This article deals with worm gearings in cases when the worm-gear pair is composed either of plastic or of powder-metal made components. The technology used in production of plastic gears, as well as powder-metal technology, imposes certain constraints on the design parameters of a worm and of a mating gear. Gears of this kind are better suited for manufacturing, and, thus, they can be cheaper in production, while gears of other kinds are inconvenient in production, which makes them costlier. Certain advantages of gearing theory accomplishments can be taken in order to combine the benefits of plastic-gear technology, and of powder-metal technology, with benefits of particular designs of worm-gear pairs.

### Introduction

In recent decades, plastic gears, as well as powder-metal gears (PM gears) got an extensive application in a variety of industries. In many cases, gears of regular design can be successfully replaced either by plastic gears, or by PM gears that are usually less expensive and of a higher quality. Unfortunately, not every kind of gears can be made of plastic or use conventional methods adopted in powder-metal industry. For example, spur gears are easier in production compared to worm gears, helical gears, double-helical and/or herring-bone gears, gears with circular arc geometry in the lengthwise direction of the gear tooth, and so forth.

In “worm-to-spur gear” mesh, it is a common and well-established practice to set the worm in relation to a mating spur gear at an angle that corresponds to the worm pitch helix angle, *ψ*_{w}. The shaft angle, Σ, in this case equals Σ = 90° – *ψ*_{w}. Such a case is illustrated in Figure 1. It is evident that the shaft angle in this particular application is not a right angle (Σ ≠ 90°).

However, in particular applications, it is preferred to have worm gearings (a) with a shaft angle equal to a right angle, and (b) with a spur (not helical) gear in mesh with a mating worm. Under such a scenario, the spur gear is manufactured using a technology adopted easier in production of plastic gears or PM gears. Manufactured this way, the gears get cheaper. The replacement of a helical gear with an equivalent spur gear is beneficial as the axial thrust pointed along the spur gear axis of rotation in this case is eliminated, and the shaft can be mounted on bearings of a cheaper design.

It should be mentioned here that right-angle gear pairs with an involute worm engaged in mesh with a spur involute gear are known. An example of such a gear pair is depicted in Figure 2. Here, a “worm-to-spur gear” gear pair with a right shaft angle is shown (Σ = 90°). It is right point to stress here that originally shown in the Figure 2 worm was designed to be engaged in mesh with a helical gear. Later on, width of the space between the worm threads was enlarged, and in this way, an additional room was created in order to get the worm engaged in mesh with a spur gear. This is made clear from a comparison of the thickness of the worm top land, t_{o.w}, and of the space width, w_{f.w}, in the bottom land. The first is much smaller than the second (that is, an inequality t_{o.w} << w_{f.w} is observed). There is no need to explain here that the obtained mesh is far from perfect, as the contact conditions between the spur gear tooth flank, G, and the worm threads, W, are violated. In this particular case, the helical gear pitch helix angle, “*ψ*_{w}”, can be interpreted as the axes misalignment deviation, “ΔΣ” in the “worm-to-spur gear” mesh. All perfect gears (with no exclusion, have to meet the three fundamental laws of gearing [1]).

Ultimately, it can be concluded from the earlier discussion that the two requirements, that is, (a) to keep the shaft angle equal to a right angle, and (b) to use a spur gear instead of helical gear, are contradictory to one another.

The developments in the theory of gearing [1] make designing worm gear pairs with a right shaft angle (Σ = 90°) possible, and with a spur gear engaged in mesh with the worm.

### Elements of the Theory of Enveloping Surfaces

When a worm and a mating spur gear rotate in mesh, a virtual straight-sided rack is engaged in mesh with each of the components. This rack is commonly referred to as the “basic rack, R.” Tooth flanks of the spur gear may be viewed as envelopes to the consecutive position of a tooth flank of the basic rack, R, in its motion in relation to a reference system associated with the spur gear. Generated this way, the worm is referred to as the “involute worm” (that differs from “Archimedean worm,” as well as from “convolute worm”). The same is also valid with respect to the mating worm: The worm threads may be construed as envelopes to the consecutive position of a tooth flank of that same basic rack, R, in its motion in relation to a reference system associated with the worm. With that said, one can conclude that the geometry of the tooth flank of the spur gear, as well as the threads of the worm, are generated as envelopes to a plane surface that performs a screw motion along and about a straight line, which, in nature, is the axis of rotation of the spur gear (of the worm).

Following [1], the generation of an enveloping surface to consecutive positions of a plane that performs a screw motion is briefly outlined:

In a more general case, that is, in a case of an involute helical gear, the generating basic rack, R, of a helical gear is tilted at a certain angle, *ψ*_{b.g}, in relation to the axis of rotation of the gear as shown in Figure 3. When the gears rotate, the basic rack, R, travels in the direction specified by the vector, V_{r}. The magnitude, V_{r}, of the vector, V_{r}, of the linear velocity is synchronized with a rotation, *ω*_{g}, of the gear in a timely manner.

The gear tooth flank, G, is an envelope to consecutive positions of the lateral plane when the rack, R, performs the screw motion along and about the axis, O_{g}.

To derive the geometry of a gear tooth flank, G, and of mating worm threads, W, consider a plane, R, that performs a screw motion, shown in Figure 4. The plane, R, forms a certain angle *ψ*_{b.w} with X_{0}– axis of the “Cartesian” coordinate system, X_{0}Y_{0}Z_{0} (shown later in this article, angle *ψ*_{b.w} in nature is the base helix angle of a helical gear). The axis, X_{0}, is the axis of the screw motion.

The screw motion of the plane, R, is composed of two elementary motions:

(1) The rotation with an angular velocity, m_{g}, about the X_{0}– axis.

(2) The translation, V_{r}, along the X_{0}– axis is another motion.

The magnitudes, *ω*_{g} and V_{r}, of the rotation vector, m_{g}, and the linear velocity vector, V_{r}, respectively, are synchronized with one another in Equation 1:

Here, the pitch diameter of the gear is denoted by d_{g}.

The linear velocity vector, V_{r}, can be expressed as the sum of two vectors in Equation 2:

The component V_{1} of the translation vector, V_{r}, is within the plane, R. This component does not affect the geometry of the enveloping surface, G, and, thus, the component V_{1} can be omitted from further analysis. The component V_{2} is perpendicular to the plane, R. The geometry of the gear tooth flank strongly depends on the magnitude (V_{2} = V_{r}sin*ψ*_{b.g}) and direction of this component.

Based on the schematic shown in Figure 4, the following expression in Equation 3:

was derived [1]. In this expression the worm base helix angle, *ψ*_{b.w}, is expressed in terms of the normal profile angle, *φ*_{n}, of the basic rack, R, and of the angle, *ξ*_{r}, that the basic rack, R, forms with the perpendicular to the gear axis of rotation (see Figure 4).

The expression in Equation 3 can be transformed and represented in the form of Equation 4:

known from many advanced sources [3], and others.

It is important to stress that the angle (which is commonly referred to as the “worm setting angle, *ξ*_{r }”), and the worm pitch helix angle, *ψ*_{b.g}, are not identical to one another as they are of a different nature.

Equation 3 is the core for understanding “spur gear-to-worm” right-angle gearing.

The involute gear tooth flank, G, generated this way perfectly correlates with involute gears’ tooth flank geometries generated using other method. A well-known schematic (available from the public domain) is used here intentionally (Figure 5). This schematic is adopted by many gear experts, which makes it easier to demonstrate the configuration of the rolling plane in relation to the involute gear tooth flank. The schematic is overlapped by the plane that performs a screw motion. It is instructive to stress that the plane, R, shown in Figure 4 is that same newly introduced plane in Figure 5.

The earlier discussion on generation of a screw involute surface can be summarized with the following statement:

If we screw, with pitch, p, a plane about an axis fixed in space, which axis makes an angle, *α*, with the said plane, we describe in space a continuum of planes that successively intersect one another in a continuum of straight lines. These lines sweep out the involute helicoid, (a, *α*), where a = ptan*α*.

It might be correspondingly said that the screw of the plane “sweeps” in the same involute helicoid. The involute helicoid is, in any event, the so-called “envelope” of the continuum of planes.

This is a well-known theorem not found in most books about gearing. It may however be found in some books about kinematics [9]. See also page 335 in the book by Dr. Phillips [10].

Another solution to the problem of determining the envelope of a plane that performs a screw motion is given by Cormac [8].

### ‘Spur Gear-to-Worm’ Orthogonal Gearing

At hand, the problem designing the worm can be formulated as follows: To design an involute worm that can be engaged in a proper right-angle mesh with a spur involute gear.

For an involute worm of a specified module, m, the number of starts, Z_{w}, normal profile angle, *φ*_{n}, and the worm-setting angle, *ξ*_{w}, in Equation 5 (see Figure 6):

is used for the calculation of the base diameter of the worm, d_{b.w}.

Once the shaft angle in a “spur gear-to-worm” pair is a right angle, then the “worm setting angle, *ξ*_{r}” has to be equal to zero, and, thus, an equation *ξ*_{r} = 0° has to be valid. Under such a scenario, the equality *ψ*_{b.w} = *φ*_{n} (see Equation 3) is observed.

In this particular case (that is, when *ξ*_{r} = 0°) Equation 5 reduces to:

Having the base helix angle, *ψ*_{b.w}, of the worm calculated, the rest of the design parameters of the worm are calculated as follows [1], [4]:

1. Axial pitch of the worm, p_{x}:

Here, the module of the worm is designated as “m.”

2. Base diameter of the worm, d_{b.w}:

With the axial pitch, p_{x}, the base diameter, d_{b.w}, and the design parameters of the basic rack, R, known, the rest of the design parameters of the worm can be calculated using known formulae [1], [4].

Having calculated the design parameters of a helical rack, the corresponding design parameters of a helical gear with a given tooth number, N_{g}, can be calculated as well. Standard equations [1], [4], are used for the calculation of the design parameters of a helical gear.

Compare the design parameters of the worm of the proposed design (see Figure 6) with that shown in Figure 2. The performed comparison reveals that, for a spur gear with a standard tooth profile, tooth thickness and space width in the worm of the proposed design are equal to one another (w_{w} = t_{w}, measured on the pitch cylinders), while in the current worm design they are not (w_{f.w} > t_{o.w}, measured on the bottom-land and outer cylinders correspondingly). The thicker threads in the worm depicted in Figure 6 are stronger, and — what is also of critical importance for highly-rotating worms — provide significantly better heat removal from the friction zone between the worm threads and the gear tooth flanks.

The discussed approach for the calculation of the design parameters of an involute worm for a right-angle mesh with an involute spur gear also was used in designing hobs for cutting spur and helical cluster gears [5], [6], [7], and others.

### Conclusion

Orthogonal worm gear pairs may be designed so as to engage a spur gear in mesh with a mating involute worm under a right shaft angle in the worm gear pair. The same is also valid with respect to two-starts worm gear pairs. Axial thrust in the spur gear is eliminated under such a scenario. The involute worm gear may be mounted on cheaper bearings. No increase in design and production of a worm of such a design can be anticipated.

The designed involute worm can be approximated either by a corresponding Archimedean worm or by a corresponding convolute worm (that is, by a worm, straight generating lines of which are not tangent to the base cylinder, as no base cylinder can be constructed to a convolute worm).

Multiple-start worms can be designed using the disclosed approach.

The discussed approach is validated for designing hobs for hobbing spur and helical cluster gears.

### References

- Radzevich, S.P., Theory of Gearing: Kinematics, Geometry, and Synthesis, 2nd edition, revised and expanded, Boca Raton, FL, 2018, 934 pages.
- Radzevich, S. P., Design and Investigation of Skiving Hobs for Finishing of Hardened Gears, Ph.D. thesis, Kiev Polytechnic Institute, Kiev, 1982, 298p.
- Radzevich, A.P., A Study on Efficient Hobbing of Cylindrical Involute Gears, Ph.D. Thesis, Kuybishev, Kuybishev Polytechnic Institute, 1988, 300p.
- Radzevich, S.P., Dudley’s Handbook of Practical Gear Design and Manufacture, 3rd Edition, CRC Press, Boca Raton, FL, 2016, 629 pages.
- Radzevich, S.P., Gear Cutting Tools: Science and Engineering, 2nd Edition, CRC Press, Boca Raton, FL, 2017, 606 pages.
- Radzevich, S.P., “About Hob Idle Distance in Gear Hobbing Operation,” ASME Journal of Mechanical Design, Vol. 124, December 2002, pp.772-786.
- Radzevich, S.P., “On a Possible Way of Size and Weight Reduction of a Car Transmission,” Gear Technology magazine, July/August, 2003, pp. 44-50.
- Cormac, P., A Treatise on Screws and Worm Gear, Their Mills and Hobs, London, Chapman & Hall, Ltd., 1936, 138p.
- Prudhomme, R., and Lemasson, G., Cinématique, École Nationale Supérienre d’Arts et Métiers, École d’Engenieurs, Donod, Paris, 1906, 1955.
- Phillips, J., General Spatial Involute Gearing, Springer, New York, 2003, 498 pages.