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RESEARCH PAPERS

A Novel Design of Cylindrical Hob for Machining of Precision Involute Gears

[+] Author and Article Information
Stephen P. Radzevich

Department of Mechanical Engineering,  Eaton Automotive Innovation Center, 26201 Northwestern Highway, Southfield, MI 48037stephenpradzevich@eaton.com

The DG∕K approach is based on fundamental results obtained in differential geometry of surfaces, and in kinematics of multi-parametric motion of a rigid body in E3 space. (For details see Refs. 4 and 1. A copy of the monograph is available from The Library of Congress.) A perfect example of application of the DG∕K approach is disclosed in Ref. 5.

Olivier disclosed two principal methods of generation of enveloping surfaces in his work published as early as 1842.

The DGB approach proved to be useful for solving a variety of gear related problems (see, for example, the monograph by Buckingham (11), as well as more recent publications (8,12)).

The DG∕K approach is based on fundamental results obtained in differential geometry of surfaces, and in kinematics of multiparametric motion of a rigid body in E3 space. The method is disclosed in two monographs (1,4) by the author. Both the monographs are available from the Library of Congress.

In order to clarify the transition from the auxiliary rack R with straight-line tooth profile to the auxiliary rack Rm with curved tooth profile, it is convenient to consider the following analogy. When a straight-line is rotating about an axis that is parallel to the line, a surface of circular cylinder is generated. Then consider the straight-line that is inclined to the axis of rotation. Consecutive positions of the straight-line form a surface of a single-sheet hyperboloid of revolution.

Descriptive-geometry-based methods serve as a perfect “filter” for detection and elimination of rough errors of the analysis.

J. Mech. Des 129(3), 334-345 (Mar 06, 2006) (12 pages) doi:10.1115/1.2406105 History: Received October 24, 2005; Revised March 06, 2006

This paper aims at development of a novel design of precision gear hob for machining involute gear on a conventional gear-hobbing machine. The reported research is based on use of fundamental results obtained in analytical mechanics of gearing. For solving the problem, both the descriptive-geometry-based (DGB) methods together with pure analytical methods have been employed. The use of DGB methods is insightful for solving most of the principal problems, which consequently were analytically solved. The analytical methods used provide an example of application of the DG∕K-method of surface generation developed earlier by the author. For interpretation of the results of research, several computer codes in the commercial software MathCAD∕Scientific were composed. Ultimately, a method of computation of parameters of design of a hob with straight-line lateral cutting edges for machining of precision involute gears is developed in the paper. Coincidence of the straight-line lateral cutting edges of the hob with the straight-line characteristics of it generating surface eliminates the major source of deviations of the hobbed involute gears. The relationships between major principal design parameters that affect the gear hob performance are investigated with the use of vector algebra, matrix calculus, and elements of differential geometry. Gear hobs of the proposed design yield elimination of the principal and major source of deviation of the desired hob tooth profile from the actual hob tooth profile. The reported results of this research are ready to put in practice.

Copyright © 2007 by American Society of Mechanical Engineers
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References

Figures

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Figure 5

Cylindrical generating surface T of an involute hob

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Figure 6

Determining the orientation of the rake-face of an involute hob teeth

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Figure 7

Orientation of rake-face of the precision gear hob

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Figure 9

Hob-setting angle ζh of an involute hob

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Figure 10

The involute gear hob after been reground

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Figure 11

Impact of number of starts Nh of the involute hob onto the actual orientation of the rake plane determined by the angle ξ (m=10mm, ϕn=20deg, ζh=3deg, nh=10, αt=12deg)

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Figure 1

The auxiliary phantom rack R of an involute hob

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Figure 2

Generation of the auxiliary rack R of an involute hob

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Figure 3

Determining base helix angle ψb.h of an involute hob

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Figure 4

Determining the involute hob base diameter db.h

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Figure 8

The employed characteristic vectors

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Figure 12

Impact of the involute hob normal pressure angle ϕn onto the actual orientation of the rake plane determined by the angle ξ (m=10mm,Ng=1, ζh=3deg, nh=10, αt=12deg)

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Figure 13

Impact of the hob-setting angle ζh onto the actual orientation of the rake plane determined by the angle ξ (m=10mm, ϕn=20deg, Ng=1, nh=10, αt=12deg)

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Figure 14

Modification of the gear hob tooth profile

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Figure 15

Distribution of normal curvature in the differential vicinity of a point on the surface T of the FETTE gear hob (DIN 8002A, Cat.-No 2022, Ident. No 1202055)

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Figure 16

Normal radii of curvature RT(υ) (a); and normal curvature kT(υ) (b) of the machining surface T of the gear hob versus υ=dVT∕dUT

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Figure 17

Computation of the desired value of normal radius of curvature RT of the hob surface T

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Figure 18

Computation of parameters db.h* and R* of the precision involute hob having modified tooth profile

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