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Gear Forces and Strength & Durability of Gears, Elements of Metric Gear Technology (Cont.)

Figure 15-2 Figure 15-2
Table 15-3 Figure 15-3
Shown in Figure 15-2 is an example of a chart measuring tooth profile error and lead error using a Zeiss UMC 550 tester. Also see Table 15-3.

15.1.5. Outside Diameter Runout and Lateral Runout
To produce a high precision gear requires starting with an accurate gear blank. Two criteria are very important:
  1. Outside diameter (OD) runout.
  2. Lateral (side face) runout.
Table 15-4 Table 15-4
The lateral runout has a large impact on the gear tooth accuracy. Generally, the permissible runout error is related to the gear size. Table 15-4 presents equations for allowable values of OD runout and lateral runout.

Table 15-5 Table 15-5
15.2 Accuracy Of Bevel Gears
JIS B 1704 regulates the specification of a bevel gear's accuracy. It also groups bevel gears into 9 grades, from 0 to 8.

There are 4 types of allowable errors:
  1. Single Pitch Error.
  2. Pitch Variation Error.
  3. Accumulated Pitch Error.
  4. Runout Error of Teeth (pitch circle).
These are similar to the spur gear errors.
  1. Single Pitch Error, (fpt)
    The deviation between actual measured pitch value between any adjacent teeth and the theoretical circular pitch at the central cone distance.
  2. Pitch Variation Error, (fpu)
    Absolute pitch variation between any two adjacent teeth at the central cone distance.
  3. Accumulated Pitch Error, (Fp)
    Difference between theoretical pitch sum of any teeth interval, and the summation of actual measured pitches for the same teeth interval at the central cone distance.
  4. Runout Error of Teeth, (Fr)
    This is the maximum amount of tooth runout in the radial direction, measured by indicating a pin or ball placed between two teeth at the central cone distance. It is the pitch cone runout.
Table 15-5 presents equations for allowable values of these various errors.
Table 15-6 Table 15-6
The equations of allowable pitch variations are in Table 15-6.

The equations of allowable pitch variations are in Table 15-6.

Besides the above errors, there are seven specifications for bevel gear blank dimensions and angles, plus an eighth that concerns the cut gear set:
  1. The tolerance of the blank outside diameter and the crown to back surface distance.
  2. The tolerance of the outer cone angle of the gear blank.
  3. The tolerance of the cone surface runout of the gear blank.
  4. The tolerance of the side surface runout of the gear blank.
  5. The feeler gauge size to check the flatness of blank back surface.
  6. The tolerance of the shaft bore dimension deviation of the gear blank.
  7. The contact band of the tooth mesh.
Table 15-7 Table 15-7
Figure 15-3 Figure 15-3
Item 8 relates to cutting of the two mating gears' teeth. The meshing tooth contact area must be full and even across the profiles. This is an important criterion that supersedes all other blank requirements.

15.3 Running (Dynamic) Gear Testing
An alternate simple means of testing the general accuracy of a gear is to rotate it with a mate, preferably of known high quality, and measure characteristics during rotation. This kind of tester can be either single contact (fixed center distance method) or dual (variable center distance method). This refers to action on one side or simultaneously on both sides of the tooth. This is also commonly referred to as single and double flank testing. Because of simplicity, dual contact testing is more popular than single contact. JGMA has a specification on accuracy of running tests.

  1. Dual Contact (Double Flank) Testing
    In this technique, the gear is forced meshed with a master gear such that there is intimate tooth contact on both sides and, therefore, no backlash. The contact is forced by a loading spring. As the gears rotate, there is variation of center distance due to various errors, most notably runout. This variation is measured and is a criterion of gear quality. A full rotation presents the total gear error, while rotation through one pitch is a tooth-to-tooth error. Figure 15-3 presents a typical plot for such a test.

  2. For American engineers, this measurement test is identical to what AGMA designates as Total Composite Tolerance (or error) and Tooth-to-Tooth Composite Tolerance. Both of these parameters are also referred to in American publications as "errors", which they truly are. Tolerance is a design value which is an inaccurate description of the parameter, since it is an error. Allowable errors per JGMA 116-01 are presented on the next page, in Table 15-7.

  3. Single Contact Testing
    In this test, the gear is mated with a master gear on a fixed center distance and set in such a way that only one tooth side makes contact. The gears are rotated through this single flank contact action, and the angular transmission error of the driven gear is measured. This is a tedious testing method and is seldom used except for inspection of the very highest precision gears.


In designing a gear, it is important to analyze the magnitude and direction of the forces acting upon the gear teeth, shaft, bearings, etc. In analyzing these forces, an idealized assumption is made that the tooth forces are acting upon the central part of the tooth flank.

16.1 Forces In A Spur Gear Mesh
Figure 16-2 Figure 16-2
Figure 16-1The spur gear's transmission force Fn, which is normal to the tooth surface, as in Figure 16-1, can be resolved into a tangential component, Fu, and a radial component, Fr. Refer to Equation (16-1). The direction of the forces acting on the gears are shown in Figure 16-2. The tangential component of the drive gear, Fu1, is equal to the driven gear's tangential component, Fu2, but the directions are opposite. Similarly, the same is true of the radial components.

Equation 16-1

Figure 16-3 Figure 16-3
16.2 Forces In A Helical Gear Mesh
The helical gear's transmission force, Fn which is normal to the tooth surface, can be resolved into a tangential component, F1, and a radial component, Fr. See Figure 16.3
Equation 16-2

The tangential component, F1, can be further resolved into circular subcomponent, Fu, and axial thrust subcomponent, Fa.
Equation 16-3

Substituting and manipulating the above equations result in:
Equation 16-4

Figure 16-4 Figure 16-4
The directions of forces acting on a helical gear mesh are shown in Figure 16-4. The axial thrust sub-component from drive gear, Fa1, equals the driven gear's, Fa2, but their directions are opposite. Again, this case is the same as tangential components Fu1, Fu2 and radial components Fr1, Fr2.

16.3 Forces On A Straight Bevel Gear Mesh
Figure 16-5 Figure 16-5
The forces acting on a straight bevel gear are shown in Figure 16-5. The force which is normal to the central part of the tooth face, Fn, can be split into tangential component, Fn, and radial component, F1, in the normal plane of the tooth.
Equation 16-5

Again, the radial component, F1, can be divided into an axial force, Fa, and a radial force,Fr, perpendicular to the axis.
Equation 16-6

And the following can be derived:
Equation 16-7

Table 16-2 Table 16-2
Let a pair of straight bevel gears with a shaft angle Σ = 90°, a pressure angle αn = 20° and tangential force, Fu, to the central part of tooth face be 100. Axial force, Fa, and radial force, Fr, will be as presented in Table 16-2.

Figure 16-6 Figure 16-6
Figure 16-6 contains the directions of forces acting on a straight bevel gear mesh. In the meshing of a pair of straight bevel gears with shaft angle Σ = 90°, all the forces have relations as per Equations (16-8).

Equation 16-8

Figure 16-7 Figure 16-7
Table 16-3 Table 16-3
16.4 Forces In A Spiral Bevel Gear Mesh
Spiral gear teeth have convex and concave sides. Depending on which surface the force is acting on, the direction and magnitude changes. They differ depending upon which is the driver and which is the driven. Figure 16-7 presents the profile orientations of rightand left-hand spiral teeth. If the profile of the driving gear is convex, then the profile of the driven gear must be concave. Table 16-3 presents the concave/convex relationships.

16.4.1 Tooth Forces On A Convex Side Profile
The transmission force, Fn, can be resolved into components F1 and Ft as:
Equation 16-9

Figure 16-8 Figure 16-8
See Figure 16-8. Then F1 can be resolved into components Fu and Fs:
Equation 16-10

On the axial surface, Ft and Fs can be resolved into axial and radial subcomponents.
Equation 16-11

By substitution and manipulation, we obtain:
Equation 16-12

Figure 16-9 Figure 16-9
16.4.2 Tooth Forces On A Concave Side Profile
On the surface which is normal to the tooth profile at the central portion of the tooth, the transmission force, Fn, can be split into F1 and Ft as (see Figure 16-9):
Equation 16-13

And F1 can be separated into components Fu and Fs on the pitch surface:
Equation 16-14

So far, the equations are identical to the convex case. However, differences exist in the signs for equation terms. On the axial surface, Ft and Fs can be resolved into axial and radial subcomponents. Note the sign differences.
Equation 16-15

The above can be manipulated to yield:
Equation 16-16

Table 16-4 Table 16-4
Table 16-4 cont. Table 16-4
Let a pair of spiral bevel gears have a shaft angle Σ = 90°, a pressure angle αn = 20°, and a spiral angle βm = 35°. If the tangential force, Fu, to the central portion of the tooth face is 100, the axial thrust force, Fa, and radial force, Fr, have the relationship shown in Table 16-4, Table 16-4 (cont.)

The value of axial force, Fa, of a spiral bevel gear, from Table 16-4, could become negative. At that point, there are forces tending to push the two gears together. If there is any axial play in the bearing, it may lead to the undesirable condition of the mesh having no backlash. Therefore, it is important to pay particular attention to axial plays. From Table 16-4(2), we understand that axial thrust force, Fa, changes from positive to negative in the range of teeth ratio from 1.5 to 2.0 when a gear carries force on the convex side. The precise turning point of axial thrust force, Fa, is at the teeth ratio z1 / z2 = 1.57357.

Figure 16-10 Figure 16-10
Figure 16-10 describes the forces for a pair of spiral bevel gears with shaft angle Σ = 90°, pressure angle αn = 20°, spiral angle βm = 35° and the teeth ratio, u, ranging from 1 to 1.57357.

Figure 16-11 Figure 16-11
Figure 16-11 expresses the forces of another pair of spiral bevel gears taken with the teeth ratio equal to or larger than 1.57357.

16.5 Forces In A Worm Gear Mesh

Figure 16-12 Figure 16-12
16.5.1 Worm as the Driver
For the case of a worm as the driver, Figure 16-12, the transmission force, Fn, which is normal to the tooth surface at the pitch circle can be resolved into components F1 and Fr1.
Equation 16-17

At the pitch surface of the worm, there is, in addition to the tangential component, F1, a friction sliding force on the tooth surface, μFn. These two forces can be resolved into the circular and axial directions as:

and by substitution, the result is:
Equation 16-19

Figure 16-13 Figure 16-13
Figure 16-13 presents the direction of forces in a worm gear mesh with a shaft angle Σ = 90°. These forces relate as follows:
Equation 16-20

The coefficient of friction has a great effect on the transmission of a worm gear. Equation (16-21) presents the efficiency when the worm is the driver.
Equation 16-21

Figure 16-14 Figure 16-14
16.5.2 Worm Gear as the Driver
For the case of a worm gear as the driver, the forces are as in Figure 16-14 and per Equations (16-22).

Equation 16-22

When the worm and worm gear are at 90° shaft angle, Equations (16-20) apply. Then, when the worm gear is the driver, the transmission efficiency ηΙ is expressed as per Equation (16-23).

Equation 16-23

The equations concerning worm and worm gear forces contain the coefficient μ. This indicates the coefficient of friction is very important in the transmission of power.

16.6 Forces In A Screw Gear Mesh
The forces in a screw gear mesh are similar to those in a worm gear mesh. For screw gears that have a shaft angle Σ = 90°, merely replace the worm's lead angle γ, in Equation (16-22), with the screw gear's helix angle β1.

Figure 16-15 Figure 16-15
In the general case when the shaft angle is not 90°, as in Figure 16-15, the driver screw gear has the same forces as for a worm mesh. These are expressed in Equations (16-24).

Equation 16-24

Forces acting on the driven gear can be calculated per Equations (16-25).

Equation 16-25

Figure 16-16 Figure 16-16
If the Σ term in Equation (16-25) is 90°, it becomes identical to Equation (16-20). Figure 16-16 presents the direction of forces in a screw gear mesh when the shaft angle Σ = 90° and β1 = β2 = 45°.


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: 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:
Equation 17-1

Equation 17-2

Equation 17-3


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:
Equation 17-4

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:

σF = actual stress on dedendum at root
σF lim = allowable stress

Then Equation (17-4) becomes Equation (17-5)
Equation 17-5

Equation (17-6) presents the calculation of Ft lim:
Equation 17-6

Equation (17-6) can be converted into stress by Equation (17-7):
Equation 17-7

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:
bwbsmn,   bw and bs can be put directly into Equation (17-6).
bwbs > 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,Yε
Load distribution factor is the reciprocal of radial contact ratio.
Equation 17-8

Figure 17-1 Figure 17-1
Table 17-1 Table 17-1
Table 17-1 shows the radial contact ratio of a standard spur gear.

17.1.4 Helix Angle Factor, Yβ
Helix angle factor can be obtained from Equation (17-9).
= Equation 17-9

Table 17-2 Table 17-2
17.1.5 Life Factor, KL
We can choose the proper life factor, KL, from Table 17-2. The number of cyclic repetitions means the total loaded meshings during its lifetime.

17.1.6 Dimension Factor of Root Stress, KFX
Generally, this factor is unity.
Equation 17-10

Table 17-3 Table 17-3
17.1.7 Dynamic Load Factor, KV
Dynamic load factor can be obtained from Table 17-3 based on the precision of the gear and its pitch line linear speed.

Table 17-4 Table 17-4
17.1.8 Overload Factor, K0
Overload factor, K0, is the quotient of actual tangential force divided by nominal tangential force, Ft. If tangential force is unknown, Table 17-4 provides guiding values.

Equation 17-11

17.1.9 Safety Factor for Bending Failure, SF
Safety factor, SF, is too complicated to be decided precisely. Usually, it is set to at least 1.2.

Table 17-5 Table 17-5
Table 17-6 Table 17-6
Table 17-7 Table 17-7
Table 17-8 Table 17-8
Table 17-8A Table 17-8a
Table 17-8B Table 17-8b
17.1.10 Allowable Bending Stress At Root, δF lim
For the unidirectionally loaded gear, the allowable bending stresses at the root are shown in Tables 17-5 to 17-8. In these tables, the value of δF lim is the quotient of the tensile fatigue limit divided by the stress concentration factor 1.4. If the load is bidirectional, and both sides of the tooth are equally loaded, the value of allowable bending stress should be taken as 2/3 of the given value in the table. The core hardness means hardness at the center region of the root.

See Table 17-5 for δF lim of gears without case hardening. Table 17-6 gives δF lim of gears that are induction hardened; and Tables 17-7 and 17-8 give the values for carburized and nitrided gears, respectively. In Tables 17-8A and 17-8B, examples of calculations are given.

» Surface Strength Of Spur And Helical Gears - Continued on page 9