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    Technical Resources   Contact Ratio of Gears, Tooth Modifications, and the Relationship Among The Gears In A Planetary Gear System, Elements of Metric Gear Technology, Page 6

Contact Ratio of Gears, Tooth Modifications, and the Relationship Among The Gears In A Planetary Gear System, Elements of Metric Gear Technology (Cont.)

Table 10-17 Table 10-17
Table 10-18 Table 10-18
Table 10-19 Table 10-19
Figure 10-1010.3.3 Internal Gears
As shown in Figure 10-10, measuring an internal gear needs a proper pin which has its tangent point at d + 2xm circle. The equations are in Table 10-17 for obtaining the ideal pin diameter. The equations for calculating the between pin measurement, dm, are given in Table 10-18.

Table 10-19, lists ideal pin diameters for standard and profile shifted gears under the condition of module m = 1 and pressure angle α = 20°, which makes the pin tangent to the pitch circle d + 2xm.

Table 10-20 Table 10-20
10.3.4 Helical Gears
The ideal pin that makes contact at the d + 2xm mn pitch circle of a helical gear can be obtained from the same above equations, but with the teeth number z substituted by the equivalent (virtual) teeth number zv. Table 10-20 presents equations for deriving over pin diameters.

Table 10-21 Table 10-21
Table 10-22 Table 10-22
Table 10-21 presents equations for calculating over pin measurements for helical gears in the normal system.

Table 10-22 and Table 10-23 present equations for calculating pin measurements for helical gears in the radial (perpendicular to axis) system.

Table 10-23 Table 10-23
10.3.5 Three Wire Method Of Worm Measurement
The teeth profile of Type III worms which are most popular are cut by standard cutters with a pressure angle αc = 20°. This results in the normal pressure angle of the worm being a bit smaller than 20°. The equation below shows how to calculate a Type III worm in an AGMA system.

Equation 10-14

Table 10-24 Table 10-24
Table 10-25 Table 10-25
r = Worm Pitch Radius
rc = Cutter Radius
zw = Number of Threads
ɣ = Lead Angle of Worm

The exact equation for a three wire method of Type III worm is not only difficult to comprehend, but also hard to calculate precisely. We will introduce two approximate calculation methods here:

    (a) Regard the tooth profile of the worm as a linear tooth profile of a rack and apply its equations. Using this system, the three wire method of a worm can be calculated by Table 10-24.

These equations presume the worm lead angle to be very small and can be neglected. Of course, as the lead angle gets larger, the equations' error gets correspondingly larger. If the lead angle is considered as a factor, the equations are as in Table 10-25.

Table 10-26 Table 10-26
Table 10-27 Table 10-27
Table 10-28 Table 10-28
Table 10-29 Table 10-29
    (b) Consider a worm to be a helical gear. This means applying the equations for calculating over pins measurement of helical gears to the case of three wire method of a worm. Because the tooth profile of Type III worm is not an involute curve, the method yields an approximation. However, the accuracy is adequate in practice.

Tables 10-26 and 10-27 contain equations based on the axial system. Tables 10-28 and 10-29 are based on the normal system.

Table 10-28 shows the calculation of a worm in the normal module system. Basically, the normal module system and the axial module system have the same form of equations. Only the notations of module make them different.

Table 10-30 Table 10-30
Table 10-30 cont. Table 30 cont.
Table 10-30 cont. Table 10-30 cont.
Table 10-30 cont. Table 10-30 cont.
Table 10-30 cont. Table 10-30 cont.
Table 10-30 cont. Table 10-30 cont.
Table 10-30 cont. Table 10-30 cont.
Table 10-30 cont. Table 10-30 cont.
Table 10-30 cont. Table 10-30 cont.
10.4 Over Pins Measurements For Fine Pitch Gears With Specific Numbers Of Teeth
Table 10-30, Table-30 part 2, Table-30 part 3, Table-30 part 4, Table-30 part 5, Table-30 part 6, Table-30 part 7, Table-30 part 8 and Table-30 part 9 presents measurements for metric gears. These are for standard ideal tooth thicknesses. Measurements can be adjusted accordingly to backlash allowance and tolerance; i.e., tooth thinning.


Figure 11-1
To assure continuous smooth tooth action, as one pair of teeth ceases action a succeeding pair of teeth must already have come into engagement. It is desirable to have as much overlap as is possible. A measure of this overlap action is the contact ratio. This is a ratio of the length of the line-of-action to the base pitch. Figure 11-1 shows the geometry for a spur gear pair, which is the simplest case, and is representative of the concept for all gear types. The length-of-action is determined from the intersection of the line-of-action and the outside radii. The ratio of the length-of-action to the base pitch is determined from:

Equation 11-1

It is good practice to maintain a contact ratio of 1.2 or greater. Under no circumstances should the ratio drop below 1.1, calculated for all tolerances at their worst case values.

A contact ratio between 1 and 2 means that part of the time two pairs of teeth are in contact and during the remaining time one pair is in contact. A ratio between 2 and 3 means 2 or 3 pairs of teeth are always in contact. Such a high ratio is generally not obtained with external spur gears, but can be developed in the meshing of internal gears, helical gears, or specially designed nonstandard external spur gears.

When considering all types of gears, contact ratio is composed of two components:
  1. Radial contact ratio (plane of rotation perpendicular to axes), εα
  2. Overlap contact ratio (axial), εβ
The sum is the total contact ratio, εɣ.

The overlap contact ratio component exists only in gear pairs that have helical or spiral tooth forms.

11.1 Radial Contact Ratio Of Spur And Helical Gears, εα
The equations for radial (or plane of rotation) contact ratio for spur and helical gears are given in Table 11-1, with reference to Figure 11-2.

When the contact ratio is inadequate, there are three means to increase it. These are somewhat obvious from examination of Equation (11-1).
Table 11-1 Table 11-1
Figure 11-2
  1. Decrease the pressure angle. This makes a longer line-of-action as it extends through the region between the two outside radii.
  2. Increase the number of teeth. As the number of teeth increases and the pitch diameter grows, again there is a longer line-of-action in the region between the outside radii.
  3. Increase working tooth depth. This can be done by adding addendum to the tooth and thus increase the outside radius. However, this requires a larger dedendum, and requires a special tooth design.

An example of helical gear:
Helical Gear Example

Note that in Table 11-1 only the radial or circular (plane of rotation) contact ratio is considered. This is true of both the spur and helical gear equations. However, for helical gears this is only one component of two. For the helical gear's total contact ratio, εɣ, the overlap (axial) contact ratio, εβ, must be added. See Paragraph 11.4.

11.2 Contact Ratio Of Bevel Gears, εα
The contact ratio of a bevel gear pair can be derived from consideration of the equivalent spur gears, when viewed from the back cone. See Figure 8-8.

With this approach, the mesh can be treated as spur gears. Table 11-2 presents equations calculating the contact ratio.

Figure 8-8 Figure 8-8
Table 11-2 Table 11-2
An example of spiral bevel gear (see Table 11-2):
Spiral Bevel Gear Example

11.3 Contact Ratio For Nonparallel And Nonintersecting Axes Pairs, ε
This group pertains to screw gearing and worm gearing. The equations are approximations by considering the worm and worm gear mesh in the plane perpendicular to worm gear axis and likening it to spur gear and rack mesh. Table 11-3 presents these equations.

Example of worm mesh:
Worm Mesh Example

11.4 Axial (Overlap) Contact Ratio, εβ
Helical gears and spiral bevel gears have an overlap of tooth action in the axial direction. This overlap adds to the contact ratio. This is in contrast to spur gears which have no tooth action in the axial direction.

Figure 11-3 Figure 11-3
Table 11-3 Table 11-3
Table 11-4 Table 11-4
Thus, for the same tooth proportions in the plane of rotation, helical and spiral bevel gears offer a significant increase in contact ratio. The magnitude of axial contact ratio is a direct function of the gear width, as illustrated in Figure 11-3. Equations for calculating axial contact ratio are presented in Table 11-4.

It is obvious that contact ratio can be increased by either increasing the gear width or increasing the helix angle.


Intentional deviations from the involute tooth profile are used to avoid excessive tooth load deflection interference and thereby enhances load capacity. Also, the elimination of tip interference reduces meshing noise. Other modifications can accommodate assembly misalignment and thus preserve load capacity.

Figure 12-1 Figure 12-1
12.1 Tooth Tip Relief
There are two types of tooth tip relief. One modifies the addendum, and the other the dedendum. See Figure 12-1. Addendum relief is much more popular than dedendum modification.

12.2 Crowning And Side Relieving
Figure 12-2 Crowning and side relieving are tooth surface modifications in the axial direction. See Figure 12-2.

Crowning is the removal of a slight amount of tooth from the center on out to reach edge, making the tooth surface slightly convex. This method allows the gear to maintain contact in the central region of the tooth and permits avoidance of edge contact with consequent lower load capacity. Crowning also allows a greater tolerance in the misalignment of gears in their assembly, maintaining central contact.

Relieving is a chamfering of the tooth surface. It is similar to crowning except that it is a simpler process and only an approximation to crowning. It is not as effective as crowning.

12.3 Topping And Semitopping
Figure 12-3 In topping, often referred to as top hobbing, the top or outside diameter of the gear is cut simultaneously with the generation of the teeth. An advantage is that there will be no burrs on the tooth top. Also, the outside diameter is highly concentric with the pitch circle. This permits secondary machining operations using this diameter for nesting.

Semitopping is the chamfering of the tooth's top corner, which is accomplished simultaneously with tooth generation. Figure 12-3 shows a semitopping cutter and the resultant generated semitopped gear. Such a tooth tends to prevent corner damage. Also, it has no burr. Figure 12-4 The magnitude of semitopping should not go beyond a proper limit as otherwise it would significantly shorten the addendum and contact ratio. Figure 12-4 specifies a recommended magnitude of semitopping.

Both modifications require special generating tools. They are independent modifications but, if desired, can be applied simultaneously.


The objective of gears is to provide a desired motion, either rotation or linear. This is accomplished through either a simple gear pair or a more involved and complex system of several gear meshes. Also, related to this is the desired speed, direction of rotation and the shaft arrangement.

13.1 Single-Stage Gear Train
A meshed gear is the basic form of a singlestage gear train. It consists of z1 and z2 numbers of teeth on the driver and driven gears, and their respective rotations, n1 & n2. The speed ratio is then:

Equation 13-1

13.1.1 Types Of Single-Stage Gear Trains
Gear trains can be classified into three types:
  1. Speed ratio > 1, increasing: n1 < n2
  2. Speed ratio = 1, equal speeds: n1 = n2
  3. Speed ratio < 1, reducing: n1 > n2

Figure 13-1 Figure 13-1
Figure 13-1a Figure 13-1a
Figure 13-1b Figure 13-1b
Figure 13-1 illustrates four basic types. For the very common cases of spur and bevel meshes, Figures 13-1(a) and 13-1(b), the direction of rotation of driver and driven gears are reversed. In the case of an internal gear mesh, Figure 13-1(c), both gears have the same direction of rotation. In the case of a worm mesh, Figure 13-1(d), the rotation direction of z2 is determined by its helix hand.

In addition to these four basic forms, the combination of a rack and gear can be considered a specific type. The displacement of a rack, l, for rotation θ of the mating gear is:

Equation 13-2

πm is the standard circular pitch
z1 is the number of teeth of the gear

Figure 13-2 Figure 13-2
Figure 13-3 Figure 13-3
13.2 Two-Stage Gear Train
A two-stage gear train uses two single-stages in a series. Figure 13-2 represents the basic form of an external gear two-stage gear train.

Let the first gear in the first stage be the driver. Then the speed ratio of the two-stage train is:

Equation 13-3

In this arrangement, n2 = n3

In the two-stage gear train, Figure 13-2, gear 1 rotates in the same direction as gear 4. If gears 2 and 3 have the same number of teeth, then the train simplifies as in Figure 13-3. In this arrangement, gear 2 is known as an idler, which has no effect on the gear ratio. The speed ratio is then:

Equation 13-4

Figure 13-4 Figure 13-4

13.3 Planetary Gear System
The basic form of a planetary gear system is shown in Figure 13-4. It consists of a Sun Gear (A), Planet Gears (B), Internal Gear (C) and Carrier (D). The input and output axes of a planetary gear system are on a same line. Usually, it uses two or more planet gears to balance the load evenly. It is compact in space, but complex in structure. Planetary gear systems need a high-quality manufacturing process. The load division between planet gears, the interference of the internal gear, the balance and vibration of the rotating carrier, and the hazard of jamming, etc. are inherent problems to be solved.

Figure 13-4 is a so called 2K-H type planetary gear system. The sun gear, internal gear, and the carrier have a common axis.

13.3.1 Relationship Among The Gears In A Planetary Gear System
In order to determine the relationship among the numbers of teeth of the sun gear A, (za), the planet gears B, (zb), and the internal gear C, (zc), and the number of planet gears, N, in the system, the parameters must satisfy the following three conditions:
Equation 13-5

This is the condition necessary for the center distances of the gears to match. Since the equation is true only for the standard gear system, it is possible to vary the numbers of teeth by using profile shifted gear designs.

To use profile shifted gears, it is necessary to match the center distance between the sun A and planet B gears, ax1, and the center distance between the planet B and internal C gears, ax2.
Equation 13-6

Equation 13-7

This is the condition necessary for placing planet gears evenly spaced around the sun gear. If an uneven placement of planet gears is desired, then Equation (13-8) must be satisfied.

Equation 13-8

θ = half the angle between adjacent planet gears

Condition No. 3:
Equation 13-9

Satisfying this condition insures that adjacent planet gears can operate without interfering with each other. This is the condition that must be met for standard gear design with equal placement of planet gears. For other conditions, the system must satisfy the relationship:

Equation 13-10

dab = outside diameter of the planet gears
ax = center distance between the sun and planet gears

Besides the above three basic conditions, there can be an interference problem between the internal gear C and the planet gears B. See SECTION 5 that discusses more about this problem.

Figure 13-6a Figure 13-6a
Figure 13-6b Figure 13-6b
Figure 13-6c Figure 13-6c
13.3.2 Speed Ratio Of Planetary Gear System
In a planetary gear system, the speed ratio and the direction of rotation would be changed according to which member is fixed. Figures 13-6(a), 13-6(b) and 13-6(c) contain three typical types of planetary gear mechanisms, depending upon which member is locked.

Table 13-1 Table 13-1
(a) Planetary Type

In this type, the internal gear is fixed. The input is the sun gear and the output is carrier D. The speed ratio is calculated as in Table 13-1.

Equation 13-11

Note that the direction of rotation of input and output axes are the same. Example: za = 16, zb = 16, zc = 48, then speed ratio = 1/4.

(b) Solar Type

Table 13-2 Table 13-2
In this type, the sun gear is fixed. The internal gear C is the input, and carrier D axis is the output. The speed ratio is calculated as in Table 13-2, on the following page.

Equation 13-12

Note that the directions of rotation of input and output axes are the same.

Example: za = 16, zb = 16, zc = 48, then the speed ratio = 1/1.3333333.

(c) Star Type

This is the type in which Carrier D is fixed. The planet gears B rotate only on fixed axes. In a strict definition, this train loses the features of a planetary system and it becomes an ordinary gear train. The sun gear is an input axis and the internal gear is the output. The speed ratio is:

Equation 13-13

Figure 13-6c Figure 13-6c
Referring to Figure 13-6(c), the planet gears are merely idlers. Input and output axes have opposite rotations.

Example: za = 16, zb = 16, zc = 48; then speed ratio = –1/3.

Figure 13-5 Figure 13-5
13.4 Constrained Gear System
A planetary gear system which has four gears, as in Figure 13-5, is an example of a constrained gear system. It is a closed loop system in which the power is transmitted from the driving gear through other gears and eventually to the driven gear. A closed loop gear system will not work if the gears do not meet specific conditions.

Figure 13-7 Figure 13-7
Let z1, z2 and z3 be the numbers of gear teeth, as in Figure 13-7. Meshing cannot function if the length of the heavy line (belt) does not divide evenly by circular pitch. Equation (13-14) defines this condition.

Equation 13-14

Figure 13-8 Figure 13-8
Figure 13-8 shows a constrained gear system in which a rack is meshed. The heavy line in Figure 13-8 corresponds to the belt in Figure 13-7. If the length of the belt cannot be evenly divided by circular pitch then the system does not work. It is described by Equation (13-15). Equation 13-15


Up to this point the discussion has implied that there is no backlash. If the gears are of standard tooth proportion design and operate on standard center distance they would function ideally with neither backlash nor jamming.

Backlash is provided for a variety of reasons and cannot be designated without consideration of machining conditions. The general purpose of backlash is to prevent gears from jamming by making contact on both sides of their teeth simultaneously. A small amount of backlash is also desirable to provide for lubricant space and differential expansion between the gear components and the housing. Any error in machining which tends to increase the possibility of jamming makes it necessary to increase the amount of backlash by at least as much as the possible cumulative errors. Consequently, the smaller the amount of backlash, the more accurate must be the machining of the gears. Runout of both gears, errors in profile, pitch, tooth thickness, helix angle and center distance – all are factors to consider in the specification of the amount of backlash. On the other hand, excessive backlash is objectionable, particularly if the drive is frequently reversing or if there is an overrunning load. The amount of backlash must not be excessive for the requirements of the job, but it should be sufficient so that machining costs are not higher than necessary.

» Gear Backlash - Continued on page 7