The main goal of gear tooth flank modification is a reduction of sensitivity of gearing to the variation of the tooth flank geometry, and to various displacements of the gears from their desirable configuration under operating load, especially when the power density through the gear pair increases. For this purpose, a determination of favorable geometry of the tooth flanks is required. Then, the determined geometry of a gear and a mating pinion tooth flanks is approximated by surface patches easy to be machined on available, in-house gear generators. Ultimately, the proposed approach for the determination of the design parameters of gear tooth flank modification enables the highest possible power density through a gear pair, along with a significant reduction of noise excitation and vibration generation.

### Introduction

Gears are used with an intent to transmit and transform a rotary motion from a driving shaft to a driven shaft. Either alteration of the direction of rotation of the output shaft (including, but not limited to a variation of the shaft angle), or alteration of the output shaft angular velocity, or both, are observed when a rotation is transmitted by a gear pair.

An increase of input rotation along with an increase of an input torque is a main trend in the development of gears and gear transmissions. The chief reason for this is a necessity to increase power density transmitted by means of gears. As the power density increases, gears and gear transmissions are getting more sensitive to the accuracy of the tooth flanks geometry, as well as to the displacements of the tooth flanks from their nominal position. The gear performance is significantly affected by (a) the inaccuracies of the gears and of the gear housing, (b) the elastic deformation of the gear set under operating load, and (c) because of thermal expansion of the gear material, as well as of the gear housing material, and so forth. Gear tooth flank modification can be an effective means for significant reduction of sensitivity of a gear set from most sources of gear failure. Here and below, the term *“gear tooth flank modification”* stands for various intentionally introduced (i) deviations of the actual tooth flank geometry that form in the corresponding geometrically-accurate gearing, and (ii) displacements of the gear tooth flanks from their nominal location and orientation.

A novel efficient approach for gear tooth flank modification is briefly disclosed below. This approach is illustrated primarily with respect to gearing with displaced gear tooth flanks and with non-deformed gear tooth.

### Motivation

Almost an infinite number of modified gear tooth flank geometries are known in the present-day practice. A few examples of the most extensively used geometries of the modified tooth flanks are illustrated in Figure 1 and Figure 2.

The proposal in this article for gear tooth flank modification makes possible a significant reduction in the number of possible geometries of the modified tooth flanks: Just a few of them depend on the geometry of the surface patch used for the approximation.

The determination of the design parameters of the required gear tooth flank modification for a particular application of a gear set has to be properly engineered, which is a must. This gear problem is a sophisticated one, and it has no reliable solution yet. Use of a *“trial-and-error method”* is recommended when solving problems of this sort. There is no evidence that this problem has a chance to get solved as long as conventional approaches are used for these purposes.

It is instructive to notice in this regard that the origination of the idea of gear tooth flank modification can be traced back to the first half of the 20th century. It is likely Walker, H. [1] was the first to point out the importance of tooth flank corrections for heavily loaded spur gears. The concept of a gear tooth flank modification is illustrated in Figure 3. The concept of gear tooth flank modification gained the attention of gear experts, and many papers and articles are published on the topic.

Despite that, the problem of correct gear tooth flank modification has yet to be solved. That being said, it makes it clear that gear science (and gear engineering) has no clue about how this problem can be solved.

Many similarities can be easily found between the unsolved problem of correct gear tooth flank modification and the well-known challenge of a “perpetual motion machine.”

Hundreds of claims have been made in the past (just a couple centuries ago) stating that: (a) a “perpetual motion machine” is invented, (b) a “perpetual motion machine” is designed, (c) a “perpetual motion machine” is manufactured, (d) a “perpetual motion machine” has been tested, and, finally, (e) a “perpetual motion machine” is ready to be used in the industry. Where we are now? Do we have any examples of application of a “perpetual motion machine”? The answer is, of course, no. Do we have a chance to solve the problem of a “perpetual motion machine”? The answer is still no. Is there a reason to continue wasting efforts and funds trying to solve an engineering problem that has no solution at all due to the natural constraints? Per the author’s opinion, the answer to that question is also no. However, certain enthusiastic gear designers still look for a solution to his insolvable problem.

Let’s go back to the problem of gear tooth flank modification: There is no chance to get the problem solved using the “trial-and-error” approach, as this approach is endless, and is affordable only to those who have access to an infinite resource of funding. This article will look at developing scientific/engineering methods for determining the design parameters of correct gear tooth flank modification. It is likely the scientific theory of gearing is the only leverage available in today’s practice, using it to solve the problem of gear tooth flank modification.

### The Core of the Problem

As proposed by H. Walker [1], the concept of gear tooth flank modification targeted the accommodation for the gear tooth deflection under the operating load. No gear axes displacements under the load were considered by H. Walker. Eventually, gear and mating pinion axes displacements also were involved in the analysis, as these displacements affected the mesh of involute tooth profiles.

The key advantages of the geometry of tooth flanks in *S** _{pr}* —

*gearing*are used below in the proposed method of gear tooth flank modification. An in-depth understanding of all root causes of high sensitivity of gearing to tooth flank displacements under operating loads is required if one is about to derive equations for the tooth flanks of the geometrically-accurate real gears.

The design parameters of a geometrically-accurate gear pair are exactly equal to their desirable (calculated) values. In reality, however, gear pairs undergo bending by loads — overheating may result in heat distortion of gears, shafts, and housing. The shafts of a gear and of a mating pinion are displaced from their desirable positions by manufacturing and mounting errors, as well as by the flexibility of the housing, and so forth.

In Figure 4, the configuration of the axes of rotation of the gears in the ideal parallel-axes gearing (see Figure 4a), and the displaced axes of the rotation of each gear under the operating load (see Figure 4b) are shown. It should be noticed that the initially straight neutral center line of the shafts in Figure 4a becomes a spatial three-dimensional (3D) curved center-line under the load applied (see Figure 4b).

The deviations of the shaft bearings from their nominal configuration due to the manufacturing errors and displacements under the load should be recognized as the potential root cause for displacement of the axes of a gear and of a mating pinion. Larger deviations result in larger axes displacements and vice versa. Longer shafts are less sensitive to the displacements of this kind. However, the stiffness of longer shafts is lower than that of short shafts. For a particular application, a favorable combination of the allowed bearing displacements and the shaft length can be determined. This issue could be of critical importance for high-power-density gearboxes, for which the shafts should be of the shortest possible length. The shorter the shafts the tighter the tolerances on bearings displacements.

As it is clear from this brief discussion, the axes of rotations of the gears are displaced from their desirable position. The actual configuration of the axis of rotation of a gear in relation to its desirable configuration is of critical importance.

The concept, adopted for tooth flank generation of gears in “*S** _{pr}* —

*gearing system*,” can be viewed in alternative manner.

Consider the plane of action, *PA*, for example, in a geometrically-accurate parallel-axes gear pair, as schematically illustrated in Figure 5. The effective face width in the gear pair is designated as *F** _{pa}*. Construct a desirable line of contact,

*LC*

*, in the nominal configuration of the axes of rotation,*

_{des}*O*

*and*

_{g}*O*

*, of a gear and a mating pinion. After that, two extreme configurations of the desirable line of contact,*

_{p}*LC*

*, are constructed. One of them is corresponded to the maximal positive, and another one — to the extremal negative deviation of the desirable line of contact,*

_{des}*LC*

*, from its nominal configuration. All three of the desirable lines of contact are situated within the plane of action,*

_{des}*PA*.

The plane of action, *PA*, can be sliced on an infinite number of slices, each of which is perpendicular to the gear axis of rotation, *O** _{g}*. The width of each elementary slice is designated as Δ

*F*

*.*

_{pa}For the slices at both ends of the face width, *F** _{pa}*, the elementary lines of contact,

*lc*

*and*

_{g}*lc*

*, for the gear and that for the mating pinion, are constructed so, as for a geometrically-accurate crossed-axes gear pair having the maximal linear, {δ*

_{p}*} and {δ*

_{g}*}, and the maximal angular, {ϕ*

_{p}*} and {ϕ*

_{g}*}, displacements of the gear axis of rotation,*

_{p}*O*

*, and that,*

_{g}*O*

*, of the pinion from their nominal configuration in relation to the plane of action,*

_{p}*PA*. The displacements, {δ

*} and {ϕ*

_{g}*}, in nature, are the tolerances for the elementary displacements, {δ*

_{g}*} and {ϕ*

_{g}*}, respectively.*

_{g}For the slice at the middle of the face width, *F** _{pa}*, the elementary lines of contact,

*lc*

*and*

_{g}*lc*

*, are constructed so, as for a geometrically-accurate crossed-axes gear pair having a zero linear, {δ*

_{p}*} and {δ*

_{g}*}, and a zero angular, {ϕ*

_{p}*} and {ϕ*

_{g}*}, displacements of the gear axis of rotation,*

_{p}*O*

*, and that,*

_{g}*O*

*, of the pinion from their nominal configuration in relation to the plane of action,*

_{p}*PA*.

For the rest of the slices, Δ*F** _{pa}*, in between the middle slice, and the slices at the both ends of the face-width,

*F*

*, the elementary lines of contact,*

_{pa}*lc*

*and*

_{g}*lc*

*, are constructed so, as for a geometrically-accurate crossed-axes gear pair having intermediate values of the linear, {δ*

_{p}*} and {δ*

_{g}*}, and the angular, {ϕ*

_{p}*} and {ϕ*

_{g}*}, displacements of the gear axis of rotation,*

_{p}*O*

*, and that,*

_{g}*O*

*, of the pinion from their nominal configuration in relation to the plane of action,*

_{p}*PA*.

It is right to point out here that the instantaneous plane-of-action apex, *A** _{pa}*, as well as the instantaneous base cone apexes,

*A*

*and*

_{g}*A*

*, of a gear and of a mating pinion occupy different locations for different slices. The central point,*

_{p}*P*, is remained stationary. There is certain freedom for the gear designer to choose functions of the distribution of the elementary displacements, [δ

*], [ϕ*

_{g}*], and [δ*

_{g}*], [ϕ*

_{p}*], within the face-width,*

_{p}*F*

*.*

_{pa}Two envelope curves, *a*_{g}*Pc** _{g}* and

*a*

_{p}*Pc*

*, can be constructed to each of two families of the elementary lines of contact,*

_{p}*lc*

*and*

_{g}*lc*

*. In*

_{p}*S*

_{pr}*— gearing*, the tooth flanks, G and P, of a gear and of a mating pinion are generated by these two curves,

*a*

_{g}*Pc*

*and*

_{g}*a*

_{p}*Pc*

*, one of which is constructed for the gear, and another one is constructed for the pinion.*

_{p}For a specified gear pair, the distance, *r** _{pa}*, is within the interval as seen in Equation 1:

The tolerances, {Δ*r*^{(-)}* _{pa}*} and {Δ

*r*

^{(+)}

*}, for the accuracy of the distance,*

_{pa}*r*

*, are shown in Figure 6.*

_{pa}An operating angular base pitch, ϕ* ^{r}_{b.op}*, constructed for a nominal configuration of the rotation vectors, ω

*and ω*

_{g}*, in an*

_{p}*S*

_{pr}*— gearing*is shown in Figure 7. The plane-of-action pitch circle of a radius,

*r*

*, is intersected by the sides of the angle, ϕ*

_{w.pa}*, at points,*

^{r}_{b.op}*a*and

*b*, that is, the points,

*a*and

*b*, are at an angular distance, ϕ

*, form one another. One of the three design parameters, ϕ*

^{r}_{b.op}*,*

_{b.op}*r*

*, and*

_{w.pa}*ab*, can be expressed in terms of the other two. Moreover, the linear distance,

*ab*, also can be calculated in Equation 2:

For a gear pair with the displaced tooth flanks, G and P, the actual values of the design parameters, *j** _{b.op}* and

*r*

^{inst}

*, alter, while the linear distance,*

_{pa}*ab*, is remained of that same value (

*ab*=

*const*). For a specified value of the distance, Δ

*r*

^{inst}

*, the current value of the angular operating base pitch, ϕ*

_{pa}^{inst}

*, is calculated as:*

_{b.op}Here, in Equation 3, the distance, Δ*r*^{inst}* _{pa}*, is a signed value.

In *S*_{pr}* — gearing*, a gear and a mating pinion feature an “interval” of angular base pitches, and not a fixed angular base pitch of a certain value. The “interval” of the angular base pitches makes the gear pair capable of accommodating for a prescribed range of permissible values of the axes displacements.

The values, ϕ^{min}* _{b.op}* and ϕ

^{max}

*, are the minimum and the maximum values of the angular operating base pitch, ϕ*

_{b.op}^{inst}

*, correspondingly, calculated from Equation 3. The maximum “negative” deviation, and the maximum “positive” deviation of the angular operating base pitch, ϕ*

_{b.op}^{inst}

*, from its nominal value, ϕ*

_{b.op}*, are designated as Δϕ*

_{b.op}^{min}

*and Δϕ*

_{b.op}^{max}

*, correspondingly. The extremal deviations, Δϕ*

_{b.op}^{min}

*and Δϕ*

_{b.op}^{max}

*, form an interval for permissible variation in Equation 4:*

_{b.op}of the operating base pitch, ϕ^{inst}* _{b.op}*, in a gear pair.

The intervals of permissible variation of the angular base pitches of a gear, ϕ^{inst}* _{b.g}*, and of a mating pinion, ϕ

^{inst}

*, in*

_{b.p}*S*

_{pr}*— gearing*are identical to the permissible variation of the operating angular base pitch, ϕ

^{inst}

*, of the gear pair (see Equation 4). No transmission error is observed once the base pitches of a gear and its mating pinion are equal to one another, and both of them are equal to a current value of the operating base pitch of the gear pair. Ultimately,*

_{b.op}*S*

_{pr}*— gears*are capable of transmitting a rotation from a driving shaft to a driven shaft smoothly with no vibration generation and no noise excitation.

In a particular application, when the axes of rotation of a gear and of a mating pinion are exactly parallel to one another, the base cones reduce to corresponding base cylinders. Because of this, the angular operating base pitch, ϕ^{inst}* _{b.op}*, approaches zero (that is, ϕ

^{inst}

*→ 0°), and the corresponding radius,*

_{b.op}*r*

^{inst}

*, approaches an infinity, that is,*

_{pa}*r*

^{inst}

*= (*

_{pa}*r*

*+ Δ*

_{w.pa}*r*

^{inst}

*) → ∞ [as well as the axial distance,*

_{pa}*A*, approaches an infinity (

*A*→ ∞)]. The L’Hôpital’s rule (French mathematician Guillaume François Antoine, Marquis de Ll’Hôpital, 1661 – February 2, 1704) is used to resolve the issue.

Contact points trace lines on the tooth flanks. As an example, the location and orientation of the trace of contact point, *T** _{r}*, within tooth flank of a spur

*S*

_{pr}*— gearing*is schematically shown in Figure 8. In a case of zero axes displacement, the trace of contact point on the left flank,

*Tr*

^{0}

*, of the gear tooth, as well as on the right flank,*

_{l}*Tr*

^{0}

*, of the gear tooth, goes through the middle of the face of the gear (see Figure 8a). In a case when positive axes displacement is observed, the trace of contact point for the left side,*

_{r}*Tr*

^{+}

*, and the right side,*

_{l}*Tr*

^{+}

*, of the gear tooth are shifted oppositely toward the ends of the gear, as schematically illustrated in Figure 8b. Similarly, negative axes displacement results in the trace of contact point for the left,*

_{r}*Tr*

^{–}

*, and for the right,*

_{l}*Tr*

^{–}

*, sides of the gear tooth being shifted oppositely toward the opposite ends of the gear, as illustrated in Figure 8c. Schematics in Figure 8 pertain to*

_{r}*S*

_{pr}*— gearing*that features constant in time values of all the displacements. When the displacements alter in time, then the paths of contact of a more complex geometry are observed.

The use of an *S*_{pr}*— gear system* (or just *S*_{pr}* —gearing*, for simplicity) is the only way that is developed in the modern theory of gearing to transmit a rotation smoothly [3, 4, 5]. *S*_{pr}* —gearing* is a kind of geometrically-accurate gearing that is insensitive to the axe’s displacements. At any and all permissible configurations of the gears, the angular base pitch of the gear, ϕ* _{b.g}*, and that of the mating pinion, ϕ

*, each of them, still equal to operating base pitch, ϕ*

_{b.p}*, in the gear pair: the equalities ϕ*

_{b.op}*≡ ϕ*

_{b.g}*and ϕ*

_{b.op}*≡ ϕ*

_{b.p}*are always valid in*

_{b.op}*S*

_{pr}*— gearing*. Due to this property,

*S*

_{pr}*— gearing*features the tooth flank geometry that provides the gears with the capability to be “insensitive” to any and all displacements of reasonable values of the tooth flanks from their nominal disposition. With tooth flank geometries of this kind, the displacements are simply absorbed due to the specific geometry of the tooth flanks.

### Gearing-Based Method of Modification of Gear Tooth Flanks

Design and machining of the perfectly modified gears can be executed in the following way:

•** 1: **An appropriate (geometrically-accurate) *S*_{pr}* — gearing* is designed as a replacement to the original design of the gear pair for a particular application.

•** 2: **The developed geometry of the tooth flank in *S*_{pr}* — gearing* is approximated by (preferably) a single patch of a surface that is easy for machining (Commonly, these are surface patches that enable the minimal deviation of the actual value of the angular base pitch from its desirable value.)

•** 3: **In production, the approximated (and not geometrically-accurate) tooth flank is generated on the gear generator.

•** 4: **After being machined, the gears and the pinions that have the smallest possible difference in the actual values of angular base pitches can be paired with one another. This results in a reduction of amount of poorly finished gears.

Two advantages are inherited to *S*_{pr}* — gearing*. The first one is due to *S*_{pr}* — gearing* is geometrically-accurate (as angular base pitches of a gear, and of a mating pinion, both, are equal to operating angular base pitch of the gear pair). The second one due to *S*_{pr}* — gearing* is insensitive to displacements of the axes of rotation of the gear and that of the mating pinion.

Complex geometry of tooth flanks of the gear, and of the mating pinion, entail the main disadvantage of *S*_{pr}* — gearing* — tooth flanks in gears for *S*_{pr}* — gearing* are inconvenient to get finished accurately. This latter problem can be resolved by means of an appropriate approximation of the original tooth flanks by surface patches that are easy to finish cut. The surfaces of revolution, the surfaces of translation, screw surfaces of a constant pitch are promising candidates for these surfaces.

Been correctly approximated, the actual tooth flanks feature the angular base pitches (ϕ* _{b.g}*, ϕ

*, and ϕ*

_{b.p}*) of almost equal value; gears modified this way can be correctly engineered and are easy to produce.*

_{b.op}It is clear now why the proposed kind to gear tooth flank modification is referred to *S*_{pr}* — gearing*–*based modification of tooth flanks.*

In the meantime, use of gears with *S*_{pr}* — gearing-based modification of tooth flanks* is the only way, to modify gear tooth flanks so as to keep the angular base pitches (ϕ* _{b.g}*, ϕ

*, and ϕ*

_{b.p}*) of almost equal value.*

_{b.op}### Conclusion

A novel method of gear tooth flank modification is disclosed in this article.

It is shown from the beginning that a correct solution to the problem of gear tooth flank modification has a chance to get solved on the premise of scientific theory of gearing [3], along with in-detail understanding of modern methods of gear finishing processes [2].

No base pitches are considered when designing gears with tooth flanks modified in a conventional manner; gears modified this way cannot be correctly engineered.

Gear tooth flanks have to be modified so as to minimize the difference between the angular base pitch of a gear and that of a mating pinion from operating angular base pitch of the gear pair. The tooth flank modification, in both cases, is aiming to accommodate for deflections of the gear drive components, as well as for the deflections of gear teeth. As the linear kind of contact of tooth flanks in geometrically-accurate gears is substituted with the point kind of contact in the modified gears, the degree of conformity of the interacting tooth flanks of the gear and the mating pinion at the point of their contact is reduced. However, it must remain the highest possible.

The use of this proposed gear tooth flank modification makes possible a reduction of the accuracy requirements to the gear in order to keep the accuracy tolerances wider and, thus, to use less accurate and cheaper gears instead of more precise and costly gears. This gear tooth flank modification is developed to accommodate for the manufacturing errors, as well as for other displacements of the tooth flanks from their desirable location and orientation. Also, the modified gears are capable of accommodating for the displacements of the tooth flanks under the operating load.

After being machined, the gears and the pinions that have the smallest possible difference in the actual values of angular base pitches can be paired with one another. This results in a reduction of the amount of poorly finished gears.

Initially developed for spur gears, the concept of tooth flank modification has been applied to helical gears.

### References

- Walker, H., “Gear Tooth Deflection and Profile Modification.,” Engineer, Vol. 166, 1938, pp. 434-436. [also: Walker, H., “Gear Tooth Deflection and Profile Modification,” The Engineer, August 1940, Vol. 170, pp. 102.]
- 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.
- Radzevich, S.P., “On the kinematics and tooth flank geometry of gearing,” International scientific journal “Machines. Technologies. Materials,” 2020, Year XIV, Issue 4, pages 142-153. WEB ISSN 1314-507X; Print ISSN 1313-0226. (Bulgaria).
- Radzevich, S.P., “On the Kinematics and Tooth Flank Geometry of gearing,” pages 154-167 in: Proceedings of the 7th International BAPT Conference “Power Transmissions 2020,” Balkan Association of Power Transmission, 10-13.06.2020, Borovets, Bulgaria, 2020, 168 pages. ISBN 978-619-7383-18-8.

### Bibliography

- Pat. No. 407.437, (USA). Machine for Planing Gear Teeth./G.B. Grant, Filed: January 14, 1887 (serial No. 224,382), Patented: July 23, 1889.
- Radzevich, S.P., (Editor), Advances in Gear Design and Manufacture, CRC Press, Boca Raton, Florida, 2019, 549 pages.
- Radzevich, S.P., “An Examination of High-Conformal Gearing,” Gear Solutions, February, 2018, pages 31-39.
- 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.
- Radzevich, S.P., High-Conformal Gearing: Kinematics and Geometry, 2nd edition, Elsevier, Amsterdam, 2020, 506 pages.
- Radzevich, S.P., Geometry of surfaces: A Practical Guide for Mechanical Engineers, 2nd edition, Springer International Publishing, 2019, © Springer Nature Switzerland AG (2020), XXVI, 329 pages, 182 illustrations. [ISBN-10: 3030221830, ISBN-13: 978-3030221836].
- 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. [Upon request, a .pdf of this article can be ordered (for free) from the author].
- Radzevich, S.P., (Editor), Recent Advances in Gearing: Scientific Theory and Applications, Springer, 1st ed., 2022 edition (June 25, 2021), 569 pages.
- Radzevich, S.P. (Editor), Storchak, M.G. (Editor), Advances in Gear Theory and Gear Cutting Tool Design, Springer, 2022, 500 pages.