The article deals with the kinematics and the geometry of geometrically-accurate gearing (with perfect gearing). Considered at the beginning for parallel-axes involute gearing, conjugate action law later on is formulated for intersected-axes gearing (straight bevel gearing, spiral bevel gearing, and so forth), as well as for crossed-axes gearing (hypoid gearing, spiroid gearing, worm gearing, and so forth). Examples of violation of the conjugate action law in gearing of all three kinds (that is, in parallel-axes gearing, in intersected-axes gearing, and in crossed-axes gearing) are considered.

### Introduction

The origination of geometrically-accurate gears can be traced back to the mid of the 18th century and can be credited to Leonhard Euler, who proved that involute tooth profile fits the best gears that operate on parallel axes of rotation of a gear and a mating pinion. It is proven that in order to be referred to as “geometrically-accurate,” the gears have to fulfill three fundamental laws of gearing [1]:

**1: **At every instant of rotation of the gears, the condition of contact of the interacting tooth flanks, G and P, of a gear and a mating pinion, has to be fulfilled. The condition of contact has been known by gear engineers for a long while. For the analytical description of this condition, the “Shishkov equation of contact” n* _{g}* • V

_{Σ}= 0 (∼1948) is commonly used. It is designated: n

*is the unit vector of a common perpendicular at contact point of the tooth flanks, G and P; V*

_{g}_{Σ}is the vector of relative motion of the tooth flanks, G and P, at the point of their contact.

**2: **At every instant of rotation of the gears, the condition of conjugacy of the interacting tooth flanks, G and P, of a gear and a mating pinion, have to be fulfilled^{1}. The condition of conjugacy is analytically described (2017) by the “equation of conjugacy” in the form of a triple scalar product p* _{ln}* x V

*• n*

_{m}*= 0. It is designated: p*

_{g}*is the unit vector along the axis of instant rotation, P*

_{ln}*, of a gear and a mating pinion; n*

_{ln}*is the unit vector of a perpendicular to the gear tooth flank, G (to the pinion tooth flank, P); V*

_{g}*is the vector of linear velocity of a certain gear/pinion tooth flank point, m, together with the plane of action, PA (that is, V*

_{m}*= V*

_{m}*).*

_{pa}**3: **At every instant of rotation of the gears, the gear angular base pitch, ϕ* _{b.g}*, has to be equal to the operating base pitch of the gear pair, ϕ

*(that is, an equality, ϕ*

_{b.op}*= ϕ*

_{b.g}*, has to be valid), and the pinion angular base pitch, ϕ*

_{b.op}*, has to be equal to the operating base pitch of the gear pair, ϕ*

_{b.p}*(that is, an equality, ϕ*

_{b.op}*= ϕ*

_{b.p}*, has to be valid).*

_{b.op}In this article, the discussion will be focused mainly of the fulfillment of the condition of conjugacy of the interacting tooth flanks of a gear and a mating pinion.

### Conjugate Action Law: State-of-the-Art

In today’s practice, the analysis of the gear kinematics and geometry is limited to the fulfillment only of the “condition of contact” (or, in other words, of the “enveloping condition”) of the interacting tooth flanks of a gear, G, and its mating pinion, P. It is loosely assumed that once the condition of contact, n_{g} • V* _{Σ}* = 0, between the interacting tooth flanks of a gear and a mating pinion is fulfilled, then the tooth flanks, G

**and**

**P, are conjugate. The later statement is incorrect: Non conjugate tooth flanks of a gear and a mating pinion also commonly meet the condition of contact, n**

_{g}• V

*= 0.*

_{Σ}To be conjugate, common perpendiculars at every point of the line of contact, LC, between the tooth flanks, G** **and** **P, have to pass through the axis of instant rotation of the interacting surfaces every time, that is, for any and all possible configurations of the surfaces, G** **and** **P, relative to each other.

The conjugate action law in parallel-axes gearing was discovered by L. Euler, and (later and independently, by Felix Savary) in the mid-18th century. A lot of effort was undertaken on the conjugate action law by Charles Camus. Unfortunately, Camus failed to discover/finalize the conjugate action law; however, his input is important. With that said, it is clear now why the conjugate action law for parallel-axes gearing is referred to as “Camus-Euler-Savary main theorem of gearing” (or, just “CES — main theorem of gearing,” for simplicity). In Europe, the conjugate action law for parallel-axes gearing is referred to as “Willis theorem” due to the input of Robert Willis of the United Kingdom, who introduced this theorem to the public [2].

Conjugate action law (according to Willis): The angular velocities of the two pieces are to each other inversely as the segments into which the “line of action” divides the line of centers, or inversely as the perpendiculars from centers of motion upon the line of action.

According to the “conjugate action law,” the center-distance, C, is divided by the pitch point, P, onto two segments, O_{g}P = r_{g} and O_{p}P = r_{p}, so that a proportion:

is valid.

The conjugate action law is illustrated in Figure 1. At the beginning, it is necessary to stress here the difference between the line of action, LA, and the path of contact, P_{c}. In geometrically-accurate parallel-axes involute gearing, the path of contact is a straight-line segment. This straight-line segment is aligned with the line of action, LA. Despite the line of action, LA, and the path of contact, P_{c}, aligning with each other, it is important to realize they are two different lines. These two lines align to one another in parallel-axes involute gearing and do not align to one another in gearing of other systems.

In today’s practice, fulfillment of the condition of conjugacy of the interacting tooth flanks of a gear and its mating pinion is commonly limited to the analysis of the simplest case of perfect (with zero axes misalignment) parallel-axes gears, that is, to the case of perfect involute gearing. The condition of conjugacy of the tooth flanks for gears with another tooth flank geometry is not discussed at all^{2}. Examples can be readily found for gears that feature a cycloidal tooth profile, a circular-arc tooth profile, as well as tooth profiles of other geometries different from the involute of a circle. Moreover, no analysis of the condition of conjugacy of the interacting tooth flanks of a gear and its mating pinion for gears that operate on intersected axes of rotation, as well as for gears that operate on crossed axes of rotation, has ever been performed.

This is all that is known to this end on the conjugate action law. There is no evidence of understanding of the conjugate action law (a) in gearing with modified tooth flanks, (b) in intersected-axes gearing, and (c) in crossed-axes gearing. It is important to bridge this gap in modern gear theory and practice, as the conjugate action of a gear and a mating pinion tooth flanks is vital in both gear design and gear production.

### Conjugate Action Law in Cases of Intersected-Axes Gearing, and Crossed-Axes Gearing

Prior to beginning the discussion on how the conjugate action law works in intersected-axes gearing and crossed-axes gearing, let’s look at the specific features of this law of gearing in case of parallel-axes gearing.

Referring to Figure 2, consider a schematic of meshing in geometrically-accurate parallel-axes involute gearing.

With gearing of this particular kind, the tooth flanks, G and P, are designed so as to fulfill both the condition of contact and the condition of conjugacy. Therefore, at every instant of meshing of the gears, the line of action, LA, intersects the center-line, ** CL**, at the stationary pitch point, P. Due to this, the condition of contact (“Shishkov equation of contact”):

and the condition of conjugacy:

of the tooth flanks, G and P, are fulfilled every time.

The “equation of conjugacy” (see Equation 3) is derived based on the following consideration.

The plane of action, PA (as well, as the unit normal vector, n_{pa}, to the plane of action, PA: n_{pa} = p_{ln} x V_{m}), can be specified in terms of two vectors: p_{ln} and V_{m}. The unit normal vectors, n_{g} and n_{pa}, have to be perpendicular to one another. Therefore, n_{g} • n_{pa} = 0. This expression can be represented in the form of Equation 2.

In the event a gear tooth flank, G, the pinion tooth flank, P, or both are designed improperly, an instant pitch point migrates along the center-line, ** CL**, from a left extreme position, P

_{l}, to a right extreme position, P

_{r}. An arbitrary position of the pitch point, P, in such a reciprocation is labeled as P

_{i}. Commonly, the current value of the transverse pressure angle, φ

_{t.i}, differs from it nominal value, φ

_{t}(that is, an inequality φ

_{t.1}≠ φ

_{t}is observed). Ultimately, the reciprocation of the pitch point, P, and the variation of the transverse pressure angle, φ

_{t.i}, cause an excessive unfavorable noise excitation and vibration generation in gearing.

Once the importance of fulfillment of the condition of conjugacy of the interacting tooth flanks in parallel-axes gearing is realized, then one can proceed with an analysis of the fulfillment of the condition of conjugacy of the interacting tooth flanks, G and P, in intersected-axes gearing, and in crossed-axes gearing.

The provided verbal description of the condition of conjugacy of the interacting tooth flanks, G and P, of a gear and a mating pinion can be complemented with an analytical description.

To meet the requirements imposed by the equation of conjugacy (see Equation 2), that is, in order to have a gear tooth flank, G, and a mating pinion tooth flank, P, conjugate, the unit normal vector, n_{g}, has to be entirely within the plane of action, PA.

A line of action of the unit normal vector, n_{g}, intersects the axis of instant rotation, P_{ln}, if the condition of conjugacy, p_{ln} x V_{m} • n_{g} = 0, is fulfilled.

An example of fulfillment of the condition of conjugacy of a gear, G, and a mating pinion, P, tooth flanks in crossed-axes gearing is illustrated in Figure 3 (in a case of intersected-axes gearing, the lengths of the distances A_{g}A_{pa} = A_{p}A_{pa} = 0, and all three apexes, A_{g}, A_{p}, and A_{pa}, are snapped together: A_{g} ≡ A_{p} ≡ A_{pa}). A desirable line of contact, LC_{des}, between the tooth flanks, G and P, is initially specified in a reference system, X_{pa}Y_{pa}Z_{pa}, associated with the plane of action, PA. The desirable line of contact, LC_{des}, is entirely within the plane of action, PA, and travels with the PA as the gears rotate. The unit normal vector, n_{g}, to the gear tooth flank, G , always intersects the axis of instant rotation along the unit vector, p_{ln} (the pitch-line, P_{ln}). This immediately results in the condition of conjugacy of the interacting tooth flanks, G and P, in intersected-axes gearing, and in crossed-axes gearing is fulfilled.

Generated this way, the tooth flank, G, of a geometrically-accurate straight bevel gear for intersected-axes gear set is shown in Figure 4. Depending on the gear tooth count and the actual values of the rest of the design parameters, different portions of the generated gear tooth surface are used to shape the bevel gear teeth. The concept of the geometrically-accurate gears for intersected-axes gearing can be traced back to 1887 when George Grant filed the patent named as “Machine for Planing Gear Teeth” [3].

Constructed for a intersected-axes gear set, a schematic similar to that shown in Figure 4 also can be constructed for a crossed-axes gear set with any desirable geometry of the line of contact, LC_{des}. Geometrically-accurate crossed-axes gearing with true-line contact between the tooth flanks, G and P, is referred to as “R — gearing” [1]. “R — gearing” was proposed (∼2008) by Dr. S. Radzevich.

A consideration similar to that above is valid with respect to the pinion tooth flank, P (not illustrated in Figure 3). When the gears rotate, the desirable line of contact, LC_{des}, describes the gear and the pinion tooth flanks. The generation of the gear tooth flank, G, is considered in a reference system, X_{g}Y_{g}Z_{g}, associated with the gear, and the generation of the pinion tooth flank, P, is considered in a reference system, X_{p}Y_{p}Z_{p}, associated with the pinion. The so-called “describing method” is used for the generation of the tooth flanks, G and P.

The second fundamental law of gearing (the conjugate action law) is helpful when approximate gearing is analyzed. A point of a gear tooth flank, G, can be a contact point of the tooth flanks, G and P, when the condition of conjugacy is met. For a corresponding angular configuration of a gear and a mating pinion point of a pinion tooth flank, P, that is anticipated to be in contact with the gear tooth flank, G, can actually be a contact point only when the condition of conjugacy is met. A possibility of contact of the teeth flanks, G and P, in approximate gearing can be verified by means of comparison of coordinates of the “contact” point on the gear tooth flank, G, and that on the pinion tooth flank, P. If the coordinates of the points are identical to one another, then a point really is a contact point (the unit normal vectors, n_{pl} and n_{m}, must be aligned with each other). Otherwise, contact of the teeth flanks, G and P, in these points is not possible.

Gears for intersected-axes gear pairs and crossed-axes gear pairs have to be designed so as to fulfill the conjugate action law as discussed in [1] — this is a must. Moreover, gears for intersected-axes gear pairs and crossed-axes gear pairs have to be finish-cut or ground/honed so as to fulfill the conjugate action in the gear generating process [4]. Currently, this is the only way to eliminate the gear lapping process, which is required to finish poorly designed and poorly finish-cut gears for intersected-axes, as well as for crossed gear pairs. In today’s industry, none of the gear companies is concerned with the fulfillment of the conjugate action law when designing and finishing gears for intersected-axes gear pairs and for crossed-axes gear pairs.

It is proven [5] that the contact pattern of a favorable geometry and location can be ensured only in conjugate intersected-axes and crossed-axes gear pairs.

An in-depth understanding of conjugate action law in cases of I_{a} — gearing, and C_{a} — gearing is vital for the development of software for gear-measuring machines (GMMs). Accurate and reliable gear inspection is not possible in cases when the conjugate action law (in cases of I_{a} — gearing, and C_{a} — gearing) is ignored.

Gear experts familiar with the basics of modern gear theory realize how vital the conjugate action law for gearing is in a general sense, and the importance of the conjugate action law for intersected-axes gear sets — crossed-axes gear sets in particular — as well as for gear sets with the axes misalignment.

### Examples of Violation of Conjugate Action Law

For finishing gears for geometrically-accurate intersected-axes and crossed-axes gearing, appropriate methods of finishing gear tooth flanks are outlined in [4]. Unfortunately, in today’s industry, when finishing gears for I_{a} — gearing and C_{a} — gearing, the applied methods and means are based on the application of a crown-generating rack with straight-sided teeth. Two schematics of the crown-generating rack are shown in Figure 5.

Straight bevel gears are finish-cut using methods and means based on the application of the crown generating rack shown in Figure 5a that features a straight-sided tooth profile (trapezoidal tooth profile). Spiral bevel gears are finish-cut/ground using methods and means based on the application of the crown generating rack shown in Figure 5b that also features a straight-sided tooth profile. No accurate gears for intersected-axes gearing, as well as for crossed-axes gearing, can be generated by the crown racks depicted in Figure 5, as none of these racks is capable of generating conjugate tooth flanks, G and P, of two mating gears.

**Straight bevel gears: **The crown generating rack (see Figure 5a) is extensively used for generating tooth flanks of straight bevel gears. In a gear-planing operation, the tooth flank of the crown-generating rack is reproduced by means of straight cutting edges of two cutters that reciprocate toward the axis of rotation of the gear to be machined. A close up of the gear planing operation is shown in Figure 6a.

Milling straight bevel gears by means of dual interlocking circular cutters is another method of finish-cut gears for intersected-axes gear sets. In a gear-milling operation, the tooth flank of the crown generating rack is reproduced by means of straight-cutting edges of two mill cutters that rotate about their axis of rotation. Cutting edges of the mill cutters are perpendicular to the axes of their rotation. In some applications, the straight cutting edges of the mill cutters slightly deviate from the orthogonal position. In this case, the gear-tooth flanks are generated by means of an internal cone reproduced by the cutting edges. A closeup of the gear milling operation is shown in Figure 6b.

Only approximate gears can be finish-cut in both cases shown in Figure 6 and in similar cases where a crown generating with straight-sided teeth are used.

**Spiral bevel gears: **The crown-generating rack (see Figure 5b) is extensively used for generating tooth flanks in spiral bevel gears. In a gear-machining operation, the tooth flank of the crown-generating rack is reproduced by means of straight cutting edges of multiple cutters that rotate about the axis of rotation of the cutter-head. A close up of the face-milling operation is depicted in Figure 7.

Only approximate gears can be face-milled in face-milling operation shown in Figure 7, where a crown generating with straight-sided teeth are used.

**Spiral bevel gear grinding operation: **The crown generating rack (see Figure 5b) is used for generating tooth flanks of spiral bevel gears. In a gear-grinding operation, the tooth flank of the crown-generating rack is reproduced by means of abrasive grains of the abrasive wheel that rotates about the axis of rotation of the grinding wheel. A close up of the gear grinding operation is shown in Figure 8.

Only approximate gears can be finished up in gear grinding operation shown in Figure 8, where a crown generating with straight-sided teeth are used.

**Machining/finishing of spur/helical gears: **When skiving spur gears (Figure 9a), honing internal helical gears (Figure 9b), as well as in numerous other gear-finishing operations, the conjugate action law is violated, and, thus, no geometrically-accurate gears can be produced.

An in-depth analysis of gear sets of various gear systems reveals that mainly approximate gears are designed and manufactured today. Double-enveloping gearing (cone drive) is one of the examples in this regard. Known for more than a century, the so-called “octoid gears” is another example of approximate gearing in which the conjugate action law is violated.

### Conclusion

This article deals with geometrically-accurate gears with parallel-axes gear sets (P_{a} — gearing), intersected-axes gear sets (I_{a} — gearing), and with crossed-axes gear sets (C_{a} — gearing). The “enveloping condition” (namely, the “Shishkov equation of contact”) is briefly outlined. The “condition of conjugacy” of the interacting tooth flanks of a gear and a mating pinion is discussed in more detail mainly with a focus on intersected-axes gearing, as well as on crossed-axes gearing. Examples of applying the results of the analysis with respect to approximate gearing are provided. The readers’ attention should focus on the violation of the “condition of conjugacy” in today’s design practice and the production of gears for I_{a} — gearing and C_{a} — gearing. It is also mentioned that the so-called “describing method” of the generation of tooth flanks of a gear and a mating pinion have to be used instead of the “generating method” commonly used in the today’s practice. Interchangeable bevel gears can be produced if the conjugate action law is fulfilled when the gears for I_{a} — gearing and C_{a} — gearing are designed and finish-cut/ground. The interchangeable gears for I_{a} — gearing and C_{a} — gearing do not need to be paired, and the broken gears can be replaced individually and not as a gear set.

Gears for intersected-axes gear pairs and crossed-axes gear pairs have to be designed to fulfil the conjugate action law — this is a must. Moreover, gears for intersected-axes gear pairs and crossed-axes gear pairs have to be finish-cut or ground/honed to fulfil the conjugate action in the gear generating process. Currently, this is the only way to eliminate the gear-lapping process, which is required to finish poorly designed and poorly finish-cut gears for intersected-axes, as well as for crossed gear pairs. No lapping is required when the gears are finish-cut/ground geometrically-accurate (in accordance to the conjugate action law).

It is proven that the contact pattern of a favorable geometry and location can be ensured only in conjugate intersected-axes and crossed-axes gear pairs.

An in-depth understanding of conjugate action law in cases of I_{a} — gearing and C_{a} — gearing is vital for the development of software for gear-measuring machines (GMMs). No accurate and reliable gear inspection is possible in cases when the conjugate action law (in cases of I_{a} — gearing and C_{a} — gearing) is ignored.

In summary, it becomes evident the following statement is valid: Properly engineered gears for I_{a} —, and C_{a} — gear sets fulfil the conjugate action law.

When the condition of conjugacy is violated, the gears can not be used in the design of high-power-density gear transmissions, in gear transmissions with high-rotated gears, in the design of low-tooth-count gears, and so forth.

### Footnotes

^{1} In case of parallel-axes gearing, the condition of conjugacy (the conjugate action law) was known for L. Euler. In the time of Euler and later on, only verbal formulation of the conjugate action law was known; no analytical expression for this important law of gearing has been derived till the beginning of third millennium.

^{2} Wrong practice is commonly adopted when analyzing whether the condition of conjugacy of the tooth flanks, G and P, is fulfilled, or it is not. “Shishkov equation of contact”, n* _{g}* • V

*= 0, is loosely used for this purpose (here, in the equation, n*

_{Σ}*is the unit normal vector to the tooth flanks G and P at point, K, of their contact, and V*

_{g}*is the velocity vector of the resultant relative motion of the surfaces G and P at the contact point, K). The condition of contact of the tooth flanks G and P, and not the condition of conjugacy of the interacting tooth flanks of a gear and its mating pinion is analytically described by “Shishkov equation of contact”. Unfortunately, often this difference is not recognized at all.*

_{Σ}### References

- Radzevich, S.P., Gear Cutting Tools: Science and Engineering, 2nd Edition, CRC Press, Boca Raton, FL, 2017, 606 pages.
- Radzevich, S.P., Theory of Gearing: Kinematics, Geometry, and Synthesis, 2nd Edition, revised and expanded, CRC Press, Boca Raton, FL, 2018, 934 pages.
- U.S. Pat. No. 407.437. Machine for Planing Gear Teeth. /G.B. Grant, Filed: January 14, 1887 (serial No. 224,382), Patent issued: July 23, 1889.
- Willis, R., Principles of Mechanisms, Designed for the Use of Students in the Universities and for Engineering Students Generally, London, John W. Parker, West Stand, Cambridge: J. & J.J. Deighton, 1841, 446 pages.
- Radzevich, S.P., et al, “Preliminary Results of Testing of Low-Tooth-Count Bevel Gears of a Novel Design”, Gear Solutions, Part 1, October 2014, pp. 25-26; Part 2, December 2014, pp. 20-21; Part 3, January 2015, pp. 20-23.