Design features of hobbed bevel gears for crossed-axes gear sets

This article deals with hobbed bevel gears for crossed-axes gear pairs. The discussion is primarily focused on the kinematics and geometry of the generated tooth flanks of the machined gears. A possibility of hobbing of the bevel gears is the key feature of the gear design under consideration. The discussion begins with a brief overview of hobbed gears for crossed-axes gear pairs. In this section of the article, the focus is mainly on the earliest accomplishments by the pioneers of gear design and gear production, namely, on the achievements of Nikola Trbojevich (also known as Nicholas Terbo) and Heinrich Schicht. Then, the possibility of geometrically-accurate Palloid-like gear system is discussed in more detail.

1 On the Origination of Hobbed Gears for Crossed-Axes Gear Pairs

Gear manufacturers have focused their attention on the possibility of generating of tooth flanks in bevel gears primarily due to the high productivity of such a gear machining operation. Two gear experts are credited with the key accomplishments in the field of hobbing bevel gears with the conical hob: Nikola Trbojevich (1886-1973) — also known as Nicholas Terbo — and Heinrich Schicht (1886-1962). The success with the design, manufacture, and application of hobbed gears for crossed-axes gearing is due to these two famous gear experts.

Chronologically, Trbojevich was the first to propose a solution to the problem of hobbing of bevel gears.

Trbojevich, a world-renowned research engineer, mathematician, and inventor, was a nephew and friend of Nikola Tesla. Trbojevich is the first key developer of the methods of and the means for hobbing bevel gearing. He held nearly 200 U.S. and foreign patents, principally in the field of gear design.

1.1 The contribution by Nikola Trbojevich

Trbojevich’s most notable work that brought him international recognition was the invention of the hobbed bevel gear for crossed-axes gearing, along with a method and design of a cutting tool to cut gears of this novel design.

On January 3, 1922, Trbojevich filed a patent application titled “Gear” [3]. The invention related to spiral bevel gears and how they can be successfully manufactured by a hobbing process, as illustrated in Figure 1a. It was a new type of spiral bevel gear using previously unexploited mathematical techniques. The process employed in hobbing, as well as the construction of the hob used, are also inventions of Trbojevich. It is also claimed that the bevel gear teeth are of longitudinally modified involute outwardly divergent curvature (see Figure 1a), all of the said curves being substantially such as may be traced by a point on a line in fixed spaced parallel relation to a line rolling, without sliding, upon a base circle concentric with the axis of the crown gear. The bevel gear tooth flanks are generated by means of a conical hob, schematically shown in Figure 2.

The invention (Figure 2) relates to the manufacture of spiral bevel gears and is designed to construct a hob capable of generating such gears. The discovery by Trbojevich showed that spiral bevel gears of a novel type can be manufactured by a hobbing process employing a hob of simple construction, which may be easily manufactured and maintained in operative condition [5], [7], and others. This concept received further development in [4], [6], and others.

The accomplishments of Trbojevich in the design of hobbed bevel gears, as well as the methods of and means for their production, have been granted numerous U.S. and International patents.

1.2 The contribution by Heinrich Schicht

In the 1920s, two engineers, Schicht and G. Preis, invented a novel method for the generation of bevel gears by means of conical hobs. Schicht is commonly considered as the second key developer of the methods for and means of machining bevel gears for crossed-axes gear sets. He is credited first of all with the invention of Palloid gearing [8]. This method is illustrated in Figure 3.

In the Palloid gear cutting method, two hobs with opposite threads are required to produce a pinion and wheel. Schicht took up the idea of hobbing cylindrical gears and transferred it to bevel gears using a conical hob instead of a cylindrical hob to manufacture spiral bevel gears. In this way, he developed a continuous-indexing generator for spiral bevel gears. The first machine that used the Palloid method for generating spiral bevel gears and hypoid gears with a conical hob was built in1923. (See Figure 4)

Made by Schicht and Preis improvements in the method of, and apparatus for, hobbing herringbone bevel gears are known from the British 1924 patent [10].

Later on, this method was replaced by the face-milled and face-hobbed methods.

1.3 Hobs for the generation of bevel gears

Hobs of various designs can be used to cut bevel gears for crossed-axes gearing. (A method of bevel gear machining was presented in Germany in November 17, 1923, using a conical hob. Schicht and Preiss may have applied for a patent with this idea in 1921; however, this data needs to be verified).

As an illustrative example, a hob with a concave generating rack is shown in Figure 5.

The hob (see Figure 5) may have a single thread or a plurality of screw threads. The pitch surface line on which the pitch is measured is concave. For convenience of manufacture, the top and root line of the hob teeth are concave. The hob surface is a curvilinear surface of revolution generated by a concave line.

The hob of such a design is used in the bevel gear hobbing process according to the methods in [11] and [12].

Hob cutters of other designs for the bevel gear generation are also known. Tooth flanks in Palloid gears are generated using specially designed conical hobs. Numerous Great Britain patents [No. 230885 (1925), and No. 234131 (1926)], as well as U.S. patents [No. 2,037,930 (1936), and No. 2,146,232 (1939)] were granted to Schicht on the Palloid gear system, the methods of and means for their production.

In today’s industry, Palloid hob is the only hob used in spiral bevel gear generation. The conical single-start hob has a pitch cone angle of  30°. On its conical body is a single-start worm with a lead of  π〈m₀ (see Figure 6) [1]. This tool allows continuous indexing; however, the generation motion requires a superimposed pivoting motion of the hob around the axis of the virtual crown gear. Depending on the required cutting length, SF, which results from the face width of the bevel gear to be cut, a hob from series A, B, or C is selected. The hobs are also graded according to module size and tooth thickness modifications and differ in their cutting direction and hand or lead. (See Figure 6)

Figure 7 illustrates an example of the set up of a Palloid hob on spiral bevel gear generator.

The cutting teeth of a Palloid hob are relief ground, and only its front faces need regrinding in straight flutes. So, the geometry of the individual hob tooth is preserved except for a slight reduction in the hob diameter. To prolong the tool life, the Palloid hob can be coated and, after regrinding, even recoated.

Intermediate conclusion: It is often loosely claimed that Palloid gears, which are cut by hobs, feature a tooth form that follows a true involute curve. This is incorrect. It should be stressed here that the Palloid gears also are a kind of approximate gear; that is, they are not capable of transmitting a rotation uniformly.

It can be shown that no geometrically-accurate bevel gears can be generated by the methods proposed by Trbojevich nor Schicht. The discussed methods and the gear cutting tools are applicable only for the generation of approximate gears.

In order to prove this statement, three meshes have to be considered. They include:

• A gear-to-hob mesh.
• A pinion-to-hob mesh.
• The gear-to-pinion mesh.

In the gear-to-hob mesh, angular base pitch of the gear, ϕ_b.g, angular base pitch of the hob, ϕ_b.h, and operating base pitch in the gear-to-hob mesh, must be equal to the operating base pitch, ϕ_b.op.gh (ϕ_b.g = ϕ_op.gh, and ϕ_b.h = ϕ_b.op.gh).

In the pinion-to-hob mesh, angular base pitch of the pinion, ϕ_b.p, angular base pitch of the hob, ϕ_b.h, and operating base pitch in the pinion-to-hob mesh, must be equal to the operating base pitch, ϕ_b.op.ph  (ϕ_b.p = ϕ_b.op.ph, and ϕ_b.h = ϕ_b.op.ph).

In the gear-to-pinion mesh, angular base pitch of the gear, ϕ_b.g, angular base pitch of the pinion, ϕ_b.p, and operating base pitch in the gear-to-pinion mesh, must be equal to the operating base pitch, ϕ_b.op.gp (ϕ_b.g = ϕ_b.op.gp, and ϕ_b.p = ϕ_b.op.gp).

The equality of the angular base pitches: ϕ_b.g, ϕ_b.p, ϕ_b.h, ϕ_b.op.gh, ϕ_b.op.ph, and ϕ_b.op.gp = ϕ_b.op, is the necessary and sufficient condition all precision gears must fulfill [1].

2 A Possibility of Geometrically-Accurate (Precision) Palloid-like Gears

Palloid gearing is a kind of gearing with crossing axes of rotation of a gear and a mating pinion. Therefore, there are many similarities between Palloid gearing and conventional C_a — gearing [13], [14], [16]. The latter is illustrated in Figure 8, where a crossed-axes gear pair is shown. In this illustration, the gear pair is overlapped by a corresponding gear vector diagram.

Referring to Figure 8, consider a crossed-axes gear pair (C_a — gearing) together with the associated rotation vectors ω_g and ω_p. A Cartesian coordinate system, XYZ, is associated with the crossed-axes gear pair. The rotation vectors, ω_g and ω_p, of a gear and its mating pinion are at a center-distance, C. The center-distance, C, is a straight-line segment measured along the center-line, CL, and equals to the closest distance of approach between the axes of rotation, O_g and O_p, of the gear and its mating pinion, correspondingly. In the particular case under consideration, the crossed-axes angle, Σ, equals 90°. The crossed-axes angle, Σ, can be either acute (Σ < 90°) or obtuse (Σ > 90°).

The principal difference of Palloid gearing from C_a — gearing of conventional design is due to, in a Palloid gear pair, a hobbed gear is engaged in mesh with a hobbed pinion.

Palloid gearing is a kind of approximate gearing; that is, gearing of this particular kind is not capable of transmitting a steady motion uniformly with a constant angular velocity ratio u_ω = const. It is reasonable to question whether a geometrically-accurate Palloid-like gearing is possible. To answer this question, the gear-to-hob mesh (namely, the mesh of the gear to be generated with the conical hob) needs to be thoroughly investigated.

Analysis of the mesh bevel gear-to-conical hob begins with the gear vector diagram. An example of the gear vector diagram for gearing of this particular kind is illustrated in Figure 9.

In the case under consideration, the key feature of gear vector diagram is due to a gear ratio in bevel gear-to-conical hob mesh. As the conical hobs are designed either single-start or they feature just a few number of starts, the hob center-distance, C_h, is drastically reduced when either the bevel gear tooth flanks, G, or the bevel pinion tooth flanks, P, are generated. Remember, the following two equalities C_gm = C_g + C_h, and C_pm = C_p + C_h, are valid when machining a gear or a pinion, correspondingly (here, C_gm and C_pm are the center-distances in the bevel gear-to-conical hob mesh, and in the bevel gear-to-conical hob, correspondingly; C_g and C_p are the gear center-distance, and the pinion center-distance, correspondingly) [1]. Note, the inequality C_g >> C_h is always valid in Palloid gearing, as well as in all designs of palloid-like gearing.

The rotation vector of the hob in the gear-to-hob mesh is labeled ω_h. The vector, ω_h, is pointed along the axis of rotation of the hob, O_h. Considered as an element of gear-to-hob mesh, the rotation vector, ω_h, is synchronized with the rotation vector, ω_g, of the gear.

No sliding is observed when the gear and the hob rotate about their axes of rotation, O_g and O_h.

The number of starts, N_h, of the hob is significantly smaller compared to tooth count, N_g, in the gear: N_h << N_g. The rotations of the hob, ω_h, and of the gear, ω_g, are reciprocal to the tooth ratio in the gear-to-hob mesh (N_h/N_g = ω_g/ω_h). The hob rotation vector, ω_h, is determined so the actual values of the rotation vectors ω_g and ω_pl in the gear-to-pinion mesh can be retained. The projection of the rotation vector, ω_h, onto the axis, O_pa, equals to the ω_pa.

In further analysis, the advantage can be taken from the following fact. In a crossed-axes gear set of conventional design, the tooth flanks of a bevel gear, G (as well as the tooth flanks of the mating pinion, P), is generated by means of a desirable line of contact, LC_des: In the bevel gear-to-pinion mesh, the tooth flanks G and P interact with one another along the desirable line of contact, LC_des. This line is entirely situated in the plane of action, PA, of the bevel gear-to-pinion mesh. The plane of action, PA, forms the transverse pressure angle, φ₁, with the normal plane (N_ln — plane).

The desirable line of contact, LC_des, always spans over the face width, F_pa, and is entirely located within the active portion of the plane of action, PA, at list at one instant of time. The line-of-contact-span-angle, ζ_lc, is important for further discussion [15].

In gear-to-hob mesh of crossed-axes gearing (see Figure 10), the conical hob apex, A_h, is displaced in relation to the plane-of-action apex, A_pa. The conical hob axis of rotation, O_h, does not intersected the plane-of-action axis of rotation, O_pa. A distance, C_h, is the closest distance of approach of the crossing axes O_pa and O_h.

In crossed-axes gearing, actual value of the line-of-contact-span-angle, ζ__lc, is greater than that in intersected-axes gearing at a certain angle, ζ_cd. This is due to the displacement at the center-distance, C_h.

Determination of actual value of the angle, ζ_cd, is based on the concept that in the worst-case scenario, the radial length of desirable line of contact, LC_des, exactly equals the plane-of-action face width (ef = F_pa) [15]. Therefore, (a) the configuration ac of the desirable line of contact, LC_des, is critical; (b) the configuration ad of the desirable line of contact, LC_des, is acceptable; and (c) the configuration ab of the desirable line of contact, LC_des, is not acceptable at all.

The determination of the actual value of the angle, ζ_cd, is outlined immediately below [15].

Consider a triangle ΔA_paA_hf. This triangle is constructed in the plane of action, PA. In the triangle, ΔA_paA_hf, the side A_paA_hf = r_o.pa – is the outer radius of the plane of action, PA, and the side A_hf = r_o.p – is the outer radius of the base cone of the conical hob.

The actual value of the angle ∠A_pag in the triangle ΔfgA_pa equals to the component, ζ_cd, contributed to the total angle, ζ__lc, by the displacement, C_h, and is calculated in Equation 1:

where r_o.pa and r_f.pa are the outer and the inner radii of the plane of action, PA. These design parameters are known from the gearset layout.

In the triangle, ΔA_apA_hf, the length of the side A_hf = √(A_apf)² − C²_p = √r²_o.ap − C²_p.

In the triangle, ΔA_apA_hg, the length of the side A_hg = √(A_apg)² − C²_p = √r²_f.ap − C²_p.

Having the lengths of the straight-line segments A_hf and A_hg calculated, the actual value of the conical hob face width, F_h, is calculated in Equation 2:

Ultimately, the total value of the angle, ζ__lc, equals to summa in Equation 3:

Having the angle, ζ__lc, determined, one can proceed with the determination of the actual value of the conical hob base cone angle, γ_b.

The actual value of the base cone angle, γ_b, of the conical hob having the outer cone distance r_o.p = fA_h, and the face width F_h, is seen in Equation 4:

For the calculations, the base cone angle, γ_b, can be expressed in terms of design parameters of the gear component — this is a routine procedure.

By convention, a so-called worm factor, k_w, can be taken into account. In this latter case, the two expressions: γ_b > ζ__lc /2π and γ_b ≤ ζ__lc /2π, are substituted with equivalent expressions: γ_b > k_w 〈ζ__lc /2π and γ_b ≤ k_w 〈ζ__lc /2π. The actual value of the “worm factor, k_w” can be determined empirically based on the experience accumulated in the gear industry.

The constructed gear vector diagram of the bevel gear-to-pinion mesh (see Figure 9) together with the specified value of the transverse pressure angle, φ_t, make the calculation of the design parameters of the conical hob base cone possible.

In the bevel gear-to-conical hob mesh (the gear machining mesh, in other words), (a) the gear base cone, (b) the mating conical worm base cone, and (c) the plane of action, PA, are rotated ( ω_g, ω_p, and ω_pl) about their axes of rotation, and all the rotations are synchronized with one another.

When the plane of action, PA, turns through the operating angular base pitch, ϕ_b.op, the mating gear turns through the gear angular base pitch, ϕ_b.g. The equality ϕ_b.g = ϕ_b.op is valid at every instant (for any and all angular configurations of the pinion and of the plane of action, PA).

When the plane of action, PA, turns through the operating angular base pitch, ϕ_b.op, the mating conical worm also turns through the angular base pitch, ϕ_b.h, of the conical hob. The equality ϕ_b.h = ϕ_b.op is valid at every instant (for any and all angular configurations of the conical hob, and of the plane of action, PA).

When the plane of action, PA, turns through the operating angular base pitch, ϕ_b.op, the mating conical worm:

• Makes a full rotation, 360°, about its axis of rotation (single-start conical worm).

• Makes a rotation through the angle of 180°, about its axis of rotation (double-start conical worm).

• Makes a rotation through the angle of 120°, about its axis of rotation (triple-start conical worm), and so forth.

Calculation of the design parameters of the conical hob is based on the considered features of the bevel gear-to-conical hob mesh.

The base cone of the hob determined this way is used consequently to calculate the design parameters of the hob.

In the bevel gear-to-pinion mesh, the tooth flanks, G and P, of the bevel gear and its mating pinion are generated by a moving desirable line of contact, LC_des. The gear tooth flank, G, is represented as a family of consecutive positions of the desirable line of contact, LC_des, in its motion together with the plane of action, PA, in relation to a Cartesian reference system, X_gY_gZ_g, associated with the gear, while the pinion tooth flank, P, is represented as a family of consecutive positions of that same desirable line of contact, LC_des, in its motion together with the plane of action, PA, in relation to a Cartesian reference system, X_pY_pZ_p, associated with the pinion [1].

In the bevel gear-to-conical hob mesh, that same plane of action, PA, is employed. This mesh is shown in Figure 9. Moreover, that same desirable line of contact, LC_des, is used to generate the threads, H, of the conical hob. The conical hob threads, H, are represented as a family of consecutive positions of the desirable line of contact, LC_des, in its motion together with the plane of action, PA, in relation to a Cartesian reference system, X_hY_hZ_h, associated with the hob.

The equation for the position vector of point, r _H, of the conical hob threads, H, immediately follows from the discussed kinematics of generation of this surface, seen in Equation 5:

Here, r_lc – is the designated position vector of the point of the desirable line of contact, LC_des, specified in a Cartesian reference system, X_paY_paZ_pa, associated with the plane of action, PA; Rs(pa|→h) – is the operator of linear transformation (the operator of the transition from the reference system, X_hY_hZ_h, to the reference system, X_paY_paZ_pa) [1].

Generated this way, the conical hob is geometrically-accurate, and, therefore, it is capable of cutting geometrically-accurate bevel gears for precision palloid-like gearing. It is important to stress here the generate gear and its mating pinion are always in line contact with one another. Contact stress and bending stress are reduced in gears with the line contact between tooth flanks. Moreover, bevel gears of such a design quit when operating.

An analysis similar to the gear-to-hob mesh can be also performed with respect to the pinion-to-hob mesh.

Conclusion

In numerous applications, Palloid gearing is advantageous over gearing of other gear systems.

Present-day designs of conical hobs for cutting bevel gears for Palloid gearing are capable of generating only approximate gears. Therefore, the input uniform rotation is transmitted to the output shaft with fluctuation; that is, the angular velocity ratio is inevitably variable. Due to this fact, an excessive noise excitation and vibration generation are common if Palloid gearing is used to transmit high rotation.

It is shown that geometrically-accurate palloid-like gearing is possible. Gearing of this gear system is capable of uniformly transmitting a steady rotation from the driving shaft to the driven shaft. Moreover, a gear and a mating pinion tooth flanks can be designed so as to maintain line contact between the interacting tooth flanks, G and P. Contact stress and bending stress are reduced in gears with line contact between tooth flanks. Moreover, bevel gears of such a design quit when operating. 

References

1. Klingelnberg-Palloid-Spiralkegelräder: Ihre Berechnung ihre Herstellung und ihr Einbau (German Edition) 1941st Edition, German Edition by Walter Krumme (Author), Springer; 1941st edition (January 1, 1941), 126 pages.
2. Müller, H., “Optimal Flank Forms for Large Bevel Gears,” Gear Technology magazine, November/December issue, 2016, pages 96-102.
3. Pat. 1,465,149 (USA), Gear, N. Trbojevich, Filed: January 3, 1922, Patented: August 14, 1923.
4. Pat. 1,465,150 (USA), Method of Forming Spiral Bevel Gears, N. Trbojevich, Filed: January 3, 1922, Patented: August 14, 1923.
5. Pat. 1,465,151 (USA), Hob, N. Trbojevich, Filed: January 3, 1922, Patented: August 14, 1923.
6. Pat. 1,575,396 (USA), Method of Producing Gears, N. Trbojevich, Filed: January 21, 1921, Patented: March 2, 1926.
7. Pat. 1,607,218 (USA), Gear Cutter, N. Trbojevich, Filed: January 21, 1921, Patented: November 16, 1926.
8. Pat. No. 230,884 (Great Britain), Improvements in the Method of Milling Helicoidal Bevel Gears, and Appliances Therefore, by Heinrich Schicht, Filed: November 17, 1923, Published: March 17, 1925.
9. Pat. 230,885 (Great Britain), Improvements in Milling Machines for Cutting Bevel Gears, H. Schicht, and G. Preis, Filed: November 17, 1923, Complete Accepted: March 17, 1925.
10. Pat. 234,131 (Great Britain), Improvements in the Method of, and Apparatus for, Hobbing Herringbone Bevel Gears, H. Schicht, and G. Preis, Filed (Germany): May 16, 1924, Filed (in United Kingdom): May 15, 1925, Complete Accepted: June, 1926.
11. Pat. 2,037,930 (USA), Machine for Cutting Bevel Gears with Longitudinally Curved Teeth, H. Schicht, Filed: December 17, 1930, Patented: April 21, 1936.
12. Pat. 2,146,232 (USA), Hob, H. Schicht, Filed: December 17, 1930, Patented: February 7, 1939.
13. Radzevich, S.P., “Design Features of Perfect Gears for Crossed-Axes Gear Pairs,” Gear Solutions magazine, February, 2019, pp. 36-43.
14. 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.
15. 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.
16. 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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2. Radzevich, S.P., (Editor), Advances in Gear Design and Manufacture, CRC Press, Boca Raton, Florida, 2019, 549 pages.
3. Radzevich, S.P., “An Examination of High-Conformal Gearing,” Gear Solutions, February, 2018, pages 31-39.
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.
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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. [Upon request, a PDF of this article can be ordered (for free) from the author].
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