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Research Papers: Design of Direct Contact Systems

Revisiting Generation and Meshing Properties of Beveloid Gears

[+] Author and Article Information
Alessio Artoni

Dipartimento di Ingegneria Civile e Industriale,
Università di Pisa,
Largo Lucio Lazzarino 2,
Pisa 56122, Italy
e-mail: alessio.artoni@ing.unipi.it

Massimo Guiggiani

Dipartimento di Ingegneria Civile e Industriale,
Università di Pisa,
Largo Lucio Lazzarino 2,
Pisa 56122, Italy
e-mail: massimo.guiggiani@ing.unipi.it

1Corresponding author.

Contributed by the Power Transmission and Gearing Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received April 6, 2017; final manuscript received June 30, 2017; published online July 26, 2017. Assoc. Editor: Hai Xu.

J. Mech. Des 139(9), 093301 (Jul 26, 2017) (9 pages) Paper No: MD-17-1251; doi: 10.1115/1.4037345 History: Received April 06, 2017; Revised June 30, 2017

The teeth of ordinary spur and helical gears are generated by a (virtual) rack provided with planar generating surfaces. The resulting tooth surface shapes are a circle-involute cylinder in the case of spur gears, and a circle-involute helicoid for helical gears. Advantages associated with involute geometry are well known. Beveloid gears are often regarded as a generalization of involute cylindrical gears involving one additional degree-of-freedom, in that the midplane of their (virtual) generating rack is inclined with respect to the axis of the gear being generated. A peculiarity of their generation process is that the motion of the generating planar surface, seen from the fixed space, is a rectilinear translation (while the gear blank is rotated about a fixed axis); the component of such translation that is orthogonal to the generating plane is the one that ultimately dictates the shape of the generated, envelope surface. Starting from this basic fact, we set out to revisit this type of generation-by-envelope process and to profitably use it to explore peculiar design layouts, in particular for the case of motion transmission between skew axes (and intersecting axes as a special case). Analytical derivations demonstrate the possibility of involute helicoid profiles (beveloids) transmitting motion between skew axes through line contact and, perhaps more importantly, they lead to the derivation of designs featuring insensitivity of the transmission ratio to all misalignments within relatively large limits. The theoretical developments are confirmed by various numerical examples.

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Figures

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Fig. 1

Basic geometric layout for generation of spur gear teeth

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Fig. 2

Spur gear scenario: front view of involute tooth surfaces, their base cylinder, and the plane of action (obtained with θ ranging between θ = −70 deg and θ+ = 70 deg)

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Fig. 3

Generation with the generating plane displaced not orthogonally to itself (negative θ only)

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Fig. 4

A straight beveloid gear pair for motion transmission between intersecting axes

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Fig. 5

Basic geometric layout for generation with helicality

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Fig. 6

Planes of action and base cylinders for the cases δ = 0 (planar, spur gear case) and δ = 20 deg, the generation ratio r being equal

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Fig. 7

Involute helicoid tooth surfaces extending out of their base cylinders for the cases δ = 0 (planar, spur gear case) and δ = 20 deg

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Fig. 8

A pair of spur gears in mesh: (a) gears to be synchronized and (b) synchronized gears in mesh

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Fig. 9

Helicality rotations δp and δg of pinion and gear blanks and pinion spin σ

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Fig. 10

Three-dimensional setup for involute helicoid gears transmitting motion between skew axes (no spin)

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Fig. 11

Three-dimensional setup for involute helicoid gears transmitting motion between skew axes in the presence of spin (σ)

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Fig. 12

Pitch cones expressing the general relative position between hypoid pinion and gear, in the presence of misalignments E, P, G, and β. Here, deviations from the common 90 deg shaft angle layout are depicted. (Other symbols: e, shaft offset or nominal shortest distance; Σ, nominal shaft angle; and C.P., crossing point. Adapted from Ref. [15].)

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Fig. 13

Involute helicoid tooth surfaces for motion transmission between skew axes, with spin

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Fig. 14

Involute helicoid tooth surfaces for motion transmission between skew axes, zero spin

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Fig. 15

Involute helicoid tooth surfaces for motion transmission between intersecting axes (with spin)

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Fig. 16

Influence of misalignments on the line-of-action (all units in millimeters)

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