Research Papers

A Spatial Version of Octoidal Gears Via the Generalized Camus Theorem

[+] Author and Article Information
Giorgio Figliolini

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
Department of Mechanical Engineering and CIM,
McGill University,
817 Sherbrooke Street W,
Montreal, QC H3A 03C, Canada
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

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.

Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.


Blaschke, W. , 1960, Kinematik und Quaternionen, VEB Deutscher Verlag der Wissenschaften, Berlin.
Veldkamp, G. R. , 1976, “ On the Use of Dual Number, Vectors and Matrices in Instantaneous Spatial Kinematics,” Mech. Mach. Theory, 11(2), pp. 141–156. [CrossRef]
Figliolini, G. , Stachel, H. , and Angeles, J. , 2007, “ A New Look at the Ball-Disteli Diagram and Its Relevance to Spatial Gearing,” Mech. Mach. Theory, 42(10), pp. 1362–1375. [CrossRef]
Disteli, M. , 1911, “ Über die Verzahnung der Hyperboloidräder mit geradlinigem Eingriff,” Z. Math. Phys., 59(3), pp. 244–298.
Figliolini, G. , Stachel, H. , and Angeles, J. , 2013, “ On the Synthesis of Spatial Cycloidal Gears,” Meccanica, 48(5), pp. 1239–1249. [CrossRef]
Figliolini, G. , Stachel, H. , and Angeles, J. , 2013, “ On Martin Disteli's Spatial Cycloidal Gearing,” Mech. Mach. Theory, 60(1), pp. 73–89. [CrossRef]
Pottmann, H. , and Wallner, J. , 2001, Computational Line Geometry, Springer-Verlag, Berlin.
Stachel, H. , 2000, “ Instantaneous Spatial Kinematics and the Invariants of the Axodes,” Ball 2000 Symposium, Cambridge, UK, Paper No. 23, pp. 1–14.
Angeles, J. , 2014, Fundamentals of Robotic Mechanical Systems: Theory, Methods, Algorithms, 4th ed., Springer, New York.
Disteli, M. , 1914, “ Über das Analogon der Savaryschen Formel und Konstruktion in der kinematischen Geometrie des Raumes,” Z. Math. Phys., 62(3), pp. 261–309.
Do Carmo, M. P. , 1976, Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliffs, NJ.
Wang, D. L. , Liu, J. , and Xiao, D. Z. , 2000, “ Geometrical Characteristics of Some Typical Constraints,” Mech. Mach. Theory, 35(10), pp. 1413–1430. [CrossRef]
Dong, H. , Ting, K.-L. , Yu, B. , Liu, J. , and Wang, D. , 2012, “ Differential Contact Path and Conjugate Properties of Planar Gearing Transmission,” ASME J. Mech. Des., 134(6), p. 061010. [CrossRef]
Yu, B. , and Ting, K.-L. , 2013, “ Manifold Conjugation and Discrete Gear Design,” ASME Paper No. DETC2013-13401.
Müller, H. R. , 1963, Kinematik, Sammlung Göschen, Walter de Gruyter, Berlin.
Figliolini, G. , Stachel, H. , and Angeles, J. , 2015, “ The Role of the Orthogonal Helicoid in the Generation of the Tooth Flanks of Involute-Gear Pairs With Skew Axes,” ASME J. Mech. Rob., 7(1), p. 011003. [CrossRef]
Litvin, F. L. , and Fuentes, A. , 2004, Gear Geometry and Applied Theory, Cambridge University Press, Cambridge, UK.
Figliolini, G. , and Angeles, J. , 2005, “ Algorithms for Involute and Octoidal Bevel-Gear Generation,” ASME J. Mech. Des., 127(4), pp. 664–672. [CrossRef]
Wang, D. , and Wang, W. , 2015, Kinematic Differential Geometry and Saddle Synthesis of Linkages, Wiley, Singapore, Singapore.
Ting, K.-L. , and Soni, A. H. , 1983, “ Instantaneous Kinematics of a Plane in Space Motion,” ASME J. Mech. Des., 105(3), pp. 552–559.
Wang, W. , and Wang, D. , 2014, “ Curvature Theory of the Envelope Curve in Two-Dimension and Envelope Surface in Three-Dimension Motion,” ASME J. Mech. Rob., 7(3), p. 031019. [CrossRef]
Phillips, J. , 2003, General Spatial Involute Gearing, Springer-Verlag, New York.
Dooner, D. B. , and Griffis, M. W. , 2007, “ On Spatial Euler-Savary Equations for Envelopes,” ASME J. Mech. Des., 129(8), pp. 865–875. [CrossRef]
Müller, H. R. , 1962, Sphärische Kinematik, VEB Deutscher Verlag der Wissenschaften, Berlin.


Grahic Jump Location
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.

Grahic Jump Location
Fig. 2

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

Grahic Jump Location
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

Grahic Jump Location
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).

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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.

Grahic Jump Location
Fig. 9

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

Grahic Jump Location
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

Grahic Jump Location
Fig. 8

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

Grahic Jump Location
Fig. 11

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

Grahic Jump Location
Fig. 12

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



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In