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Journal for Geometry and Graphics 7 (2003), No. 2, 201--210
Copyright Heldermann Verlag 2003
Explicit Calculation Methods for Conjugate Profiles
Gert Bär
Institute for Geometry, Dresden University of Technology, Zellescher Weg 12-14,
Willersbau B 119, 01062 Dresden, Germany,
baer@math.tu-dresden.de
baer@math.tu-dresden.de
The tooth profiles of two meshing spur gears are considered. A given
profile may envelop another gear profile. The paper presents an explicit
parametrization of the gear profile. It is also shown that the
corresponding conjugate rack profile is explicitly determined.
Analogously, the line of action is described. Afterwards, the inverse
problem is studied. If the gear centers, the gear ratio, and a line of
action are given then conjugate gear profiles are determined by the
solution of an ordinary differential equation of first order. An example
of application is shown. Finally, this differential equation is
geometrically interpreted by introducing the pole distance. At any
instant the rate of change of the pole distance relates to the
inclination of the rack profile. It is shown that for a specific linear
relation, corresponding conjugate profiles are trochoids of 2nd or 3rd
order.
Keywords: plane kinematics, conjugate tooth profiles, line of action, rack profile.
MSC: 53A17.
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