Publication: A New Method for Bevel Gear Tooth Flank Computation
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A New Method for Bevel Gear Tooth Flank Computation

- incollection -
 

Author(s)
Ulf Hutschenreiter

Published in
Computational Differentiation: Techniques, Applications, and Tools

Editor(s)
Martin Berz, Christian Bischof, George Corliss, Andreas Griewank

Year
1996

Publisher
SIAM

Abstract
The use of automatic differentiation for an exact computation of geometric properties of bevel gear tooth flanks is discussed. Known models of the kinematics of the machine tools generating bevel gear tooth flanks have been improved. Using that result, we compute curvature properties of the generated envelope surfaces directly. From an analytical point of view, the envelope condition used in differential geometry can be interpreted as a singularity of a vector function in 3-space. By an analytical characterization of this singularity we obtain a two-dimensional parameterization of the generated flank for any machine parameter set. The necessary partial derivatives of the vector function are computed with the help of automatic differentiation. For the first time it is possible to compute the curvature properties and the resulting undercut geometry between tooth and fillet surface of the tooth flanks of any spiral bevel or hypoid gear without any approximation.

Cross-References
Berz1996CDT

BibTeX
@INCOLLECTION{
         Hutschenreiter1996ANM,
       author = "Ulf Hutschenreiter",
       editor = "Martin Berz and Christian Bischof and George Corliss and Andreas Griewank",
       title = "A New Method for Bevel Gear Tooth Flank Computation",
       booktitle = "Computational Differentiation: Techniques, Applications, and Tools",
       pages = "329--341",
       publisher = "SIAM",
       address = "Philadelphia, PA",
       key = "Hutschenreiter1996ANM",
       crossref = "Berz1996CDT",
       abstract = "The use of automatic differentiation for an exact computation of geometric
         properties of bevel gear tooth flanks is discussed. Known models of the kinematics of the machine
         tools generating bevel gear tooth flanks have been improved. Using that result, we compute curvature
         properties of the generated envelope surfaces directly. From an analytical point of view, the
         envelope condition used in differential geometry can be interpreted as a singularity of a vector
         function in 3-space. By an analytical characterization of this singularity we obtain a
         two-dimensional parameterization of the generated flank for any machine parameter set. The necessary
         partial derivatives of the vector function are computed with the help of automatic differentiation.
         For the first time it is possible to compute the curvature properties and the resulting undercut
         geometry between tooth and fillet surface of the tooth flanks of any spiral bevel or hypoid gear
         without any approximation.",
       keywords = "Spiral bevel gears, hypoid gears, tooth flank computation, undercutting, edge of
         regression, envelope condition, derivative tensors, higher order differentiation, cusp
         singularities.",
       year = "1996"
}


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