Gearings that operate on crossing axes of rotation of a gear and a mating pinion have been known for centuries. Common sense and accumulated practical experience are the two foundations for the adopted practice of designing gears in ancient times. Such a practice was permissible as the input and output rotations at that time were low, and the power being transmitting was also low: The gears were driven by animals, water flow, wind flow, and so forth.

Because the input and output rotations are permanently rising and the transmitting power is rising as well, the tolerances for the gear accuracy are getting tighter.

Nowadays, gears cannot be designed based only on common sense along with the tremendous experience accumulated by the industry. The insufficiency of such an approach is evident because poorly engineered gears vibrate when operating, producing excessive noise. To reduce noise and vibration and to increase the power density, gears need to be lapped and to get paired when manufacturing.

### Introduction

Gear pairs with crossing axes of rotation of a gear and a mating pinion (Figure 1) are discussed in this article. Gearings of this kind are extensively used, first of all, in the automotive industry, but they are used in other industries as well. High power density of crossed-axes gear pairs (or just C_{a} — gearings, for simplicity), low noise excitation, and low vibration generation, are among the requirements of prime importance C_{a} — gearings have to comply with. To achieve these goals, gears for C_{a} — gearings have to be properly designed, accurately manufactured, and correctly assembled in the housing. If designed, manufactured, and assembled properly, crossed-axes gearings feature line contact between the tooth flanks of a gear, G, and a mating pinion, P. Perfect gearings that operate on crossing axes of rotation of a gear and a mating pinion, and those featuring line contact between the tooth flanks, were proposed (in 2008) by Dr. S. Radzevich [1]. Gearing of this particular kind is commonly referred to as “R — gearing.” Due to the line contact between the tooth flanks, R — gearing features the highest possible power density and are nearly-noiseless.

### 1. Requirements to Perfect Gear Design

When designing gears for perfect C_{a} — gearings, the gears have to align with three fundamental laws of gearing:

**The first fundamental law of gearing.** This requires a proper contact of a gear tooth flank, G, and a mating pinion tooth flank, P. The “Shishkov equation of contact” (Figure 2):

is commonly used to describe analytically this law of gearing (Prof. V.A. Shishkov, 1948 [2, 3]). Here it is designated: n_{g} and is the unit vector of a common perpendicular at a point of contact of the tooth flanks, G and P, and V_{Σ} is the linear velocity vector of the resultant relative motion of the tooth flanks, G and P.

The “Shishkov equation of contact, n_{g }• V_{Σ} = 0 has been known to proficient gear people since 1948, therefore there is no need to discuss this equation here in more detail.

**The second fundamental law of gearing.** To fulfill this law of gearing at every point of the line of contact, a straight line along the common perpendicular vector, n_{g}, must intersect the axis of instant rotation, P_{ln}, of the tooth flanks, G and P. The “equation of conjugacy” (Figure 3):

is proposed by Prof. S.P. Radzevich (2017) [1] to describe analytically this law of gearing. Here it is designated (Figure 3): p_{ln} and is the unit vector along the axis of instant rotation, P_{ln} (as the angular velocity vector, ω_{pl}, is also along the axis of instant rotation, the unit vector, p_{ln}, can be substituted with the vector, ω_{pl}), V_{m} is the linear velocity vector along an instant line of action, LA_{inst}, through the point of interest, m. In addition, the condition

n_{pl} • n_{g} ≠ 0 has to be fulfilled (here, n_{pl} is the unit normal vector to the axis of instant rotation, P_{ln}; the vector, n_{pl}, is located within the plane of action, PA, of the gear pair).

In the simplest case of perfect P_{a} — gearing, the second fundamental law of gearing reduces to conjugate action law, known as “CES — theorem of parallel-axes gearing” (where “CES” stands for “Camus-Euler-Savary theorem of parallel-axes gearing”).

**The third fundamental law of gearing.** In order to meet this law of gearing, the angular base pitch of a gear, *ϕ*_{b.g}, must be equal to the operating base pitch of the gear pair, *ϕ*_{b.op}, and the angular base pitch of a mating pinion, *ϕ*_{b.p}, also must be equal to the operating base pitch of the gear pair, *ϕ*_{b.op} (Prof. S.P. Radzevich, 2008, [1]). The third fundamental law of gearing is analytically expressed by a set of two equations (Figure 4):

or simply as *ϕ*_{b.g }= *ϕ*_{b.p }= *ϕ*_{b.op}.

In 2008, the concept of the “operating base pitch, *ϕ*_{b.op}, of a gear pair” was introduced by Prof. S.P. Radzevich [1]. The proposed concept is specified for all three possible kinds of gearings, that is, for: (a) P_{a} — gearings, (b) I_{a} — gearings, and (c) C_{a} — gearings [1].

Only in perfect P_{a} — gearing, all three base pitches, that is, p_{b.g}, p_{b.p}, and p_{b.op}, are liner design parameters, as the line of contact, LC^{i}_{des} (and the plane of action, PA) travels straight when the gears rotate.

In order to succeed designing perfect C_{a} — gearings, fulfillment of all three fundamental laws of gearing is a must.

### 2. Design Principles of Perfect Involute Gears (with Parallel Axes of the Gears Rotation)

The design of spur and helical gears for P_{a} — gearing is investigated to the best possible extent, while C_{a} — gearings, as well as I_{a} — gearings, are investigated more poorly. With that said, it makes sense to briefly outline the design principles of perfect involute parallel-axes gearing and then to consider how these design principles work (or have to work) in a case of crossing axes of the rotation of two gears. Finding and discussing commonalities between C_{a} — gearings and P_{a} — gearing is helpful to understand the core of the kinematics and the geometry of C_{a} — gearings. All the design principles for I_{a} — gearings then can be derived from the case for C_{a} — gearings, assuming the center-distance of a zero length.

Consider a schematic of P_{a} — gearing that is specified by the gears rotations, *ω*_{g} and *ω*_{p}, a center-distance, C, and a transverse pressure angle, *φ*_{t} (Figure 5). The axis of rotation of the gear, O_{g}, and that of the pinion, O_{p}, are parallel to each other and are at a center-distance, C, apart from one another. The plane of action, PA, in external parallel-axes gear pair, is a plane that forms the transverse pressure angle, *φ*_{t}, with the plane through the axes O_{g} and O_{p}.

The base cylinder of a diameter d_{b.g} is associated with the gear. Similarly, the base cylinder of a diameter d_{b.p} is associated with the pinion (recall the “pulley-and-belt” equivalence of the P_{a} — gearing). For the specified rotations, *ω*_{g} and *ω*_{p}, the actual values of the diameters, d_{b.g} and d_{b.p}, can be expressed in terms of the center-distance, C, and the transverse pressure angle, *φ*_{t}.

The base cylinders in external parallel-axes gear pair are tangent to the plane of action, PA, from the opposite sides. The gear and the pinion rotate, *ω*_{g} and *ω*_{p}, about their axes of rotation. The rotations, *ω*_{g} and *ω*_{p}, are synchronized with one another reciprocal to the tooth count, N_{g} and N_{p}, of the gear and the pinion, that is, the ratio

*ω*_{g }∗ N_{g} = *ω*_{p }∗ N_{p} is valid in perfect parallel-axes gearing.

The plane of action is considered as a zero thickness film that is unwrapping from the base cylinder of the gear, and is wrapping onto the base cylinder of the pinion, or vice versa. In such a motion, the plane of action, PA, travels with a linear velocity, v_{lc}. Magnitude, V_{lc}, of the linear velocity, v_{lc}, is timed with the rotations, *ω*_{g} and *ω*_{g}, to ensure rolling with no slippage of the plane of action, PA, over the base cylinders of the gear and the pinion.

Let us consider two neighboring lines of contact as illustrated in Figure 5.

In a reference system associated with the plane of action, PA, a desirable line of contact, LC^{i}_{des}, of the i — th pair of teeth of the gear and the pinion can be specified by position vector, r_{lc}, of point of the line LC^{i}_{des}. When the gears rotate, the line of contact, LC^{i}_{des}, travels in relation to a reference system associated with the gear. For analytical description of the transition from the first reference system to the second one, an operator of the resultant coordinate system transformation, Rs^{pa} (PA |γ g), is commonly used (go to [1] on details of coordinate system transformation). Therefore, a gear tooth flank, G, can be construed as a locus of consecutive positions of the line of contact, LC^{i}_{des}, in its motion in relation to the reference system of the gear. Having the expressions for r_{lc} and Rs^{pa} (PA |γ g) determined, the gear tooth flank, G, can be expressed by position vector of point, r_{g}^{pa}:

A desirable line of contact, LC^{i}_{des}, of the i — th pair of teeth of the gear and the pinion is drawn within the plane of action, PA. As an example, the desirable line of contact, LC^{i}_{des}, is shown for a case of helical involute gearing. In the case under consideration, this straight line segment forms the base helix angle, *ψ*_{b}, with the axis of instant rotation, P_{ln}, or, the same, with the axes O_{g} and O_{p} (the base helix angle, *ψ*_{b}, is not shown in Figure 5).

The desirable line of contact, LC^{i+}^{1}_{des}, of the adjacent (i+1) — th pair of teeth of the gear and the pinion is parallel to the straight line, LC^{i}_{des}, and also is located within the plane of action, PA.

Measured in a common transverse section of the gear pair, the distance, p_{b.op}, between two adjacent desirable lines of contact, LC^{i}_{des} and LC^{i+}^{1}_{des}, is referred to as the “operating base pitch” of the gear pair.

The operating base pitch of a gear pair is a calculated design parameter of a gear pair. It cannot be measured directly in a gear pair. In a “perfect” parallel-axes gearing, the operating base pitch of a gear pair is equal to: p_{b.op} = *π*∗d_{b.g}/N_{g} = = *π*∗d_{b.p}/N_{p}. Here, d_{b.g} is the gear base diameter, and N_{g} is the gear tooth count.

Note that “operating base pitch” in perfect parallel-axes gear pair is introduced “prior to (!!)” a gear and a mating pinion tooth flanks, G and P, are generated.

In cases of parallel-axes gearing, the identities p_{b.g }≡ p_{b.op }and

p_{b.p }≡ p_{b.op }(or simply p_{b.g }≡ p_{b.p}≡ p_{b.op}) are met only for “perfect gearing,” that is, for involute gearing. For any and all types of non-involute gearings, the identity cannot be fulfilled, and, moreover, base pitch of the gear and the pinion cannot be specified in non-involute gearings, while the operating base pitch of the gear pair can be easily calculated.

It is proven that involute tooth profiles can be generated according to one of two ways, that is, (a) by the considered above tracing method, and by (b) generating method as an envelope to consecutive positions of a basic rack (not considered here).

### 3. Design Features of Perfect Gears

with Crossing Axes of Gear Rotation

Having refreshed our memory on generation of tooth flanks of gears for perfect P_{a} — gearing (that is clear for proficient gear experts), let’s follow that same routing and generate tooth flanks of gears for perfect C_{a} — gearings that are not as clear for many of us.

At the beginning, a configuration of the axes of rotation, O_{g} and O_{p}, of a gear and a mating pinion has to be specified. For this purpose, the center-distance, C, and the crossing angle, Σ, of the gear’s axes have to be known. Then, the gear ratio, u (that is, u = *ω*_{g}/*ω*_{p}), and the transverse pressure angle, *φ*_{t}, must be also given. Having all these parameters determined, one can construct the plane of action, PA, for a C_{a} — gear pair. The plane of action, PA, is a plane through the axis of instant rotation, P_{ln}, that forms the transverse pressure angle, *φ*_{t}, with a perpendicular to the center-line, *C**L*.

Point of intersection of the center-line, *C**L*, by the axis of instant rotation, P_{ln}, is the plane-of-action apex, and is designated as A_{pa}. Similarly, the points of intersection of the center-line, *C**L*, by the axes of rotation, O_{g} and O_{p}, of the gear and the mating pinion, are the gear, A_{g}, and the pinion, A_{p}, base cone apexes, correspondingly.

All these design parameters are illustrated in Figure 6.

A circular-arc line of contact, LC, between a gear and a mating pinion tooth flanks, G and P, in a perfect crossed-axes gear pair is shown in Figure 7 as an example. Lines of contact of other geometries are also possible.

In a reference system associated with the plane of action, PA, a desirable line of contact, LC, can be specified by position vector, r_{lc}, of point of the line LC. When the gears rotate, the line of contact, LC, travels in relation to a reference system associated with the gear. For the purpose of analytical description of the transition from the first reference system to the second one, an operator of the resultant coordinate system transformation, Rs^{ca} (PA |γg), is commonly used. (Details on coordinate system transformation to the best possible extent are discussed in [1], and in a few more advanced sources.) Therefore, a gear tooth flank, G, can be viewed as a locus of consecutive positions of the line of contact, LC, in its motion in relation to the reference system of the gear. Having the expressions for r_{lc} and Rs^{ca} (PA |γ g) determined, the gear tooth flank, G, can be expressed by position vector, r_{g}^{ca}, of point:

Tooth flank of a gear, G, for perfect C_{a} — gearings with line contact between tooth flanks of a gear and a mating pinion, is specified by Equation 5. The tooth flanks generated this way fulfill both the condition of contact, as well as the condition of conjugacy of the interacting tooth flanks.

A plane of action, PA, in a crossed-axes gear pair is shown in Figure 8. Every two desired lines of contact, LC, are at an equal angular distance, *ϕ*_{b.pa}, from one another. This angular distance is referred to as the “angular operating base pitch, *ϕ*_{b.pa}” in a crossed-axes gear pair. As all desired lines of contact, LC, are spaced equally, the generated gear tooth flanks feature an angular base pitch of a constant value.

Ultimately, if the gears are designed in compliance with the proposed approach, they can be engaged in perfect crossed-axes mesh with line contact between the tooth flanks.

### 4. Inconsistencies in Crossed-Axes Gearings of Nowadays Design

Almost all inconsistencies in gearings of today’s design that feature crossing axes of rotation of a gear and a mating pinion are due to specific features of the tooth flanks’ generation. Because of this, only approximate gears can be generated in conventional gear machining processes.

Conventional principle of the gear tooth flank, G, generation in crossed-axes gearing is illustrated in Figure 9. In the machining process, a gear to be machined is engaged in mesh with a crowned face gear with straight-sided teeth (round basic rack). Such a schematic is not applicable for finish-cutting of gears for perfect C_{a} — gearings, as not all of three fundamental laws of gearing are fulfilled in the gear machining process.

The condition of contact of the gear and the crowned face gear with straight-sided teeth is fulfilled in most cases of gear machining.

The condition of conjugacy of the gear and the crowned face gear with straight-sided teeth is violated, as the common perpendicular to the gear and the mating pinion tooth flanks, G and P, does not intersect the axis of instant rotation, P_{ln}, at every instant of time. This is illustrated in Figure 10. All today’s technologies of cutting/finishing bevel gears for C_{a} — gearings are developed on the premise of a crowned face gear with straight-sided teeth. No accurate gears for C_{a} — gearings can be produced this way.

The just-mentioned violation of the condition of conjugacy means that a crowned face gear with straight-sided teeth cannot be used for finishing gears for perfect gearings. A basic rack of a more complex geometry can be used for this purpose. However, neither gear-cutting tools, nor bevel-gear generators of modern design fit the basic rack of a complex geometry, that is, the basic rack of this type is impractical.

The angular base pitches, *ϕ*_{b.g} and *ϕ*_{b.p}, not only don’t equal to an operating base pitch of the gear pair, *ϕ*_{b.op}, but they (that is the angular base pitches *ϕ*_{b.g} and *ϕ*_{b.p}) cannot be constructed for approximate C_{a} — gearings.

Therefore, if the gears are generated by a crown gear (by the basic rack) with straight-sided teeth, they cannot be engaged in a perfect crossed-axes mesh with line contact between the tooth flanks.

### 5. A Few Complementary Comments

As it is impossible to discuss all the design features of crossed-axes gear pairs in a single article, a few more important considerations are noted:

- Precision gears for perfect crossed-axes gear pairs designed in compliance to the discussed approach and then manufactured to the blueprint feature better performance.
- No lapping process is required for finishing the gears. The gears simply can be finish-cut or ground on conventional bevel gear generators/grinders.
- The gears do not need to be paired. The gears become self-replaceable: There is no need to replace an entire gear pair. In necessary, only one broken bevel gear is replaced by a new one.
- In production of gears for C
_{a}— gearings, multiple-axes NC machines can be used for machining gears themselves, electrodes for EDM, dies for net-forging, and tools for extruding plastic gears. - Even gears with a few teeth (12 and fewer) can be designed and manufactured precisely (with conjugate tooth flanks and with the angular base pitches equal to operating base pitch of the gear pair).
- The derived equations for a gear and a mating pinion tooth flank can be used for the purposes of gear inspection as the reference surfaces. This makes the gear inspection procedure more reliable.
- When assembling gear drives, no shift-in and no shift-out for the adjustment of the gears are permissible, as all three apexes: A
_{g}, A_{p}, and A_{pa}, must be snapped together. This can be ensured when the gears are designed, manufactured, inspected, and put together in the gear housing. - The C
_{a}— gearings can be used at the first stage of a long gear train [where the input rotation is high, and the input torque is low (here a low axial thrust acts against bearings)], and not at the last stage, where the axial thrust is high. Conventional, and alternative arrangements of gear pairs in a gear drive become possible (as illustrated in Figure 11) if the gears are produced in compliance to the three fundamental laws of gearing. Huge and heavy roller bearings in the first case (Figure 11a) can be replaced with light and small roller bearings as illustrated in the second case (Figure 11b). - Gears for I
_{a}— gearings, and for C_{a}— gearings, are not identical to one another (the angular base pitch, and so forth). Therefore, gears designed and manufactured for C_{a}— gearings cannot be used in gearings, and vice versa. - As the theory of gearing evolves, the term “perfect/precision hypoid gearing” with line contact between the tooth flanks will become more common in all industries.

An intensive research in the field of crossed-axes gearings (and of intersected-axes gearings, as a reduced case of C_{a} — gearings) on the premise of the scientific theory of gearing has started just a decade ago [1].

### Conclusion

In today’s practice of design and manufacture of gears for crossed-axes gear pairs (hypoid gears in particular) no attention is paid to:

(a) the compliance of the designed gears to the conjugate action law (that is commonly referred to as the main law of gearing),

(b) to equality of the gear base pitch to the operating base pitch of the gear pair,

(c) proper axial location of the gears,

and a few others to be mentioned. When designing gears, as well as when the gears are inspected and manufactured, all these important gear design parameters are simply ignored by all the gear designers/manufacturers (including the world leaders in production gears for precision crossed-axes gear pairs).

To improve the accuracy and performance of gears that operate on intersected axes of rotation, it is recommended to design the gears so they are aligned to the three fundamental laws of gearing discussed in the article.

When machining (and especially when finishing) gears for C_{a} — gearings, the parameters of the kinematics and of the geometry of the machining process and the cutting tool have to follow the three fundamental laws of gearing.

### References

- 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: Radzevich, S.P., Theory of Gearing: Kinematics, Geometry, and Synthesis, CRC Press, Boca Raton, Florida, 2012, 743 pages].
- Shishkov, V.A., “Elements of Kinematics of Generation and Meshing in Gearings,” in: Theory and Calculation of Gears, Leningrad, Lonitomash, Vol. 6, 1948.
- Shishkov, V.A., Generation of Part Surfaces in Continuous-Indexing Machining, Moscow, Mashgiz, 1951, 152 pages.

### Bibliography

- Radzevich, S.P., “An Examination of High-Conformal Gearing,”
*Gear Solutions,*February 2018, pages 31-39. - Radzevich, S.P., Gear Cutting Tools: Science and Engineering, 2nd Edition, CRC Press, Boca Raton, FL, 2017, 606 pages.
- Radzevich, S.P., “AGMA/ANSI/ISO Standards on Bevel and Hypoid Gears,”
*Gear Solutions,*September 2017, pages 45-56. - Radzevich, S.P. et al, “Preliminary Results of Testing of Low-Tooth-Count Bevel Gears of a Novel Design. Part 3,”
*Gear Solutions,*January 2015, pp. 20-23. - Radzevich, S.P., et al, “Preliminary Results of Testing of Low-Tooth-Count Bevel Gears of a Novel Design. Part 2,”
*Gear Solutions,*December 2014, pp. 20-21. - Radzevich, S.P., et al, “Preliminary Results of Testing of Low-Tooth-Count Bevel Gears of a Novel Design. Part 1,”
*Gear Solutions,*October 2014, pp. 25-26. - Radzevich, S.P., Irigireddy, V.V., Precision Bevel Gears with Low Tooth Count, AGMA Technical Paper 14FTM18, American Gear Manufacturers Association, 2014, 12 pages.