The article deals with geometrically-accurate worm gear pairs that feature line contact between tooth flanks of a worm gear and thread surfaces of a mating worm. Worm gearing of this kind can be viewed as a reduced case of R — gearing (R — gearing was proposed by Prof. Radzevich in about 2008). At the beginning, a variety of known designs of worm gearing are briefly outlined. Most known designs of worm gearing are approximate, and all of them feature point contact between tooth flanks of a worm gear and thread surfaces of a mating worm. A principal feature of worm gearing with line contact is disclosed. A criterion to distinguish worm in worm gearing from helical low-tooth-count gear is proposed. Worm gearing with a line contact has a high potential for application in high-power-density gear transmissions.

### Introduction: State-of-the art

Worm gearing has a long history. As an example, a primitive worm gear pair is depicted in a 1493 book by Leonardo da Vinci [1]. No comments on the tooth flank geometry can be found in [1].

A plurality of designs of worm gear pairs is used in the industry.

Shown in Figure 1a, single-enveloping worm gear pair is comprised of a cylindrical worm and an enveloping worm gear. (A cylindrical worm can also be engaged in mesh with a spur or helical gear. Worm gearing of this kind has a limited application in the today’s industry.) Either single-start or multiple-start worms are used in the design of single-enveloping worm gear pairs.

Three different kinds of straight-sided worms are used in the design of single-enveloping worm gearing:

**• Archimedean worms: **Worms of this design feature a straight-sided thread profile in a section of the worm by a plane through the worm centerline. Worm gearing of this design is an example of approximate gearing; that is, it is not capable of smoothly transmitting an input uniform rotary motion to an output shaft.

**• Involute worms: **Worms of this design feature a straight-sided thread profile in a section of the worm by a plane tangent to the worm base cylinder. Only involute worm gear pairs are capable of smoothly transmitting an input uniform rotary motion to an output shaft.

**• Convolute worms: **Worms of this design feature a straight-sided thread profile in a section of the worm by a plane perpendicular either to the worm line of threads or to the worm line of thread spaces. Worm gearing of this design is an example of approximate gearing; that is, it is not capable of smoothly transmitting an input uniform rotary motion to an output shaft.

Worms with a curved section of the worm thread by a plane of various configurations are also known.

The contact ratio in single-enveloping worm gearing is approximately equal to “gear-to-rack” mesh. Commonly, the contact ratio does not exceed two

A double-enveloping worm gear pair (or “globoidal gearing”) is comprised of a globoidal worm and an enveloping worm gear, as illustrated in Figure 1b. The contact ratio in double-enveloping worm gearing is slightly greater than that in single-enveloping worm gearing and commonly does not exceed two. It is loosely claimed in almost all advance sources on gearing that globoidal gearing is stronger compared to single-enveloping gearing, as it features a higher contact ratio (due to the worm it is enveloping). Actually, an enveloping worm contributes almost nothing to the actual value of contact ratio in double-enveloping worm gearing, as the length of the line of action in globoidal gearing slightly exceeds that in single-enveloping worm gearing.

Spiroid gearing (see Figure 2a) is comprised of a conical worm and a face-gear. Gearing of this design features point contact between tooth flanks of a face-gear and thread surfaces of a conical worm. The contact ratio in spiroid gearing is commonly under two. The latter is illustrated in Figure 3 where contact patterns are localized at the middle of the face-gear face and not overlap it. Even though the face width of the pinion overlaps the face width of the gear, the contact ratio in the gear pair does not exceed two.

Helicon gearing (see Figure 2b) is comprised of a cylindrical worm and a face-gear. The contact ratio in helicon gearing is approximately the same value as in spiroid gearing.

Numerous other designs of worm gearing are known, including, but not limited to, internal worm gearing.

Worms can also be used in the design of intersected-axes gearing as well.

### Motivation

The necessity of increasing the power density of gear transmissions is one of the main trends of evolution of gear design in the today’s industry. As a result, gears with a low tooth count (the so-called “low-tooth-count gears” or “ LTC — gears”) are extensively used in the design of modern machinery. On the other hand, the evolution of worm gearing tends to get more extensive applications of worm gearsets with multiple-start worms.

One may conclude that eventually the tooth number in helical gears will get smaller, while the number of starts in worm gearsets will become greater. Eventually, the tooth count in a gear and the number of starts in a worm will equal one another. The difference between “low-tooth-count gear” and “multiple-start worm” will get blurry. A natural question arises in this regard: How can a gear with a small number of teeth and a multiple-start worm be distinguished from one another?

This is illustrated in Figure 4. Only a highly skilled and experienced gear expert can identify whether the shown gear component is a worm or it is a gear. Design, calculation, manufacture, inspection, and application of worms and of low-tooth-count gears are not identical to one another. Therefore, in some cases, it could be critical to precisely specify the actual kind of a gear component.

The necessity of a criterion to distinguish a multiple-start worm from a low-tooth-count gear is clarified by an example of a gear component shown in Figure 4.

### Criterion to distinguish ‘multiple-start worm’ from helical ‘low-tooth-count gear’

The first known attempt to develop a criterion to distinguish a “multiple-start worm” from a helical “low-tooth-count gear” was undertaken by the author in ∼2008 [2]. This was an approximation to what is proposed later.

Design features of an involute helical gear is a perfect start-point for the discussion of the criterion to distinguish a “multiple-start worm” from a helical “low-tooth-count gear.” This discussion is schematically illustrated in Figure 5.

A helical involute pinion base diameter, d_{b.p}, base helix angle, ψ_{b.p}, and the pinion face width, F_{p}, are taken into account in the following analysis.

The base helix lead, L_{b.p}, can be expressed in terms of d_{b.p} and ψ_{b.p}:

As the actual value of the lead in a helical involute gear is of the same value regardless of the diameter of a cylinder at which the lead is calculated, the base helix lead, L_{b.p}, and that measured on pitch cylinder, L, are identical to one another, therefore Equation 1 can be represented in the form:

A correlation between the lead, L, and the face width, F_{p}, can be employed as a criterion to distinguish a “multiple-start worm” from a helical “low-tooth-count gear.”

It is assumed that, if in a helical gear component one 360-degree-thread spans over the pinion face width, F_{p}, (in this case the inequality L ≥ F_{p} is valid), this gear component is referred to as a “low-tooth-count gear.” Otherwise, the gear component is referred to as a helical “multiple-start worm.”

With that said, a gear component for which an inequality (L < F_{p}) is valid:

is referred to as “multiple-start worm”. Otherwise, when an inequality ( ) is valid:

the gear component is referred to as “low-tooth-count gear”.

It is important to stress here that the desirable line of contact, LC_{des}, has to be entirely within the zone of action, ZA. To meet this requirement, a projection, F_{lc} (where F_{lc} = L • sinψ_{b.p}), of the desirable line of contact, LC_{des}, onto the transverse plane must be longer compared to the perimeter of the base circle, that is, in helical involute gearing, an inequality:

must be fulfilled in order to refer the gear component as to “multiple-start worm.”

Once the concept of the criterion to distinguish a “multiple-start worm” from a helical “low-tooth-count gear” is clear with respect to the helical involute gear, this concept can be evolved to the most general case of gearing, namely, to the case of crossed-axes gear pair.

The desirable line of contact, LC_{des}, of various geometries is used in design of crossed-axes gearing. Moreover, a plurality of various configurations of the line of contact, LC_{des}, is known. Figure 6 illustrates examples of geometries and configurations of the desirable line of contact, LC_{des}, in relation to the plane of action, PA, in a crossed-axes gear pair. [It is important to stress here that a pinion base-cone-apex, A_{p}, coincides with the plane-of-action apex, A_{pa}, in intersected-axes gearing (when the identity, A_{p} ≡ A_{pa}, is valid), and a pinion base-cone-apex, A_{p}, is displaced from the plane-of-action apex, A_{pa}, in crossed-axes gearing (when the identity, A_{p} ≠ A_{pa}, is observed)].

A desirable line of contact, LC_{des}, shaped in the form of a straight line through the plane-of-action apex, A_{pa}, is shown in Figure 6a. The line of contact, LC_{des}, overlaps the effective face width, F_{pa}, in a gear pair and is entirely within the zone of action, ZA. The line-of-contact-span-angle, ζ _{lc}, is of a zero value in the case under consideration (ζ _{lc} = 0°).

In a bevel gear with skew teeth, a desirable line of contact, LC_{des}, is shaped in the form of a straight line that crosses the plane-of-action axis of rotation, O_{pa}. In Figure 6b, the axis of rotation, O_{pa}, is a straight line through the plane-of-action apex, A_{pa}, and that is perpendicular to the plane of drawing. The line of contact, LC_{des}, overlaps the effective face width, F_{pa}, and is entirely within the zone of action, ZA. The line-of-contact-span-angle, ζ _{lc}, is of a certain value (ζ _{lc} > 0°).

In a spiral bevel gear, a desirable line of contact, LC_{des} (see Figure 6c), is shaped in form of a circular arc segment of a certain radius, R_{lc}. The line of contact, LC_{des}, overlaps the effective face width, F_{pa}, and is entirely within the zone of action, ZA. The line-of-contact-span-angle, ζ _{lc}, is of a certain value (ζ _{lc} > 0°).

A desirable line of contact, , in the form of a circular arc segment of a certain radius, R_{lc}, can be configured so as to be tangent to radial direction of the bevel gear (see Figure 6d). The line of contact, LC_{des}, overlaps the effective face width, F_{pa}, and is entirely within the zone of action, ZA. The line-of-contact-span-angle, ζ _{lc}, is of a certain value (ζ _{lc} > 0°).

In all the cases in Figure 6, the line of contact, LC_{des}, spans over the face width, F_{pa}, and is entirely within the active portion of the plane of action, PA, at least at one instant of time. The line-of-contact-span-angle, ζ _{lc}, is important for further discussion.

Because of the crossed-axes gearing (see Figure 7), the pinion apex, A_{p}, is displaced in relation to the plane-of-action apex, A_{pa}, the pinion axis of rotation, O_{p}, does not intersected the plane-of-action axis of rotation, O_{pa}. A distance, C_{p}, is the closest distance of approach of the crossing axes O_{pa} and O_{p}.

In crossed-axes gearing, the actual value of the line-of-contact-span-angle, ζ _{lc}, is greater than that in intersected-axes gearing at a certain angle, ζ _{c d}, caused by the displacement a center-distance, C_{p}. The center-distance, C_{p}, is a portion of the center-distance, C. The displacement of the pinion axis of rotation, O_{p}, from the axis of instantaneous rotation, P_{ln}, is specified by the displacement, C_{p}.

The determination of the actual value of the angle, ζ _{c d}, is outlined below.

Consider a triangle ∆A_{pa}A_{p}f. This triangle is constructed in the plane of action, PA. In the triangle, ∆A_{pa}A_{p}f, the side A_{pa}f = r_{o.pa} — is the outer radius of the plane of action, PA, and the side A_{p}f = r_{o.p} — is the outer radius of the base cone of the pinion.

The actual value of the angle ∠ fA_{pa}g in the triangle ∆fgA_{pa} equals to the component, ζ _{cd}, contributed to the total angle, ζ _{lc}, by the displacement, C_{p}, and is calculated from a formula:

where r_{o.pa} and r_{f.pa} are the outer and the inner radii of the plane of action, PA. These design parameters are known from the gearset layout.

In the triangle, ∆A_{ap}A_{p}f, the length of the side A_{p}f =

√(A_{ap}f)^{2} − C_{p}^{2} = √ r^{2}_{o.ap} − C_{p}^{2}.

In the triangle, ∆A_{ap}A_{p}g, the length of the side A_{p}g =

√(A_{ap}g)^{2} − C_{p}^{2} = √ r^{2}_{f.ap} − C_{p}^{2}.

Having the lengths of the straight-line segments A_{p}f and A_{p}g calculated, the actual value of the pinion face width, F_{p}, is calculated form the formula:

Ultimately, the total angle, ζ _{lc}, equals to the summa:

Having the angle, ζ _{lc}, determined, one can proceed with the determination of the actual value of the pinion base cone angle, γ_{b}.

The actual value of the base cone angle, γ_{b}, of the pinion having the outer cone distance r_{o.p} = fA_{p}, and the face width F_{p}, equals:

A gear component that features the base cone angle γ_{b} > ζ _{lc}/2π, is referred to as “low-tooth-count gear.” Otherwise, when γ_{b} ≤ ζ _{lc}/2π, a gear component is referred to as “multiple-start worm.” For the calculations, the base cone angle, γ_{b}, can be expressed with the design parameters of the gear component — this is a routine procedure.

By convention, a so-called “worm factor, k_{w}” can be taken into account. In this latter case, the two expressions: γ_{b} > ζ _{lc}/2π and γ_{b} ≤ ζ _{lc}/2π, are substituted with equivalent expressions: γ_{b} > ζ _{lc}/2π and γ_{b} ≤ ζ _{lc}/2π. The actual value of the “worm factor, k_{w}” can be determined based on the experience accumulated in the gear industry.

### Conclusion

Worm gearing that features line contact between tooth flanks of a worm gear and thread surfaces of a mating worm is discussed in the article. Worm gearing of this particular kind has a tremendous potential of application in today’s industry as it perfectly fits the requirements of high-power-density transmissions.

The commonality between helical “low-tooth-count gear” and “multiple-start worm” is discussed with respect to crossed-axes gearing ( C — gearing). A certain commonality can be noted between intersected-axes gearing (I — gearing) and worm gearing. No commonality between parallel-axes gearing (P_{a} — gearing) and worm gearing is discussed here.

An objective numerical criterion to distinguish a helical “low-tooth-count gear” from a “multiple-start worm” is proposed in this article. The proposed criterion to distinguish “worm vs. LTC — gear” (that is, the desirable line of contact, LC_{des}, spans over the angle of 360 degrees) does not depend on gear ratio in the gearset for which the gear component is designed. The criterion can be referred to as “worm criterion.”

The highest possible power density in a gear transmission is anticipated in the area where helical “low-tooth-count gear” merges “multiple-start worm” (or vice versa).

### References

- da Vinci, Leonardo, The Madrid Codices, Volume 1, 1493, Facsimile Edition of “Codex Madrid 1,” original Spanish title: Tratado de Estatica y Mechanica en Italiano, McGraw Hill Book Company, 1974.
- Radzevich, S.P., Theory of Gearing: Kinematics, Geometry, and Synthesis, 2nd Edition, revised and expanded, CRC Press, Boca Raton, FL, 2018, 934 pages. [1st edition: CRC Press, Boca Raton, Florida, 2012, 743 pages].