table 17-12-2 Table 17-12
Table 17-13 Table 17-13
Table 17-14 Table 17-14
Table 17-14A Table 17-14a
Table 17-15 Table 17-15
Table 17-16 Table 17-16
Table 17-16A Table 17-16a
Table 17-16B Table 17-6b
Figure 17-6 Table 17-6
17.2.14 Overload Factor, Ko
The overload factor is obtained from either Equation (17-11) or from Table 17-4.

17.2.15 Safety Factor For Pitting,SH
The causes of pitting involves many environmental factors and usually is difficult to precisely define. Therefore, it is advised that a factor of at least 1.15 be used.

17.2.16 Allowable Hertz Stress, σH lim
The values of allowable Hertz stress for various gear materials are listed in Tables 17-12, 17-2 part 2, 17-13, 17-14, 17-14A, 17-15, 17-16, 17-16A and 17-16B. Values for hardness not listed can be estimated by interpolation. Surface hardness is defined as hardness in the pitch circle region. Also see Figure 17-6.

17.3 Bending Strength Of Bevel Gears
This information is valid for bevel gears which are used in power transmission in general industrial machines. The applicable ranges are:

Module: m, 1.5 to 25 mm
Pitch Diameter: d, less than 1600 mm for straight bevel gears, less than 1000 mm for spiral bevel gears
Linear Speed: v, less than 25 m/sec
Rotating Speed: n, less than 3600 rpm

17.3.1 Conversion Formulas
In calculating strength, tangential force at the pitch circle, Ftm, in kgf; power, P, in kW, and torque, T, in kgf • m, are the design criteria. Their basic relationships are expressed in Equations (17-23) through (17-25).

Equation 17-23
Equation 17-24
Equation 17-25

17.3.2 Bending Strength Equations
The tangential force, Ftm, acting at the central pitch circle should be equal to or less than the allowable tangential force, Ftm lim, which is based upon the allowable bending stress σF lim. That is:
Equation 17-26

The bending stress at the root, σF, which is derived from Ftm should be equal to or less than the allowable bending stress σF lim.
Equation 17-27

The tangential force at the central pitch circle, Ftm lim (kgf), is obtained from Equation (17-28).
Equation 17-28

where: βm : Central spiral angle (degrees)
m : Radial module (mm)
Ra : Cone distance (mm)
And the bending strength σF (kgf/mm2) at the root of tooth is calculated from Equation (17-29).

Equation 17-29

17.3.3 Determination of Factors in Bending Strength Equations


17.3.3.A Tooth Width, b (mm)
The term b is defined as the tooth width on the pitch cone, analogous to face width of spur or helical gears. For the meshed pair, the narrower one is used for strength calculations.

Figure 17-7 Figure 17-7
Figure 17-8 Figure 17-8
Figure 17-9 Figure 17-9
17.3.3.B Tooth Profile Factor, YF
The tooth profile factor is a function of profile shift, in both the radial and axial directions. Using the equivalent (virtual) spur gear tooth number, the first step is to determine the radial tooth profile factor, YFO, from Figure 17-8 for straight bevel gears and Figure 17-9 for spiral bevel gears. Next, determine the axial shift factor, K, with Equation (17-33) from which the axial shift correction factor, C, can be obtained using Figure 17-7. Finally, calculate YF by Equation (17-30).
Equation 17-30

Should the bevel gear pair not have any axial shift, then the coefficient C is 1, as per Figure 17-7. The tooth profile factor, YF, per Equation (17-31) is simply the YFO. This value is from Figure 17-8 or 17-9, depending upon whether it is a straight or spiral bevel gear pair. The graph entry parameter values are per Equation (17-32).

Equation 17-31
Equation 17-32

where: ha = Addendum at outer end (mm)
ha0 = Addendum of standard form (mm)
m = Radial module (mm)

The axial shift factor, K, is computed from the formula:
Equation 17-33

17.3.3.C Load Distribution Factor, Yε
Load distribution factor is the reciprocal of radial contact ratio.
Equation 17-34

The radial contact ratio for a straight bevel gear mesh is:
Equation 17-35

Table 17-17 Table 17-17
Table 17-18 Table 17-18
Table 17-19 Table 17-19
See Tables 17-17, 17-18 and 17-19 for some calculating examples of radial contact ratio for various bevel gear pairs.

17.3.3.D Spiral Angle Factor, Yβ
The spiral angle factor is a function of the spiral angle. The value is arbitrarily set by the following conditions:
Equation 17-36

Table 17-20 Table 17-20
Table 17-2 Table 17-2
17.3.3.E Cutter Diameter Effect Factor, YC
This factor of cutter diameter, YC, can be obtained from Table 17-20 by the value of tooth flank length, b / cos βm (mm), over cutter diameter. If cutter diameter is not known, assume YC = 1.00.

17.3.3.F Life Factor, KL
We can choose a proper life factor, KL, from Table 17-2 similarly to calculating the bending strength of spur and helical gears.

Table 17-21 Table 17-21
Table 17-22 Table 17-22
Table 17-23 Table 17-23
17.3.3.G Dimension Factor Of Root Bending Stress, KFX
This is a size factor that is a function of the radial module, m. Refer to Table 17-21 for values.

17.3.3.H Tooth Flank Load Distribution Factor, KM
Tooth flank load distribution factor, KM, is obtained from Table 17-22 or Table 17-23.

Table 17-24 Table 17-24
Table 17-24A Table 17-24a
Table 17-24B Table 17-24b
17.3.3.I Dynamic Load Factor, KV
Dynamic load factor, KV, is a function of the precision grade of the gear and the tangential speed at the outer pitch circle, as shown in Table 17-24. Also see Table 17-24A and Table 17-24B.

17.3.3.K Reliability Factor, KR
The reliability factor should be assumed to be as follows:
  1. General case: KR = 1.2
  2. When all other factors can be determined accurately: KR = 1.0
  3. When all or some of the factors cannot be known with certainty: KR = 1.4

Table 17-5 Table 17-5
Table 17-8 Table 17-8
17.3.3.L Allowable Bending Stress at Root, σF lim
The allowable stress at root σF lim can be obtained from Tables 17-5 through 17-8, similar to the case of spur and helical gears.

17.4 Surface Strength Of Bevel Gears
This information is valid for bevel gears which are used in power transmission in general industrial machines. The applicable ranges are:

Radial Module: m, 1.5 to 25 mm
Pitch Diameter: d, Straight bevel gear under 1600 mm
Spiral bevel gear under 1000 mm
Linear Speed: v, less than 25 m/sec
Rotating Speed: n, less than 3600 rpm

17.4.1 Basic Conversion Formulas
The same formulas of SECTION 17.3 apply.

17.4.2 Surface Strength Equations
In order to obtain a proper surface strength, the tangential force at the central pitch circle, Ftm, must remain below the allowable tangential force at the central pitch circle, Ftm lim, based on the allowable Hertz stress σF lim.
Equation 17-37
Alternately, the Hertz stress σH, which is derived from the tangential force at the central pitch circle must be smaller than the allowable Hertz stress σH lim.

Equation 17-38
The allowable tangential force at the central pitch circle, Ftm lim, in kgf can be calculated from Equation (17-39).
Equation 17-39

The Hertz stress, σH (kgf/mm2) is calculated from Equation (17-40).
Equation 17-40

17.4.3 Determination of Factors In Surface Strength Equations


17.4.3.A Tooth Width, b (mm)
This term is defined as the tooth width on the pitch cone. For a meshed pair, the narrower gear's "b" is used for strength calculations.

17.4.3.B Zone Factor, ZH
The zone factor is defined as:
Equation 17-41

Figure 17-10 Figure 17-10
If the normal pressure angle αn is 20°, 22.5° or 25°, the zone factor can be obtained from Figure 17-10.

17.4.3.C Material Factor, ZM
The material factor, ZM, is obtainable from Table 17-9.

17.4.3.D Contact Ratio Factor, Zε
The contact ratio factor is calculated from the equations below.
Equation 17-42
where: εα = Radial Contact Ratio
εβ = Overlap Ratio

17.4.3.E Spiral Angle Factor, Zβ
Little is known about these factors, so usually it is assumed to be unity.
Equation 17-43

Figure 17-11 Figure 17-11
17.4.3.F Life Factor, KHL
The life factor for surface strength is obtainable from Table 17-10.

17.4.3.G Lubricant Factor, ZL
The lubricant factor, ZL, is found in Figure 17-3.

17.4.3.H Surface Roughness Factor, ZR
The surface roughness factor is obtainable from Figure 17-11 on the basis of average roughness, Rmaxm, in µ m. The average surface roughness is calculated by Equation (17-44) from the surface roughnesses of the pinion and gear (Rmaxm1 and Rmaxm2), and the center distance, a, in mm.

Equation 17-44

17.4.3.I Sliding Speed Factor, ZV
The sliding speed factor is obtained from Figure 17-5 based on the pitch circle linear speed.

17.4.3.J Hardness Ratio Factor, ZW
The hardness ratio factor applies only to the gear that is in mesh with a pinion which is quenched and ground. The ratio is calculated by Equation (17-45).
Equation 17-45

where Brinell hardness of the gear is: 130 ≤ HB2 ≤ 470

If the gear's hardness is outside of this range, ZW is assumed to be unity.
Equation 17-46

17.4.3.K Dimension Factor, KHX
Since, often, little is known about this factor, it is assumed to be unity.
Equation 17-47

Table 17-25 Table 17-25
Table 17-26 Table 17-26
Table 17-26A Table 17-26a
Table 17-26B Table 17-26b
17.4.3.L Tooth Flank Load Distribution Factor, KH β
Factors are listed in Tables 17-25 and 17-26. If the gear and pinion are unhardened, the factors are to be reduced to 90% of the values in the table. Also see Tables 17-26A and 17-26B.

17.4.3.M Dynamic Load Factor, KV
The dynamic load factor can be obtained from Table 17-24.

17.4.3.N Overload Factor, K0
The overload factor can be computed by Equation 17-11 or found in Table 17-4.

17.4.3.O Reliability Factor, CR
The general practice is to assume CR to be at least 1.15.

17.4.3.P Allowable Hertz Stress, σH lim
The values of allowable Hertz stress are given in Tables 17-12 through 17-16.

17.5 Strength Of Worm Gearing
This information is applicable for worm gear drives that are used to transmit power in general industrial machines with the following parameters:
Axial Module: mx, 1 to 25 mm
Pitch Diameter of Worm Gear: d2, less than 900 mm
Sliding Speed: vs, less than 30 m/sec
Rotating Speed, Worm Gear: n2, less than 600 rpm

17.5.1 Basic Formulas:
Equation 17-48

17.5.2 Torque, Tangential Force and Efficiency
(1) Worm as Driver Gear (Speed Reducing)
Equation 17-49

where:
T2 = Nominal torque of worm gear (kg • m)
T1 = Nominal torque of worm (kgf • m)
Ft = Nominal tangential force on worm gear's pitch circle (kgf)
d2 = Pitch diameter of worm gear (mm)
u = Teeth number ratio = z2 /zw
ηR = Transmission efficiency, worm driving (not including bearing loss, lubricant agitation loss, etc.)
µ= Friction coefficient

(2) Worm Gear as Driver Gear (Speed Increasing)
Equation 17-50

where: ηI = Transmission efficiency, worm gear driving (not including bearing loss, lubricant agitation loss, etc.)

Figure 17-12 Figure 17-12
Table 17-27 Table 17-27
17.5.3 Friction Coefficient, µ
The friction factor varies as sliding speed changes. The combination of materials is important. For the case of a worm that is carburized and ground, and mated with a phosphorous bronze worm gear, see Figure 17-12. For some other materials, see Table 17-27.

For lack of data, friction coefficient of materials not listed in Table 17-27 are very difficult to obtain. H.E. Merritt has offered some further information on this topic. See Reference 9.

17.5.4 Surface Strength of Worm Gearing Mesh
(1) Calculation of Basic Load
Provided dimensions and materials of the worm pair are known, the allowable load is as follows:
Equation 17-51
Equation 17-52

(2) Calculation of Equivalent Load
The basic load Equations (17-51) and (17-52) are applicable under the conditions of no impact and the pair can operate for 26000 hours minimum. The condition of "no impact" is defined as the starting torque which must be less than 200% of the rated torque; and the frequency of starting should be less than twice per hour.

An equivalent load is needed to compare with the basic load in order to determine an actual design load, when the conditions deviate from the above.

Equivalent load is then converted to an equivalent tangential force, Fte, in kgf:

Equation 17-53

and equivalent worm gear torque, T2e, in kgf • m:
Equation 17-54

(3) Determination of Load
Under no impact condition, to have life expectancy of 26000 hours, the following relationships must be satisfied:
Equation 17-55
For all other conditions:
Equation 17-56

NOTE: If load is variable, the maximum load should be used as the criterion.

17.5.5 Determination of Factors in Worm Gear Surface Strength Equations


Figure 17-13

17.5.5.A Tooth Width of Worm Gear, b2 (mm)
Tooth width of worm gear is defined as in Figure 17-13.

17.5.5.B Zone Factor, Z
Equation 17-57

Table 17-28 Table 17-28
where: Basic Zone Factor is obtained from Table 17-28.
Equation

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