The mounting distance in intersected-axes gearing and in crossed-axes gearing will be explored in this article. It is shown the plane of action in a gear pair can be employed for the calculation of tolerance for the accuracy of the mounting distance in gearing of both kinds. The tolerance for the accuracy of the mounting distance is expressed in terms of the maximum permissible angle when the tooth flanks of a gear and of a mating pinion make edge contact. The obtained result makes possible design and production of intersected-axes and of crossed-axes gear pairs with interchangeable gears. The necessity of pairing of gears is eliminated, namely, only a broken gear can be replaced with a new one, instead of the replacement of the whole gear pair.

### Introduction

The performance of intersected-axes and of crossed-axes gear pairs strongly depends on the accuracy of the axial location of the gears — on the accuracy of the mounting distance. Precision gearing is especially sensitive to the accuracy of the mounting of gears in the gear housing.

The mounting distance in gears for intersected-axes gear pairs, as well as for crossed-axes gear pairs, is a design parameter that cannot be inspected directly, as the base cone apex of the gear does not exist physically. This is the root cause for the significant inconveniences when inspecting the accuracy of the mounting distance. Therefore, indirect methods for the inspection of the accuracy of the mounting distance are used in practice.

No reliable methods for the calculation of the tolerances for the accuracy of the mounting distance in intersected-axes and in crossed-axes gearing are developed yet.

Instead, the required configuration of the gears in a gear pair is commonly obtained by means of shims (“trial and error method”).

In this article, a method for the calculation of tolerance for the accuracy of the mounting distance in intersected-axes, as well is in crossed-axes gearing, is proposed. This result is of critical importance as it allows to the design and production of the gears to be properly engineered.

### Preliminary Comments

It is well known that the tolerance for the accuracy of the mounting distance in intersected-axes gearing and in crossed-axes gearing is a critical design parameter of gearing of this design. As of 1993, the author had plenty of opportunities to visit gear manufacturers including, but not limited to, the automotive industry sector, aerospace, and so forth. The calculation of the tolerance for accuracy of the mounting distance in intersected-axes gearing, as well as in crossed-axes gearing, was one of the key author’s interests in these visits. The author failed to get from representatives of the industry a proficient answer to the question of how this tolerance is calculated in present-day practice.

A performed in-detail analysis of the input gear-design parameters entered in a software for the gear calculation makes it possible to conclude that the applied methods of the calculation are inappropriate for the calculation of tolerance for the accuracy of the mounting distance in gearing.

The input set of gear-design parameters is incomplete to calculate the tolerance for the accuracy of the mounting distance in gearing.

The readers’ attention in the following text is focused primarily on the mounting distance in intersected-axes gearing and in crossed-axes gearing.

In intersected-axes gear pair (see Figure 1a), the axis of rotation of a gear, O_{g}, and that of a mating pinion, O_{p}, intersect each other at point. By nature, the point of intersection (a) is the base-cone-apex, A_{g}, of the gear, (b) is the base-cone-apex, A_{p}, of the mating pinion, and (c) is the plane-of action-apex, A_{pa}. In other words, three points: A_{g}, A_{p}, and A_{pa}, are snapped together at the point of intersection of the axes of rotation O_{g} and O_{p}. Such a configuration of the points A_{g}, A_{p}, and A_{pa}, must be retained all the time when the intersected-axes gear pair operates.

In crossed-axes gear pair (see Figure 1b), the axis of rotation of a gear, O_{g}, and that of a mating pinion, O_{p}, cross each other. By nature, (a) the base-cone-apex, A_{g}, of the gear, (b) the base-cone-apex, A_{p}, of the mating pinion, and (c) the plane-of action-apex, A_{pa}, all three of them are situated within the center-line, CL. Such a configuration of the points A_{g}, A_{p}, and A_{pa}, must be retained all the time when the crossed-axes gear pair operates.

In both cases, violation of the required configuration of the points, A_{g}, A_{p}, and A_{pa}, results in edge contact between tooth flanks, G and P, of a gear and a mating pinion, as illustrated in Figure 2.

In geometrically-accurate gears, at every point K within the line of contact, LC, the unit normal vectors, **n**_{g} and **n**_{p}, to the interacting tooth flanks, G and P, align to one another as depicted in Figure 2a. In this way, the unit normal vectors, **n**_{g} and **n**_{p}, form an angle of 180°.

When the gears are displaced from their nominal configuration, as shown in Figure 2b, the unit normal vectors, **n**_{g} and **n**_{p}, form an angle that equals (180° − θ). This schematic (see Figure 2b) is the key to understand how a tolerance for the accuracy of the mounting distance can be calculated. No calculations of the tolerance for the accuracy of the mounting distance can be performed if a tolerance [θ] for the angle θ is not specified.

The total deviation, θ, can be viewed as a superposition of the two components, namely, of θ_{g} and θ_{p} (that is θ = θ_{g} + θ_{p}). The first one is due to inaccuracy of the gear, while the second one is due to inaccuracy of the pinion. The tolerance, [θ], for the accuracy of the angle θ can be somehow shared between the components [θ_{g}] and [θ_{p}]. Except of miter gears, usually, the inequality θ_{g} < θ_{p} is observed.

An important conclusion can be drawn up from the above discussion: No axial shift of the gears from their nominal position in the gear housing (see Figure 3), is permissible neither in intersected-axes gearing, nor in crossed-axes gearing. It is a wrong practice to adjust for the backlash in intersected-axes gearing by means of “inward-outward” axial shift of the gears.

### Tolerance for the accuracy of mounting distance in intersected-axes gearing

To operate properly, gears in intersected-axes gear pairs must be properly configured in relation to one another [1], [2], [3].

The axis of rotation of a gear, O_{g}, and that, O_{pa}, of the plane of action, PA, in a geometrically-accurate intersected-axes gear pair intersect as illustrated in Figure 4a. The base cone apex of the gear, A_{g}, and the plane-of-action apex, A_{pa}, both, coincide with point of intersection of the axes O_{g} and O_{pa}. A rotation of the gear is designated as ω_{g}, and rotation of the plane of action is designated as ω_{pa}, correspondingly. When the gears rotate, the base cone of the gear and the plane of action roll over each other with no sliding.

In reality, an error in mounting distance, ε_{g}, is always observed. The latter is shown in Figure 4b. In such a scenario, the gear base cone apex, A*_{g}, does not coincide with the plane-of-action apex, A_{pa}, as the apex, A*_{g}, is displaced axially in relation to A_{pa} at a distance ε_{g}. (Here, we are not going into a detailed analysis of the gap between the gear base cone and the plane of action). In a case depicted in Figure 4b, the outward displacement, ε_{g}, is of a positive value. An inward displacement is of a negative value (not shown in Figure 4b).

The performed analysis in Figure 4 allows for the derivation of an equation for the calculation of the tolerance for the accuracy of the mounting distance in intersected-axes gear pair.

Consider the intersection of two teeth in contact by the plane of action, PA, as illustrated in Figure 5a. These teeth contact one another along a straight-line segment ab. (Here, for simplicity, but without loss of generality, a pair of geometrically-accurate straight bevel gears is considered). The configuration of the gears in Figure 5a corresponds to a case when the center line of the gear tooth in a section by the pitch plane is aligned with the axis of instant rotation, P_{ln}, in the gear pair.

The schematic shown in Figure 5 [1], [3], is constructed for straight bevel gears. Similar schematics can be constructed for gears with another geometry of the tooth flanks in the lengthwise direction of the gear tooth.

Under any and all circumstances, edge contact between the gear and the pinion teeth has to be avoided. That is why, in order to keep the actual value of the angle θ within a very tight tolerance for the accuracy of this parameter, the tolerances for the accuracy of the rest of the design parameters, which the axes misalignment depends on, have to be set very tight.

Solely within the gear mesh, only the unit normal vectors **n**_{g} and **n**_{p} are taken into account, while the unit normal perpendiculars to the gear, and the pinion faces are ignored.

Moreover, no edge roundness, chamfers, etc. are considered in this analysis.

A case of outward displaced gear is illustrated in Figure 5b. As a result of the displacement, a gap between the teeth is observed. This gap is shown by two straight-line segments, ab and a*b*, each of which is entirely located on the tooth flanks, G and P, of the two interacting teeth.

A schematic for the derivation of a formula for the calculation of a tolerance for the accuracy of the mounting distance is depicted in Figure 5c. It should be stressed that, from the very beginning, point b within the straight-line segment ab is the closest point to the straight-line segment a*b*. Therefore, an angle, θ_{g}, through which the plane of action, PA, has to be turned about the axis of rotation O_{pa} depends on the actual distance of point b to the straight-line segment a*b*.

Assuming that the gear is motionless, and the plane of action, PA, turns about its axis of rotation, O_{pa}, through an angle at which point, b, touches the straight-line segment, A*_{g}b*, then consider a triangle, ΔA_{pa}A*_{g}b. In this triangle, A_{pa}b = r_{o.pa}, A_{pa}A*_{g} = ε_{g}, and A*_{g}b = r*_{o.pa}.

Law of cosine is used for the determination of the actual value of the angle, θ_{g}:

Then, either the actual value of the angle, θ_{g}, can be expressed in terms of the mounting distance error, ε_{g}, and the design parameters of the gear pair, or a maximum permissible values of the displacement, ε_{g}, can be expressed in terms of the maximum permissible value of the angle θ_{g}, and the design parameters of the gear pair:

Here, [ε_{g}] is the tolerance for the accuracy of the gear mounting distance, ε_{g}.

Here, [θ_{g}] is the tolerance for the accuracy of the gear angle, θ_{g}.

It can be shown (see Figure 5c) that the equality:

is valid.

When the pinion is fully aligned, the equalities ε = ε_{g} and θ = θ_{g}, and Equation 2 and Equation 3 can be used for the calculation of tolerance on the mounting distance in the intersected-axes gear pair.

An analysis similar to that above, can be performed for a mating bevel pinion:

Here, [ε_{p}] is the tolerance for the accuracy of the pinion mounting distance, ε_{p}; and [θ_{p}] is the tolerance for the accuracy of the gear angle, θ_{p}.

Further, when the gear is fully aligned, the equalities ε = ε_{p} and θ = θ_{p}, and Equation 5 and Equation 6 can be used for the calculation of tolerance on the mounting distance in the intersected-axes gear pair.

Finally, in a more general case, a bevel gear and a mating bevel pinion, both, are misaligned. In such a scenario, either the actual value of the angle θ can be expressed in terms of the mounting distance errors, ε_{g} and ε_{p}, and the design parameters of the gear pair or a maximum permissible values of the displacements, *ε*_{g} and ε_{p}, can be expressed in terms of the maximum permissible value of the angle θ, and the design parameters of the gear pair.

The angle θ is the angle that is formed by two perpendiculars, **n**_{g} and **n**_{p} [that is, θ = ∠(**n**_{g},**n**_{p})], constructed at a point within the edge contact of a gear, G, and a mating pinion, P, tooth flanks, correspondingly: **n**_{g} is the unit normal vector to the gear tooth flank, G, and **n**_{p} the unit normal vector to the pinion tooth flank, P.

For the determination of the tolerances, [ε_{g}] and [ε_{p}], for the accuracy of the permissible axial displacements, ε_{g} and ε_{p}, of the gear and the mating pinion, either one of the tolerances (either the tolerance [ε_{g}], or the tolerance [ε_{p}]), or a ratio of the tolerances, [ε_{g}]/[ε_{p}], has to be pre-specified.

For the accurate calculation of the tolerances, [ε_{g}] and [ε_{p}], for the accuracy of the permissible axial displacements, ε_{g} and ε_{p}, of the gear and the mating pinion, the angle, θ, is viewed as a summa θ = θ_{g}+θ_{p}. Here the angle θ_{g} equals to θ_{g} = 90° − ∠(**n**_{g},**n**_{pa}), and the angle θ_{p} equals to θ_{p} = 90° − ∠(**n**_{p},n_{pa}). The angles, θ_{g} and θ_{p}, are entered into the equations for the calculation of the tolerances, [ε_{g}] and [ε_{p}], correspondingly.

The performed analysis reveals the actual value of the angle, θ, changes when the gears rotate. The maximum value of the angle, θ, is observed at the very beginning of the meshing of two gear teeth. As the rotation proceeds, the angle, θ, reduces to its minimum value. A minimum value of the angle, θ, is observed within a plane through the axis of instant rotation, P_{ln}, oriented perpendicular to the plane of action, PA. Further, the angle, θ, increases to its maximum value at the very end of the meshing of two gear teeth.

A more in detail analysis is not presented here as the equations are bulky.

To calculate the tolerance band [ε] for intersected-axes gearing, the maximum and the minimum value of the angle θ is entered into Equation 5.

In addition to the discussed approach, another approach for the calculation of the tolerance for the accuracy of the mounting distance in intersected-axes gearing is developed.

When the displacements ε_{g} (or the displacement ε_{p}) is of a negative value, this results in that the edge contact occurs at the opposite face of the gear.

### Tolerance for the accuracy of mounting distance in crossed-axes gearing

In crossed-axes gearing, interaction of tooth flanks, G and P, of a gear and of a mating pinion occurs within the plane of action, PA. Therefore, it makes sense to consider disposition of the base cone of a gear in relation to the plane of action. Later on, the results of the analysis obtained in this way can be applied to the disposition of the base cone of a mating pinion in relation to the gear housing.

In a geometrically-accurate crossed-axes gear pair, the axis of rotation of a gear, O_{g}, and that, O_{pa}, of the plane of action, PA, are crossed, as illustrated in Figure 6a. The base cone apex of the gear, A_{g}, and the plane-of-action apex, A_{pa}, are at the gear center-distance, C_{g}, apart from one another. A rotation of the gear is designated as ω_{g}, and a rotation of the plane of action is designated as ω_{pa}, correspondingly. Sliding between the base cone of the gear, and the plane of action only along the axis, P_{ln}, is observed when the gears rotate — no sliding in a direction perpendicular to the axis, P_{ln}, is observed.

In reality, an error, ε_{g}, in the mounting distance of the gear is always observed. The latter is shown in Figure 6b. In such a scenario, the gear base cone apex, A*_{g}, does not coincide with the point, A_{h.g}, as A*_{g} is displaced axially in relation to A_{h.g} at a distance ε_{g}. In a case shown in Figure 6b, the outward displacement, ε_{g}, is of a positive value. An inward displacement is of a negative value (not shown in Figure 6b).

This discussion proceeds with the derivation of an equation for the calculation of the tolerance for the accuracy of the mounting distance in a crossed-axes gear pair.

Consider two teeth in contact intersected by the plane of action, PA, as illustrated in Figure 7a. The teeth contact one another along a straight-line segment ab. (Here, for simplicity, but without loss of generality, a pair of geometrically-accurate straight bevel gears is considered).

The schematic shown in Figure 7 [1-2], are constructed for straight bevel gears. Similar schematics can be constructed for gears with another geometry of the tooth flanks in the lengthwise direction of the gear tooth.

A case of outward displaced gear is illustrated in Figure 7b. As a result of the displacement, a gap between the teeth is observed. This gap is shown by two straight-line segments, ab and a*b*, each of which is entirely located on the tooth flanks, G and P, of the two interacting teeth.

A schematic for the derivation of a formula for the calculation of a tolerance for the accuracy of the mounting distance is depicted in Figure 7c. It should be stressed that point b within the straight-line segment ab is the closest point to the straight-line segment a*b*. Therefore, an angle, θ_{g}, through which the plane of action, PA, has to be turned about the axis of rotation O_{pa} depends on the actual distance of point b to the straight-line segment a*b*.

Assume that the gear is motionless, and the plane of action, PA, turns about its axis of rotation, O_{pa}, through an angle at which point, b, touches the straight-line segment, A*_{g}b*. Then, consider a triangle, ΔA_{pa}A*_{g}b. In this triangle, A_{pa}b = r_{o.pa}, A_{pa}A*_{g} = √C^{2} + ε^{2}_{g}, , and A*_{g}b = r*_{o.pa}.

Law of cosine can be used for the determination of the actual value of the angle, θ_{g}:

Then, either the actual value of the angle, θ_{g}, can be expressed in terms of the mounting distance error, ε_{g}, and the design parameters of the gear pair, or a maximum permissible values of the displacement, ε_{g}, can be expressed in terms of the maximum permissible value of the angle θ_{g}, and the design parameters of the gear pair:

Here, [ε_{g}] is the tolerance for the accuracy of the gear mounting distance, ε_{g}.

Here, [θ_{g}] is the tolerance for the accuracy of the gear angle, θ_{g}.

It can be shown (see Figure 7c) that the equality:

is valid.

When the pinion is fully aligned, the equalities ε = ε_{g} and θ = θ_{g}, and Equation 9 and Equation 10 can be used for the calculation of tolerance on the mounting distance in the crossed-axes gear pair.

An analysis similar to that above, can be performed for a mating bevel pinion:

Here, [ε_{p}] is the tolerance for the accuracy of the pinion mounting distance, ε_{p}; and [θ_{p}] is the tolerance for the accuracy of the gear angle, θ_{p}.

Further, when the gear is aligned, the equalities ε = ε_{p} and θ = θ_{p}, and Equation 12 and Equation 13 are used for the calculation of tolerance on the mounting distance in the crossed-axes gear pair.

Finally, in a more general case, both a bevel gear and a mating bevel pinion are misaligned. Under such a scenario, either the actual value of the angle θ can be expressed in terms of the mounting distance errors, ε_{g} and ε_{p}, and the design parameters of the gear pair, or a maximum permissible values of the displacements, ε_{g} and ε_{p}, can be expressed in terms of the maximum permissible value of the angle θ, and the design parameters of the gear pair.

The angle *θ* is formed by two perpendiculars, **n**_{g} and **n**_{p} [that is, θ = ∠(**n**_{g},**n**_{p})], constructed at point of edge contact of a gear, G, and a mating pinion, P, tooth flanks, correspondingly: **n**_{g} is the unit normal vector to the gear tooth flank, G, and **n**_{p} the unit normal vector to the pinion tooth flank, P.

For the determination of the tolerances, [ε_{g}] and [ε_{p}], for the accuracy of the permissible axial displacements, ε_{g} and ε_{p}, of the gear and the mating pinion, either one of the tolerances (either the tolerance [ε_{g}], or the tolerance [ε_{p}]), or a ratio of the tolerances, [ε_{g}]/[ε_{p}], has to be pre-specified.

For the accurate calculation of the tolerances, [ε_{g}] and [ε_{p}], for the accuracy of the permissible axial displacements, ε_{g} and ε_{p}, of the gear and the mating pinion, the angle, θ, is viewed as a summa θ = θ_{g} + θ_{p}. Here the angle θ_{g} equals to θ_{g} = 90° − ∠(**n**_{g},**n**_{pa}), and the angle θ_{p} equals to θ_{p} = 90° − ∠(**n**_{p},**n**_{pa}). The angles, θ_{g} and θ_{p}, are entered into the equations for the calculation of the tolerances, [ε_{g}] and [ε_{p}], correspondingly.]

The performed analysis reveals that the actual value of the angle, θ, alters when the gears rotate. The maximum value of the angle, θ, is observed at the very beginning of meshing of two gear teeth. As the rotation proceeds, the angle, θ, reduces to its minimum value. A minimum value of the angle, θ, is observed within a plane through the axis of instant rotation, P_{ln}, perpendicular to the plane of action, PA. Further, the angle, θ, increases to its maximum value at the very end of meshing of two gear teeth. (A more in-detail analysis is not presented here as the equations are bulky.)

When the displacements ε_{g} (or the displacement ε_{p}) is of a negative value, this results in that the edge contact occurs at the opposite face of the gear.

To calculate the tolerance band [ε] for crossed-axes gearing, the maximum, and the minimum value of the angle θ is entered into Equation 12.

### Conclusion

No reliable methods for calculation tolerances for the mounting distance in intersected-axes and in crossed-axes gearing have been developed yet.

Instead, the required configuration of the gears in a gear pair is commonly obtained by means of shims (“trial and error method”). This reveals the design and production of gears for I_{a} — axes and for C_{a} — axes gearing are poorly engineered. As a result, pairing of gears is required, and the gears are not interchangeable.

The proposed method of calculation tolerance for the accuracy of the mounting distance in intersected-axes gearing, as well as in crossed-axes gearing, is helpful to resolve the issue. The gears for I_{a} — axes, and for C_{a} — axes gearing do not need in pairing. Moreover, they can be designed and manufactured interchangeably.

### References

- Radzevich, S.P., Theory of Gearing: Kinematics, Geometry, and Synthesis, 2nd Edition, revised and expanded, CRC Press, Boca Raton, FL, 2018, 898 pages.
- Radzevich, S.P., “Understanding the Mounting Distance: Crossed-Axes Gearing (Hypoid Gearing),” Gear Solutions magazine, February 2020, pp. 38-43.
- Radzevich, S.P., “Understanding the Mounting Distance: Intersected-Axes Gearing (Bevel Gearing),” Gear Solutions magazine, December 2019, pages 42-47.

### 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., “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.