This article deals with the accuracy of precision gears for gear pairs that operate on crossing axes of rotation of a gear and of a mating pinion. A few general comments are given regarding the features of gear-inspection procedure. In particular, the readers’ attention is stressed on the importance of the accurate gear datum surface for the purpose of gear inspection. Key elements of the kinematics of precision crossed-axes gearing are briefly outlined, and the principal reference planes in a gear pair are discussed. Generation of tooth flank of precision gear for crossed-axes gearing is disclosed. The right reference surface for inspection of precision gears for crossed-axes gearing is identified.

### Introduction

Gears are the machine elements of complex shape. To design and manufacture gears, gear drawings with a set of all necessary dimensions are commonly used. In the past, these were mainly hard (paper) copies of the gear images, which later on were replaced with their digital images.

Two groups of the gear design parameters are recognized. Interaction of the functional gear tooth surfaces depends on the actual values of the design parameters of the first group. The rest of the parameters of the performance of the gears depends on the actual values of the design parameters of the second group.

In this article, attention should be focused mainly on the gear design parameters of the first group. Smoothness of rotation of the output shaft strongly depends on the actual values of the design parameters of this group of parameters.

The methods of gear inspection for crossed-axes gear pairs used in the present-day practice are poor and inconsistent — almost all of them do not align to the modern theory of gearing. The approximate datum surface used for inspection purposes is the main reason for that. The datum surface is not conjugate to the gear being inspected, as it is generated as an envelope surface to a family of consecutive positions of the straight-sided crown rack in its motion in relation to the gear being inspected. This becomes especially evident when gears with a low tooth count are inspected — in this case the inconsistency of known methods of gear inspection is getting more evident. As the datum surface itself deviates from the desirable datum surface, the output of the gear inspection cannot be accurate.

When inspecting gears, the actual values of the gear design parameters are compared with the corresponding design parameters of the reference surface of the gear. The reference surface of the gear can be viewed as a virtual surface of a gear tooth that feature zero deviations from the ideal one.

True readings when inspecting gears can be obtained if and only if the reference surface of the gear is correct. No true readings can be obtained if the reference surface of the gear deviates from its desired geometry.

### Reference Surface of the Gear

Commonly, the reference surface of a gear to be inspected is generated by means of a virtual straight-sided crown rack, R. For this purpose, the gear vector diagram of the mesh “gear-to-crown rack” is employed. When rolling in relation to the reference system, X_{g}Y_{g}Z_{g}, associated with the gear to be inspected, the auxiliary crown-rack, R, occupies a plurality of consecutive positions. Envelope surface to a family of consecutive positions of the auxiliary crown-rack, R, is assumed to be the reference surface of the gear.

Several published works (see [1], [2] and others) disagree. The auxiliary crown-rack, R, and the reference surface of the gear are the two different surfaces of a completely different nature. They are not congruent, and, moreover, the deviations of the envelope surface from the desired reference surface are not known. Neither the actual value of the deviations nor the directions (inward or outward the bodily side) in which the deviations are measured are not known. Such a surface cannot be used as a reference surface when inspecting precision gears.

Without going into details, the following statement can be drawn up: No better alternatives are known for reference surfaces for the inspection of precision gears that operate on crossing axes of rotation. Instead of a strictly defined reference surface, another poorly defined surface is extensively used in computer software for modern CMMs and GMMs. This is because the straight-sided crown rack, R, is not conjugate to the reference surface of the gear.

Having no reference surface, it is impossible to make correct measurements, and all readings when inspecting precision gears for gear pairs that operate on crossing axes of rotation cannot be correct.

In order to proceed with the further analysis, the key elements of the kinematics of precision crossed-axes gearing are required to be concisely outlined.

### Key Elements of the Kinematics of Precision Crossed-Axes Gearing

Crossed-axes gear pairs are used to transmit a rotary motion from an input shaft to an output shaft, axes of which intersect. Commonly, the pinion is a driver, and the gear is an epy-driven component of the gear pair.

As it illustrated in Figure 1a, the gear is rotated, ω_{g}, about its axis of rotation, O_{g}, while the pinion is rotated, ω_{p}, about its axis of rotation, O_{p}. The axes of rotation, O_{g} and O_{p}, are at a center-distance, C, apart from one another. In other words, the distance between the gear base-cone-apex, A_{g}, and between the pinion base-cone-apex, A_{p}, equals C. The axes, O_{g} and O_{p}, form the shaft angle, Σ. More precisely, the shaft angle, Σ, is commonly specified as Σ = ∠(ω_{g},ω_{p}).

Here, in Figure 1a, an image of crossed-axes gear pairs is overlapped by a corresponding gear vector diagram. The gear vector diagram, constructed for this gear pair, is shown in Figure 1b.

The axis of instantaneous rotation, P_{ln}, is pointed along the vector of instantaneous rotation, ω_{pl}, where ω_{pl} = ω_{p} — ω_{g}.

Three rotation vectors: ω_{g}, ω_{p}, and w_{pl}, form the core of the gear-vector diagram. These rotation vectors are employed to specify four principal planes associated with the gear pair. They are (see Figure 2):

The pitch-line plane (or just P_{ln}—plane, for simplicity). The plane, is a plane through the vector of instantaneous rotation, ω_{pl}, and through the center-line, CL;

The center-line plane (or just C_{ln}—plane, for simplicity). The C_{ln}—plane, is a plane through the center-line, CL, perpendicular to the vector of instantaneous rotation, ω_{pl};

The normal-line plane (or just N_{ln}—plane, for simplicity). The N_{ln}—plane, is a plane through the plane-of-action-apex, A_{pa}, perpendicular to the center-line, CL;

The plane-of-action plane (or just PA—plane, for simplicity). The PA-—plane, is a plane through the vector of instantaneous rotation, ω_{pl}, that forms a transverse pressure angle, φ_{t.}_{ω}, with the N_{ln}—plane.

Usage of gear vector diagrams (see Figure 1) together with the set of principal planes associated with the gear pair (see Figure 2) significantly simplify the further discussion.

### Tooth Flank Geometry in Precision Gears for Gear Pairs that Operate on Crossing Axes of Rotation

The proposed method of inspection of precision gears for gear pairs that operate on crossing axes of rotation of the gears has many similarities with the method of generation of the gear tooth flanks in R—gearing. R—gearing is the only kind of geometrically-accurate crossed-axes gearing with line contact between the tooth flanks G and P. Therefore, it would be helpful to outline the key points of derivation of the gear tooth flank geometry in R—gearing.

Designing of gears for R—gearing, begins with the specification of a desirable line of contact, LC_{des}. As an example, let us assume that the desirable line of contact, LC_{des}, is shaped in the form of a circular arc, LC_{circ}, of certain radius R_{lc}, that is, LC_{des} ≡ LC_{circ}, as shown in Figure 3. By definition, the line of contact, LC_{circ}, is entirely situated within the plane of action, PA. Even though the desirable line of contact, LC_{des}, is already specified, no tooth flanks, G and P, of the gear, and of the mating pinion teeth are generated yet.

In crossed-axes gearing, the plane of action, PA, is tangent from the opposite sides to base cones of the gear, and that of the mating pinion. It is helpful to consider the sketch in Figure 3 together with the schematic of the set of principal planes and the gear vector diagram depicted in Figure 2. It is also recommended to include in this analysis the gear vector diagram along with the schematic of the gear pair shown in Figure 1.

Consider a Cartesian coordinate system, X_{g}Y_{g}Z_{g}, associated with the gear. Another Cartesian coordinate system, X_{p}Y_{p}Z_{p}, is associated with the mating pinion. The base-cone-apexes, A_{g} and A_{p}, are at distances C_{g} and C_{p} from the plane-of-action-apex, A_{pa}, correspondingly, as illustrated in Figure 4.

When the plane of action, PA, is rotated, ω_{pa}, about the plane-of-action-apex, A_{pa}, the Cartesian coordinate systems, X_{g}Y_{g}Z_{g} and X_{p}Y_{p}Z_{p}, are rotated, and the rotations, ω_{g} and ω_{p}, are properly timed with one another. In such a relative motion two family of the lines of contact, LC_{circ}, are generated. The first family of the lines LC_{circ} is generated in the reference system X_{g}Y_{g}Z_{g}. These lines are entirely situated within the gear tooth flank, G, and, therefore can be employed as reference for inspection the gear tooth flank, G. Another family of the lines LC_{circ} is generated in the reference system X_{p}Y_{p}Z_{p}. These lines are entirely situated within the pinion tooth flank, P, and, therefore can be employed as reference for inspection the pinion tooth flank, P.

### Kinematics of Inspection of Precision Gears for Crossed-Axes Gearing

When inspecting precision gears for crossed-axes gearing, stylus of the dial type indicator has to travel properly in relation to the gear to be inspected. For this purpose, the gear is rotated about its axis of rotation, O_{g}. The plane of action, PA, is rotated about its axis of rotation, O_{pa} (the axis of rotation, O_{pa}, is a straight line through the plane-of-action-apex, constructed perpendicular to the plane of action, PA). The rotations, ω_{g} and ω_{pa}, are synchronized with one another. In addition to the rotations, ω_{g} and ω_{pa}, of the gear, and of the plane of action, the stylus travels either along the gear tooth (in the lengthwise direction of the gear tooth), or across the tooth.

The concept of inspection of gear in lengthwise direction of gear tooth. In the course of inspection of a gear for real crossed-axes gear pair, intersection of the tooth flank, G, by the plane of action, PA, is considered. In Figure 5a, the line of intersection through arbitrary point within the tooth flank is labeled as LG_{i}. When the gear is rotated simultaneously with the plane of action, PA, the geometry of the line of intersection, LG_{i}, of the gear to be inspected alters. A plurality of the lines of intersection, LG_{i}, on the actual gear tooth flank, those generated during the meshing cycle of a pair, is situated within a bend of an angular width, δ, in the plane of action, PA. The angular deviation, δ_{i}, at a current configuration of the gear in relation to the plane of action, PA, can be physically measured in a gear/pinion. The current value of the angular deviation, δ_{i}, can be also calculated.

When inspecting tooth flank geometry of a gear in the lengthwise direction of the gear tooth (see Figure 5b), the stylus tip travels along the line of intersection of the desirable tooth flank by the plane of action, PA. At the current point, m, the line, a, along which the deviation is measured, is situated within the plane of action, PA, and it is perpendicular to the bevel gear tooth flank at this point, m.

The considered concept can be employed for the inspection of the variation of the angular base pitch in a gear.

The concept of inspection of the gear tooth profile. When inspecting a gear for intersected-axes gear pair (see Figure 6a), the gear is rotated about its axis of rotation, O_{g}. The plane of action, PA, is rotated about its axis of rotation, O_{pa}. When inspecting crossed-gears (see Figure 6b), the axes of rotation, O_{g} and O_{pa}, are at a certain distance, C_{g}, from one another.

The rotations, ω_{g} and ω_{pa}, are synchronized with one another so as to keep the stylus at current point, m, on the desirable gear tooth flank, G. Point, m, is situated within the plane of action, PA. The line a, along which the deviation is measured is entirely located within the plane of action, PA, and it is perpendicular to the gear tooth flank at the point, m. The stylus tip is motionless in relation to the plane of action, PA, when inspecting the tooth profile error of the gear tooth flank.

When inspecting the gear tooth flank, G, the reference surface in both cases is a kind of conjugate surface, namely, it can be conjugate to both: to the tooth flank of the crown rack, R (not straight-sided rack in this particular case), and to the tooth flank of a mating pinion, P.

The tooth flank of a precision gear for gear pairs that operate on crossing axes of rotation of a gear and of a mating pinion is schematically shown in Figure 7.

The net formed of stylus tip traces in the lengthwise direction of the gear tooth, and in its transverse section, is entirely situated on the reference surface.

**The concept of inspection of operating angular base pitch in crossed-axes gearing: **Operating base pitch, ϕ* _{b.op}*, is defined as the angular distance between points of intersection of two adjacent line of contact, LC

*and LC*

_{i}*, by a common circular arc that is centered at the plane-of-action apex, A*

_{i+1}*. This is illustrated in Figure 8.*

_{pa}The actual value of the operating base pitch, ϕ_{b.op}, does not depend on the value of the circular arc radius: r_{pa.1}, r_{pa.2}, and so forth. For a specified gear pair, the operating base pitch, ϕ_{b.op}, is of a constant value (j_{b.op} = const).

The angular base pitch in a gear, ϕ_{b.g}, must be equal to the operating base pitch, ϕ_{b.op}, in a gear pair (ϕ_{b.g} ≡ ϕ_{b.op}). Similarly, the angular base pitch in a mating pinion, ϕ_{b.p}, must be equal to the operating base pitch, ϕ_{b.op}, in a gear pair (ϕ_{b.p} ≡ ϕ_{b.op}).

Three angular base pitches, namely: ϕ_{b.g}, ϕ_{b.p}, and ϕ_{b.op}, are of fundamental importance for geometrically-accurate gearing.

No conjugate reference surface can be generated by means of a straight-sided crown rack, as recommended in [4], [5], [6], [7]. The surface generated this way deviates from the true references surface. This difference adds to the total error of the gear tooth flank inspection. This error is getting more substantial when gears with a low tooth count are inspected: the lower the tooth count, the greater the error, and vice versa [3].

### Conclusion

Gear inspection is an important part of the gear production process. Modern CMMs and GMMs are extensively used in the industry for the validation of the accuracy of machined gears. Computer software is a core of correct application of CMMs and GMMs. Such software must be developed based on the correct analytical description of the tooth flanks of a gear and a mating pinion and their interaction.

The results of the research in the theory of gearing obtained to this end enable one solving vital problems that pertain to gear cinematics and to gear geometry. In particular, this includes the development of computer software for gear inspection.

Commercial software somehow resembles a meat grinder: to get fresh ground meat (at the output from the meat grinder), fresh meat at the input is required — this is a must. There is no chance to get fresh ground meat if the meat to be ground is rotten. The meat grinder itself is not capable of improving the quality of the ground meat.

Something similar is observed when the design parameters of a gear pair are treated by computer software. The mathematical description of the gear pair (the mathematical foundation of the software) must be accurate, complete, and reliable; otherwise, no accurate output from the computer can be obtained, which is evident.

Certain means and possibility of validation of the software for CMMs and GMMs are a must.

Present day software for CMMs and GMMs offers no reliable means to validate the accuracy of the produced gears. Fortunately, this problem has a solution — CMMs can be effectively used to check the accuracy of the base pitch in the gear. This is the most reliable way to validate the accuracy of gears for C—gearing, as the angular base pitch of a gear is the only design parameter on which the performance of the gear pair depends. Such an approach is capable of determining separately a contribution of:

- The design errors caused by the inconsistencies inherited to the gear software itself.
- The errors caused in the gear machining processes.
- The errors caused by the distortion of the gear after heat treatment to the resultant deviation of the gear (pinion) tooth flank.

Inspection of the gear base pitch can be performed by a CMM’s operator of average skills and experience.

### References

- Radzevich, S.P., Theory of Gearing: Kinematics, Geometry, and Synthesis, 2nd Edition, revised and expanded, CRC Press, Boca Raton, FL, 2018, 934 pages. [1st edition: Radzevich, S.P., Theory of Gearing: Kinematics, Geometry, and Synthesis, CRC Press, Boca Raton, Florida, 2012, 743 pages.].
- 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. ISBN-13: ISBN-10: 0367649020, 978-0367649029
- 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. [This article can be requested from the author at no charge].
- Stadtfeld, H.J., Gleason Bevel Gear Technology: The Science of Gear Engineering and Modern Manufacturing Methods for Angular Transmissions, The GLEASON Works, Rochester, New York, 2014, 491p.
- Stadtfeld, H.J., Gleason Kegelradtechnologie: Ingenieurwissenschaftliche Grundlagen und modernste Herstellunsverfahren für Winkelgetribe, Renningen, Expert-Verlag, 2013, 491p.
- Stadtfeld, H.J., Practical Gear Engineering: Answers to Common Gear Manufacturing Questions, Gleason Corporation, Rochester NY, 2019, 395 pages.
- Stadtfeld, H.J., “Why are Today’s Hypoids the Perfect Crossed-Axes Gear Pairs?,” Gear Solutions magazine, May 2019, pages 42-50.