Inspection of the accuracy of gear tooth flanks: An alternative approach

This article deals with inspection of accuracy of functional portions of gear tooth flanks. The functional portions of the tooth flanks are those that interact with one another when the gears rotate. Deviations of the actual gear tooth flank, G^act, from the desirable tooth flank, G, in geometrically-accurate gears are considered below only within the functional portion of the tooth flanks of mating gears. It is shown in the article that in geometrically-accurate gearing, (a) the angular base pitch of the gear, ϕ_b.g, must be equal to the operating angular base pitch, ϕ_b.op, of the gear pair (i.e., the equality ϕ_b.g = ϕ_b.op must be observed), and (b) the angular base pitch of the mating pinion, ϕ_b.p, must be equal to the operating angular base pitch, ϕ_b.op, of the gear pair (i.e., the equality ϕ_b.p = ϕ_b.op must be observed). (These two equalities are equivalent to the following: ϕ_b.g = ϕ_b.p = ϕ_b.op.) Gears of all designs, namely, gears for parallel-axes gearing (or just  P_a — gearing, for simplicity), for intersected-axes gearing (or just I_a — gearing, for simplicity), as well as for crossed-axes gearing (or just  C_a — gearing, for simplicity), are briefly covered in this article.

1 Inspection of Accuracy of Gear Tooth Flanks: State-of-the-Art

Gear accuracy is an important consideration in designing, in production, and in application of gears and gear pairs, in design of various mechanisms and machines. The performance of gearboxes, the gear-noise emission, and vibration generation, the dynamics of gearing, and so forth, are strongly correlated with gear accuracy, including, but not limited to, the accuracy of functional portions of tooth flank of the gear.

Numerous design parameters are inspected in production of precision gears. The transverse gear tooth profile angle, φ_t, the tooth pitch helix angle, ψ_g, and the gear circular pitch, p, are among these design parameters. Each of them (namely: φ_t, ψ_g, and p) is inspected to a high level of accuracy in order to ensure the actual values of these gear-design parameters are within corresponding tight tolerance bands. In this article, attention is focused on the accuracy mainly of the just-listed gear design parameters.

Two involute gears that feature equal base pitches (base pitch of the gear, p_b.g, and that of the mating pinion, p_b.p, must be equal to each other) can be engaged in proper mesh with one another. Therefore, instead of inspecting three design parameters (φ_t, ψ_g, and p) in a gear, it is sufficient to inspect the equality of the base pitches of mating gears (p_b.g = p_b.p) [1]. Inspection of a single design parameter (p_b.g) is preferred rather than several design parameters (φ_t, ψ_g, and p).

This statement only covers geometrically-accurate involute gears. It is not true when the gears are displaced in relation to one another. Moreover, this statement is not valid with respect to intersected-axes gearing (or just I_g — gearing, for simplicity), nor with respect of crossed-axes gearing (or just C_g — gearing, for simplicity). Therefore, equality of base pitches (p_b.g = p_b.p) cannot be used when inspecting  I_g — gearing, nor C_g — gearing. (In these two later cases, angular base pitches, ϕ_b.g and ϕ_b.p, of a gear and of the mating pinion are compared with the angular base pitch of the gear pair, j_b.op.) See Figure 1.

Contact pattern is extensively used to control the quality of bevel ears for I_g — gear pairs, as well as for C_g — gear pairs. Unfortunately, this latter approach is poorly engineered, first of all, because the use of contact pattern returns more qualitative rather than quantitative results of the analysis of the gear accuracy. When the accuracy of bevel gears either for I_g — gear pairs, or for C_g — gear pairs, is inspected by means of contact pattern, application of a trial-and-error method becomes inevitable. Having the contact pattern of a particular geometry, dimensions, and orientation within the gear tooth flank, it is not evident as to how the gear tooth flank has to be corrected in order to get an appropriate contact pattern. Only skilled and experienced gear personnel can do this job successfully.

On top of that, only the accuracy of paired bevel gears can be inspected this way.

The inspection should be replaced by means of contact pattern with a more properly engineered method of gear inspection.

2 Root causes for base pitch deviation in a gear pair

The uniform transmission of rotation from the input shaft to the output shaft is possible only by means of geometrically-accurate gears. In geometrically-accurate gears of all designs (i.e., in P_a —, in I_g —, and in C_g — gearing), the base pitch of a gear and that of a mating pinion are exactly equal to the operating base pitch of the gear pair. It is important to stress here that the base pitch in a gear, p_b.g (and, p_b.p, in a mating pinion) can be specified only in cases when the operating base pitch in the gear pair, p_b.op, is known. When the operating base pitch in the gear pair, p_b.op, for some reason alters the design parameter, p_b.g (and the design parameter, p_b.p) is no longer equal to the base pitch in a gear (and in the mating pinion), as the gear/pinion gets an approximate one.

No set of gears can ever be made to absolute precision, and each of the individual gear teeth will have geometrical variations. In reality, in gears of all designs (i.e., in P_a — gearing, in I_g — gearing, as well as in C_g — gearing), the base pitch of a gear and that of a mating pinion deviates from the operating base pitch of the gear pair. Because of this, uniform transmission of rotation from the input shaft to the output shaft becomes impossible: The equality of the base pitches of a gear and of a mating pinion to the operating base pitch in the gear pair is of critical importance to guarantee the high performance of gears.

Two main root causes for base pitch variation in a gear pair are considered in the following analysis:

Manufacturing errors (when machining/finishing the gear and the mating pinion tooth flanks) are the first root cause for base pitch variation. When the actual gear tooth flank, G^act, deviates from the desirable tooth flank geometry, G, then a violation of the condition of equal base pitches is inevitable.

Linear and angular displacements of the gear and the mating pinion in relation to one another under the influence of various factors are the second root cause for base-pitch variation.

In reality, the accuracy of gear tooth flanks is inspected within functional portions of the surfaces G and P of a gear and of a mating pinion.

As illustrated in Figure 2, the functional portion of tooth flanks, G, in a gear, is bound by (a) the start-of-active profile circle (or just  SAP — circle, for simplicity), (b) the end-of-active profile circle (or just EAP — circle, for simplicity), and (c) the functional face width, F_pa, in the gear pair (see Figure 3).

The construction of the design parameter Z_pa is illustrated in Figure 2.

The following designations are adopted in Figure 2 and Figure 3:

O_g and O_p are the axes of rotation of the gear and of the mating pinion, correspondingly.

CL is the center line.

C is the center distance.

P is the pitch point.

φ_t is the transverse pressure angle.

r_b.g and r_b.p are the base radii of the gear and of the mating pinion, correspondingly.

LA is the line of action in the gear pair.

r_g and r_p are the pitch radii of the gear and of the mating pinion, correspondingly.

r_sap.g is the start-of-active-profile radius of the gear.

r_sap.p is the start-of-active-profile radius of the gear.

r_eap.g is the end-of-active-profile radius of the gear.

r_eap.p is the end-of-active-profile radius of the gear.

p_b.op is the operating base pitch in the gear pair.

ρ_g is the radius of curvature of the gear tooth profile at the r_eap.g circle of the gear (at the point G_eap within the circle of the radius r_eap.g).

ρ_p is the radius of curvature of the pinion tooth profile at the r_eap.p circle of the gear (at the point P_eap within the circle of the radius r_eap.p).

ω_g and ω_p are the rotations of the gear and of the mating pinion, correspondingly.

F_g and F_p are the face widths of the gear and of the mating pinion, correspondingly.

Z_pa is the length of the plane of action, PA.

F_pa is the face width of the plane of action, PA.

In geometrically-accurate parallel-axes gearing, the mating gears are rotated about the axes O_g and O_p with angular velocities, ω_g and ω_p, correspondingly. The angular velocities, ω_g and ω_p, are synchronized with one another to meet the ratio u = ω_p/ω_g (in reduction gears). The axes of rotation, O_g and O_p, are situated at a center distance, C, apart from each other. The center distance, C, is divided by the pitch point, P, onto two straight line segments O_gP and O_pP. Relation of the lengths of the straight line segments, O_gP and O_pP, to one another is reciprocal to the ratio of the angular velocities, ω_g and ω_p:

Base circle radii, r_b.g and r_b.p, of the gear and of the mating pinion, correspondingly, relate to one another reciprocal to the ratio of the angular velocities, ω_g and ω_p:

The line of action, LA, is a straight line through the pitch point, P, that is externally tangent to the base circles of the radii, r_b.g and r_b.p. Points N_g and N_p are the points of tangency of the line of action, LA, with the base circles.

The line of action, LA, forms a transverse pressure angle, φ_t, with a perpendicular to the center-line, CL.

The functional portion of the line of action, LA, is terminated by points G_eap and P_eap.

The point G_eap is the point of intersection of the line of action, LA, by the gear circle of a radius r_eap.g (here r_eap.g is the radius of the end-of-active-profile of the gear), or (in other words), this is the point of intersection of the line of action, LA, by the pinion circle of a radius r_sap.p. (Here r_sap.p is the radius of the start-of-active-profile of the pinion.)

The point P_eap is the point of intersection of the line of action, LA, by the pinion circle of a radius r_eap.p. (Here r_eap.p is the radius of the end-of-active-profile of the pinion,) Or (in other words), this is the point of intersection of the line of action, LA, by the gear circle of a radius r_sap.g. (Here r_sap.g is the radius of the start-of-active-profile of the gear.)

The length of the functional portion of the line of action, LA, equals to Z_pa (see Figure 2).

The operating base pitch, p_b.op, in geometrically-accurate parallel-axes gearing is measured within section by a transverse plane (in the plane of drawing in Figure 2).

The construction of the design parameter F_pa is illustrated in Figure 3.

Deviations of the actual gear tooth flank, G^act, from the desirable tooth flank geometry, G, of a geometrically-accurate gear are considered below only within the functional portion of the gear tooth flank [1].

The schematic shown in Figure 2 pertains to ideal P_a — gearing with zero displacement of the gear and of the mating pinion in relation to one another. In reality, the gear and the mating pinion are displaced in relation to one another. The linear displacements (δ_x, δ_y, δ_z) of the axes of rotation, O_g and O_p, are commonly measured along the center-line, CL (δ_x), in the direction parallel to the axes of rotation, O_g and O_p (δ_y), and in the direction perpendicular to the center-line, CL (δ_z). The angular displacements θ_x, θ_y, θ_z  of the axes of rotation, O_g and O_p, are commonly measured about those same axes, correspondingly. (It is assumed here, that, due to small values of the deviations, δ_x, δ_y, δ_z, and θ_x, θ_y, θ_z, transverse pressure angle, φ_t, in geometrically-accurate gearing, and in real gearing, remains the same value.)

Involute gears are insensitive to the displacements, ±δ_x, measured along the center-line, CL. This particular statement is known from many sources that can be found out in the public domain. (It can be shown that gearing of this design is also insensitive to the linear displacements ±δ_y and ±δ_z.)

An incorrect conclusion is almost always drawn from that: It is loosely stated that real involute gears are insensitive to the variation of center-distance (C ± δ_x). This statement is incorrect.

When linear displacements, δ_x, δ_y, δ_z, are incorporated into the analysis of parallel-axes involute gearing, this gearing that remains is a kind of ideal gearing (that does not exist in reality). This is because no parallel-axes involute gearing can feature linear displacements (δ_x, δ_y, δ_z) and, at that same time, have zero angular displacements (θ_x, θ_y, θ_z). The angular displacements (θ_x, θ_y, θ_z) are inevitable in real gearing similar to the linear displacements (δ_x, δ_y, δ_z). Therefore, real parallel-axes gearing are sensitive to the linear displacements (δ_x, δ_y, δ_z) in that they are sensitive to the angular displacements (θ_x, θ_y, θ_z). This was never taken into consideration in any analyses carried out with respect to real parallel-axes involute gearing.

Only geometrically-accurate involute gears are covered by this statement. When the gears are displaced in relation to one another (the linear displacements δ_x, δ_y, δ_z, and the angular displacements, θ_x, θ_y, θ_z, of the gear axes O_g and O_p) this statement is not true. Moreover, this statement is not valid with respect to intersected-axes gearing nor with crossed-axes gearing.

In geometrically-accurate gears a tooth flank of a gear is intersected by the plane of action, PA, along a line. When the gears rotate, at certain times this line of intersection becomes a desirable line of contact, LC_des.

As no set of gears can ever be made to absolute precision, each of the individual gear teeth will have geometrical variations. (Reminder: The gear tooth flank, G, and the pinion tooth flank, P, both are generated by a common planar curve, LC. The planar curve, LC, is entirely situated within the plane of action, PA. When the gears operate, the planar curve, LC, is congruent with the line of contact between the interacting tooth flanks, G and P.)

In geometrically-inaccurate gears, a tooth flank of a gear is intersected by the plane of action, PA, along a line, LG_i (and LP_i in geometrically-inaccurate pinions). This line of intersection, LG_i, never coincides with a desirable line of contact, LC_des, when the gears rotate. The line of intersection, LG_i, always deviates from the desirable line of contact, LC_des. By nature, the distances between every two corresponding points (measured along the i — th path of contact, P_c⁽ⁱ⁾) of the lines LC_des and LG_i equals to the variation of the base pitch in the gear pair. The proposed alternative approach of inspection of accuracy of gear tooth flanks is based on the evaluation of deviation of the line LG_i from the desirable line of contact, LC_des.

As an example, consider the approach in a case of crossed-axes gear pairs.

3 Inspection of Gears for Crossed-Axes Gear Pairs

The methods of inspection gears for crossed-axes gear pairs used in practice today are poor and inconsistent. First of all, this is because an approximate datum surface is always used for inspection purposes. An inappropriate datum surface is the main reason for poor quality of inspection. The actually used 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 is especially evident when gears with a low tooth count are inspected — in this case, the inconsistency of known methods for inspecting gears is getting clearer. As the actual datum surface deviates from the desirable datum surface, the output of the gear inspection cannot be accurate.

3.1 Inspection of gear tooth flank geometry

When inspecting a gear for crossed-axes gear pair (see Figure 4), the relative motion of the gear to be inspected and the plane of action are performed. (The center-distance in Figure 4 is set to zero when bevel gears of intersected-axes gear pair are inspected.) The gear is rotated, ω_g, about its axis of rotation, O_g. The plane of action, PA, is rotated, ω_pa, about its axis of rotation, O_pa. The rotations, ω_g and ω_pa, are synchronized with one another.

Inspection of a gear tooth flank geometry in a gear for a crossed-axes gear pair is similar to that of a gear for an intersected-axes gear pair. When inspecting a gear tooth flank geometry, the rotations, ω_g and ω_pa, are synchronized so as to keep the stylus at the current point   on the desirable bevel gear tooth flank. The axes, O_g and O_pa, are at a certain distance, C_g, from one another when inspecting crossed-gears. The rotations, ω_g and ω_pa, are synchronized with one another so as to keep the stylus at a current point m on the desired bevel gear tooth flank.

Point, m, is within the plane of action, PA. The straight line, a, along which the deviation is measured, is within the plane of action, PA, and it is perpendicular to the gear tooth flank at point, m. The stylus tip is motionless in relation to the plane of action, PA, when the bevel gear tooth flank, G^act, has no deviations from the datum tooth flank, G. When inspecting real gears (with non-zero deviations of the tooth flank, G^act), the readings of the indicator of dial type are proportional to the actual deviations.

3.2 Inspection of gear in the lengthwise direction of gear teeth

When inspecting the tooth flank geometry of a gear in the lengthwise direction of the gear tooth, the stylus tip travels along the line of intersection of the desirable tooth flank by the plane of action, PA. At 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.

In the course of inspection of a gear for real crossed-axes gear pair, tooth flank of the gear, G, is intersected in the plane of action, PA, along a line of intersection, LG_i. In Figure 5, the line of intersection through an 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 of teeth, is 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, also can be calculated.

Point, m, is within the plane of action, PA. The straight line, a, along which the deviation, is measured and is within the plane of action, PA, and it is perpendicular to the gear tooth flank at point, m. The stylus tip is motionless in relation to the plane of action, PA, when the bevel gear tooth flank, G^act, has no deviations from the datum tooth flank, G. When inspecting real gears (with non-zero deviations of the tooth flank, G^act), the readings of the indicator of dial type are proportional to the actual deviations.

In a similar manner, the variation of the angular base pitch in a pinion can be inspected.

When the gear-center distance, C_g, (or the pinion-center distance, C_p), is set equal to zero, the disclosed approach for inspection of accuracy of gear tooth flanks for crossed-axes gear pairs (see Figure 4 and Figure 5) becomes applicable for inspection of the accuracy of tooth flanks in bevel gears for intersected-axes gear pairs.

Conclusion

This article deals with various methods of the inspection of accuracy of gear tooth flanks. A brief overview of known methods of the inspection is performed, and the key disadvantages of them are pointed out. The inspection of the accuracy of bevel gears for intersected-axes gearing (contact pattern; paired gears), as well as for crossed-axes gearing, cannot be engineered properly. This inspection method should be considered the first to be replaced with inspection of the angular base pitch in bevel gears.

In geometrically-accurate gearing, (a) the angular base pitch of the gear, j_b.g, must be equal to the operating angular base pitch, ϕ_b.op, of the gear pair (i.e., the equality ϕ_b.g = ϕ_b.op must be observed), and (b) the angular base pitch of the mating pinion, ϕ_b.p, must be equal to the operating angular base pitch, ϕ_b.op, of the gear pair (i.e., the equality ϕ_b.p = ϕ_b.op must be observed). These two equalities are equivalent to the following: ϕ_b.g = ϕ_b.p = ϕ_b.op.

An alternative approach for the inspection of the accuracy of gear tooth flanks is discussed in the article. The approach is based on the evaluation of the variation of the angular base pitch of the gear (and that of a mating pinion) from the operating angular base pitch in the gear pair. The smaller the variation of the base pitch of the gear, the smaller the deviation of the actual tooth flank of the gear, G^act, from its desirable geometry, G, and vice versa.

When the gear center-distance, C_g, (or the pinion center-distance, C_p), is set to zero, the disclosed approach for the inspection of accuracy of gear-tooth flanks for crossed-axes gear pairs becomes applicable for the inspection of accuracy of tooth flanks in bevel gears for intersected-axes gear pairs.

Gears/pinions having the smallest possible difference in the angular base pitch can be paired with one another, resulting in the reduction of poorly finished gears. 

References

Radzevich, S.P., Theory of Gearing: Kinematics, Geometry, and Synthesis, 3rd Edition, revised and expanded, CRC Press, Boca Raton, FL, 2022, 1208 pages.

Bibliography

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