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TECHNICAL PAPERS

# Synthesis of the Base Curves For $N$-Lobed Elliptical Gears

[+] 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 127(5), 997-1005 (Oct 15, 2004) (9 pages) doi:10.1115/1.1901707 History: Received July 03, 2003; Revised October 15, 2004

## Abstract

Motivated by the need to synthesize the tooth profiles of noncircular gears, we approach the synthesis of the tooth profile of circular spur gears using their pitch circle, rather than their base circle. We do this by means of envelope theory. The proposed formulation gives the involute tooth profile and its well-known base circle for any pitch radius and profile angle of the rack cutter, which coincides with the pressure angle for circular gears. Then, the foregoing approach applies to the synthesis of the base curves of noncircular gears with involute tooth profiles and of their rack. We do this by resorting to basic differential geometry using the Euler–Savary Theorem, rather than to envelope theory. In particular, the formulation of both base curves for the right and left involute tooth profiles is obtained, for the first time, for $N$-lobed elliptical gears and their rack through the formulation of the pitch curves and their evolutes. The proposed formulation is illustrated with numerical results.

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Topics: Gears

## Figures

Figure 3

Generation for envelope of the base curve B

Figure 4

Base circle B of a circular gear with involute tooth profile

Figure 6

Sketch of the pitch curves of an elliptical pinion and its rack

Figure 7

N-lobed elliptical pitch curves for N=1 (continuous line), N=2 (dashed line), N=3 (dash-dot line), N=4 (dotted line) and pitch radii rA=300mm and rB=100mm

Figure 8

Pitch curves of pinion rack for N=1, 2, 3 and pitch radii rA=200mm and rB=100mm

Figure 1

Pure-rolling motion of the epicycle line ε on the pitch circle P

Figure 2

Generation of the envelope of the involute tooth profile I for a circular gear

Figure 5

Involute tooth profile and base circle for a pitch radius rp=50mm and p=30mm: (a) pressure angle ϕ=20°; (b) pressure angle ϕ=40°

Figure 16

Pitch, evolute and base curves of rack for a 3-lobed elliptical pinion with rA=200mm, rB=100mm and αc=20°: (a) base curve for the right involute tooth profile; (b) base curve for the left involute tooth profile

Figure 17

Sketch of a basic ellipse or one-lobe pitch curve

Figure 9

N-lobed elliptical pitch curves and their evolutes: (a) N=3, rA=200mm, rB=175mm and θ1=25°; (b) N=2, rA=200mm, rB=50mm and θ1=320°

Figure 10

Sketch for the synthesis of the base curve for N-lobed elliptical gear with involute tooth profile

Figure 11

Graphical application of the Euler–Savary theorem

Figure 12

Pitch, evolute and base curves for a 1-lobed elliptical gear with rA=200mm and rB=50mm: (a) base curves for αc=20°; (b) base curves for αc=40°. Pitch curve (dash-dotted line); evolute curve (continuous line); base curve for the right profile (dashed line); base curve for the left profile (dotted line)

Figure 13

Pitch, evolute and base curves for a 3-lobed elliptical gear with rA=200mm: (a) rB=180mm and αc=20°; (b) rB=185 and αc=30°. Pitch curve (dash-dotted line); evolute curve (continuous line); base curve for the right profile (dashed line); base curve for the left profile (dotted line)

Figure 14

Pitch curve P1 and left base curve B1 for a 3-lobed elliptical gear, where I1′ is a straight tooth profile and I1″ is a convex tooth profile

Figure 15

Pitch, evolute and base curves for a 5-lobed elliptical gear with rA=500mm: (a) rB=500mm; (b) rB=490mm; (c) rB=480mm; (d) rB=470mm; (e) rB=460mm; (f ) rB=400mm

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