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Worm Gear Mesh Geometry, Elements of Metric Gear Technology (Cont.)

Figure 9-2 Figure 9-2

9.1 Worm Mesh Geometry

Figure 9-1 Although the worm tooth form can be of a variety, the most popular is equivalent to a V-type screw thread, as in Figure 9-1. The mating worm gear teeth have a helical lead. (Note: The name "worm wheel" is often used interchangeably with "worm gear".) A central section of the mesh, taken through the worm's axis and perpendicular to the worm gear's axis, as shown in Figure 9-2, reveals a rack-type tooth of the worm, and a curved involute tooth form for the worm gear. However, the involute features are only true for the central section. Sections on either side of the worm axis reveal nonsymmetric and noninvolute tooth profiles. Thus, a worm gear mesh is not a true involute mesh. Also, for conjugate action, the center distance of the mesh must be an exact duplicate of that used in generating the worm gear.

To increase the length-of-action, the worm gear is made of a throated shape to wrap around the worm.

Figure 9-3
9.1.1 Worm Tooth Proportions
Worm tooth dimensions, such as addendum, dedendum, pressure angle, etc., follow the same standards as those for spur and helical gears. The standard values apply to the central section of the mesh. See Figure 9-3a. A high pressure angle is favored and in some applications values as high as 25° and 30° are used.

9.1.2 Number Of Threads
The worm can be considered resembling a helical gear with a high helix angle.

For extremely high helix angles, there is one continuous tooth or thread. For slightly smaller angles, there can be two, three or even more threads. Thus, a worm is characterized by the number of threads, zw.

9.1.3 Pitch Diameters, Lead and Lead Angle
Referring to Figure 9-3:
Equation 9-1

Equation 9-2

9.1.4 Center Distance

Equation 9-3

9.2 Cylindrical Worm Gear Calculations
Cylindrical worms may be considered cylindrical type gears with screw threads. Generally, the mesh has a 90° shaft angle. The number of threads in the worm is equivalent to the number of teeth in a gear of a screw type gear mesh.

Figure 9-4 Figure 9-4
Thus, a one-thread worm is equivalent to a one-tooth gear; and two-threads equivalent to two-teeth, etc. Referring to Figure 9-4, for a lead angle ɣ, measured on the pitch cylinder, each rotation of the worm makes the thread advance one lead.

There are four worm tooth profiles in JIS B 1723, as defined below.

Type I Worm: This worm tooth profile is trapezoid in the radial or axial plane.
Type II Worm: This tooth profile is trapezoid viewed in the normal surface.
Type III Worm: This worm is formed by a cutter in which the tooth profile is trapezoid form viewed from the radial surface or axial plane set at the lead angle. Examples are milling and grinding profile cutters.

Type IV Worm: This tooth profile is involute as viewed from the radial surface or at the lead angle. It is an involute helicoid, and is known by that name.

Type III worm is the most popular. In this type, the normal pressure angle αn has the tendency to become smaller than that of the cutter, αc.

Per JIS, Type III worm uses a radial module mt and cutter pressure angle αc = 20° as the module and pressure angle. A special worm hob is required to cut a Type III worm gear. Standard values of radial module, mt, are presented in Table 9-1.

Table 9-1 Table 9-1
Table 9-2 Table 9-2
Because the worm mesh couples nonparallel and nonintersecting axes, the radial surface of the worm, or radial cross section, is the same as the normal surface of the worm gear.

Similarly, the normal surface of the worm is the radial surface of the worm gear. The common surface of the worm and worm gear is the normal surface. Using the normal module, mn, is most popular. Then, an ordinary hob can be used to cut the worm gear.

Table 9-2 presents the relationships among worm and worm gear radial surfaces, normal surfaces, axial surfaces, module, pressure angle, pitch and lead.

Figure 9-4 Figure 9-4
Reference to Figure 9-4 can help the understanding of the relationships in Table 9-2. They are similar to the relations in Formulas (6-11) and (6-12) that the helix angle β be substituted by (90° – ɣ). We can consider that a worm with lead angle ɣ is almost the same as a screw gear with helix angle (90° – ɣ).

Table 9-3 Table 9-3
Figure 9-5 Figure 9-5
9.2.1 Axial Module Worm Gears
Table 9-3 presents the equations, for dimensions shown in Figure 9-5, for worm gears with axial module, mx, and normal pressure angle αn = 20°.

Table 9-4 Table 9-4
9.2.2 Normal Module System Worm Gears
The equations for normal module system worm gears are based on a normal module, mn, and normal pressure angle, αn = 20°. See Table 9-4.

Figure 9-6 Figure 9-6
9.3 Crowning Of The Worm Gear Tooth
Crowning is critically important to worm gears (worm wheels). Not only can it eliminate abnormal tooth contact due to incorrect assembly, but it also provides for the forming of an oil film, which enhances the lubrication effect of the mesh. This can favorably impact endurance and transmission efficiency of the worm mesh. There are four methods of crowning worm gears:

    1. Cut Worm Gear With A Hob Cutter Of Greater Pitch Diameter Than The Worm.

A crownless worm gear results when it is made by using a hob that has an identical pitch diameter as that of the worm. This crownless worm gear is very difficult to assemble correctly. Proper tooth contact and a complete oil film are usually not possible.

However, it is relatively easy to obtain a crowned worm gear by cutting it with a hob whose pitch diameter is slightly larger than that of the worm. This is shown in Figure 9-6. This creates teeth contact in the center region with space for oil film formation.

    2. Recut With Hob Center Distance Adjustment.

The first step is to cut the worm gear at standard center distance. This results in no crowning. Then the worm gear is finished with the same hob by recutting with the hob axis shifted parallel to the worm gear axis by ±Δ h. This results in a crowning effect, shown in Figure 9-7.

Figure 9-7 Figure 9-7
    3. Hob Axis Inclining Δθ From Standard Position.

Figure 9-8 Figure 9-8
Figure 9-9 Figure 9-9
In standard cutting, the hob axis is oriented at the proper angle to the worm gear axis. After that, the hob axis is shifted slightly left and then right, Δθ, in a plane parallel to the worm gear axis, to cut a crown effect on the worm gear tooth. This is shown in Figure 9-8. Only method 1 is popular. Methods 2 and 3 are seldom used.

    4. Use A Worm With A Larger Pressure Angle Than The Worm Gear.

This is a very complex method, both theoretically and practically. Usually, the crowning is done to the worm gear, but in this method the modification is on the worm. That is, to change the pressure angle and pitch of the worm without changing the pitch line parallel to the axis, in accordance with the relationships shown in Equations 9-4:

Equation 9-4

In order to raise the pressure angle from before change, αx', to after change, αx, it is necessary to increase the axial pitch, px', to a new value, px, per Equation (9-4). The amount of crowning is represented as the space between the worm and worm gear at the meshing point A in Figure 9-9.

This amount may be approximated by the following equation:

Equation 9-5
Figure 9-10 Figure 9-10
d1 = Pitch diameter of worm
k = Factor from Table 9-5 and Figure 9-10
px = Axial pitch after change
px' = Axial pitch before change

Table 9-5

Table 9-6 Table 9-6
An example of calculating worm crowning is shown in Table 9-6.

Because the theory and equations of these methods are so complicated, they are beyond the scope of this treatment. Usually, all stock worm gears are produced with crowning.

9.4 Self-Locking Of Worm Mesh
Self-locking is a unique characteristic of worm meshes that can be put to advantage. It is the feature that a worm cannot be driven by the worm gear. It is very useful in the design of some equipment, such as lifting, in that the drive can stop at any position without concern that it can slip in reverse. However, in some situations it can be detrimental if the system requires reverse sensitivity, such as a servomechanism.

Self-locking does not occur in all worm meshes, since it requires special conditions as outlined here. In this analysis, only the driving force acting upon the tooth surfaces is considered without any regard to losses due to bearing friction, lubricant agitation, etc. The governing conditions are as follows:

Let Fu1 = tangential driving force of worm

Equation 9-6
αn = normal pressure angle
ɣ = lead angle of worm
µ = coefficient of friction
Fn = normal driving force of worm

If Fu1 > 0 then there is no self-locking effect at all. Therefore, Fu1 ≤ 0 is the critical limit of self-locking.

Let αn in Equation (9-6) be 20°, then the condition:

Fu1 ≤ 0 will become:
(cos 20° sin ɣ – µ cos ɣ) ≤ 0

Figure 9-11 Figure 9-11
Figure 9-11 shows the critical limit of self-locking for lead angle ɣ and coefficient of friction µ. Practically, it is very hard to assess the exact value of coefficient of friction µ. Further, the bearing loss, lubricant agitation loss, etc. can add many side effects. Therefore, it is not easy to establish precise self-locking conditions. However, it is true that the smaller the lead angle ɣ, the more likely the self-locking condition will occur.


There are direct and indirect methods for measuring tooth thickness. In general, there are three methods:
  • Chordal Thickness Measurement
  • Span Measurement
  • Over Pin or Ball Measurement

Figure 10-1 Figure 10-1
10.1 Chordal Thickness Measurement
This method employs a tooth caliper that is referenced from the gear's outside diameter. Thickness is measured at the pitch circle. See Figure 10-1.

Table 10-1 Table 10-1
10.1.1 Spur Gears
Table 10-1 presents equations for each chordal thickness measurement.

Table 10-2 Table 10-2
10.1.2 Spur Racks And Helical Racks
The governing equations become simple since the rack tooth profile is trapezoid, as shown in Table 10-2.

Table 10-3 Table 10-3
Table 10-4 Table 10-4
Table 10-5 Table 10-5
Figure 10-2 Figure 10-2
10.1.3 Helical Gears
The chordal thickness of helical gears should be measured on the normal surface basis as shown in Table 10-3. Table 10-4 presents the equations for chordal thickness of helical gears in the radial system. See Figure 10-2 and Table 10-5 for Gleason straight bevel gears.

Table 10-6 Table 10-6
Table 10-6 presents equations for chordal thickness of a standard straight bevel gear.

If a standard straight bevel gear is cut by a Gleason straight bevel cutter, the tooth angle should be adjusted according to:

Equation 10-1

Table 10-7 Table 10-7
This angle is used as a reference in determining the circular tooth thickness, s, in setting up the gear cutting machine.

Table 10-7 presents equations for chordal thickness of a Gleason spiral bevel gear.

Figure 10-3 Figure 10-3
Figure 10-3 shows how to determine the circular tooth thickness factor K for Gleason spiral bevel gears.

The calculations of circular thickness of a Gleason spiral bevel gear are so complicated that we do not intend to go further in this presentation.

Table 10-8 Table 10-8
10.1.5 Worms And Worm Gears
Table 10-8 presents equations for chordal thickness of axial module worms and worm gears.

Table 10-9 Table 10-9
Table 10-9 contains the equations for chordal thickness of normal module worms and worm gears.

10.2 Span Measurement Of Teeth
Span measurement of teeth, sm, is a measure over a number of teeth, zm, made by means of a special tooth thickness micrometer. The value measured is the sum of normal circular tooth thickness on the base circle, sbn, and normal pitch, pen (zm – 1).

10.2.1 Spur And Internal Gears
Table 10-10 Table 10-10
The applicable equations are presented in Table 10-10.

Equation 10-2

Figure 10-4 Figure 10-4
Figure 10-4 shows the span measurement of a spur gear. This measurement is on the outside of the teeth. For internal gears the tooth profile is opposite to that of the external spur gear. Therefore, the measurement is between the inside of the tooth profiles.

Table 10-11 Table 10-11
Table 10-12 Table 10-12
10.2.2 Helical Gears
Tables 10-11 and 10-12 present equations for span measurement of the normal and the radial systems, respectively, of helical gears.

Equation 10-3

Equation 10-4
Figure 10-5
There is a requirement of a minimum blank width to make a helical gear span measurement. Let bmin be the minimum value for blank width. Then
Equation 10-5

where βb is the helix angle at the base cylinder,
Equation 10-6

From the above, we can determine that at least 3mm of Δb is required to make a stable measurement of sm.

Figure 10-6 Figure 10-6
Figure 10-7 Figure 10-7
10.3 Over Pin (Ball) Measurement
As shown in Figures 10-6 and 10-7, measurement is made over the outside of two pins that are inserted in diametrically opposite tooth spaces, for even tooth number gears; and as close as possible for odd tooth number gears.

Figure 10-8 Figure 10-8
Figure 10-9 Figure 10-9
Figure 10-10 Figure 10-10
Figure 10-11 Figure 10-11
The procedure for measuring a rack with a pin or a ball is as shown in Figure 10-9 by putting pin or ball in the tooth space and using a micrometer between it and a reference surface. Internal gears are similarly measured, except that the measurement is between the pins. See Figure 10-10. Helical gears can only be measured with balls. In the case of a worm, three pins are used, as shown in Figure 10-11. This is similar to the procedure of measuring a screw thread. All these cases are discussed in detail in the following sections.

Note that gear literature uses "over pins" and "over wires" terminology interchangeably. The "over wires" term is often associated with very fine pitch gears because the diameters are accordingly small.

10.3.1 Spur Gears In measuring a gear, the size of the pin must be such that the over pins measurement is larger than the gear's outside diameter. An ideal value is one that would place the point of contact (tangent point) of pin and tooth profile at the pitch radius. However, this is not a necessary requirement. Referring to Figure 10-8, following are the equations for calculating the over pins measurement for a specific tooth thickness, s, regardless of where the pin contacts the tooth profile:

For even number of teeth:

Equation 10-7

For odd number of teeth:

Equation 10-8

where the value of φ1 is obtained from:

Equation 10-9

When tooth thickness, s, is to be calculated from a known over pins measurement, dm, the above equations can be manipulated to yield:
Equation 10-10

Equation 10-11

Equation 10-12

Equation 10-13

In measuring a standard gear, the size of the pin must meet the condition that its surface should have the tangent point at the standard pitch circle. While, in measuring a shifted gear, the surface of the pin should have the tangent point at the d + 2xm circle.

Table 10-13
Table 10-14
Table 10-15
The ideal diameters of pins when calculated from the equations of Table 10-13 may not be practical. So, in practice, we select a standard pin diameter close to the ideal value. After the actual diameter of pin dp is determined, the over pin measurement dm can be calculated from Table 10-14.

Table 10-15 is a dimensional table under the condition of module m = 1 and pressure angle α = 20° with which the pin has the tangent point at d + 2xm circle.

Table 10-16
Table 10-16A
10.3.2 Spur Racks And Helical Racks
In measuring a rack, the pin is ideally tangent with the tooth flank at the pitch line. The equations in Table 10-16 can, thus, be derived. In the case of a helical rack, module m, and pressure angle α, in Table 10-16, can be substituted by normal module, mn, and normal pressure angle, αn, resulting in Table 10-16A.

» Internal Gears - Continued on page 6