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RESEARCH PAPERS

# The Synthesis of the Pitch Surfaces of Internal and External Skew-Gears and Their Racks

[+] Author and Article Information
Giorgio Figliolini

DiMSAT, University of Cassino, G. Di Biasio 43,03043 Cassino (FR), Italyfigliolini@unicas.it

Jorge Angeles

Department of Mechanical Engineering & Centre for Intelligent Machines, McGill University, 817 Sherbrooke Street, Montreal, Quebec H3A 2K6, Canadaangeles@cim.mcgill.ca

J. Mech. Des 128(4), 794-802 (Oct 14, 2005) (9 pages) doi:10.1115/1.2202875 History: Received March 31, 2005; Revised October 14, 2005

## Abstract

The synthesis of the pitch surfaces of any pair of external and internal skew gears, using dual algebra and the principle of transference, is the subject of this paper. The spatial motion of the Euclidean space is transferred to the dual space in order to obtain a simplified dual spherical motion, thus emulating the motion of bevel gears. The relative screw motion is hence analyzed by determining the position of the instant screw axis and the angular and sliding velocities. Moreover, the hyperboloid pitch surfaces of the driving and driven gears are synthesized, along with the helicoid pitch surface of their rack. Several numerical results are reported.

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Copyright © 2006 by American Society of Mechanical Engineers
Topics: Motion , Screws , Gears
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## Figures

Figure 15

Hyperboloid pitch surfaces of skew gears and their helicoidal rack for a1=100mm, α1=30deg, and k=2

Figure 14

Hyperboloid pitch surfaces of skew gears and their helicoidal rack for a1=100mm, α1=90deg, and k=−2

Figure 13

Hyperboloid pitch surfaces of skew gears and their helicoidal rack for a1=100mm, α1=90deg, and k=−1

Figure 12

Plücker conoid for a1=100mm and α1=45deg, along with both hyperboloid pitch surfaces for k=0.707

Figure 11

Plücker conoid for a1=100mm and α1=90deg, along with both hyperboloid pitch surfaces for k=0

Figure 10

Plücker conoid for a1=100mm and α1=30deg, along with both hyperboloid pitch surfaces, for k=2

Figure 9

Plücker conoid for a1=100mm and α1=90deg, along with both hyperboloid pitch surfaces, for k=−2

Figure 8

Plücker conoid for a1=100mm and α1=90deg, along with both hyperboloid pitch surfaces for k=−1

Figure 7

Sketch to determine the Plücker conoid in dual space

Figure 6

Plot of b2 versus α1, with a1=100mm and for k varying in the range (+∞,−∞)

Figure 5

Hyperboloid pitch surfaces for a1=100mm, α1=30deg, and k=2

Figure 4

Hyperboloid pitch surfaces for a1=100mm, α1=90deg, and k=−2

Figure 3

Hyperboloid pitch surfaces for a1=100mm, α1=90deg, and k=−1

Figure 2

Relative screw motion: (a) external gears; (b) internal gears

Figure 1

Relative screw motion between skew gears in: (a) Euclidean space; (b) dual space

## Errata

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