Mounting distance is one of the most critical design parameters in intersected-axes gearing (in straight bevel gearing, spiral bevel gearing, and so forth). In this article, the mounting distance is discussed with focus on “geometrically-accurate gearing” (that is, with focus on “perfect” gearing). The term “geometrically-accurate gearing” means gearing with zero deviations of the actual values of all the design parameters from their desired values. The disclosed results of the analysis are applicable to bevel gears machined on the bevel gear generators available on the market.

### Introduction

Bevel gearing is extensively used in today’s industry. Gears for bevel gearing have to be manufactured and assembled in a housing in such a way that ensures smooth running and favorable load distribution between the gear teeth. High power density (or, in other words, the “power-to-weight ratio”) in bevel gearing along with quietness when operating is strongly desired. If engineered properly, bevel gears can be installed in the same manner as spur or helical gears and behave and perform just as well. To achieve these goals, certain requirements have to be fulfilled when designing, manufacturing, and assembling bevel gear pairs. Accuracy of the mounting distance is one of the most critical design parameters in this regard.

### 1 Elements of Kinematics, and Geometry in Intersected-Axes Gearing

The concept of “geometrically-accurate” intersected-axes gearing can be traced back to 1887 when George B. Grant filed an invention disclosure titled Machine for Planing Gear Teeth [1].

A straight bevel gear pair is depicted in Figure 1 as an example. When bevel gearing operates, the gear spins, *ω*_{g}, about its axis of rotation, O_{g}. The mating pinion spins, *ω*_{p}, about its axis of rotation, O_{p}. The axes of rotation, O_{g} and O_{p}, intersect one another, and the point of intersection coincides with the plane-of-action apex, A_{pa} — this is a must for perfect performance of bevel gearing.

A base cone is associated with the gear. The gear base cone apex, A_{g}, is located within the gear axis of rotation, O_{g}. Another base cone is associated with the pinion. The pinion base cone apex, A_{p}, is located within the pinion axis of rotation axis, O_{p}. The gear and the pinion must be configured in relation to each other to ensure the apexes, A_{g} and A_{p}, snap together with the plane-of-action apex, A_{pa}. This is a must [2]. Neither axial displacement of the gear nor axial displacement of the pinion is allowed from the position shown in Figure 1. The plane of action, PA, is a plane through the plane-of-action apex, A_{pa}, that is tangential to the base cones of the gear and the mating pinion.

### 2 Mounting Distance

To operate properly, gears in intersected-axes gearing must be properly configured in relation to one another. Mounting distance is a key design parameter that helps keep gears properly configured and is the most important parameter for ensuring correct operation. Mounting distance can be specified as the distance from a locating surface on the back of one gear (most commonly a bearing seat) to the plane-of-action apex, A_{pa}.

Theoretically, in an intersected-axes gear pair, a gear base cone apex, A_{g}, a mating pinion base cone apex, A_{p}, and the plane-of-action apex, A_{pa}, mast be snapped together, as schematically illustrated in Figure 2a.

In reality, in order to transmit power, gears need to be supported. A gear housing (see Figure 2b) is used for this purpose. The gear housing features a corresponding number of bores that are used to support gear shafts in the housing. A centerline of a bore for the gear shaft is labeled as O_{h.g}. Correspondingly, a centerline of a bore for the pinion shaft is designated by O_{h.p}. The centerlines, O_{hg} and O_{hp}, intersect one another at point A_{h}. Strong constraints on the configuration of the gear and its mating pinion in the gear housing are imposed by this point, A_{h}.

The gear must be configured in the gear housing so as to keep the gear axis of rotation, O_{g}, aligned with the axis, O_{h.g}, in the gear housing (that is, O_{g} ≡ O_{h.g}), as illustrated in Figure 2c. The gear base cone apex, A_{g}, must be coincident with the point, A_{h}, of intersection of the centerlines, and (that is, A_{g} ≡ A_{h}). Then, the rest of the important components — bearings, shims, fasteners, and so forth — are put together so as to keep the gear mounting distance, MD_{g}, to the blueprint. The gear mounting distance, MD_{g}, is measured between the gear back face, and the gear base cone apex, A_{g} ≡ A_{pa} ≡ A_{h}), as shown in Figure 2d.

Similarly, the pinion must be configured in the gear housing so as to keep the pinion axis of rotation, O_{p}, aligned with the axis, O_{hp} (that is, O_{p} ≡ O_{h.p}), in the gear housing, as illustrated in Figure 2e. The pinion base cone apex, A_{p}, must be coincident with the point, A_{h}, of intersection of the centerlines, O_{hg} and O_{hp} (that is, A_{p} ≡ A_{h}). Then, the rest of the important components — bearings, shims, fasteners, and so forth — are put together so as to keep the pinion mounting distance, MD_{p}, to the blueprint. The pinion mounting distance, MD_{p} is measured between the pinion back face and the pinion base cone apex, as shown in Figure 2f.

After the pinion is in position, the tapered roller bearings of the gear assembly are pre-loaded; at the same time, the gear assembly is placed axially in the housing.

Finally, once the gear and the pinion are properly mounted in the gear housing, their axes are aligned to the centerlines of the corresponding bores in the gear housing (that is, O_{g} ≡ O_{hg}, and O_{p} ≡ O_{hp}), and the apexes, A_{g}, A_{p}, and A_{pa}, all of them are snapped together with the point of intersection, A_{h}, of the centerlines, O_{hg} and O_{hp}, in the gear housing, as schematically illustrated in Figure 2g.

An important conclusion can be drawn up from the above discussion (see Figure 2):

#### Intermediate conclusion

In intersected-axes gearing, NO axial displacements of the gears from the position when the base cone apexes, A_{g} and A_{p}, of a gear and a mating pinion coincide with the point of intersection, A_{h} (≡ A_{pa}), of the centerlines, O_{hg} and O_{hp}, in the gear housing are allowed.

Therefore, it is a wrong practice to adjust for the backlash in intersected-axes gearing by means of “inward-outward” axial displacement of the gears as shown in Figure 3, as recommended in [3], as well as in many other sources.

In precision intersected-axes gearing, it is recommended to inspect the mounting distance at a nominal operating load, as well as the preloaded bearings. Tolerance for the accuracy of the mounting distance is tight.

### 3 Motivation

Consider a geometrically-accurate intersected-axes gear pair (I_{a} — gear pair). A rotation from a driving member to a driven member in the gear pair is transmitted through a line of contact, LC, between the tooth flanks, G and P, of the gear and the mating pinion.

Lines of contact, LC, of various geometries are practically used in design of intersected-axes gear pairs. For simplicity, but without loss of generality, a straight line of contact ab is shown in Figure 4a. In reality, due to various factors affecting the real configuration of the gear and the pinion in relation to each other, the desirable line contact between the tooth flanks, G and P, can be substituted with their “edge contact,” as illustrated in Figure 4b. When edge contact is observed^{1}, the tooth flanks, G and P, interact with each other at point. The edge is formed by two surfaces: G (or P), and one of the gear/pinion faces. Under such an assumption, the edge can be considered as a line.

Lines of intersection of the tooth flanks, G and P, by the plane of action, PA, form an angle, *θ*.

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 5a. In this way, the unit normal vectors, n_{g} and n_{p}, form an angle of 180°.

When the gears are misaligned, as shown in Figure 5b, the unit normal vectors, n_{g} and n_{p}, form an angle that equals (180° − *θ)*. This schematic (see Figure 5b) is a 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 accuracy of the angle *θ* is specified.

Even a small axes misalignment in intersected-axes gearing (of about three angular minutes or so) results in a severe contact pattern change. The contact area should cover the entire flank surface (without edge contact concentration) if the nominal load rating is reached. An offset error of ±50 micrometers leads to a contact pattern change of about the same magnitude.

Under any and all circumstances, edge contact between the gear and 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, tolerances for the accuracy of the design parameters — the axes misalignment depends on — have to be set very tightly.

With only the gear mesh considered, only the unit normal vectors, n_{g} and n_{p}, are taken into account, while the unit is normally perpendicular to the gear, and the pinion faces are ignored.

In this analysis, no edge roundness, chamfers, etc., are considered.

It should be mentioned here that a proper shape along with a desired location and orientation of the contact pattern in a gear pair is the No. 1 priority from the standpoint of the actual value of the mounting distance.

### 4 Accuracy of the Mounting Distance in Intersected-Axes Gearing

In intersected-axes gearing, interaction of the gear and mating pinion tooth flanks, G and P, takes place within the plane of action, PA. Therefore, it makes sense to consider the 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.

#### 4.1 Disposition of base cone of a gear in relation of the plane of action

In a geometrically-accurate intersected-axes gear pair (that is, in a perfect intersected-axes gear pair) the axis of rotation of a gear, O_{g}, and that, O_{pa}, of the plane of action, PA, intersect as illustrated in Figure 6a. 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 6b. In such a scenario, the gear base cone apex, A*_{g}, does not coincide with the plane-of-action apex, A_{pa}, as A*_{g} is displaced axially in relation to A_{pa} at a distance *ε*_{g}. (Here, we are not going into details of the analysis of the gap between the gear base cone and the plane of action). In a case depicted 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).

The performed analysis in Figure 6 allows us to proceed with the derivation of an equation for the calculation of the tolerance for the accuracy of the mounting distance in an intersected-axes gear pair.

#### 4.2 Derivation of the equation for the calculation of tolerance for the accuracy of the mounting distance

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).

A case of an 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 from the very beginning 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, Δ A_{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 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 axial displacement, *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 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 axial displacement, *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 are both 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 the angle that 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, G.

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 ([*ε*_{g}] or [*ε*_{p}]) or a ratio of the tolerances, [*ε*_{g}]/[*ε*_{p}], has to be pre-specified.

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 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}, 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 detailed analysis is not presented here as the equations are overly complicated.

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 displacement *ε*_{g} (or *ε*_{p}) is of a negative value, this results in the edge contact occurring at the opposite face of the gear.

### 5. Permissible Alteration to Bevel Gear Flank Surface Geometry

As a gear and a mating pinion tooth only interact with one another within the plane of action, PA, the geometry of the tooth flanks, G and P, allows for a slight modification (a few examples are illustrated in Figure 8) aimed at avoiding edge contact in gearing. The modification is allowed only for lines of intersection of the tooth flanks, G and P, by the plane of action. Under such a scenario, only the angular base pitch, *ϕ*_{b.g}, of a gear equals to the operating base pitch, *ϕ*_{b.op}, of the gear pair (that is, the equality *ϕ*_{b.g} ≡ *ϕ*_{b.op} is valid); and similarly, the angular base pitch, *ϕ*_{b.p}, of a mating pinion equals to the operating base pitch, *ϕ*_{b.op}, of the gear pair (that is, the equality *ϕ*_{b.p} ≡ *ϕ*_{b.op} is valid). No violation of the equalities *ϕ*_{b.g} ≡ *ϕ*_{b.op} and *ϕ*_{b.p} ≡ *ϕ*_{b.op} is permissible in precision intersected-axes gearing.

### Conclusion

A favorable configuration of bevel gears in intersected-axes gearing is discussed in this article. It is stressed that the mounting distance is one of the most important design parameters. In order to have intersected-axes gearing engineered properly, the tolerance for the accuracy of the mounting distance (and the tolerance for the accuracy of the axial displacements of the gears) has to be tight and needs to be calculated. Calculation of the tolerance for the accuracy of the mounting distance in intersected-axes gearing is a challenging problem. No equations for such calculations are in the public domain.

The disclosed approach for the calculation of the tolerance for the accuracy of the mounting distance in intersected-axes gearing is focused on geometrically-accurate bevel gears (that is, on perfect bevel gears). However, it is also valid for the cases of the approximate gear cut in a regular way on gear generators and so forth.

Bevel gears for intersected-axes gearing have to be designed so (see [2]) as to eliminate the necessity of shimming-in/shimming-out when assembling. These gears do not need to be lapped when finishing the tooth flanks. The gears do not need to be paired, as they are interchangeable and can be replaced individually (not as a whole gear set). If designed, machined [4], and assembled properly, no severe contact pattern changes are observed.

Accuracy of the mounting distance has to be inspected on the correct preload on the pinion shaft and gear carrier bearings. The gearbox housing accuracy and stiffness must be assured accordingly. No testing is a must to verify the accuracy of the mounting distance if the gears and the housing are designed, machined, and assembled properly [2].

^{1}^{ }Edges are considered here as lines of intersection of the tooth flanks of a gear, G, and one of two gear faces.

### References

- 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.
- Radzevich, S.P., Theory of Gearing: Kinematics, Geometry, and Synthesis, 2nd Edition, revised and expanded, CRC Press, Boca Raton, FL, 2018, 934 pages.
- khkgears.net/new/gear_knowledge/gear_technical_reference/gear_backlash.html.
- Radzevich, S.P., Gear Cutting Tools: Science and Engineering, 2nd Edition, CRC Press, Boca Raton, FL, 2017, 606 pages.