This article deals with gears. The consideration is mainly focused on a fundamental design parameter — the base pitch. In addition to commonly used base pitches, (a) base pitch in a gear and (b) base pitch in a mating pinion, one more base pitch is discussed. This is the so-called “operating base pitch in a gear pair.” Even though this article focuses mainly on parallel-axes gearing, the obtained result is valid for gears of all designs, namely for parallel-axes gearing, intersected-axes gearing, and crossed-axes gearing.

### Introduction

The base pitch of a gear (and that of a mating pinion) is one of the fundamental design parameters of involute gears; those operate on parallel axes of rotation. The required geometry of the effective portion of an involute gear tooth profile can be completely specified in terms of the base pitch of the gear. In order to engage two involute gears in a proper mesh with one another, it is a common practice to keep the base pitches of mating gears equal to one another (p_{b.g} = p_{b.p}).

Gears of all designs, namely parallel-axes gears, intersected-axes gears, and crossed-axes gears, feature base pitch. In this article, the readers’ attention will be focused mainly on the base pitch in parallel-axes gearing. In Figure 1, an external parallel-axes involute gear pair is taken as an example (for illustrative purposes). Without loss of generality, the obtained results of the analysis can be enhanced to the area of gearing of other designs: to internal involute gearing, intersected-axes gearing, crossed-axes gearing, and so forth.

In geometrically-accurate involute gears, the base pitch, p_{b.g} and p_{b.p}, is measured in linear units: either in inches or in millimeters. In geometrically-accurate gears for intersected-axes gear pairs and in geometrically-accurate gears for crossed-axes gear pairs, the base pitch, ϕ_{b.g} and ϕ_{b.p}, has angular design parameters, so they are measured in angular units (in both cases).

### Preliminary Comments

Traditionally, the concept of “base pitch” is considered only with respect to parallel-axes involute gearing. The recent developments in the field of gearing [2] reveal that geometrically-accurate gears that operate either on intersected-axes of rotation of the mating gears or on crossed-axes of rotation of the gears, also feature “base pitch.” In these latter two cases, the “angular base pitch” is considered instead of the linear “base pitch.”

For simplicity, but without loss of generality, the readers’ attention should be focused mainly on the concept of “base pitches” in parallel-axes involute gearing. The concept of the “angular base pitch” is just mentioned in a few appropriate occasions.

For any and all parallel-axes involute gear pairs (see Figure 1), a corresponding “equivalent pulley-and-belt transmission” can be constructed — an example is shown in Figure 2.

In Figure 2, the pulleys are tightly connected to one another by means of a belt — by a plane of action, PA, that is tangent to the pulleys. By means of the moving, V_{lc}, belt, an input rotation, ω_{p}, from the driver is transmitted to the driven, ω_{g}. A desirable line of contact between the tooth flanks, G and P (in the gear pair to be modeled), can be either a straight line, parallel to the axes of rotation of the pulleys, LC_{spur} (like in spur gears), or an inclined straight line, LC_{helical} (like in helical gears), or a circular arc, LC_{circ}, or it can be shaped even in a form of an arbitrary planar curve, LC_{arbitr}. Regardless of the actual geometry of the desirable line of contact, LC_{des}, this line is always entirely situated within the plane of action — which is a must.

The “equivalent pulley-and-belt transmission” is helpful to perform the analysis of parallel-axes involute gearing. Numerous kinematic and design parameters of involute gearing (namely, smooth transmission of the input rotation, sliding of involute tooth profiles, and so forth) can be investigated using the simple “equivalent pulley-and-belt transmission,” rather than the gear tooth flanks, the geometry of which is complex. The direction of rotation of the mating involute gears, and of the pulleys, is the only the key difference between an involute gear pair and between the “equivalent pulley-and-belt transmission.” In a gear pair, the driver tooth flank “pushes” over the tooth flank of the driven. Therefore, the mating gears are rotated in the opposite directions. In the “equivalent pulley-and-belt transmission,” the driven pulley is “pulled” by the driver. Thus, both pulleys are rotated in the same direction.

The “equivalent pulley-and-belt transmission” is of fundamental importance for the in-depth understanding of the design and of the functioning of parallel-axes involute gearing. Unfortunately, so far it is not known who should be credited with this significant accomplishment in the scientific theory of gearing.

### Operating Base Pitch: Parallel-Axes Gearing

Consider a gear pair that operates on parallel axes of rotation of a gear and of a mating pinion (P_{a} — gearing). The displacements of the gears from their nominal configuration are of zero values. Parallel-axes gearing of this particular kind is commonly referred to geometrically-accurate P_{a} — gearing.

In Figure 3, a section of a gear pair by a transverse plane is depicted. The transverse plane is chosen here as the power (the rotation and the torque) is transmitted in a section of a transverse plane.

At the beginning, only two axes of rotation of a gear, O_{g}, and of a mating pinion, O_{p}, are given, as illustrated in Figure 3. The axes of rotation, O_{g} and O_{p}, are parallel and are at a center-distance, C, apart from one another (C = O_{g}O_{p}).

The line of action, LA, is a straight line through the pitch point, P. The line of action, LA, forms transverse pressure angle, φ_{t}, with the perpendicular to the center-line, CL, at P.

The base circle of the gear and that of the mating pinion are centered at points, O_{g} and O_{p}, correspondingly, and are tangent to the line of action, LA.

The operating base pitch, p_{b.op}, in parallel-axes gearing equals:

Straight-line segments of the length, p_{b.op}, are constructed within the line of action, LA. The operating base pitch in parallel-axes involute gearing is measured along the line of action, LA.

In a gear for geometrically-accurate parallel-axes gearing, the base pitch, p_{b.g}, is measured along a straight line that is perpendicular to the involute gear tooth profiles, as shown in Figure 4.

This straight line is tangent to the base cylinder of the gear of a diameter d_{b.g}. At the pitch point, P, the straight line forms a transverse pressure angle, φ_{t}, with the radial direction, O_{g}P.

In a mating pinion for geometrically-accurate parallel-axes gearing, the base pitch, p_{b.p}, is measured along a straight line perpendicular to the involute pinion tooth profiles, as shown in Figure 5.

This straight line is tangent to the base cylinder of the pinion of a diameter d_{b.p}. At the pitch point, P, the straight line forms a transverse pressure angle, φ_{t}, with the radial direction, O_{p}P.

As the base pitches, p_{b.g}, p_{b.p}, and p_{b.op}, are equal to one another — that is, the equalities p_{b.g} = p_{b.op}, and p_{b.p} = p_{b.op} are valid — the involute gear can be engaged in a proper mesh with a corresponding involute pinion (see Figure 6). Equality of the base pitches, p_{b.g}, p_{b.p}, and p_{b.op}, (namely, p_{b.g} = p_{b.op}, and p_{b.p} = p_{b.op}), is a fundamental law of geometrically-accurate parallel-axes gearing.

At the pitch point, P, the straight line forms a transverse pressure angle, φ_{t}, with a perpendicular to the center-line, CL.

The base pitch of a gear, p_{b.g}, and that of a mating pinion, p_{b.p}, can be constructed only for involute gears. Gears of no other geometries feature base pitch.

### Non-Coincident Lines of Intersection of Gear and Pinion Tooth Flanks by the Plane of Action

Real parallel-axes gearing is composed of two real gears, that is, two gears manufactured either on the machine tool or using other equipment. Tooth flanks of actual gears deviate from those in geometrically-accurate involute gears. As the tooth flanks in real gears are not identical to the tooth flanks in geometrically-accurate involute gears, the line of intersection, LI_{g}, of the real tooth flank, G _{real}, by the tangent plane differs from that in geometrically accurate involute gears (see Figure 7a). The same is valid with respect to the line of intersection, LI_{p}, of the real tooth flank, P _{real}, by the tangent plane, as illustrated in Figure 7b. Moreover, often the geometry of the lines of intersection, LI_{g} and LI_{p}, vary when the gears rotate.

Different geometry of the lines of intersection, LI_{g} and LI_{p}, results in these two lines cannot form a desirable line of contact, LC_{des}, of the tooth flanks, G and P, in parallel-axes gearing, as it is required (see Figure 7c) for geometrically-accurate gearing. This is shown in the Figure 7c scenario and is equivalent to that shown in Figure 6.

In reality, the lines of intersection, LI_{g} and LI_{p}, are not congruent to one another, but instead they are somehow configured in relation to each other. A possible configuration of the lines of intersection, LI_{g} and LI_{p}, is illustrated in Figure 7d. It should be realized here that the configuration (see Figure 7d) is not the only one permissible.

The non-coincidence of the line of intersection, LI_{g} and LI_{p}, results in violation of the fundamental law of gearing, as the equality p_{b.g} = p_{b.p} is no longer valid. Once the equality p_{b.g} = p_{b.p} is violated, either one of two, or both base pitches, p_{b.g} and p_{b.p}, are not equal to the operating base pitch, p_{b.op}, in the gear pair.

In the example shown in Figure 8, the violation of the fundamental law of gearing ( p_{b.g} = p_{b.op}, and p_{b.p} = p_{b.op}) is due to the specific features of the tooth flank geometry.

Gears with a circular-arc tooth flank geometry (in the lengthwise direction of the gear teeth) are convenient to be machined either by the face-mill cutters or by the face hobs (see Figure 8a). Cut in this way, gears have a constant normal base pitch in the sections by normal planes through the center of the circular arc of a radius, R_{g}. The transverse base pitch in different plane sections of the gear is of a different value within the gear face width, F_{g}. Once the transverse base pitch is not of a constant value, it cannot be equal to the operating base pitch in a gear pair, which is of a constant value (p_{b.g} ≠ p_{b.op}). Because of this, the fundamental law of gearing is violated.

In order to design and to manufacture geometrically-accurate gears of the kind under consideration, it is necessary to keep the transverse base pitch of a constant value in all the sections by the transverse planes, as illustrated in Figure 8b. Once the transverse base pitch is of a constant value, it can be equal to the operating base pitch in a gear pair. In this way, the violation of the fundamental law of gearing (p_{b.g} ≠ p_{b.op}) can be eliminated.

**Operating Angular Base Pitch – Intersected-Axes Gearing. **The concept of the “base pitch,” along with the concept of the “operating base pitch,” can be applied to gears that operate on intersected axes of rotation of a gear and of a mating pinion. An example is illustrated in Figure 9. Here, in Figure 9, A* _{g}* is the gear base cone apex; A

*is the pinion base cone apex; A*

_{p}*is the plane-of-action apex; ϕ*

_{pa}*is the angular operating base pitch in an intersected-axes gear pair. In intersected-axes gearing, the apexes A*

_{b.op}*, A*

_{g}*, and A*

_{p}*, are snapped together (which is a must in geometrically-accurate intersected-axes gearing).*

_{pa}It is important to stress here that the idea of geometrically-accurate intersected-axes gearing can be traced back to 1887, when George B. Grant invented a machine for planing gear teeth [1]. At this moment the geometry of the tooth flank of a gear for geometrically-accurate gearing was determined, and G. Grant is credited with this important invention. However, neither the concept of the “base pitch” nor the concept of the “operating base pitch” were investigated by G. Grant — the importance of these fundamental concepts was not realized by G. Grant. This analysis has been performed much later (around 2008) by Prof. S.P. Radzevich [2].

For proper operation of an intersected-axes gear pair, the angular base pitch of a gear, ϕ_{b.g}, and that of a mating pinion, ϕ_{b.p}, must be equal to the operating angular base pitch, ϕ_{b.op}, in the gear pair; that is, the equalities ϕ_{b.g} = ϕ_{b.op}, and ϕ_{b.p} = ϕ_{b.op} must be valid at every instant of time when the gears operate.

Considered in a reference system X_{g}Y_{g}Z_{g}, associated with the gear, a locus of consecutive positions of the line of contact, LC, represent tooth flank, G, of the geometrically-accurate gear for intersected-axes gear pair (see Figure 9). Similarly, considered in a reference system X_{p}Y_{p}Z_{p}, associated with the mating pinion, a locus of consecutive positions of the line of contact, LC, represent tooth flank, P, of the geometrically-accurate pinion for intersected-axes gear pair.

**Operating Angular Base Pitch – Crossed-Axes Gearing. **The concept of the “base pitch” along with the concept of the “operating base pitch” can be applied to gears that operate on crossing axes of rotation of a gear and of a mating pinion, as illustrated in Figure 10. Here, in Figure 10, A* _{g}* is the gear base cone apex; A

*is the pinion base cone apex; A*

_{p}*is the plane-of-action apex; ϕ*

_{pa}*is the angular operating base pitch in crossed-axes gear pair.*

_{b.op}Geometrically-accurate crossed-axes gearing with line contact between the tooth flanks, G and P, of a gear and that of a mating pinion, were invented (around 2008) and investigated by Prof. S.P. Radzevich [2]. The gearing of this design is commonly referred to as “R — gearing.” “R — gearing” is the only design of crossed-axes gearing that features line contact between the tooth flanks of a gear, G, and that of a mating pinion, P.

For proper operation of crossed-axes gear pair, the angular base pitch of a gear, ϕ_{b.g}, and that of a mating pinion, ϕ_{b.p}, must be equal to the operating base pitch, ϕ_{b.op}, in the gear pair, that is, the equalities ϕ_{b.g} = ϕ_{b.op}, and ϕ_{b.p} = ϕ_{b.op} must be valid at every instant of time when the gears operate.

Considered in a reference system X_{g}Y_{g}Z_{g}, associated with the gear, a locus of consecutive positions of the line of contact, LC, represent the tooth flank, G, of the geometrically-accurate gear for crossed-axes gear pair (see Figure 10). Similarly, considered in a reference system X_{p}Y_{p}Z_{p}, associated with the mating pinion, a locus of consecutive positions of the line of contact, LC, represent the tooth flank, P, of the geometrically-accurate pinion for crossed -axes gear pair.

Compare as shown in Figure 9, a schematic of intersected-axes gearing with a corresponding schematic of crossed-axes gearing, shown in Figure 10.

In intersected-axes gearing, all three apexes (the gear base-cone-apex, A_{g}; the pinion base-cone-apex, A_{p}; and the plane-of-action apex, A_{pa}) are snapped together (see Figure 9). Such a configuration of the apexes, A_{g}, A_{p}, and A_{pa}, is reflected in the actual tooth flank geometry of the gear, G, and of the mating pinion, P, in I_{a} — gearing.

In crossed-axes gearing, all three apexes (the gear base-cone-apex, A_{g}; the pinion base-cone-apex, A_{p}; and the plane-of-action apex, A_{pa}) are situated along the center-line, CL (see Figure 10). Such a configuration of the apexes, A_{g}, A_{p}, and A_{pa}, is also reflected in the actual tooth flank geometry of the gear, G, and of the mating pinion, P, in C_{a} — gearing.

As in I_{a} — gearing, the actual configuration of the apexes, A_{g}, A_{p}, and A_{pa}, is different from that in C_{a} — gearing, the actual tooth flank geometry of the gear, G, and of the mating pinion, P, in both cases is also different. This means the gears, those finish cut for I_{a} — gearing and those finish cut for C_{a} — gearing, are not interchangeable and cannot be replaced with one another. Moreover, different methods of gear finishing and different gear-finishing tools are required to be used when finishing precision gears for I_{a} — gearing and for C_{a} — gearing. The methods and means for gear inspection are also different.

### Conclusion

In today’s practices, base pitches, p_{b.g} and p_{b.p}, are defined only in the design of parallel-axes gearing. To this end, no base pitches are defined for intersected-axes and for crossed-axes gearing.

The base pitch of a gear, p_{b.g}, and that of a mating pinion, p_{b.p}, can be constructed only for involute gears. Gears of no other geometries feature a base pitch.

The concept of the “operating base pitch” is not considered at all.

The operating base pitch can be calculated for gear pairs of any and all designs, despite the actual tooth profile geometry.

In geometrically-accurate parallel-axes gearing, the “operating base pitch” is a linear design parameter (it is measured in inches, mm, and so forth). In geometrically-accurate intersected-axes gearing, as well as in crossed-axes gearing, the “angular operating base pitch” is an angular design parameter (and, thus, measured in degrees, angular minutes, angular seconds, and so forth).

The operating base pitch in a gear pair also can be considered as the reference for base pitches of a gear and that of a mating pinion.

It is recommended to incorporate the concepts: (a) of the operating base pitch in gearing of all designs and (b) of the angular base pitches in both intersected-axes gearing, as well as in crossed-axes gearing, into present-day practice of the design and manufacture of precision gears. This is an easy and almost costless way to significantly improve gear accuracy.

### References

- 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., Theory of Gearing: Kinematics, Geometry, and Synthesis, 2nd Edition, revised and expanded, CRC Press, Boca Raton, FL, 2018, 898 pages.

### Bibliography

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., “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.