In this article, a few of the most critical accomplishments in the scientific theory of gearing will be discussed.

The now-famous equation, “knowledge is power” (“scientia potestas est”), was coined by Francis Bacon (January 22, 1561-April 9, 1626). Formulated as early as 1597, this famous equation was proven to be correct for centuries. Knowledge in any and all areas of human activity gives one a strong advantage over those less knowledgeable ones. Gearing is no exception. Knowledgeable and skilled gear engineers always have a greater chance for success doing business, rather than less knowledgeable and less skilled ones. This article is aimed at the promotion of knowledge (namely, the promotion of the “scientific theory of gearing”) to gear practitioners, and, in this way, to equip them with a means to achieving success running a business in the gear industry.

Introduction

The necessity of scientific foundations of gearing for design and manufacturing has long been realized by gear practitioners. Gears that operate on parallel axes of rotation of a gear and a mating pinion are the first to attract attention of mathematicians and gear theoreticians. It was Leonhard Euler who, in the mid-18th century, proved that an involute tooth profile fits the best gears that operate on parallel axes of rotation of a gear and a mating pinion.

Having started his own career in gearing in 1975, the author had many opportunities to get familiar with state-of-the-art innovations in both gear design and gear production. Shortly after graduating from college, the author was faced with: (a) numerous specific problems in gearing, and (b) the absence of an adequate means to solve these problems. The decades of research in the field of gearing theory by the author has resulted in a 2018 monograph titled “Theory of Gearing: Kinematics, Geometry, and Synthesis” [10]. It definitely makes sense to the gear community to get familiar with the capabilities that the proposed theory of gearing provides gear practitioners. In this article, a few of the most critical accomplishments in the scientific theory of gearing will be briefly discussed. The author concludes many in the gear community may be unfamiliar with the key accomplishments in the scientific theory of gearing. A recently published (2019) Gear Solutions article [15] (and not this article only) is taken as a representative example that proves this conclusion. Unfortunately, there are plenty of publications of this sort in current journals and magazines for gear engineers and scientists. Moreover, the article [15] (as well as other similar publications) may mislead gear practitioners, offering wrong approaches in the field of gear design and gear manufacturing. All the ambiguities in the practice of gear design and gear production should be eliminated. An old Chinese proverb, “The beginning of wisdom is to call things by their right names,” is a good place to start.

Motivation

This article was necessitated by the recently published article: Stadtfeld, H.J., “Why are Today’s Hypoids the Perfect Crossed-Axes Gear Pairs?” Gear Solutions magazine, May 2019, pp. 42-50 [15]. These comments pertain to the key accomplishments in the scientific theory of gearing. The comments could be confusing to inexperienced gear community members, especially those who attempt to study the gear theory on their own and apply the theory in their practical work.

Here and below in this text, the article [15], is used solely as a representative example of an improper understanding of the scientific theory of gearing by a majority of gear practitioners, including those who may call themselves “gear theoreticians.” An additional reason to reference the article [15] is that it claims to be written on behalf of established gear theory and manufacturing (see page 43 in [15]): “This write-up is meant to be a response on behalf of established gear theory and manufacturing.”) No reference is provided in [15] as to whether this “established gear theory” is available to the reader in the public domain.

Recommendation to the readers of the current article: Except for section 1, the rest of the sections in this article intentionally reflect the article [15]. This is to help the reader better understand what these two articles are about, their differences, and why the approach outlined in the current article is advantageous over the approach recommended in the article [15]. When reading the current article, it is recommended to have the article [15] for comparison.

1 The introductory section: Novikov Gearing (Conformal Gearing)

The evolution of gear art is heavily associated with Ernest Wildhaber. No doubt, Wildhaber is a gifted inventor and a famous gear engineer. Despite that, Wildhaber has never been recognized as a gear scientist, as is loosely claimed in [15] (page 43). Because of that, Wildhaber cannot be granted with the title of “the father of modern gear theory” (see page 43 in [15]). Both these statements are incorrect: Wildhaber was NOT a gear scientist at all, and, therefore, it is incorrect to refer to him as the father of modern gear theory. Wildhaber is credited with NO key accomplishments in the scientific theory of gearing (see [10] for details). A difference between the terms “a gifted inventor and a famous gear engineer” and “a gear scientist” must be noted.

The earlier performed comparison of “Novikov gearing” [2], and Wildhaber “Helical gearing” [1] reveals these two gear systems are completely different, and, moreover, these gear systems are not compatible with one another [10, 7, 9]. On top of that, the term “Wildhaber-Novikov gearing” is incorrect by nature, as these two gear systems (namely, Wildhaber “Helical gearing” [1] and “Novikov gearing” [2]) cannot be combined into a common gear system (see [10, 7, 9], and others, for details). It is a huge mistake to adopt the term “Wildhaber-Novikov gearing.”

“Novikov gearing” can be viewed as a reduced case of a parallel-axes involute gear pair, while Wildhaber “Helical gearing” cannot ([3, 6] and numerous other sources).

Today’s gear professionals should know the difference between “Novikov gearing” [2] and Wildhaber “Helical gearing” [1].

2 News About Hypoid Gears?

The article [15] proves today’s hypoid gearsets are approximate by nature, and, thus, they are not geometrically-accurate gearsets. No alterations can be even anticipated.

Further, no attempt is taken in the article [15] to “question the creditability of the scientists and gear engineers who worked on the theory and its improvement over approximately 100 years.” No gear theory was developed during these 100 years, and no key scientific accomplishments are attained at this period of time. “Trial-and-error method” is the main — and perhaps only — tool used for “over approximately 100 years.” But this is not a theory of gearing. Instead, this is a collection of more-or-less reasonable solutions to separate gear problems. Combined together, these solutions do not form a gear theory.

3 The Three Fundamental Laws of Gearing

Leonhard Euler can be credited for originating geometrically-accurate gears (even though the term “geometrically-accurate gears” was introduced much later [10]). It is proven [10] that, in order to be referred to as “geometrically-accurate,” the gears have to fulfill three fundamental laws of gearing. The fulfillment of three fundamental laws of gearing is necessary and sufficient to refer to a gearset as a “geometrically-accurate gearset.” To the best possible extent, all three fundamental laws of gearing are discussed in [10]. These laws of gearing are briefly outlined here:

The first fundamental law of gearing: “At every point of contact of tooth flanks of a gear and a mating pinion, the vector of their instantaneous relative motion has to be orthogonal to the common perpendicular at every instant of time.”

In this form, the first fundamental law of gearing is valid for gearing of all designs that operate on parallel axes, intersecting axes, and crossing axes of rotation of a gear and its mating pinion.

Numerous interpretations of the first fundamental law of gearing are known. The most practical and extensively adopted way of interpretation of the first fundamental law of gearing is to analytically describe this law of gearing in the form of the “Shishkov equation of contact, ng VΣ = 0.” The “Shishkov equation of contact, ng VΣ = 0” is known to practically every gear engineer. Professor V.A. Shishkov was the first to express (1948) the first fundamental law of gearing in the form of the dot product of a unit vector of a common perpendicular, ng, by the vector of the resultant linear velocity, VΣ, of the tooth flanks in relation to one another. Both the vectors, ng and VΣ, are calculated at the contact point of a gear and its mating pinion tooth flanks.

The second fundamental law of gearing is often referred to as the “conjugate action law.” When the “Shishkov equation of contact, ng VΣ = 0” is fulfilled, the “conjugate action law” can either be fulfilled, or it can be violated. In Figure 1, an example of interaction of local patches of the tooth flanks, G and P, is schematically illustrated. The tooth flanks, G and P, make contact at point, K. The radii of curvature of the interacting tooth flanks at contact point, K, equal to Rg and Rp, correspondingly (see Figure 1a). The centers of curvature of the tooth profiles,  G and P, are denoted by Og and Op, correspondingly. In the instantaneous motion of the pinion, P, in relation to the gear, G, the pinion performs an instantaneous rotation, ωp/g, about the point Og. The radius of curvature of the generated actual gear tooth flank, G act, equals to Rp/g Rg. In this scenario, the second fundamental law of gear is fulfilled, and the actual tooth flank, G act, is identical to the desirable gear tooth flank, G , as shown in Figure 1a.

Figure 1: Examples of (a) fulfillment, and violation: (b) interference, and (c) separation, of a gear, G, and its mating pinion, P, tooth flanks: the “Shishkov equation of contact, ng • VS = 0” is fulfilled in all three cases (a) through (c), while the law of conjugacy is violated in the cases (b) and (v).

Note: In Figure 1a, the rotation, ωn, of the pinion, P, in relation to the gear, G, about the contact perpendicular, ng, is not prohibited by the second fundamental law of gearing.

If the instantaneous rotation is performed either about the center Op/g (when Rp/g < Rg, see Figure 1b), or about the center Op/g (when Rp/g > Rg, see Figure 1c), the second fundamental law of gear is violated, and the actual tooth flank, G act, differs from the desirable gear tooth flank, G .

Reminder: The first fundamental law of gearing (as well as the “Shishkov equation of contact, ng VΣ = 0”) is fulfilled in all three cases shown in Figure 1, while the second fundamental law of gearing is fulfilled only in the first case illustrated in Figure 1a. The schematic in Figure 1 is helpful for understanding the difference between the first and second fundamental laws of gearing and prevents the making of unsubstantiated conclusions in this regard.

Important: When the second fundamental law of gearing is fulfilled, the first fundamental law of gearing is always fulfilled, and not vice versa.

With that said, it makes sense to go back to the article [15] and compare the results illustrated in Figure 1 with a loosely made statement (see page 44 in [15]): “The second fundamental law of gearing, … , is a redundant relationship to the first gearing law, and it is limited to cylindrical gears with parallel axes and straight bevel gears without hypoid offset. In this case, it adds nothing to the first gearing law; conjugacy is already given by the relationships required in the first gearing law.” It is correct to question whether a gear practitioner has a chance for success in the design and production of today’s sophisticated gear systems if he or she strictly follows the afore mentioned quote (see page 44 in [15]).

In the most general case of gearing (namely, in the case of “crossed-axes gearing”, or “ Ca — gearing”, for simplicity) the second fundamental law of gearing is formulated as:

The second fundamental law of gearing: “In order to smoothly transmit a uniform rotary motion from a driving shaft to a driven shaft by means of gear teeth, perpendiculars to the tooth flanks of the interacting teeth at all points of their contact must intersect the axis of instantaneous rotation in the gear pair.”

In a reduced case of gears that operate on parallel axes of rotation of a gear and its mating pinion (namely, in the case of “parallel-axes gearing”, or “Pa — gearing”, for simplicity) this fundamental law of gearing is formulated as:

The second fundamental law of (parallel-axes) gearing: “In parallel-axes gearing, in order to smoothly transmit a uniform rotary motion from a driving shaft to a driven shaft by means of gear teeth, perpendiculars to the tooth flanks of the teeth at all points of their contact must pass through a stationary point located on the line of centers, namely, the pitch point P; the pitch point subdivides the center-distance reciprocal to the angular velocities of the gear and the pinion.” (Robert Willis, 1841 [16])

These days, the second fundamental law of gearing (in a case of parallel-axes gearing) is commonly referred to as “Camus-Euler-Savari fundamental law of gearing” (or “ fundamental law of gearing,” for simplicity).

The second fundamental law of gearing is also often referred to as “conjugate action law” — and this is also correct.

The design of “Alpha Worm Gearing” (New Venture Gear, Syracuse, NY; US Pat. No. 6,148,683, Nov. 21, 2000) is a good example of the violation of the second fundamental law of gearing. Violation of the second fundamental law of gearing is the main reason for this design fail, and it has caused major financial damage to the gear industry.

The third fundamental law of gearing relates to the distribution of the gear teeth over the periphery of a gear and a mating pinion. In the most general case of gearing (in the case of crossed-axes gearing), the third fundamental law of gearing is formulated as:

The third fundamental law of gearing: “In order to smoothly transmit a uniform rotary motion from a driving shaft to a driven shaft by means of gear teeth, at every instant of time the angular base pitch of a gear, and that of a mating pinion, both have to be equal to the operating angular base pitch in the gear pair.”

In a reduced case of “parallel-axes gearing,” this fundamental law of gearing is formulated as:

The third fundamental law of gearing (in parallel-axes gearing): “In parallel-axes gearing, in order to smoothly transmit a uniform rotary motion from a driving shaft to a driven shaft by means of gear teeth, at every instant of time base pitch of a gear and that of a mating pinion, both must be equal to the operating base pitch in the gear pair.”

All the geometrically-accurate gears (that is, all the precision gears — with no exception) meet all three fundamental laws of gearing — this is a must.

4 Perfect Conjugacy in Straight Bevel Gearsets

The title of the section “Perfect Conjugacy in Straight Bevel Gearsets” in the article [15] may be confusing to readers, as “conjugacy” by nature cannot be “perfect” or “imperfect.” Gearsets of all kinds are either conjugate, or they are not conjugate. The term “perfect conjugacy” could be considered analogous to saying “a bit pregnant.”

It is loosely claimed (see page 44 in [15]) that: “The plane of action cannot be extended beyond its tangential contacting line with the base element as shown incorrectly … . The plane of action exists where tooth engagement is possible.” Gearsets that feature the plane of action that extends far beyond its tangential contacting line with the base element have been known for decades. As an example, a parallel-axes gear set of this particular kind is depicted in Figure 2 [14]. Later on, this concept was evolved (2008) by Prof. S.P. Radzevich to the cases of intersected-axes gearing, as well as to crossed-axes gearing [10].

Figure 2: Rotationally-positive external parallel-axes gear pair (Pat. No. 163857, USSR, A Helical Gearing./B.V. Shitikov, N.A. Bayazitov, Int. Cl. F06h, Filed: February 25, 1963, Published: July 22, 1964.).

In a set of articles [11], [12], and [13], the preliminary results of testing of low-tooth-count spiral bevel gears of a novel design are discussed. Without going into details of design and manufacture of spiral bevel gears of this design (proposed by Prof. S.P. Radzevich, 2008, and designed using the earlier developed scientific theory of gearing [10]), it is right to stress the following:

1. As shown in Figure 3, the predicted contact patch is perfect (see Figure 7a on page 21 in [13]).

Figure 3: Predicted contact patch in geometrically-accurate intersected-axes spiral bevel gears.

2. As shown in Figure 4, the predicted contact pattern is perfect (see Figure 9 on page 23 in [13]).

Figure 4: The predicted contact pattern in geometrically-accurate intersected-axes gearing.

3. As shown in Figure 5, the contact pattern in the roll test is also perfect (see Figure 10 on page 23 in [13]), and it indicates excellent correlation with the predicted one.

Figure 5: Contact pattern in geometrically-accurate spiral bevel gears in roll test.

4. No lapping was used to finish the gears.

5. No adjusting for the axial configuration of the gear and the mating pinion is required, and the gears are ready to run.

In today’s production of precision gears, the use of the lapping process is inevitable. Those knowledgeable in the scientific theory of gearing can eliminate dirty and obsolete lapping processes from finishing tooth flanks of gears for precision gearsets.

Concerning the statements on page 45 in [15] that state: “The Octoid is the analog function of an involute, and it provides to bevel gears the same advantage as an involute provides to cylindrical gears. … bevel gears have a trapezoidal generating profile. The straight rack … becomes a ring … as shown in Figure 6.” It has been known for a while that the tooth profile of a crown rack for generating a geometrically-accurate bevel gear has to be shaped in the form of a curve illustrated in (see Figure 6) [10], and NOT in the form of a straight-sided crown rack. Only approximate bevel gears can be generated by the just-mentioned straight-sided crown rack “shown in Figure 6” [15]. No gearing with the so-called “octoidal” path of contact is possible at all: If one assumes the path of contact shaped in a form of an octoid curve, this inevitably entails violation of the second fundamental law of gearing — it is important to bear this in mind. The so-called “octoid gearing” is a mistake that often travels from one publication to another.

Figure 6: Trace of contact point in geometrically-accurate intersected-axes gear pair.

Concerning the statement on page 46 in [15] that states: “Conjugacy is the basis of all gearsets manufactured in high volume on dedicated manufacturing machines.” All the bevel gears manufactured in high volume on dedicated manufacturing machines are generated by means of straight-sided crown rack (shown in Figure 6, page 45 in [15]), and, thus, all of them are approximate gears. Only approximate bevel gears can be generated by just mentioned straight-sided crown rack “shown in Figure 6” [15]. As the generated bevel gears are approximate, no conjugate bevel gear pairs can be composed of these gears.

Concerning the statement on page 46 in [15] that states: “Conjugate bevel gearsets cannot be used for a power transmission because manufacturing tolerances and …” Today’s gear industry is capable of manufacturing precision helical involute gears that are extensively used, for example, in the design of turbine reducers. The gear industry is NOT capable of manufacturing precision bevel gears, and this is the main reason for precision bevel gears NOT being extensively used in the industry. The bottom-line is: Precision bevel gears are NOT extensively used in the industry solely because the gear industry is not capable of producing (for a reasonable cost) these kinds of gears.

Concerning the statement on page 46 in [15] that states: “The right amount of crowning makes a gearset quiet and gives it the required load carrying capacity”. How can the “right amount” be determined for a particular bevel gear application? The desirable amount of crowning can be determined by means of a “trial and error method” – this is the only tool developed by the “established gear theory” in [15]. Today’s scientific theory of gearing not only gives an answer to the question of how the “right amount” can be determined, but it also provides an in-detail specification of the entire modified tooth flank. Properly engineered gears (both design and production) do not need to use a “trial and error method.”

5 Perfect Conjugacy in Hypoid Gearsets

The discussion on “perfect conjugacy in hypoid gearsets” in [15] is similar to “perfect conjugacy in straight bevel gearsets” discussed earlier (see Section 4). Again, the title of the section “Perfect Conjugacy in Straight Bevel Gearsets” in the article [15] may be confusing to readers as “conjugacy,” by nature, can neither  be “perfect”’ nor “imperfect.” Gearsets of all kinds are either conjugate, or they are not conjugate.

Concerning the statement on page 46 in [15] that states: “… the pitch elements of crossed axes hypoid gears are drawn as cones.” It fails to describe one of the key features of the modern scientific theory of gearing [10]: In order to design gears for hypoid gearsets, no pitch elements are required at all. The term: “… the pitch elements are hyperboloids” is obsolete and is no longer used in the modern scientific theory of gearing [10].

Concerning the statement on page 47 in [15] that states: “No plane of action exists between two hyperbolic base elements.” It is correct, as it proves there is no place for hyperboloids in the kinematics and the geometry of gears with crossing axes of rotation of a gear and its mating pinion.

Concerning the statement on page 47 in [15] that says: “The correct surface of action is curved and wrapped, as shown in Figure 10.” Surface of action in hypoid gearing (as well as in gearing of all other kinds) is a plane through the axis of instantaneous rotation, or it is a plane through the common perpendiculars constructed at all points of the line of contact between the interacting tooth flanks of a gear and its mating pinion. The normal forces of interaction in gearing are commonly described by means of vectors. A vector is a quantity possessing both magnitude and direction. Vectors are described by means of straight-line segments with a specified direction. This leaves no room for “curved surface of action.”

5.1 Conjugacy between meshing flanks

Concerning the statement on page 47 in [15] that says: “The term conjugate is used in mathematics for two or more surfaces that contact each other along a line. Since the 1970s, the term conjugate has also been employed in gear technology literature to define…” Line contact between two surfaces (namely, between tooth flanks of two mating gears) is not sufficient (and is not always necessary) to refer to the pair of surfaces as to “conjugate surfaces.” In addition, the second fundamental law of gearing has to be fulfilled for conjugate surfaces, which is a must. Moreover, it has been discovered over the last 10 years that two surfaces (two tooth flanks) can be conjugate to one another even when they are in point contact.

5.2 Definition of the conjugate gear pair

Concerning the statement on page 47 in [15] that says: “… hypoid gearset conjugacy is possible with a non-generated gear that meshes with a generated pinion.” This reveals a poor understanding of the term “conjugacy,” as no conjugate action is possible between non-generated tooth flank and generated tooth flank.

6 Why is Conjugacy Not Desirable for Real World Applications?

Most gear engineers don’t doubt whether the conjugacy of interacting tooth flanks is desirable. Conjugacy is always desirable — especially in high-power-density transmissions and in high RPM-gearsets. Unfortunately, with no lapping process in finishing the tooth flanks, the gear industry is not capable of producing precision gears for real-world gearsets. For over a decade, a concept of crossed-axes gearsets that are insensitive to the mating gears axes displacements (both, the linear displacements, as well as the angular displacements) were introduced to the public [10]. Gears of this system are commonly referred to as “Spr — gearing.” The “Spr — gearing” is a kind of point contact conjugate gearing capable of absorbing all the linear and angular displacements of a gear and its mating pinion axes of rotation, as long as the actual values of these displacements are within the prespecified tolerances for the accuracy of the gearset. At the same time, the “Spr — gearing” features the highest possible power density being transmitting by the gearset.

7 Transmission Operations

Only time will tell if it is practical “… to place the hypoid gearset between motor and transmission …?” (see page 49 in [15]). Looking to the future, one can conclude that the vehicle powertrain is not the only potential application of gears with crossing axes of rotation of a gear and a mating pinion.

8 Is Lapping an Attempt to Make Hypoid Gears Conjugate?

Eventually, the lapping process in the production of gears for bevel and hypoid gearsets will be eliminated [6]. The gear-grinding process is a No. 1 candidate for the future of gear-finishing operations. The sooner the key accomplishments in the scientific theory of gearing reaches gear practitioners, the sooner gear lapping will be replaced by gear grinding. This negates the statement on page 49 in [15] that says: “This reveals a misperception about the reason for lapping.”

An in-depth familiarity with the key accomplishments in the scientific theory of gearing will make it possible to eliminate the gear-lapping process and replace it with the gear-grinding process. This is a reliable evolution of the gear design practice and the gear-finishing processes in particular.

9 Summary

It is important to note that over the last 100 years, the gear industry has failed to:

  • Eliminate lapping process.
  • Avoid the necessity of adjustment of bevel/hypoid gears in pairs (that makes the gears not-self-replaceable).
  • Eliminate the necessity of pairing of gears that operate on intersected, as well as on crossing axes of rotation of a gear and its mating pinion.
  • Solve the problem of excessive gear noise excitation and vibration generation.

Conclusion

In conclusion, it is likely the gear industry could end up wasting funds and time following the direction of evolution outlined in the article [15].

  • In the article [15] there is no evidence of:
  • Whether the third fundamental law of gearing is understood by the author.
  • The importance of the concept of the “operating base pitch” in a gear pair is realized.
  • The concept of a gear, of a mating pinion base pitch in intersected-axes gearing, and in crossed-axes gearing, along with the concept of the “operating base pitch” in intersected-axes, and in crossed-axes gearsets as realized by the author.

Not all the inconsistences in [15] are addressed here — only some of the larger issues. However, a more detailed analysis (if necessary) can be undertaken. 

References

  1. Pat. USA, No. 1,601,750, Helical Gearing, /E. Wildhaber, Patented: October 5, 1926, Filed: November 2, 1923.
  2. Pat. USSR, No. 109,113, Gear Pairs and Cam Mechanisms Having Point System of Meshing. /M.L. Novikov, National Classification 47h, 6; Filed: April 19, 1956, published in Bull. of Inventions No.10, 1957.
  3. Radzevich, S.P., “An Examination of High-Conformal Gearing”, Gear Solutions, February, 2018, pages 31-39.
  4. Radzevich, S.P., “Conjugate Action Law in Intersected-Axes Gear Pairs and in Crossed-Axes Gear Pairs”, Gear Solutions magazine, June 2020, pages 42-48.
  5. Radzevich, S.P., “Design Features of Perfect Gears for Crossed-Axes Gear Pairs”, Gear Solutions, February, 2019, pp. 36-43.
  6. Radzevich, S.P., Gear Cutting Tools: Science and Engineering, 2nd Edition, CRC Press, Boca Raton, FL, 2017, 606 pages.
  7. Radzevich, S.P., Dudley’s Handbook of Practical Gear Design and Manufacture, Third Edition, CRC Press, Boca Raton, FL, 2016, 629 pages.
  8. Radzevich, S.P., High-Conformal Gearing: Kinematics and Geometry, 2nd edition, Elsevier, Amsterdam, 2020, 506 pages.
  9. Radzevich, S.P., “On the Inconsistency of the Term “Wildhaber-Novikov Gearing”: A New Look at the Concept of “Novikov Gearing””, Appendix A, pp. 487-501, in: Radzevich, S.P., (Editor), Advances in Gear Design and Manufacture, CRC Press, Boca Raton, Florida, 2019, 549 pages.
  10. Radzevich, S.P., Theory of Gearing: Kinematics, Geometry, and Synthesis, 2nd edition, revised and expanded, CRC Press, Boca Raton, FL, 2018, 934 pages.
  11. 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.
  12. 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.
  13. 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.
  14. Shitikov, B.V., Bayazitov, N.A., A Helical Gearing, Pat. No. 163857, USSR, Int. Cl. F06h, Filed: February 25, 1963, Published: July 22, 1964. [see also: Bayazitov, N.A., Helical Gears with a New Type of Gearing, Ph.D. Thesis, Kazan’, Kazan’ Technological & Chemical Institute, 1964.].
  15. Stadtfeld, H.J., “Why are Today’s Hypoids the Perfect Crossed-Axes Gear Pairs?”, Gear Solutions magazine, May 2019, pp. 42-50.
  16. 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.