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Research Papers

# The Role of the Orthogonal Helicoid in the Generation of the Tooth Flanks of Involute-Gear Pairs With Skew Axes

[+] Author and Article Information
Giorgio Figliolini

Associate Professor
Mem. ASME
Department of Civil and Mechanical Engineering,
University of Cassino and Southern Lazio,
Via G. Di Biasio 43,
Cassino, FR 03043, Italy
e-mail: figliolini@unicas.it

Hellmuth Stachel

Institute of Discrete Mathematics
and Geometry,
Vienna University of Technology,
Wiedner Hauptstr. 8-10/104,
Wien A-1040, Austria
e-mail: stachel@dmg.tuwien.ac.at

Jorge Angeles

Fellow ASME
Department of Mechanical Engineering and CIM,
McGill University,
817 Sherbrooke Street,
e-mail: angeles@cim.mcgill.ca

1Corresponding author.

Manuscript received September 25, 2014; final manuscript received November 24, 2014; published online December 31, 2014. Assoc. Editor: Carl Nelson.

J. Mechanisms Robotics 7(1), 011003 (Feb 01, 2015) (9 pages) Paper No: JMR-14-1257; doi: 10.1115/1.4029287 History: Received September 25, 2014; Revised November 24, 2014; Online December 31, 2014

## Abstract

Camus' concept of auxiliary surface (AS) is extended to the case of involute gears with skew axes. In the case at hand, we show that the AS is an orthogonal helicoid whose axis (a) lies in the cylindroid and (b) is normal to the instant screw axis of one gear with respect to its meshing counterpart; in general, the helicoid axis is skew with respect to the latter. According to the spatial version of Camus' Theorem, any line or surface attached to the AS, in particular any line $L$ of AS itself, can be chosen to generate a pair of conjugate flanks with line contact. While the pair of conjugate flanks is geometrically feasible, as they always share a line of contact and the tangent plane at each point of this line, they even have the same curvature, G2-continuity, when $L$ coincides with the instant screw axis (ISA). This means that the two surfaces penetrate each other, at the same common line. The outcome is that the surfaces are not realizable as tooth flanks. Nevertheless, this is a fundamental step toward the synthesis of the flanks of involute gears with skew axes. In fact, the above-mentioned interpenetration between the tooth flanks can be avoided by choosing a smooth surface attached to the AS, instead of a line of the AS itself, which can give, in particular, the spatial version of octoidal bevel gears, when a planar surface is chosen.

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## References

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## Figures

Fig. 1

Martin Disteli

Fig. 2

Positions of the I2 and I3 axes of the two skew gears, along with the I32-axis (ISA) and the IAS-axis of the orthogonal helicoid with respect to the fixed frame F0 (X,Y,Z)

Fig. 3

Hyperboloid pitch surfaces P2 and P3 along with their Plücker conoid C for a1 = 100 mm, α1 = 90 deg, and k = −1

Fig. 4

Pitch surfaces, Plücker conoid, the ISA, and the IAS of the orthogonal helicoid for a1 = 100 mm, α1 = 45 deg, and k = −1: (a) axonometric view and (b) top view

Fig. 5

Hyperboloid pitch surfaces, Plücker conoid, the ISA, and the IAS of the orthogonal helicoid for a1 = 100 mm, α1 = 90 deg, and k = −2: (a) axonometric view and (b) top view

Fig. 6

Orthogonal helicoid (AS), along with both hyperboloid pitch surfaces and their Plücker conoid for a1 = 100 mm, α1 = 45 deg and k = −1

Fig. 7

Orthogonal helicoid (AS), along with both hyperboloid pitch surfaces and their Plücker conoid for a1 = 100 mm, α1 = 65 deg and k = −3

Fig. 8

Cylindrical gears: a1 = 100 mm, α1 = 0 deg, and k = +2

Fig. 9

Bevel gears: a1 = 0, α1 = 45 deg, and k = −1

Fig. 10

Euler's sketch to find a pair of conjugate profiles (involutes of circles) that yield a constant transmission ratio

Fig. 11

Sketch of the reference frames

Fig. 12

Involute tooth flanks, orthogonal helicoid and both pitch hyperboloids for a1 = 100 mm, α1 = 30 deg, and k = −1

Fig. 13

Involute tooth flanks, orthogonal helicoid and both pitch hyperboloids for a1 = 100 mm, α1 = 60 deg, and k = −2

Fig. 14

Involute tooth flanks, orthogonal helicoid and both pitch hyperboloids for a1 = 120 mm, α1 = 70 deg, and k = −3

Fig. 15

Involute tooth flanks, orthogonal helicoid and both pitch hyperboloids for a1 = 100 mm, α1 = 0 deg, and k = −1

Fig. 16

Involute tooth flanks, orthogonal helicoid and both pitch hyperboloids for a1 = 100 mm, α1 = 0 deg, and k = +2

Fig. 17

Involute tooth flanks, orthogonal helicoid and both pitch hyperboloids for a1 = 0, α1 = 45 deg, and k = −2

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