Research Papers

Analysis of Generation Capabilities of Noncircular Cogbelt Pulleys on the Example of a Gear With an Elliptical Pitch Line

[+] Author and Article Information
Mieczysław Kujawski

Higher School of National Economy in Kutno, Lelewela 7, 99-300 Kutno, Polandmieczyslaw.kujawski@poczta.onet.pl

Piotr Krawiec

 Poznan University of Technology, Piotrowo 3, 60-965 Poznań, Polandpiotr.krawiec@put.poznan.pl

Using formulas (8.16–8.21), which are presented in postdoctoral thesis [11], one can also take small enough increments ds of rolling arc (for rack cutter ds = Rn dγN , where RN , rolling angle of cutter, dγN , angle of rotation of cutter in a feed motion) and can calculate increments 0 , 0 , K , dt and add them to values of ζ0 , η0 , γK , t. However, these formulas are more complex.

Above taken shape and dimensions of noncircular cogbelt pulley are substantiated by previously manufactured noncircular cogbelt pulleys with the same parameters, but manufactured by other methods. This situation allows to compare features and to draw conclusions concerning manufacturing methods of noncircular cogbelt pulleys.

Formulas 8,14 give the same values for γK . Relation 8 was calculated from general considerations concerning any pitch curves (details are given in Ref. [11]). Formula 14 is used for the examined ellipse with semimajor axis a = 2b and can be derived on the basis of Fig. 1 or on the transformation of formula 8. The negative value of the angle γK denotes the rotation of manufactured cogbelt pulley in a reverse direction to the direction of rolling of the angle, i.e., according to the principle applied in differential geometry.

J. Manuf. Sci. Eng 133(5), 051006 (Sep 22, 2011) (7 pages) doi:10.1115/1.4004866 History: Revised July 29, 2011; Received August 02, 2011; Published September 22, 2011; Online September 22, 2011

The aim of this paper is the analysis of the generation capabilities of noncircular cogbelt pulleys on the example of a cogbelt pulley with an elliptical pitch line. The need of elaboration of different manufacturing methods for noncircular cogbelt pulleys than existing ones for noncircular gears is shown. Relationships for the mutual position of the cutting tool and noncircular cogbelt pulley during teeth generating are given. The profile of the tool’s cutting edge, which provides the required shape of tooth spaces, is elaborated. Recommendations for designers and process engineers, who deal with elaboration of technology for noncircular cogbelt pulleys, are formulated.

Copyright © 2011 by American Society of Mechanical Engineers
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Figure 1

Geometrical relationships for a position of a noncircular cogbelt pulley in relation to a cutting tool during gear generating: K—pitch line of machined gear, Nz —pitch line of cutting tool (rack cutter or hob), N—pitch line of cutting tool (pinion cutter)

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Figure 2

A cogbelt pulley with an elliptical pitch line: zK  = 44, p = 9.525 mm

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Figure 3

Distorted profile of a tooth space, obtained as a result of an envelope motion: (a) BA—a pitch line of a machined noncircular cogbelt pulley (ellipse), 1—a pitch line of a cutting tool (circle), 2—required addendum line of a noncircular cogbelt pulley (equidistant of line BA), 3 and 4—profiles of the tool’s cutting edge in two positions: during contact of pitch lines in a point P1 (on the symmetry axis of a tooth space) and during contact in a point P2 , C and D—centers of an arc of nose radius of a tool, (b) trajectories of centers C and D in the system of a machined cogbelt pulley, 7 and 8—fragments of equidistants of trajectories 5 and 6 (chamfer – arc E1E2∧ is equidistant of arc D1D2∧)

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Figure 4

Profile forming of tool’s cutting edge with the application of a gear with the defined shape of a tooth space: (a) successive positions of a profile of a tooth space in cutting tool’s system, (b) a redesigned profile of cutting tool (1), which is necessary to cut a tooth space (2) of cogbelt pulley during generating

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Figure 5

An ellipse AB and its equidistant FG

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Figure 6

Construction of an ellipse which substitutes equidistant FG (from Fig. 5) of ellipse AB

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Figure 7

Tooth space profiles of cogbelt pulley with a small radius of a curvature as in a point A of an ellipse from Fig. 2 (rmin  = 0.5b) during machining by a tool with a straight pitch line, when the cogbelt pulley’s pitch line is displaced closer to the centre of tooth space with a value of d: (a) for d = 1.15 mm (magnification of approximate trajectories for points F and G, which are profile ends of cutting tool, are presented on the left and right side of picture), (b) for d = 1.7 mm, 1—a profile of a cutting tool, 2—a profile of a tooth space

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Figure 8

The drive of timing gear of a modern combustion engine with noncircular cogbelt pulley [16]




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