What the term ‘conjugate gear tooth flanks’ stands for

This article deals with conjugate gear tooth flanks. An evolution of the gear tooth flank geometry is briefly outlined, and the key accomplishments in the field are indicated (with the names of the corresponding researchers who deserve to be granted with a certain discovery in the area of interaction of conjugate tooth flanks). Gearing of all kinds are covered, namely: (a) parallel-axes gearing (P_a — gearing), (b) intersected-axes gearing (I_a — gearing), and (c) crossed-axes gearing (CP_a — gearing). The readers’ attention is focused here on the key features of the kinematics and geometry of P_a —, I_a —, and C_a — gearing. It is impossible neither to design, nor to implement perfect gears having no clear understanding of the kinematics, the geometry, and operating of gears of all kinds:  P_a — gearing,  I_a — gearing, and C_a — gearing. Production and application of gears with conjugate tooth flanks enables one to increase the gear accuracy and to reduce the noise emission that originates from gear transmissions.

1 Conjugate Tooth Flanks: State-of-the-Art

Gears and gear transmissions have been used by intelligent men for a long time. Centuries ago, wood was the main material used in manufacturing gears and gear transmissions (see Figure 1).

The accuracy of gears has long attracted the attention of scientists and researchers. The history of gear geometry can be traced back to the 17th century when Girard Desargus (16??), Philippe de La Hire (~1694), and C.-E.-L. Camus (~1733) published their first results into the geometry of the gear tooth flank. Despite significant efforts undertaken by the gear researchers of the time, no valuable results in this area have been recorded. It is likely the investigation into the conjugate action law by C.-E. Camus (~1733) was the first and only accomplishment that deserved the attention of gear scientists. However, even this achievement by C.-E. Camus was a mistake. Camus did not note a difference between the line of action, LA, and the path of contact, P_c, in a gear mesh. He assumed (see Figure 2) the line of action in parallel-axes gearing is a planar curve, BC (which is incorrect). Actually, the instant line of action in P_a — gearing is a straight line, (Moreover, the path of contact, P_c, is also a straight line aligned with the line of action, LA.) The line of action, LA, in gearing is the line along which a force from a driving gear is acting against the driven. Force acts only along a straight line, and it cannot act along a planar curve.

The lack of knowledge and experience in the field of gearing is the root cause for none of the gear scientists of that time to have succeeded in the discovery of the conjugate action law. This problem was waiting for L. Euler, who solved the problem of conjugate tooth flanks in parallel-axes gearing.

2 Conjugate Tooth Flanks: Parallel-Axes Gearing

The solution to the problem of conjugate tooth flanks in parallel-axes gearing was published by Leonhard Euler in his famous (1760) paper [1]. Euler derived an equation later known as the Euler-Savary equation. Félix Savary (October 4, 1797-July, 15, 1841) was a French mathematician and mechanician. About 50 years after Euler, Savary, independently of Euler, re-discovered this equation and presented it in the modern form. There is evidence the conjugate action law was known to Euler. It is not clear yet whether Euler distinguished the line of action in gear mesh from the path of contact. It could be assumed Euler valued his discovery of the involute tooth profile for geometrically-accurate  gearing more, rather than the rest of the scientific results obtained at that time. Historically, the involute gear tooth profile is a fundamental contribution of L. Euler.

Much later, around 1841, a brilliant book by Robert Willis [8] was published in the United Kingdom. In Europe, the conjugate action law is known mainly from this book by R. Willis [8] (see Figure 3). Willis’s theorem is the common name for the conjugate action law in European countries. However, R. Willis himself didn’t contribute to the conjugate action law.

Taking into account the contributions of C.-E. Camus, L. Euler, and F. Savary to solving the problem of conjugate tooth flanks in parallel-axes gearing, the second fundamental law of gearing is referred to as Camus-Euler-Savary fundamental theorem for parallel-axes gearing (or just as the  fundamental theorem for parallel-axes gearing, for simplicity). To answer the question: What does the term “conjugate gear tooth flanks” stand for, the  fundamental theorem for parallel-axes gearing is helpful.

The property of conjugacy is a specific property of two surfaces (or of two profiles) that roll over one another. The following definition to the term “Conjugate Gear Tooth Flanks” (in parallel-axes gearing) can be withdrawn from the preceding discussion:

Definition 1: Conjugate Gear Tooth Flanks in parallel-axes gearing are those whose common perpendicular always (for any and all relative configurations of the interacting tooth flanks) passes through the motionless pitch point in the gear mesh.

The proof to this statement can be found in [4], [6]. The article [4] is also available in [5].

The readers’ familiarity with the concept of equivalent pulley-and-belt transmission helpful for further discussion.

Equivalent pulley-and-belt transmission: The way that was used by L. Euler for the derivation of the involute tooth profile is long and boring. These days, when an extensive experience in the field of gear tooth profile geometry is gained, an easier way to come up with that same result can be used. A property of an equivalent pulley-and-belt transmission is often used for the derivation.

Let us begin the discussion with a trivial case of a transmission of rotary motion between two shafts that are parallel to one another. In the simplest case, a rotation from a driving shaft is transmitted to a driven shaft by means of two pulleys connected to one another by a belt, as schematically illustrated in Figure 4.

Two pulleys of diameters d₁ (driving) and d₂ (driven) are rotated about their axes, O₁ and O₂, correspondingly. The axes, O₁ and O₂, are positioned at a center-distance, C, apart from one another. The pulleys are connected to each other by a belt. The belt is tangent to the pulleys at points a and b.

The rotations, ω₁ and ω₂, of the driving and the driven pulleys are synchronized with one another so as to fulfill the ratio:

The linear velocity, V_m, of arbitrary point, m, of the belt is calculated from the formula:

Shown in Figure 4, the equivalent pulley-and-belt transmission is capable of transmitting a uniform rotation smoothly.

In a certain sense, the pulley-and-belt mechanism may be viewed as an equivalent to a parallel-axes involute gear pair, namely, as an equivalent pulley-and-belt transmission.

Three important features of the equivalent pulley-and-belt transmission should be stressed:

• First: When the pulleys rotate, an arbitrary point, m, within the portion, ab, of the belt traces a straight line in a motionless reference system associated with the transmission housing. This straight line is the path of point m. Therefore, in an equivalent pulley-and-belt transmission, every point of the belt travels straight forward. (This statement is valid only for parallel-axes gearing, and it is not valid for intersected-axes or crossed-axes gearing.)

• Second: When a uniform rotation is transmitted from the driving pulley 1 to the driven pulley 2, the torque is transmitted by a force that acts along the belt, namely, along the straight line, ab: in an equivalent pulley-and-belt transmission, a force is transmitted only along the belt. A force acts only along a straight line, and it cannot be transmitted along a curve. Therefore, in an equivalent pulley-and-belt transmission, the straight-line segment, ab, is an equivalent of the line of action, along which a force that is exerted in the driving pulley is transmitted to the driven pulley.

• Third: In an equivalent pulley-and-belt transmission, a straight path of arbitrary point, m, in the belt is aligned with the straight line of action. The line of action here is the line, along which a force that is exerted in the driving pulley is transmitted to the driven pulley.

It is important to note here that the straight-line segment, ab, at that same time serves both. Namely, it serves the path of point of interest, m, and it serves the line of action in the equivalent pulley-and-belt transmission (see Figure 4). Only in parallel-axes gearing are the path of contact, P_c, and the line of action, LA, both straight lines that align to one another. Despite the path of contact, P_c, and the line of action, LA, aligning to one another, it is critical to show the difference between the path of contact, P_c, and the line of action, LA, and never mix the two.

To the best of the author’s knowledge, it is not known who should be credited for the invention of the equivalent pulley-and-belt transmission (or, in other words, the equivalent pulley-and-belt model) of a gear pair in parallel-axes gearing.

The performed analysis of transmission of rotary motion by means of the equivalent pulley-and-belt transmission is helpful to understand the requirements that the tooth flank geometry in parallel-axes gearing have to meet.

A smooth transmission of a uniform rotary motion from one shaft to another shaft by means of gear teeth is possible only if the interacting tooth flanks, G and P, are conjugate to one another.

The fundamental requirements governing the shapes that any pair of conjugate tooth profiles may have in parallel-axes gearing states:

In parallel-axes gearing, in order to 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 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.

In other words, two planar curves are said to be conjugate to one another if a common perpendicular at point of their contact is pointed along a straight line through the pitch point, P. The center of the instant rotary motion is coincident with the pitch point, P.

It is a wrong practice to define conjugate shapes in the following manner: “A pair of transverse gear tooth profiles is said to be conjugate if a constant angular velocity of one profile produces a constant angular velocity in the meshing profile.” Constant output rotation is a consequence of conjugacy of the interacting tooth profiles. The property of conjugacy must be expressed in terms of the kinematics and the geometry of the interacting tooth flanks, G and P, of a gear and a mating pinion.

Refer to Figure 5 for more in-detail analysis of the property of conjugacy of two interacting surfaces (of two interacting profiles). Here, the directions of rotation of the driving and of the driven gears are reversed compared to that shown in Figure 4, as the driven pulley is pulled by the belt, while the driven gear is pushed by the driving gear.

For conjugate tooth flanks of a gear and a mating pinion, the contact point between the tooth flanks, G and P, traces a straight path of contact, P_c. When the friction is not taken into account, the force acts perpendicular to the common tangent plane, t – t, to the tooth flanks, G and P, through P. As long as the friction is not accounted for, the acting force is always perpendicular to the tooth flanks, G and P, at the current point of their contact. The straight line of action, LA, is aligned with the straight path of contact, P_c, namely, in geometrically-accurate parallel-axes gearing, the straight lines, LA and P_c, are congruent to one another every time the gears rotate. It is critical to bear in mind that in the theory of gearing, the path of contact, P_c, and the line of action, LA, are two completely different entities. Therefore, the difference between the line of action, LA, and the path of contact, P_c, in parallel-axes gearing needs to be firmly defined.

The preceding discussion on the conjugacy of two gear tooth flanks (tooth profiles) in parallel-axes gearing is summarized as follows:

(a) The line of action, LA, of two conjugate tooth flanks (tooth profiles) is a straight line through the motionless pitch point, P.

(b) The path of contact, P_c, for two conjugate tooth flanks (tooth profiles) is a straight line through the motionless pitch point, P.

(c) The straight lines, LA and P_c, align to one another in parallel-axes gearing, and do not align to one another in intersected-axes gearing, as well as in crossed-axes gearing.

Only conjugate gear tooth flanks (tooth profiles) in parallel-axes gearing possess the properties covered by the clauses (a) through (c). Kinematic and geometric analysis of conjugate gear tooth flanks (tooth profiles) in parallel-axes gearing (in involute gearing) can be summarized in the form graphically represented in Figure 6.

As shown in Figure 6, the diagram provides the user with a comprehensive data on the kinematics of parallel-axes gearing, and the geometry of the conjugate tooth flanks (tooth profiles), G and P, in parallel-axes gearing.

3 Conjugate Tooth Flanks: Intersected-Axes Gearing

The first valuable results on conjugate tooth flanks in intersected-axes gearing were published in the late 1880s. G.B. Grant was the first to introduce (1887) the concept of the base cone in intersected-axes gearing [7]. Readers may also wish to familiarize themselves with the 1906 book by G.B. Grant [2]. Grant considered the base cone of a gear and a plane tangent to the base cone. It was not known at the time that, by nature, the tangent plane is the plane of action in an intersected-axes gearing (of just in I_a — gearing, for simplicity). Moreover, only one base cone, and NOT two base cones of interacting tooth flanks in bevel gears were considered by G.B. Grant. In short, Grant discovered the geometry of the tooth flank of a geometrically-accurate gear for intersected-axes gearing. To answer the question: What does the term “conjugate gear tooth flanks” stand for in intersected-axes gearing, the contribution by Grant is helpful.

Based on the 1887 contribution by Grant, later on (~2008) the concept of equivalent pulley-and-belt transmission (see Figure 4) evolved through Prof. S.P. Radzevich to intersected-axes gearing [6].

Consider a plane that is tangent to a base cone as illustrated in Figure 7. The axis of rotation, O_g, of the tooth flank, G, and the axis of rotation, O_pa, of the plane, PA, intersect one another at a point. At this point, A_g ≡ A_pa, the base cone apex, A_g, and the apex, A_pa, of the plane, PA, coincide with one another. The rotation, ω_g, of the tooth flank, and the rotation, ω_pa, of the tangent plane are synchronized with one another to ensure rolling of the tangent plane over the base cone with no sliding. An arbitrary point, m, within the tangent plane, PA, travels together with this plane, and traces a circular arc, P_c, in this motion. The circular arc, P_c, is entirely situated within the tangent plane, PA, and is centered at the point, A_g ≡ A_pa. The circular arc, P_c, is the path of contact of arbitrary point, m.

A straight line, LA_m, is tangent to the path of contact, P_c, at point, m. Evidently, the tangent straight line, LA_m, is entirely within the tangent plane, PA. The line, LA_m, intersects the axis of instant rotation, P_ln, at any configuration of the base cone and the tangent plane relative to one another. By nature, the straight line, LA_m, is the instantaneous line of action in the conjugate pair of tooth flanks, G and P, that rotate about intersected axes of rotation, O_g and O_pa.

In Figure 8, the path of contact, P_c, and the instantaneous line of action, LA_m (through an arbitrary point m within the plane of action, PA), in intersected-axes gearing are constructed. As seen in the analysis of Figure 8, the path of contact, P_c, is a circular arc where all points are situated within the plane of action. The instantaneous line of action, LA_m, is tangent to the path of contact, P_c, at the point of interest, m. At any and all configurations of the intersected-axes gears relative to one another (at any configuration of the desirable line of contact, LC_des, within the plane of action, PA), the instantaneous line of action, LA_m, intersects the axis of instant rotation, P_ln, at point m_i. As shown in Figure 8, features are due to the particular pair of the tooth flanks being conjugate to one another.

The following definition to the term “Conjugate Gear Tooth Flanks” in intersected-axes gearing can be drawn up from the earlier discussion:

Definition 2: Conjugate Gear Tooth Flanks in intersected-axes gearing are those whose common perpendicular always (for any and all relative configurations of the interacting gears) passes through the motionless axis of instant rotation in the gear mesh.

The proof to this statement can be found in [4], [6].

The preceding discussion on conjugacy of two gear tooth flanks in intersected-axes gearing can be summarized as follows:

(d) The line of action, LA, can be specified not to any arbitrary intersected-axes pair of tooth flanks, G and P.

(e) Only the instant lines of action, LA_m, can be constructed for two conjugate tooth flanks.

(f) The instantaneous lines of action are straight lines that always intersect the axis of instant rotation, P_ln.

(g) The path of contact, P_c, is a circular arc that is entirely situated within the plane of action.

(h) The circular-arc path of contact is centered at the gear base cone apex, A_g (the plane-of-action apex, A_pa, is coincident with the gear base cone apex, A_g).

(i) The path of contact and the instantaneous line of action never align to one another.

Only conjugate gear tooth flanks in intersected-axes gearing possess the properties covered by the clauses (d) through (i). Kinematic and geometric analysis of conjugate gear tooth flanks in intersected-axes gearing are briefly summarized graphically in Figure 9.

4 Conjugate Tooth Flanks: Crossed-Axes Gearing

The geometrically-accurate crossed-axes gearing (or just C_a — gearing, for simplicity) with line contact between the interacting tooth flanks, G and P, is the only kind of conjugate tooth flanks for gears with crossing axes of rotation, O_g, of the gear, and, O_p, of the mating pinion. Geometrically-accurate gearing of this particular design was invented around ~2008 by Prof. S.P. Radzevich.  R — gearing is the correct name for gearing of this design. In short, Prof. S.P. Radzevich has discovered the geometry of the tooth flank of a geometrically-accurate gear for crossed-axes gearing. [Then (~2008), another design of geometrically-accurate crossed-axes gearing with point contact between the tooth flanks, G and P, was proposed by Prof. S.P. Radzevich. S_pr — gearing is the name of this novel design of crossed-axes gearing that is insensitive to the linear and angular displacements of the gears relative to one another.]

The equivalent pulley-and-belt transmission in R — gearing was proposed for the first time (~2008) by Prof. S.P. Radzevich [6]. Two base cones in tangency with the plane of action were considered [6].

To answer the question: What does the term “conjugate gear tooth flanks” stand for in crossed-axes gearing, the contribution by Prof. S.P. Radzevich is helpful.

Consider a plane that is tangent to a base cone as illustrated in Figure 10. The axis of rotation, O_g, of the tooth flank, G, and the axis of rotation, O_pa, of the plane, PA, cross one another. The axes, O_g and O_pa, intersect the center-line, CL, at points A_g and A_pa. The point, A_g, is the gear base cone apex, and the point, A_pa, is the apex, A_pa, of the plane, PA. The points, A_g and A_pa, are two distinct points that do not coincident with one another.

The rotation, ω_g, of the tooth flank, and the rotation, ω_pa, of the tangent plane are synchronized with one another so as to ensure the rolling of the tangent plane over the base cone with no sliding (sliding along the line of contact of the base cone, and of the plane of action is still permissible here). An arbitrary point, m, within the tangent plane, PA, travels together with this plane, and traces a circular arc, P_c, in this motion. The circular arc, P_c, is entirely situated within the tangent plane, PA, and is centered at the plane-of-action apex point, A_pa. The circular arc, P_c, is the path of contact of arbitrary point, m.

A straight line, LA_m, is tangent to the path of contact, P_c, at point, m. Evidently, the tangent straight line, LA_m, is entirely within the tangent plane, PA. The line, LA_m, intersects the axis of instant rotation, P_ln, at any and all configurations of the base cone and the tangent plane relative to one another. By nature, the straight line, LA_m, is the instantaneous line of action in conjugate pair of the tooth flanks, G and P, that rotate about crossing axes of rotation, O_g and O_pa.

In Figure 11, the path of contact, P_c, and the instantaneous line of action, LA_m (through an arbitrary point m within the plane of action, PA), in crossed-axes gearing are constructed. As is evident from the analysis of Figure 11, the path of contact, P_c, is a circular arc, where all points are situated within the plane of action. The instantaneous line of action, LA_m, is tangent to the path of contact, P_c, at the point of interest, m. At any configuration of the crossed-axes gears relative to one another (at any configuration of the desirable line of contact, LC_des, within the plane of action, PA) the instantaneous line of action, LA_m, intersects the axis of instant rotation, P_ln, at point m. As shown in Figure 11, these features are due to the particular pair of the tooth flanks being conjugate to one another.

The following definition to the term “Conjugate Gear Tooth Flanks” in crossed-axes gearing can be withdrawn from the earlier discussion:

Definition 3: Conjugate Gear Tooth Flanks in crossed-axes gearing are those whose common perpendicular always (for any and all relative configurations of the interacting gears) passes through the motionless axis of instant rotation through the plane-of-action apex in the C_a — gearing mesh.

The proof to this statement can be found in [4], [6].

The preceding discussion on conjugacy of two gear tooth flanks in crossed-axes gearing can be summarized as follows:

(g) No line of action, LA, can be specified to any crossed-axes pair of conjugate tooth flanks, G and P.

(h) Only the instant lines of action, LA_m, can be constructed for two conjugate tooth flanks.

(i) The instantaneous lines of action are straight lines that always intersect the axis of instant rotation, P_ln.

(j) The path of contact, P_c, is a circular arc that is entirely situated within the plane of action.

(k) The circular-arc path of contact is centered at the plane-of-action apex, A_pa (that is not coincident with the gear base cone apex, A_g).

(l) The path of contact and the instantaneous line of action never align to one another.

Only conjugate gear tooth flanks in crossed-axes gearing possess the properties covered by clauses (g) through (l). Kinematic and geometric analysis of conjugate gear tooth flanks in crossed-axes gearing are summarized and represented in the form similar to that in Figure 9 for intersected-axes gearing.

Conclusion

A brief overview on the evolution of conjugate action law is presented. In the past until the 17th century [including the research by C.-E. Camus (~1733)] no clear understanding of the geometry of conjugate tooth flanks of gears was seen. The first significant results in this field were obtained by L. Euler (~1760).

The answer to the question: “What does the Term “Conjugate Gear Tooth Flanks” Stand For?” is derived for all three cases of rotation of two smooth regular tooth flanks, namely, for (a) parallel-axes gearing (P_a — gearing), (b) intersected-axes gearing (I_a — gearing), and (c) crossed-axes gearing (C_a — gearing).

Geometrically-accurate involute gearing is no longer getting conjugate gearing if the angular displacements are taken into account (the angular displacements are inevitable in reality).

The reported results of the analysis are helpful for better understanding the kinematics, the geometry, and operation of gears of all kinds:  P_a — gearing, I_a — gearing, and C_a — gearing. 

References

1. Euler, L., De Optissima Figura Rotatum Dentibus Tribuenda. Suplementum de Figura Dentium Rotatum, Novi Commentarii Academial Petropolitanae, 1754/55, 1765.
2. Grant, G.B., A Treatise on Gear Wheels, 11th edition, Philadelphia Gear Works, Inc., Philadelphia, 1906, 105 p.
3. 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.
4. Radzevich, S.P., “Knowledge (of Gear Theory) is Power in the Design, Production, and Application of Gears,” Gear Solutions magazine, August 2020, pages 38-44.
5. Radzevich, S.P., “Poor Understanding of the Scientific Theory of Gearing by the Majority of Gear Scientists and Engineers”, Chapter 9, pages 193-214 in: Radzevich, S.P., Novikov/Conformal Gearing: Scientific Theory and Practice, Springer, 2022 (November 15, 2022), 33+493 (526) pages. ISBN-10: 3031100182, ISBN-13: 978-3031100185
6. Radzevich, S.P., Theory of Gearing: Kinematics, Geometry, and Synthesis, 3rd Edition, revised and expanded, CRC Press, Boca Raton, FL, 2022, 1208 pages.
7. 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.
8. 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, 446p.

Bibliography

1. Radzevich, S.P., (Editor), Advances in Gear Design and Manufacture, CRC Press, Boca Raton, Florida, 2019, 549 pages.
2. Radzevich, S.P., “An Examination of High-Conformal Gearing,” Gear Solutions, February, 2018, pages 31-39.
3. Radzevich, S.P., “Design Features of Perfect Gears for Crossed-Axes Gear Pairs,” Gear Solutions magazine, February, 2019, pp. 36-43.
4. Radzevich, S.P., (Editor), Dudley’s Handbook of Practical Gear Design and Manufacture, 4th edition, CRC Press, Boca Raton, FL, 2021, 1170 pages, 718 B/W Illustrations.
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