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Research Papers

# A Spatial Version of Octoidal Gears Via the Generalized Camus Theorem

[+] Author and Article Information
Giorgio Figliolini

Mem. ASME
Professor
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

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

Jorge Angeles

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

1Corresponding author.

Manuscript received March 17, 2015; final manuscript received August 28, 2015; published online November 24. Assoc. Editor: David Dooner.

J. Mechanisms Robotics 8(2), 021015 (Nov 24, 2015) (3 pages) Paper No: JMR-15-1062; doi: 10.1115/1.4031679 History: Received March 17, 2015; Revised August 28, 2015; Accepted September 03, 2015

## Abstract

Understanding the geometry of gears with skew axes is a highly demanding task, which can be eased by invoking Study's Principle of Transference. By means of this principle, spherical geometry can be readily ported into its spatial counterpart using dual algebra. This paper is based on Martin Disteli's work and on the authors' previous results, where Camus' auxiliary curve is extended to the case of skew gears. We focus on the spatial analog of one particular case of cycloid bevel gears: When the auxiliary curve is specified as a pole tangent, we obtain “pathologic” spherical involute gears; the profiles are always interpenetrating at the meshing point because of G2-contact. The spatial analog of the pole tangent, a skew orthogonal helicoid, leads to G2-contact at a single point combined with an interpenetration of the flanks. However, when instead of a line a plane is attached to the right helicoid, the envelopes of this plane under the roll-sliding of the auxiliary surface (AS) along the axodes are developable ruled surfaces. These serve as conjugate tooth flanks with a permanent line contact. Our results show that these flanks are geometrically sound, which should lead to a generalization of octoidal bevel gears, or even of bevel gears carrying teeth designed with the exact spherical involute, to the spatial case, i.e., for gears with skew axes.

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## Figures

Fig. 1

Skew axes p̂21, p̂31 of the gear wheels, the ISA p̂32 and the axis p̂41 of the AS Π4⊂Σ4 in the particular case β = φ+π/2. The Frenet frame (f̂1,f̂2,f̂3) of the axodes remains fixed to the machine frame Σ1.

Fig. 2

Frenet frame (ĝ, n̂, t̂) and striction curve of a ruled surface

Fig. 3

The tangent plane Tx at the point x of the generator ĝ is defined by the distribution parameter δ via tan ψ = −u/δ

Fig. 4

The triplet (ĝ1,ĝ2,ĝ3) is the Frenet frame for the conjugate tooth flanks Φ2 and Φ3. The corresponding Disteli axes ĝ∗ are defined by the spatial Euler–Savary equation (36).

Fig. 5

Involute bevel gearing with the pole tangent as auxiliary curve—a case which is geometrically unfeasible because the conjugate profiles c2 and c3 always penetrate each other at the meshing point Mi

Fig. 6

When the ISA coincides with the meshing line ĝ, the singular lines of the two flanks Φ2, Φ3 come together sharing the tangent plane at each point of ĝ, but the flanks open toward opposite sides

Fig. 7

Two conjugate flanks Φ2 and Φ3 with G2-contact at the common striction point Sg. The meshing line ĝ is here parallel to the ISA and a cylindric generator of Φ2 and Φ3.

Fig. 8

Snapshots of the penetrating tooth flanks with their striction curves upon meshing

Fig. 9

Planar version of the generalized Camus theorem in the particular case leading to involute gears

Fig. 10

Octoidal bevel gears: The conjugate profiles c2 and c3 are the envelopes of the great circle c4 under the motions Σ4/Σ2 and Σ4/Σ3

Fig. 11

Skew gears with torses as conjugate tooth flanks Φ2,Φ3 and permanent line contact

Fig. 12

Snapshots of the conjugate torses Φ2 and Φ3 upon meshing (ω31:ω21=−2:1)

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