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Journal of Mechanical Design | Volume 129 | Issue 3 | Research Paper
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
Article
References
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Citing Articles

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

1
Radzevich, S. P., 2001, "Fundamentals of Surface Generation", Monograph, Kiev, Rastan, Russia.
2
Bregi, B. F., Erxleben, R. F., Tersch, R. W., , 1972, "Modern Methods of Gear Manufacture", 4th ed., National Broach & Machine Division∕Lear Siegler, Inc., Detroit, MI.
3
Townsend, D. P., 1991, "Dudley’s Gear Handbook: The Design, Manufacture, and Application of Gears", 2nd ed., McGraw-Hill, New York.
4
Radzevich, S. P., 1991, "Sculptured Surface Machining on Multi-Axis NC Machine", Monograph, Kiev, Vishcha Shkola Publishing House, Kiev, Russia.
5
Radzevich, S. P., 2006, “Diagonal Shaving of an Involute Pinion: Optimization of Parameters of the Pinion Finishing Operation,” The International Journal of Advanced Manufacturing Technology , in press.
6
Radzevich, S. P., 2002, “Conditions of Proper Sculptured Surface Machining,” CAD  [CrossRef], 34 (10), pp. 727–740.
7
Olivier, T., 1842, "Theorie Geometrique des Engrenages", Bachelier, Paris.
8
Radzevich, S. P., 2002, “About Hob Idle Distance in Gear Hobbing Operation,” ASME J. Mech. Des.  [CrossRef]124 (4), pp. 772–786.
9
Radzevich, S. P., 2003, “Design of Shaving Cutter for Plunge Shaving a Topologically Modified Involute Pinion,” ASME J. Mech. Des.  [CrossRef]125 (3), pp. 632–639.
10
Radzevich, S. P., 2006, “A Novel Approach for Computation of Constraints on Parameters of Modification of the Tooth Addendum of Precision Involute Hob,” ASME J. Mech. Des.128 (6), pp. 803–811.
11
Buckingham, E., 1963, "Analytical Mechanics of Gears", Dover, New York, 546 p.
12
Radzevich, S. P., 2005, “A Descriptive-Geometry-Based-Solution to a Geometrical Problem in Rotary Shaving of Shoulder Pinion,” ASME J. Manuf. Sci. Eng.  [CrossRef]127 (4), pp. 893–900.
13
Radzevich, S. P., 1982, Design and Investigation of Skiving Hobs for finishing Hardened Gears, Ph.D. thesis, Kiev Polytechnic Institute, Kiev.
14
Radzevich, S. P., 1992, "Cutting Tools for Machining Hardened Gears", VNIITEMR, Moscow, Russia.
15
Radzevich, S. P., 1981, “A Precision Involute Hob,” USSR Patent No. 990.445.
16
Radzevich, S. P., 1982, “A Precision Involute Hob,” USSR Patent No. 1.114.505.
17
Radzevich, S. P., 1992, “A Precision Involute Hob,” USSR Patent No. 2.040.376.
18
"FETTE. Gear Cutting Tools: Hobbing, Gear Milling", Leitz Metalworking Technology Group, Nis.
19
Radzevich, S. P., Goodman, E. D., and Palaguta, V. A., 1998, “Tooth Surface Fundamental Forms in Gear Technology,” Facta Universitatis, Mechanical Engineering, 1 (5), pp. 515–525.
20
Radzevich, S. P., 2005, “A Possibility of Application of Plücker’s Conoid for Mathematical Modeling of Contact of Two Smooth Regular Surfaces in the First Order of Tangency,” Math. Comput. Modell.42 (9–10) pp. 999–1022.
21
Plücker, J., 1865, “On a New Geometry of Space,” Philos. Trans. R. Soc. London155 , pp. 725–791.
22
Radzevich, S. P., 2004, “Mathematical Modeling of Contact of Two Surfaces in the First Order of Tangency,” Math. Comput. Modell.39 (9–10), pp. 1083–1112.
23
Radzevich, S. P., 2005, “Computation of Parameters of a Form Grinding Wheel for Grinding of Shaving Cutter for Plunge Shaving of Topologically Modified Involute Pinion,” ASME J. Manuf. Sci. Eng.  [CrossRef]127 (4), pp. 819–828.
24
Tseng, J.-T., and Tsay, C.-B., 2005, “Mathematical Modeling and Surface Deviation of Cylindrical Gears with Curvilinear Shaped Teeth Cut by a Hob Cutter,” ASME J. Mech. Des.  [CrossRef]127 (5), pp. 982–987.
25
Wang, P.-Y., and Fong, Z.-H., 2006, “Fourth-Order Kinematic Synthesis for Face-Milling Spiral Bevel Gears with Modified Radial Motion (MRM) Correction,” ASME J. Mech. Des.  [CrossRef]128 (2), pp. 457–467.

Figures

Grahic Jump Location
+The auxiliary phantom rack R of an involute hob
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The auxiliary phantom rack R of an involute hob
Figure 1

The auxiliary phantom rack R of an involute hob

Grahic Jump Location
+Generation of the auxiliary rack R of an involute hob
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Generation of the auxiliary rack R of an involute hob
Figure 2

Generation of the auxiliary rack R of an involute hob

Grahic Jump Location
+Determining base helix angle ψb.h of an involute hob
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Determining base helix angle ψb.h of an involute hob
Figure 3

Determining base helix angle ψb.h of an involute hob

Grahic Jump Location
+Determining the involute hob base diameter db.h
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Determining the involute hob base diameter db.h
Figure 4

Determining the involute hob base diameter db.h

Grahic Jump Location
+Cylindrical generating surface T of an involute hob
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Cylindrical generating surface T of an involute hob
Figure 5

Cylindrical generating surface T of an involute hob

Grahic Jump Location
+Determining the orientation of the rake-face of an involute hob teeth
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Determining the orientation of the rake-face of an involute hob teeth
Figure 6

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

Grahic Jump Location
+Orientation of rake-face of the precision gear hob
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Orientation of rake-face of the precision gear hob
Figure 7

Orientation of rake-face of the precision gear hob

Grahic Jump Location
+Hob-setting angle ζh of an involute hob
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Hob-setting angle ζh of an involute hob
Figure 9

Hob-setting angle ζh of an involute hob

Grahic Jump Location
+The involute gear hob after been reground
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The involute gear hob after been reground
Figure 10

The involute gear hob after been reground

Grahic Jump Location
+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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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)
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)

Grahic Jump Location
+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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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)
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)

Grahic Jump Location
+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)
View Large  |  Save Figure  |  Download Slide (.ppt)
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)
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)

Grahic Jump Location
+Modification of the gear hob tooth profile
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Modification of the gear hob tooth profile
Figure 14

Modification of the gear hob tooth profile

Grahic Jump Location
+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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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)
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)

Grahic Jump Location
+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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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
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

Grahic Jump Location
+Computation of the desired value of normal radius of curvature RT of the hob surface T
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Computation of the desired value of normal radius of curvature RT of the hob surface T
Figure 17

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

Grahic Jump Location
+Computation of parameters db.h* and R* of the precision involute hob having modified tooth profile
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Computation of parameters db.h* and R* of the precision involute hob having modified tooth profile
Figure 18

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

Grahic Jump Location
+The employed characteristic vectors
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The employed characteristic vectors
Figure 8

The employed characteristic vectors

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