Envelope surfaces in the gear design and in the gear-machining processes

This article deals with envelope surfaces used in gear design and in gear-machining processes, those using the gear generating principle when rotations of two gears in mesh or of a gear and the gear cutting tool are timed with one another. The key features of this conventional approach of generation of envelope surfaces [developed in both, (i) in the differential geometry of surfaces, and (ii) in the kinematic geometry of surfaces] are briefly outlined. It is stressed here that the conventional approach of the generation of envelope surfaces is valid in cases when there is no correlation between the parameters of the relative rolling motion of the components and between the curvature of the interacting surfaces. It is shown that, in case of a certain correlation between the rolling motion and between the curvature of two interacting surfaces, a conventional approach of the generation of envelope surfaces is invalid. Thus, a corresponding approach for the determination of envelope surfaces is required.

This article looks at the core of the proposed method of the determination of envelope surfaces in gear design and in gear-machining processes. The problem of determining envelope surfaces in gear design and in gear-machining processes was correctly formulated and then solved by the author of this article in 2008 [2]. All three kinds of gearing [i.e.: (a) parallel-axes gearing ( P_a — gearing); (b) intersected-axes gearing ( I_a — gearing); and (c) crossed-axes gearing (C_a — gearing)] are covered by the derived solution to the problem. [(A) L. Euler (~1760) is credited with the determination of enveloping interacting tooth flanks in P_a — gearing; (B) the determination of the enveloping interacting tooth flanks in I_a -— gearing was started by G. Grant (1887) [6], and it was accomplished by Prof. S.P. Radzevich (~2008) [2]; (C) Prof. S.P. Radzevich (~2008) is credited with the determination of enveloping interacting tooth flanks in C_a — gearing [2].

Preamble

The problem of determining the envelope surface in gear design and in gear machining processes is not a new one — this problem has been around for decades. It is important to mention that, for a long time, this problem was intuitively realized by proficient gear experts, but, at the same time, it could not be correctly formulated. It was not clear where this problem originated. I remember when I was a university student (~1975), my professor mentioned (among others) about the consequences that this lack of understanding entailed for gear designers and for gear manufacturers. This problem was not correctly realized by the lead gear experts at that time (~1975).

Around 2008, the problem of determining envelope surfaces in gear design and in gear machining processes was correctly formulated and then solved by the author of this article [2]. All three kinds of gearing [i.e.: (a) parallel-axes gearing ( P_a — gearing); (b) intersected-axes gearing ( I_a — gearing); and (c) crossed-axes gearing (C_a — gearing)]  are covered by the derived solution to the problem.

1 Introduction

Consider as an example the interaction of involute tooth profiles of a gear and of a mating pinion in parallel-axes involute gearing.

Interaction of tooth flanks in parallel-axes involute gearing. A uniform rotary motion can be transmitted smoothly by means of the gear teeth of parallel-axes involute gearing. A schematic of parallel-axes gear pair associated with a corresponding equivalent pulley-and-belt transmission is shown in Figure 1). The belt stretched between the two pulleys corresponds to the line of action, a_gc_p, in Figure 1. A distance traveled by point, i, on the belt, corresponds with the one, described by the point of contact between the tooth flanks along the line of action. The angle formed by the perpendicular to the centerline, CL, with the line of action is the transverse pressure angle, φ_t.

In a case when the gear involute tooth profile, G (that starts at the point, b_g), is given, the pinion tooth profile, P (that starts at the point, b_p), can be generated as an envelope to a family of consecutive positions of the gear tooth profile, G, in its motion in relation to a reference system associated with the pinion, P, and vice versa: The gear tooth profile, G, can be generated as an envelope to a family of consecutive positions of the pinion tooth profile, P, in its motion relative to a reference system associated with the gear.

Generated in such a scenario gear and pinion tooth flanks are commonly referred to as reversibly-enveloping tooth flanks (or just as  R_e — tooth flanks, G and P, for simplicity). Conjugate tooth flanks, G and P, is another terminology that is used with respect to the tooth flanks, G and P, of such a geometry.

Important: Uniform motion transmission between two parallel axes is possible only by conjugate tooth profiles, when the line of action at any and all configurations of the mating gears passes through a stationary point known as the pitch point, P.

In another example, consider the generation of the interacting tooth profiles in the gear machining mesh in the hobbing of straight-sided splines.

Interaction of tooth flanks in straight-sided spline-spline hob mesh. One more example of the interaction of non-involute tooth profiles can be found when designing a hob for machining straight-sided splines. The hob design is based on an auxiliary rack (or basic rack, in other words), the teeth of which are engaged in a mesh with the splines of the spline-shaft. The tooth profile of the rack is commonly generated as an envelope to a family of consecutive positions of the spline profile when the pitch circle of the spline rolls with no sliding over the pitch line associated with the rack.

The determination of the coordinates of points of the tooth profile of a rack conjugate to a spline-shaft is commonly executed using the method of common perpendiculars. An example of how this kind of problem can be solved is illustrated in Figure 2 (This schematic can be traced back to the 1875 F. Reuleaux book: Reuleaux, F., Lehrbuch der Kinematik, Braunschweig, 1875).

The spline profile is associated with a pitch circle of a radius, r_w.sp. The pitch line of the rack to be determined is tangent to the pitch circle of the spline-shaft. The point of tangency of the pitch line and of the pitch circle, r_w.sp, is the pitch point in the rolling motion of the spline-shaft and the rack. The pitch point is designated as P.

The spline-shaft is rotated about its axis of rotation, O_sp. The angular velocity of this rotation is designated as ω_sp. The rack is associated with the pitch line. The rack travels straightforwardly together with the pitch line. The linear velocity of the rack is designated as V_rc.

Let’s assume that, at the initial configuration of the pitch circle and the pitch line, the profile of the spline passes through the pitch point, P. This profile is at a distance, r_sp, from the axis of rotation, O_sp, of the spline-shaft. The straight-line profile is practically tangential to a circle of the radius r_sp. The radius, r_sp, is equal to one-half the spline thickness of the spline-shaft.

When the spline-shaft rotates, the lateral spline profile is rotated together with the spline-shaft. Consequently, the spline-shaft lateral profile passes through points 1, 2, …, (i – 1), i. Point 1 is coincident with the pitch point, P. The pitch line travels straightforwardly. In this motion, the pitch point consequently occupies positions 1*, 2*, 3*, …. The distance 1* – 2*, 2* – 3*, … between consequent locations of the pitch point is equal to the lengths of the arcs ∪1-2, ∪2-3, … of the pitch circle of the spline-shaft. This is because the pitch line of the rack is rolling with no sliding over the pitch circle of the spline-shaft.

At every chosen location of the spline lateral profile, perpendiculars to the profile are constructed so as to pass through the pitch point, P. For example, a perpendicular, N_i, is normal to the spline profile at its i — th location. Point n_i is the point of tangency of the lateral profile of the spline and the rack tooth profile.

Constructed this way, the plurality of points for various configurations of the lateral spline profile are within the path of contact, P_c. In the transverse section by a plane, the path of contact, P_c, is traced by the contact point in a rolling motion of the given spline-shaft and the rack to be determined. The path of contact is determined in a stationary reference system.

In a reference system associated with the spline-shaft, all the points are situated within the lateral spline profile of the spline-shaft.

When the spline-shaft rotates, points of the lateral profile cross the path of contact, P_c. At this instance, the points coincide with the corresponding points of the rack tooth profile. An arbitrary point, n_i, within the path of contact, P_c, corresponds to the point of contact in the i — th location of the lateral profile of the spline. If point, n_i, returns to the initial position of the spline by means of rotation through the angle of the arc Pi, then this point occupies the position of the point a_i.

Similarly, in a reference system associated with the rack, contact points are within the tooth profile of the rack, which has to be determined.

Let’s assume that an arbitrary point, n_i, within the path of contact, P_c, is associated with the pitch line, P_ln. To determine the location of this point in this instance, the pitch line together with the point n_i travels through a distance equal to the arc length Pi in a direction opposite to the direction of the straight motion of the rack tooth in its rolling motion. After this transition is complete, point n_i occupies the position of point b_i. Point b_i is on the tooth profile of the rack. All points of the rack tooth profile are constructed similar to that of point b_i. By connecting the constructed points by a smooth curve, the rack tooth profile can be determined.

The aforementioned approach for the determination of the tooth profile of a rack that is enveloping to a family of a spline-shaft profile is commonly adopted. However, this method is inaccurate by nature, as it can be used only for the approximate generation of the rack tooth profile. It is assumed here the generated tooth profile of the rack can generate the spline profile of the straight-sided spline-shaft when a problem that is inverse to the original problem is considered. The latter is not correct. As these enveloping profiles do not conjugate to one another, no straight-sided spline profile can be generated in the inverse rolling of the rack in relation to spline shafts. In practice, instead of a straight-sided spline profile, a curved profile of the splines is generated (see Figure 3).

This discussion reveals the use of the method of common perpendiculars returns a tooth profile of the rack that is an envelope to a family of consecutive positions of the spline-shaft profile, but is not conjugate to it.

In a case when the gear involute tooth profile, G (that starts at the point, b_g, within the gear base circle of a diameter d_b.g), is given, the pinion tooth profile, P (that starts at the point, b_p, within the pinion base circle of a diameter d_b.p), can be generated as an envelope to a family of consecutive positions of the gear tooth profile, G, in its motion in relation to a reference system associated with the pinion, P, and vice versa: the gear tooth profile, G, can be generated as an envelope to a family of consecutive positions of the pinion tooth profile, P, in its motion relative to a reference system associated with the gear.

When generated in such a scenario, the straight-sided spline and the spline hob are commonly referred to as not-reversibly-envelope tooth flanks. Tooth flanks of such a geometry are not conjugate to one another.

2 Generation of envelope surfaces: State-of-the-art

Two approaches for the generation of envelope surfaces are used in the today’s science and engineering. One of the approaches was developed in the differential geometry of surfaces, while another one was developed in the kinematic geometry of surfaces [5].

2.1 Generation of envelope surfaces in the differential geometry.

A gear tooth flank, G, can be given vector form as seen in Equation 1:

Here, U_g and V_g are the Gaussian parameters of the gear tooth flank, G.

If the gear tooth flank, r_g, travels, and θ is the parameter of this motion, then a family of consecutive positions of the tooth flank is generated. In a certain reference system, X_gY_gZ_g, this family of the tooth flank, r_g, can be described by Equation 2:

In the reference system, X_gY_gZ_g, the envelope surface to the family of consecutive positions of the tooth flank, r_g, is described by a set of equations in Equation 3:

An equation of the pinion tooth flank, P, can be derived after the second equation in the set of Equation 3 is solved with respect to the motion parameter, θ, and the derived solution for θ is substituted into the first equation in the set of Equation 3.

2.2 Generation of envelope surfaces in the kinematic geometry

It should be mentioned that in the second half of the 19th century, Franz Reuleaux published [4] (~1875) a well-known principle, according to which relative motion of two surfaces in contact along the common perpendicular is of a zero value.

The condition of contact requires that at the contact point, both a gear and a mating pinion tooth flanks that interact with one another have to travel with equal velocities along the common perpendicular, that is, the equality V_g ≡ V_p has to be valid (see Figure 4). The relative velocity is zero in this case (V_rel = 0). In a case of V_g < V_p, the components 1 and 2 separate from one another (see Figure 5), and the relative velocity is of a positive value in this case (V_rel > 0). Inversely, in a case V_g > V_p, the components 1 and 2 interfere into one another (see Figure 6). The relative velocity is of a negative value, in this case (V_rel < 0). None of these two cases is permissible in perfect gearing.

This discussion can be summarized as follows:

The permissible instant relative motions of tooth flanks, G and P, in a gear pair are shown in Figure 7. The relative motion of the tooth flanks, G and P, along the common perpendicular, n_g, is not permissible; and the instant relative motion is permissible in any direction within the common tangent plane through contact point, K. It should be pointed out here that a swivel relative motion, ±ϕ_n, of the tooth flanks, G and P, around the axis along the common perpendicular, n_g, also meets the requirement specified by equation of contact, n_g • V_Σ = 0. However, not all kinds of the swivel motion of the tooth flanks in Figure 7 are permissible. For example, no swivel relative motion is permissible about an axis that either intersects or crosses a straight line along the common perpendicular, n_g, as the condition of contact, n_g • V_Σ = 0, in these cases is violated. The swivel motion, ±ϕ_n, of the tooth flanks is not necessary to transmit a rotation from a driving shaft to a driven shaft. However, the motion of this nature, ω_n, is observed in spatial gearing, namely, in C_a — gearing.

The condition of contact is important to the theory of gearing. Unfortunately, it is not known who should be credited with this important accomplishment (maybe this is because the condition of contact has been discovered for a more general case, and not for the purposes of gears).

Later on (in the late 1940s) Prof. V.A. Shishkov expressed this principle analytically in the form of the Shishkov equation of contact, n • V = 0. Here, n is the unit normal vector at the contact point, K, of the interacting tooth flanks, G and P; and   is the unit vector of the speed of relative motion of the gear, and of the mating pinion tooth flanks, G and P.

The accomplishments by Reuleaux and Shishkov are helpful when determining a pinion tooth flank, P, as an envelope to a family of consecutive positions of the traveling gear tooth flank, G, or vice versa, when determining a gear tooth flank, G, as an envelope to a family of consecutive positions of the traveling pinion tooth flank, P. In as such, the replacement of the conjugate tooth flanks, G_cnj and P_cnj, with the enveloping tooth flanks, G_env and P_env, is not always valid, as all conjugate tooth flanks are enveloping to one another, but not vice versa; not all the enveloping tooth flanks are conjugate to each other. As a consequence of such a replacement, the gear/pinion tooth flanks are loosely determined now as enveloping surfaces, and not conjugate surfaces. First of all, that is not permissible for precision gearing, high-power-density gearing, and as other kinds of sophisticated gearings.

The difference between conjugate gear tooth flanks and between enveloping gear tooth flanks is required to be revealed in more detail.

The determination of envelope tooth flanks (either the gear tooth flank, G, or the pinion tooth flank, P) with the help of Shishkov equation of contact, n • V = 0, makes sense when:

• The unit normal vector, n, can be determined with no differentiation of the equation of the family of the gear tooth flank, r_g = r_g(U_g,V_g,θ), with respect to the Gaussian parameters, U_g and V_g [or, the same, with no differentiation of equation of the family of the pinion tooth flank, r_p = r_p(U_p,V_p,θ) with respect to the Gaussian parameters, U_p and V_p] — this is possible for surfaces of simple geometries.

• The unit vector of the speed of relative motion, v, of the gear, and of the mating pinion tooth flanks, G and P, can be determined with no differentiation either of equation r_g = r_g(U_g,V_g,θ), or of the equation r_p = r_p(U_p,V_p,θ) with respect to the motion parameter, θ — this is possible when the kinematics of the relative motion of the tooth flanks, G and P, is simple.

In a more general case of interaction of the functional surfaces of machine elements, the use of the Shishkov equation of contact, n • V = 0, is less effective, and, in some cases, it may even be useless.

3 Generation of envelope surfaces under constraints onto the relative motion

The rolling relative motion, the swivel relative motion, and the combined rolling/swivel relative motion are the principal constraints onto the relative motion of the components when gears operate, as well as in gear-machining processes.

A functional dependence of elementary motions is the key feature of the rolling relative motion in gearing. This is due to rolling with no sliding of the axodes of the interacting gears is observed when a gear pair is operating or when gear teeth are machined.

The elementary motions of the pinion tooth flank, P, relative to the stationary gear tooth flank, G, are required to be synchronized with one another so as to permanently retain the common perpendicular to pass through a point on the axis of instant rotation, P_ln, as illustrated in Figure 8. Here, the section of the tooth flanks, G and P, by a normal plane through the vector of linear velocity, V_Σ, is shown.

Equation 4:

is based on the analysis of the instantaneous kinematics of gearing [2, 5].

Here, ω_p/g is the vector of instant rotation of the pinion tooth flanks, P, in relation to the motionless gear tooth flank, G.

Below, in Figures 9 through 11, several examples of interaction of the two local patches of tooth flanks, G and P, are schematically illustrated. [Shishkov equation of contact, n_g • V_Σ = 0, is fulfilled in all three cases].

The tooth flanks, G and P, make contact at point, K. The radii of curvature of the interacting tooth flanks at contact point, K, equal to R_g and R_p, correspondingly (see Figure 9). The centers of curvature of the tooth profiles, G and P, are denoted by o_g and o_p, correspondingly. In the instant motion of the pinion, P, in relation to the gear, G, the pinion performs an instant rotation, ω_p/g, about the point o_g. The radius of curvature of the generated actual gear tooth flank, G_act, equals to R_p/g ≡ –R_g. In this scenario, the second fundamental law of gear is fulfilled, and the actual tooth flank, G_act, is identical to the desirable gear tooth flank, G, as shown in Figure 9.

Note: Here, in Figures 9 through 11, a rotation, ω_n, of the pinion, P, in relation to the gear, G, about the contact perpendicular, n_g, is not prohibited by the second fundamental law of gearing (n_g • V_Σ = 0).

If the instant rotation is performed either about the center O_p/g (when R_p/g < R_g, see Figure 10), or about the center o_p/g (when R_p/g > R_g, see Figure 11), the second fundamental law of gear is violated, and the actual tooth flank, G_act, differs from the desirable gear tooth flank, G.

Reminder: The first fundamental law of gearing (as well as the “Shishkov equation of contact, n_g • V_Σ = 0”) is fulfilled in all three cases shown in Figures 9 through 11, while the second fundamental law of gearing is fulfilled only in the first case illustrated in Figure 11.

This discussion results in the following analytical description of the conjugate tooth flanks, G and P: When two surfaces roll in relation to one another, then the envelope to a family of consecutive positions of the traveling surface is described by a set of three equations seen in Equation 5:

Here it is designated that:

r_g (U_g,V_g,θ) — is the position vector of point of the gear tooth flank, G.

U_g,V_g — are the Gaussian parameters of the gear tooth flank, G.

R_envelope — is the radius of curvature of the traveling gear tooth flank (algebraic value of the radius of curvature).

R_traveling — is the radius of curvature of the enveloping gear tooth flank (algebraic value of the radius of curvature).

C — is the center distance.

One can play with the enveloping parameter, θ. Only “envelopes”, those, fulfilled the requirement in Equation 6:

are “real envelopes.” R_e — surfaces [5] under any kind of relative motion of the surfaces [the condition, specified by Equation 5, must be fulfilled at all points of the desirable line of contact, LC, of the tooth flanks, G and P]. The rest of the surfaces represent envelopes for the not-rolling relative motions.

This analysis is performed in the plane through the local axes of rotation, o_g and o_p.

An equivalent analysis can be performed in the plane of action, PA, that is the plane through the axes of instant rotation, o_g and o_p (the plane tangent to the base cylinders).

The schematic shown in Figures 9 through 11 is helpful for understanding the difference between the first and the second fundamental laws of gearing and prevents inaccurate conclusions in this regard.

The analysis is similar to that performed for the rolling relative motion, and it can also be performed for the swivel relative motion of the tooth flanks, G and P, as well as for combined rolling/swivel relative motion [7]. Both of these cases have not yet been thoroughly investigated.

Conclusion

This article deals with envelope surfaces used in gear design and in gear machining processes, those using the gear generating principle, when rotations of two gears in mesh or of a gear and the gear cutting tool are synchronized with one another.

The problem of the determining of envelope surfaces in gear design and in gear machining processes was correctly formulated, and then it is solved by the author of this chapter around 2008 [2]. The core of the proposed method of determination of envelope surfaces in gear design and in gear machining process is disclosed in the chapter.

The surfaces considered in this chapter of the book are commonly referred to as reversibly-enveloping surfaces, or just as R_e — surfaces, for simplicity [5].

All three kinds of gearing [i.e.: (a) parallel-axes gearing ( P_a — gearing); (b) intersected-axes gearing ( I_a — gearing); and (c) crossed-axes gearing (C_a — gearing)] are covered by the derived solution to the problem. 

References

1. Radzevich, S.P., “Envelopes in Gearing: A Novel Accomplishment in the Classical Differential Geometry”, Chapter 8 in: Gear Accuracy: A Treatise on Gear Noise Excitation, Vibration Generation, and Dynamics of Operation, S.P. Radzevich (Editor), Springer, 2025. (in press).
2. Radzevich, S.P., Theory of Gearing: Kinematics, Geometry, and Synthesis, 3rd Edition, revised and expanded, CRC Press, Boca Raton, FL, 2022, 1208 pages.
3. Willis, R., Principles of Mechanisms, Designed for the Use of Students in the Universities and for Engineering Students Generally, London, John W. Parker, West Stand, Cambridge: J. & J.J. Deighton, 1841, 446p.
4. Reuleaux, F., The Kinematics of Machinery: Outlines of a Theory of Machines, Translated and Edited by Alexander B. W. Kennedy, and with a New Introduction by Eugene S., Ferguson, Dover Publishers, Inc., New York, 1963, 622 pages. [Translated from: Reuleaux, F., Theoretische Kinematik: Grundziige einer Theori des Maschinenwesens, 1875].
5. Radzevich, S.P., Geometry of surfaces: A Practical Guide for Mechanical Engineers, 2nd edition, Springer International Publishing, 2019, © Springer Nature Switzerland AG (2020), XXVI, 329 pages, 182 illustrations.
6. 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.

Bibliography

1. Radzevich, S.P., (Editor), Advances in Gear Design and Manufacture, CRC Press, Boca Raton, Florida, 2019, 549 pages.
2. Radzevich, S.P., “An Examination of High-Conformal Gearing”, Gear Solutions, February, 2018, pages 31-39.
3. Radzevich, S.P., “Design Features of Perfect Gears for Crossed-Axes Gear Pairs”, Gear Solutions magazine, February, 2019, pp. 36-43.
4. 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.
5. Radzevich, S.P., High-Conformal Gearing: Kinematics and Geometry, 2nd edition, Elsevier, Amsterdam, 2020, 506 pages.
6. 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. [A file in .pdf format with this article can be requested from the author (at no charge)]. A reprint of the article can be found out in: Radzevich, S.P., “Poor Understanding of the Scientific Theory of Gearing by the Majority of Gear Scientists and Engineers”, Chapter 9, pages 193-214 in: Radzevich, S.P., Novikov/Conformal Gearing: Scientific Theory and Practice, Springer, 2022, 526 pages.
7. Radzevich, S.P., Novikov/Conformal Gearing: Scientific Theory and Practice, Springer, 2022 (November 15, 2022), 33+493 (526) pages.
8. Radzevich, S.P., (Editor), Recent Advances in Gearing: Scientific Theory and Applications, Springer, 1st ed., 2022 edition (June 25, 2021), 569 pages.
9. Radzevich, S.P., “The Commonalities and Differences between Helical “Low-Tooth-Count Gears” and “Multiple-Start Worms””, Gear Solutions magazine, February 2021, pp. 34-39.
10. Radzevich, S.P., Storchak, M.G. (Editors), Advances in Gear Theory and Gear Cutting Tool Design, Springer, 2022, 500 pages.