Angular velocity ratio: A novel approach for the determination of its instant value

This article deals with the angular velocity ratio in gearing. In the introduction section, it is stated the following discussion pertains to gearing of any and all designs, namely: parallel-axes gear pairs, intersected-axes gear pairs, and crossed-axes gear pairs. Attention should focus on parallel-axes gearing, as this kind of gearing can serve as an example of application of the proposed solution to the problem under consideration. In the following sections, the discussion begins with geometrically-accurate parallel-axes gearing, followed by the analysis of approximate parallel-axes gearing. The root causes for the variation of the angular velocity ratio in gearing are outlined, and the impact of the gear-tooth profile deviation on the angular velocity ratio is discussed in more detail. A novel measure of variation of the angular velocity ratio in gearing is proposed.

Introduction

Transmission of a rotation from an input shaft to an output shaft is the main purpose of gearing of all designs. In practice, a steady rotation of the input shaft is required to be transformed to a smooth rotation of the output shaft. The application of geometrically-accurate gears (or, perfect gears, in other terminology) fulfills this requirement, and under steady rotation of the input shaft, the output shaft is rotated steadily. This is true with respect to gear pairs of all designs: (a)  parallel-axes gear pairs (P_a — gearing), (b)  intersected-axes gear pairs (I_a — gearing), as well as (c)  crossed-axes gear pairs (C_a — gearing).

For simplicity, but without loss of generality, this consideration is limited to parallel-axes gearing —  P_a — gearing serves here as an illustrative example. In geometrically-accurate gearing, the actual value of the angular velocity ratio is of a constant value, and it equals to the gear ratio in the gear pair.

In contrast to geometrically-accurate gearing, real gearing always features: (a) tooth profile deviations, (b) linear and angular displacements of the axes of rotation of a gear and of a mating pinion from their nominal configuration (manufacturing errors, etc.), (c) elastic deformation of the shafts (as well as of other components) under the operating load, an increased temperature of the working environment, etc. Clauses (a-c) cause deviations of a current value of the angular velocity ratio from its nominal value.

Except that of the tooth profile deviations, the impact of the rest of the mentioned root causes of deviation of the angular velocity ratio are not discussed. The outline in the illustrative example is limited to consideration only of the impact of the tooth profile deviations on the variation of the angular velocity ratio at every instant of time.

1  Angular velocity ratio as a function of the tooth profile deviations

By definition, the angular velocity ratio, u_ϕ, equals to the ratio of an angle of rotation of the input shaft, ϕ_in, to the corresponding angle of rotation of the output shaft, ϕ_out [2]:

In geometrically-accurate parallel-axes gearing, the angular velocity ratio, u_ϕ, is of a constant value (u_ϕ = const). It is equal to the ratio of an angular velocity of the input shaft, ω_in, to the angular velocity of the output shaft, ω_out (or that is equal to ratio of tooth count of the output gear to tooth count of the input gear) [2]:

Here is designated:

N_out — is the tooth count of the output gear.

N_in —  is the tooth count of the input gear.

In real gearing, rotation of the input shaft, ω_in, is of a constant value (ω_in = const), while rotation of the output shaft, ω_out, depends on the actual values of tooth profile deviations, and, thus, it is a function of the angle of rotation of the input shaft, ω_in:

Because the rotation of the output shaft is not of a constant value (ω_out ¹ cont), the angular velocity ratio in the gear pair varies (u_ϕ = var). The variation of the angular velocity ratio, u_ϕ = var, is the root cause of an excessive noise emission, and vibration generation. Thus, it is vital to gear engineers to reduce noise excitation in the gear mesh to the lowest possible level.

To resolve this challenging problem (that is, to avoid any variation of the angular velocity ratio, u_ϕ = var, the root causes for u_ϕ = var, the impact of the tooth profile deviations onto the actual value of the angular velocity ratio, u_ϕ, is required in in-depth investigation.

2 Reciprocation of pitch point along the center-line in a gear pair

The consideration on reciprocation of the pitch point along the center-line in a gear pair is convenient to begin from the analysis of external parallel-axes geometrically-accurate gearing, also called “involute gearing.”

Referring to Figure 1, consider the schematic of an external parallel-axes geometrically-accurate gearing with transverse pressure angle, φ_t, of an arbitrary value. The gear and the pinion rotate about their axis of rotation, O_g and O_p, with angular velocities, ω_g and ω_p, correspondingly. The axes of rotations, O_p and O_g, are positioned apart from one another at a center-distance, C. The pitch point, P, is situated within the centerline, CL. The ratio C_g/C_p of the center-distances, C_g and C_p, equals to the angular velocity ratio, u_ϕ = u (the equality  C_g/C_p = u = ω_p/ω_g for the ratio C_g/C_p is observed) [here and below, is designated: C_g is the “gear center-distance,” and  C_p is the “pinion center-distance” [2]; and C_g + C_p = C; C is the center distance in the gear pair]. In geometrically-accurate parallel-axes gearing, the line of action, LA (and the path of contact, P_c), forms a transverse pressure angle, φ₁, with the perpendicular to the centerline, CL, at the pitch point, P. It is worth stressing here that the line of action, LA, and the path of contact, P_c, are two different lines, both through the pitch point, P. In a reduced case of parallel-axes gearing, these two lines, LA and P_c, align to one another (or, in other words, they are congruent to each other). The latter is not observed in more general cases of intersected-axes gearing, as well as in crosses-axes gearing.

When the gears rotate, the contact point, K, between the tooth flanks of the gear, G, and of the pinion, P, travels along the path of contact, P_c (though points a, b, c, d), with a constant linear velocity, V_K, similar to that, as in a corresponding equivalent pulley-and-belt transmission [2]. (Operating mode of a gear pair is commonly specified in terms of the linear velocity, V_K.pc, of the gear’s rotation on the pitch circle. It is common practice to specify the linear velocity, V_K.pc, of the gear rotation on the pitch circle as a product of the gear rotation, ω_g (or ω_p), times the pitch radius, r_g (or r_p) of the gear, that is, V_K.pc = ω_g 〈 r_g = ω_p. Instead of this, it makes sense to specify the linear velocity, V_K.bc, of the gear rotation on the base circle (the linear velocity of the plane of action) as a product of the rotation, ω_g (or ω_p), times the base radius, r_b.g (or r_b.p), that is, V_K.bc = ω_g 〈 r_b.g = ω_p 〈 r_b.p (in this scenario, the magnitudes of the linear velocities correlate to one another as V_K.bc = V_K.pc 〈 cosφ₁). After such an alteration is completed, the actual value of the linear velocity, V_K.bc, does not depend on the transverse pressure angle, φ₁, that is more practical.)

No reciprocation of the pitch point, P, along the center-line, CL, is observed in external parallel-axes geometrically-accurate gearing.

The pitch point, P, is motionless because of:

• The line of action, LA, and the path of contact, P_c, are two straight lines.
• The lines, LA and P_c, align to one another.
• Both the lines are the lines through the pitch point, P.

Once the key features of the transmission of a steady rotary motion by means of geometrically-accurate parallel-axes gear pair are understood, gearing with inaccurate tooth profile (approximate gearing, in other terminology) can be discussed.

Approximate parallel-axes gear pairs with tooth profile deviation always feature a path of contact, P_c, in the form of a planar curve.

Consider a schematic diagram of parallel-axes gearing that features a curved path of contact, P_c, as shown in Figure 2. The gear, and the mating pinion rotate about their axes of rotation, O_p and O_g, with angular velocities, ω_p and ω_g, correspondingly. The axes, O_g and O_p, are positioned at certain center-distance, C, apart from one another. The pitch point, P, is at a distance, r_g (here r_g = C_g), apart from the gear axis of rotation, O_g, and at a distance, r_p (here r_p = C_p), apart from the pinion axis of rotation, O_p. (The design parameters, C_g and C_p, are commonly referred to as gear center-distance, C_g, and pinion center-distance, C_p, correspondingly). The transverse pressure angle at the pitch point, P, is denoted by φ₁.

Let us assume that the gear and its mating pinion rotate so that a curved path of contact, P_c, is traced by the contact point, K. At every point of the path of contact, P_c, a corresponding instantaneous line of action, LA_inst, can be constructed.

An arbitrary contact point within the path of contact, P_c, is labeled as Kⁱ. When the gear and the mating pinion make contact at point Kⁱ, the instantaneous line of action, LAⁱ_inst, is in tangency to the path of contact, P_c, at Kⁱ. This is because the force of interaction between the tooth flanks of the driving and the driven gears is pointed along the common perpendicular to the interacting tooth profiles considered in a transverse section of a gear pair, or, the same, the force is pointed along a straight line that is tangent at contact point Kⁱ to the path of contact, P_c. [Important: Friction between the gear and the mating pinion tooth flanks, G and P, is not taken into account in this analysis.]

The line of action, LA, and the path of contact, P_c, align with one another only in geometrically-accurate parallel-axes gearing. (In the rest of cases of  P_a — gearing (cycloid gear pair, and so forth), as well as in all cases of  I_a — gearing, and of C_a — gearing, (a) the line of action, LA, does not exist (only instantaneous lines of action, LA_inst, exist), and (b) the instantaneous lines of action, LA_inst, and the path of contact, P_c, never align to one another.)

As the instantaneous line of action, LAⁱ_inst, intersects the centerline, CL, at the instantaneous pitch point, P⁽ⁱ⁾, the instantaneous angular velocity ratio, u⁽ⁱ⁾_inst, at the corresponding instant of time, can be expressed in terms of the instantaneous values of the center-distances, Cⁱ_g and Cⁱ_p, of the gear and the mating pinion:

Because of in-approximate gearing, the path of contact, P_c, is not a straight line, then the instantaneous pitch point, Pⁱ, does not coincide with the nominal pitch point, P. Therefore, the instantaneous angular velocity ratio, uⁱ_inst, differs from the nominal gear ratio:

A similar analysis can be performed with respect to other contact points, K^i-1 and Kⁱ⁺¹, within the path of contact, P_c, in the gear pair. Without going into details of the analysis, it can be concluded that in all contact points under consideration, K^i-1, Kⁱ, and Kⁱ⁺¹, the following inequality/equality is valid:

where:

It is clear from the earlier-performed analysis that no smooth transmission of a uniform input rotary motion is possible by means of approximate parallel-axes gearing that features a curved path of contact, P_c.

Only geometrically-accurate parallel-axes gearing (involute gearing) features a straight line of action, LA, and a straight path of contact, P_c, that align to one another. In other gear systems, (a) the line of action, LA, does not exist, (b) instead, a plurality of instantaneous lines of action, LA_inst (that are still straight lines) exist, and (c) the instantaneous lines of action, LA_inst, and the path of contact, P_c (that could be a planar curve) mandatorily need to be distinguished from one another.

When approximate gears rotate (Figure 2), no transmission of steady input rotary motion is possible from an input shaft to an output shaft [2].

In approximate external parallel-axes gearing (namely, in gearing with tooth profile deviations) reciprocation of the pitch point, P, up-and-down along the center-line, CL, is always observed.

The reciprocation of the pitch point, P, is due to:

• There is no line of action, LA, in approximate parallel-axes gearing; only instantaneous lines of action, LA_inst, can be constructed.
• The instant line of action, LA_inst, is a straight line through the instantaneous pitch point, Pⁱ, while the path of contact, P_c, is a planar curve through point within the center-line, CL.
• The lines, LA_inst and P_c, never align to one another, as one of them is a straight line, and the other one is a planar curve.

Once the key features of the transmission of a steady rotary motion by means of approximate parallel-axes gear pair are understood, one can proceed with an analysis and an evaluation of variation of the angular velocity ratio in approximate parallel-axes gearing.

3 Evaluation of variation of angular velocity ratio

When a gear and a mating pinion tooth profile deviate (either both of them, or, at least, one of them) from the desirable involute tooth profile, a corresponding variation of the angular velocity ratio in the gear pair becomes inevitable. Regardless of whether the variation of the angular velocity ratio falls into the corresponding tolerance band for the deviation of this sort, it is still important to the gear designer, as this variation inevitably entails an increased noise excitation and vibration generation in the gear pair.

While under a steady rotation of the driving shaft, the actual instant value of the angular velocity of the driven shaft varies, it is natural to evaluate the variation of the angular velocity ratio, Δω^max_out, by means of the difference between the maximum, ω^max_out, and the minimum, ω^min_out, values of the angular velocity of the driven shaft:

The measure, Δω^max_out, of variation of the angular velocity ratio is inconvenient to apply as it does not allow to compare the actual values of this design parameter in gear pairs of different design: gear pairs with a different tooth count, a different module/diametral pitch, etc.

In order to derive an adequate criterion for the evaluation of variation of the angular velocity ratio in a gear pair, consider a schematic of approximate parallel-axes gearing, shown in Figure 3.

In Figure 3, the nominal location of the pitch point is designated P^nom. The area of migration of the pitch point in an approximate parallel-axes gear pair is terminated by the extremal locations of the pitch point. These locations are designated by P_g^min and P_p^min, correspondingly.

The actual value of the angular velocity ratio, u_ϕ(ϕ_in), at a specified instant of time (at a given value of the rotation angle ϕ_in) equals:

As it follows from the analysis of Figure 3, the angular velocity ratio reaches its maximal value, u_ϕ^max, when the pitch point, P, coincides with the point, P_p^min:

Correspondingly, the angular velocity ratio reaches its minimal value, u_ϕ^min, when the pitch point, P, coincides with the point, P_g^min:

An actual value, uⁱ_ϕ, of the angular velocity ratio in the gear pair always falls into the interval:

Thus, instead of the design parameter, Δω_out, another parameter can be used for the evaluation of variation of the angular velocity ratio. Namely, a variation of the angular velocity ratio, uϕ, can be evaluated by means of the factor, k_uϕ:

The use of the proposed measure, k_uϕ, makes possible the evaluation of the range of variation of the angular velocity ratio, u_ϕ. It is also applicable for the comparison of gear pairs of different design: gear pairs having different tooth count, different module/diametral pitch, etc.

Here, in Equation 14, the parameter u_j^nom equals to the nominal angular velocity ratio, and is equal to: u_ϕ^nom = C_g/C_p. The design parameters, C_p and C_g, in the latter equation are shown in Figure 3.

The earlier listed formulas are simple in applications. The use of these formulas returns reliable results of the calculation.

The schematic shown in Figure 3 also reveals the proposed approach is capable of evaluating separately the contribution of the gear and  the mating pinion to the total variation of the angular velocity ratio.

For example, the contribution of an approximate gear, u_ϕ^g, to the total variation of the angular velocity ratio, Δu_ϕ, equals:

Similarly, the contribution of an approximate pinion, u_ϕ^p, to the total variation of the angular velocity ratio, Δu_ϕ, equals:

Evaluation of separate contribution of the gear, and of the mating pinion, to the total variation of the angular velocity ratio is important as it makes clear the accuracy of what of the two components (either of the gear, or of the pinion) needs to be improved.

All the design parameters of an approximate parallel-axes gear pair required to understand the variable angular velocity ratio (as well as emerging problems) are shown in Figure 4. The actual values of the base diameters, d_b.g and d_b.p, can also be functions of the angle ϕ_in: d_b.g = d_b.g(ϕ_in) and d_b.p(ϕ_in).

A possibility of accounting separately of contribution of the gear and of the mating pinion to the variation of the angular velocity ratio is one of the key advantages of the discussed approach. For example, a few different situations can be distinguished:

1: Precision gear meshes with an approximate pinion or, vice versa, approximate gear meshes with a precision pinion — the known value of the total variation of the angular velocity ratio does not give an answer to what two components are required to be improved.

2: Approximate gear meshes with an approximate pinion — the known value of the total variation of the angular velocity ratio — does not give an answer to what proportion of the two components are required to be improved.

A list of such questions can be extended.

The proposed approach provides an exact answer to which one of two components (either the gear or the pinion) is required to be improved and how much. No other approach is capable of answering this.

4 Correlation with transmission error in gear pair

Transmission error (TE) is the last problem to be considered.

The transmission error quantifies the gearbox’s imperfections when transferring energy from input to output in a metric of the gearbox’s efficiency. The higher the transmission error, the higher the risk of an amplified dynamic variation of the shaft’s rotational speed or torque.

Traditionally, transmission error (TE) is considered to be the main excitation mechanism of gear noise. The definition of TE is “the difference between the actual position of the output gear and the position it would occupy if the gear drive were perfect.”

Transmission error is the indicator of numerous effects, and is defined in the following way [1]:

Again, when transmission error is known, it is not clear which of the two components is required to be improved in order to reduce the transmission error. With the proposed approach, an exact answer to which one of the two components (either the gear or the pinion) is required to be improved and how much. No other approach can answer this.

5 On ‘history’ of variation of instant angular velocity ratio during single cycle of tooth meshing

During one cycle of meshing of a pair of gear teeth, the actual value of the angular velocity ratio, uⁱ_ϕ, in approximate gear pair varies reaches it maximum, u_ϕ^max, and minimum, u_ϕ^min, values (see Figure 5).

The “history” of variation of instant angular velocity ratio during the single cycle of gear tooth meshing depends on the actual geometry of the gear tooth profile, G, and the mating pinion tooth profile, P. The discussed approach is capable of correctly interpreting the “history” of variation of the angular velocity ratio and to investigate the influence of variation of u_ϕⁱ onto the level of noise emission in gearing.

The extremal values of the angular velocities, ω^max_out and ω^min_out, are calculated from:

The corresponding angles of rotations, ϕ^max_out and ϕ^min_out, are equal to:

Transmission error:

It should be stressed that the transmission error, TE, is not just a number. This is a function of ϕ_in. Therefore, a “history” in between  ϕ^max_out and ϕ^min_out can be accounted. An example of such a “history” is reflected in Figure 5.

Conclusion

The article deals with the angular velocity ratio in gearing.

The angular velocity ratio and the gear ratio in parallel-axes gearing in both geometrically-accurate gearing (perfect gearing) and approximate gearing are discussed.

A novel measure of variation of the angular velocity ratio in gearing is proposed. It is shown how the reported results can evolve to be applied in the calculation of transmission error in the gear pair.

The discussed results are helpful for the evaluation of the contribution to the resultant angular velocity variation of the gear and of the mating pinion separately from one another. This makes possible appropriate corrections to tooth profiles of the mating gears that were not possible earlier. 

References

1. Harris, S.L., “Dynamic Loads on the Teeth of Spur Gears,” In: Proceedings of the Institute of Mechanical Engineers, Vol. 172, 1958, pages 87-101, California, USA, Winter 1958.
2. 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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